maths problem solving difficult

These Are the 10 Hardest Math Problems Ever Solved

They’re guaranteed to make your head spin.

$pierre de fermat, math equation$

On the surface, it seems easy. Can you think of the integers for x, y, and z so that x³+y³+z³=8? Sure. One answer is x = 1, y = -1, and z = 2. But what about the integers for x, y, and z so that x³+y³+z³=42?

That turned out to be much harder—as in, no one was able to solve for those integers for 65 years until a supercomputer finally came up with the solution to 42. (For the record: x = -80538738812075974, y = 80435758145817515, and z = 12602123297335631. Obviously.)

That’s the beauty of math : There’s always an answer for everything, even if takes years, decades, or even centuries to find it. So here are nine more brutally difficult math problems that once seemed impossible, until mathematicians found a breakthrough.

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The Poincaré Conjecture

the french astronomer and mathematician henri poincare at work in his office

In 2000, the Clay Mathematics Institute , a non-profit dedicated to “increasing and disseminating mathematical knowledge,” asked the world to solve seven math problems and offered $1,000,000 to anybody who could crack even one. Today, they’re all still unsolved, except for the Poincaré conjecture.

Henri Poincaré was a French mathematician who, around the turn of the 20th century, did foundational work in what we now call topology. Here’s the idea: Topologists want mathematical tools for distinguishing abstract shapes. For shapes in 3D space, like a ball or a donut, it wasn’t very hard to classify them all . In some significant sense, a ball is the simplest of these shapes.

Poincaré then went up to 4-dimensional stuff, and asked an equivalent question. After some revisions and developments, the conjecture took the form of “Every simply-connected, closed 3-manifold is homeomorphic to S^3,” which essentially says “the simplest 4D shape is the 4D equivalent of a sphere.”

Still with us?

A century later, in 2003, a Russian mathematician named Grigori Perelman posted a proof of Poincaré’s conjecture on the modern open math forum arXiv. Perelman’s proof had some small gaps, and drew directly from research by American mathematician Richard Hamilton. It was groundbreaking, yet modest.

After the math world spent a few years verifying the details of Perelman’s work, the awards began . Perelman was offered the million-dollar Millennium Prize, as well as the Fields Medal, often called the Nobel Prize of Math. Perelman rejected both. He said his work was for the benefit of mathematics, not personal gain, and also that Hamilton, who laid the foundations for his proof, was at least as deserving of the prizes.

Fermat’s Last Theorem

Pierre de Fermat was a 17th-century French lawyer and mathematician. Math was apparently more of a hobby for Fermat, and so one of history’s greatest math minds communicated many of his theorems through casual correspondence. He made claims without proving them, leaving them to be proven by other mathematicians decades, or even centuries, later. The most challenging of these has become known as Fermat’s Last Theorem.

It’s a simple one to write. There are many trios of integers (x,y,z) that satisfy x²+y²=z². These are known as the Pythagorean Triples, like (3,4,5) and (5,12,13). Now, do any trios (x,y,z) satisfy x³+y³=z³? The answer is no, and that’s Fermat’s Last Theorem.

Fermat famously wrote the Last Theorem by hand in the margin of a textbook, along with the comment that he had a proof, but could not fit it in the margin. For centuries, the math world has been left wondering if Fermat really had a valid proof in mind.

Flash forward 330 years after Fermat’s death to 1995, when British mathematician Sir Andrew Wiles finally cracked one of history’s oldest open problems . For his efforts, Wiles was knighted by Queen Elizabeth II and was awarded a unique honorary plaque in lieu of the Fields Medal, since he was just above the official age cutoff to receive a Fields Medal.

Wiles managed to combine new research in very different branches of math in order to solve Fermat’s classic number theory question. One of these topics, Elliptic Curves, was completely undiscovered in Fermat’s time, leading many to believe Fermat never really had a proof of his Last Theorem.

The Classification of Finite Simple Groups

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From solving Rubik’s Cube to proving a fact about body-swapping on Futurama , abstract algebra has a wide range of applications. Algebraic groups are sets that follow a few basic properties, like having an “identity element,” which works like adding 0.

Groups can be finite or infinite, and if you want to know what groups of a particular size n look like, it can get very complicated depending on your choice of n .

If n is 2 or 3, there’s only one way that group can look. When n hits 4, there are two possibilities. Naturally, mathematicians wanted a comprehensive list of all possible groups for any given size.

The complete list took decades to finish conclusively, because of the difficulties in being sure that it was indeed complete. It’s one thing to describe what infinitely many groups look like, but it’s even harder to be sure the list covers everything. Arguably the greatest mathematical project of the 20th century, the classification of finite simple groups was orchestrated by Harvard mathematician Daniel Gorenstein, who in 1972 laid out the immensely complicated plan.

By 1985, the work was nearly done, but spanned so many pages and publications that it was unthinkable for one person to peer review. Part by part, the many facets of the proof were eventually checked and the completeness of the classification was confirmed.

By the 1990s, the proof was widely accepted. Subsequent efforts were made to streamline the titanic proof to more manageable levels, and that project is still ongoing today .

The Four Color Theorem

line, parallel, slope, triangle, diagram, symmetry, triangle,

This one is as easy to state as it is hard to prove.

Grab any map and four crayons. It’s possible to color each state (or country) on the map, following one rule: No states that share a border get the same color.

The fact that any map can be colored with five colors—the Five Color Theorem —was proven in the 19th century. But getting that down to four took until 1976.

Two mathematicians at the University of Illinois, Urbana-Champaign, Kenneth Appel and Wolfgang Hakan, found a way to reduce the proof to a large, finite number of cases . With computer assistance, they exhaustively checked the nearly 2,000 cases, and ended up with an unprecedented style of proof.

Arguably controversial since it was partially conceived in the mind of a machine, Appel and Hakan’s proof was eventually accepted by most mathematicians. It has since become far more common for proofs to have computer-verified parts, but Appel and Hakan blazed the trail.

(The Independence of) The Continuum Hypothesis

red, line, colorfulness, carmine, circle, rectangle, parallel, coquelicot, graphics, line art,

In the late 19th century, a German mathematician named Georg Cantor blew everyone’s minds by figuring out that infinities come in different sizes, called cardinalities. He proved the foundational theorems about cardinality, which modern day math majors tend to learn in their Discrete Math classes.

Cantor proved that the set of real numbers is larger than the set of natural numbers, which we write as |ℝ|>|ℕ|. It was easy to establish that the size of the natural numbers, |ℕ|, is the first infinite size; no infinite set is smaller than ℕ.

Now, the real numbers are larger, but are they the second infinite size? This turned out to be a much harder question, known as The Continuum Hypothesis (CH) .

If CH is true, then |ℝ| is the second infinite size, and no infinite sets are smaller than ℝ, yet larger than ℕ. And if CH is false, then there is at least one size in between.

So what’s the answer? This is where things take a turn.

CH has been proven independent, relative to the baseline axioms of math. It can be true, and no logical contradictions follow, but it can also be false, and no logical contradictions will follow.

It’s a weird state of affairs, but not completely uncommon in modern math. You may have heard of the Axiom of Choice, another independent statement. The proof of this outcome spanned decades and, naturally, split into two major parts: the proof that CH is consistent, and the proof that the negation of CH is consistent.

The first half is thanks to Kurt Gödel, the legendary Austro-Hungarian logician. His 1938 mathematical construction, known as Gödel’s Constructible Universe , proved CH compatible with the baseline axioms, and is still a cornerstone of Set Theory classes. The second half was pursued for two more decades until Paul Cohen, a mathematician at Stanford, solved it by inventing an entire method of proof in Model Theory known as “forcing.”

Gödel’s and Cohen’s halves of the proof each take a graduate level of Set Theory to approach, so it’s no wonder this unique story has been esoteric outside mathematical circles.

Gödel’s Incompleteness Theorems

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Gödel’s work in mathematical logic was totally next-level. On top of proving stuff, Gödel also liked to prove whether or not it was possible to prove stuff . His Incompleteness Theorems are often misunderstood, so here’s a perfect chance to clarify them.

Gödel’s First Incompleteness Theorem says that, in any proof language, there are always unprovable statements. There’s always something that’s true, that you can’t prove true. It’s possible to understand a (non-mathematically rigorous) version of Gödel’s argument, with some careful thinking. So buckle up, here it is: Consider the statement, “This statement cannot be proven true.”

Think through every case to see why this is an example of a true, but unprovable statement. If it’s false, then what it says is false, so then it can be proven true, which is contradictory, so this case is impossible. On the other extreme, if it did have a proof, then that proof would prove it true … making it true that it has no proof, which is contradictory, killing this case. So we’re logically left with the case that the statement is true, but has no proof. Yeah, our heads are spinning, too.

But follow that nearly-but-not-quite-paradoxical trick, and you’ve illustrated that Gödel’s First Incompleteness Theorem holds.

Gödel’s Second Incompleteness Theorem is similarly weird. It says that mathematical “formal systems” can’t prove themselves consistent. A consistent system is one that won’t give you any logical contradictions.

Here’s how you can think of that. Imagine Amanda and Bob each have a set of mathematical axioms—baseline math rules—in mind. If Amanda can use her axioms to prove that Bob’s axiom system is free of contradictions, then it’s impossible for Bob to use his axioms to prove Amanda’s system doesn’t yield contradictions.

So when mathematicians debate the best choices for the essential axioms of mathematics (it’s much more common than you might imagine) it’s crucial to be aware of this phenomenon.

The Prime Number Theorem

text, line, font, slope, parallel, design, diagram, number, plot, pattern,

There are plenty of theorems about prime numbers . One of the simplest facts—that there are infinitely many prime numbers—can even be adorably fit into haiku form .

The Prime Number Theorem is more subtle; it describes the distribution of prime numbers along the number line. More precisely, it says that, given a natural number N, the number of primes below N is approximately N/log(N) ... with the usual statistical subtleties to the word “approximately” there.

Drawing on mid-19th-century ideas, two mathematicians, Jacques Hadamard and Charles Jean de la Vallée Poussin, independently proved the Prime Number Theorem in 1898. Since then, the proof has been a popular target for rewrites, enjoying many cosmetic revisions and simplifications. But the impact of the theorem has only grown.

The usefulness of the Prime Number Theorem is huge. Modern computer programs that deal with prime numbers rely on it. It’s fundamental to primality testing methods, and all the cryptology that goes with that.

Solving Polynomials by Radicals

text, line, slope, diagram, triangle, parallel, font, plot, symmetry,

Remember the quadratic formula ? Given ax²+bx+c=0, the solution is x=(-b±√(b^2-4ac))/(2a), which may have felt arduous to memorize in high school, but you have to admit is a conveniently closed-form solution.

Now, if we go up to ax³+bx²+cx+d=0, a closed form for “x=” is possible to find, although it’s much bulkier than the quadratic version. It’s also possible, yet ugly, to do this for degree 4 polynomials ax⁴+bx³+cx²+dx+f=0.

The goal of doing this for polynomials of any degree was noted as early as the 15th century. But from degree 5 on, a closed form is not possible. Writing the forms when they’re possible is one thing, but how did mathematicians prove it’s not possible from 5 up?

The world was only starting to comprehend the brilliance of French mathematician Evariste Galois when he died at the age of 20 in 1832. His life included months spent in prison, where he was punished for his political activism, writing ingenious, yet unrefined mathematics to scholars, and it ended in a fatal duel.

Galois’ ideas took decades after his death to be fully understood, but eventually they developed into an entire theory now called Galois Theory . A major theorem in this theory gives exact conditions for when a polynomial can be “solved by radicals,” meaning it has a closed form like the quadratic formula. All polynomials up to degree 4 satisfy these conditions, but starting at degree 5, some don’t, and so there’s no general form for a solution for any degree higher than 4.

Trisecting an Angle

line, circle, parallel, diagram, drawing,

To finish, let’s go way back in history.

The Ancient Greeks wondered about constructing lines and shapes in various ratios, using the tools of an unmarked compass and straightedge . If someone draws an angle on some paper in front of you, and gives you an unmarked ruler, a basic compass, and a pen, it’s possible for you to draw the line that cuts that angle exactly in half. It’s a quick four steps, nicely illustrated like this , and the Greeks knew it two millennia ago.

What eluded them was cutting an angle in thirds. It stayed elusive for literally 15 centuries, with hundreds of attempts in vain to find a construction. It turns out such a construction is impossible.

Modern math students learn the angle trisection problem—and how to prove it’s not possible—in their Galois Theory classes. But, given the aforementioned period of time it took the math world to process Galois’ work, the first proof of the problem was due to another French mathematician, Pierre Wantzel . He published his work in 1837, 16 years after the death of Galois, but nine years before most of Galois’ work was published.

Either way, their insights are similar, casting the construction question into one about properties of certain representative polynomials. Many other ancient construction questions became approachable with these methods, closing off some of the oldest open math questions in history.

So if you ever time-travel to ancient Greece, you can tell them their attempts at the angle trisection problem are futile.

Dave Linkletter is a Ph.D. candidate in Pure Mathematics at the University of Nevada, Las Vegas. His research is in Large Cardinal Set Theory. He also teaches undergrad classes, and enjoys breaking down popular math topics for wide audiences.

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100 hard word problems in algebra

Find below a wide variety of hard word problems in algebra. Most tricky and tough algebra word problems are covered here. If you can solve these, you can probably solve any algebra problems. Teachers! Feel free to select from this list and give them to your students to see if they have mastered how to solve tough algebra problems. Find out below how you can print these problems. You can also purchase a solution if needed.

1. The cost of petrol rises by 2 cents a liter. last week a man bought 20 liters at the old price. This week he bought 10 liters at the new price. Altogether, the petrol costs $9.20. What was the old price for 1 liter?

2. Teachers divided students into groups of 3. Each group of 3 wrote a report that had 9 pictures in it. The students used 585 pictures altogether. How many students were there in all?

3. Vera and Vikki are sisters. Vera is 4 years old and Vikki is 13 years old. What age will each sister be when Vikki is twice as old as Vera?

4. A can do a work in 14 days and working together A and B can do the same work in 10 days. In what time can B alone do the work?

5. 7 workers can make 210 pairs of cup in 6 days. How many workers are required to make 450 pairs of cup in 10 days?

6. Ten years ago the ratio between the ages of Mohan and Suman was 3:5. 11 years hence it will be 11:16. What is the present age of Mohan?

7. The ratio of girls to boys in class is 9 to 7 and there are 80 students in the class. How many girls are in the class?

8. One ounce of solution X contains only ingredients a and b in a ratio of 2:3. One ounce of solution Y contains only ingredients a and b in a ratio of 1:2. If solution Z is created by mixing solutions X and Y in a ratio of 3:11, then 2520 ounces of solution Z contains how many ounces of a?

9. This week Bob puts gas in his truck when the tank was about half empty. Five days later, bob puts gas again when the tank was about three fourths full. If Bob Bought 24 gallons of gas, how many gallons does the tank hold?

10. A commercial airplane flying with a speed of 700 mi/h is detected 1000 miles away with a radar. Half an hour later an interceptor plane flying with a speed of 800 mi/h is dispatched. How long will it take the interceptor plane to meet with the other plane?

11. There are 40 pigs and chickens in a farmyard. Joseph counted 100 legs in all. How many pigs and how many chickens are there?

12. The top of a box is a rectangle with a perimeter of 72 inches. If the box is 8 inches high, what dimensions will give the maximum volume?

13. You are raising money for a charity. Someone made a fixed donation of 500. Then, you require each participant to make a pledge of 25 dollars. What is the minimum amount of money raised if there are 224 participants.

14. The sum of two positive numbers is 4 and the sum of their squares is 28. What are the two numbers?

15. Flying against the jet stream, a jet travels 1880 mi in 4 hours. Flying with the jet stream, the same jet travels 5820 mi in 6 hours. What is the rate of the jet in still air and what is the rate of the jet stream?

16. Jenna and her friend, Khalil, are having a contest to see who can save the most money. Jenna has already saved $110 and every week she saves an additional $20. Khalil has already saved $80 and every week he saves an additional $25. Let x represent the number of weeks and y represent the total amount of money saved. Determine in how many weeks Jenna and Khalil will have the same amount of money.

17. The sum of three consecutive terms of a geometric sequence is 104 and their product is 13824.find the terms.

18. The sum of the first and last of four consecutive odd integers is 52. What are the four integers?

19. A health club charges a one-time initiation fee and a monthly fee. John paid 100 dollars for 2 months of membership. However, Peter paid 200 for 6 months of membership. How much will Sylvia pay for 1 year of membership?

20. The sum of two positive numbers is 4 and the sum of their cubes is 28. What is the product of the two numbers?

21. A man selling computer parts realizes that when he sells 16 computer parts, his earning is $1700. When he sells 56 computer parts, his earning is $4300. What will the earning be if the man sells 30 computers parts?

22. A man has 15 coins in his pockets. These coins are dimes and quarters that add up 2.4 dollars. How many quarters and how many dimes does the man have?

23. The lengths of the sides of a triangle are in the ratio 4:3:5. Find the lengths of the sides if the perimeter is 18 inches.

24. The ratio of base to height of a equilateral triangle is 3:4. If the area of the triangle is 6, what is the perimeter of the triangle?

25. The percent return rate of a growth fund, income fund, and money market are 10%, 7%, and 5% respectively. Suppose you have 3200 to invest and you want to put twice as much in the growth fund as in the money market to maximize your return. How should you invest to get a return of 250 dollars in 1 year?

26. A shark was caught whose tail weighted 200 pounds. The head of the shark weighted as much as its tail plus half its body. Its body weighted as much as its head and tail. What is the weight of the shark?

27. The square root of a number plus two is the same as the number. What is the number?

28. Suppose you have a coupon worth 6 dollars off any item at a mall. You go to a store at the mall that offers a 20% discount. What do you need to do to save the most money?

29. Suppose your grades on three math exams are 80, 93, and 91. What grade do you need on your next exam to have at least a 90 average on the four exams?

30. Peter has a photograph that is 5 inches wide and 6 inches long. She enlarged each side by the same amount. By how much was the photograph enlarged if the new area is 182 square inches?

31. The cost to produce a book is 1200 to get started plus 9 dollars per book. The book sells for 15 dollars each. How many books must be sold to make a profit?

32. Store A sells CDs for 2 dollars each if you pay a one-time fee of 104 dollars. Store B offers 12 free CDs and charges 10 dollars for each additional CD. How many CD must you buy so it will cost the same under both plans?

33. When 4 is added to two numbers, the ratio is 5:6. When 4 is subtracted from the two numbers, the ratio is 1:2. Find the two numbers.

34. A store owner wants to sell 200 pounds of pistachios and walnuts mixed together. Walnuts cost 4 dollars per pound and pistachio cost 6 dollars per pounds. How many pounds of each type of nuts should be mixed if the store owner will charge 5 dollars for the mixture?

35. A cereal box manufacturer makes 32-ounce boxes of cereal. In a perfect world, the box will be 32-ounce every time is made. However, since the world is not perfect, they allow a difference of 0.06 ounce. Find the range of acceptable size for the cereal box.

36. A man weighing 600 kg has been losing 3.12% of his weight each month with some heavy exercises and eating the right food. What will the man weigh after 20 months?

37. An object is thrown into the air at a height of 60 feet. After 1 second and 2 second, the object is 88 feet and 84 feet in the air respectively. What is the initial speed of the object?

38. A transit is 200 feet from the base of a building. There is man standing on top of the building. The angles of elevation from the top and bottom of the man are 45 degrees and 44 degrees. What is the height of the man?

39. A lemonade consists of 6% of lemon juice and a strawberry juice consists of 15% pure fruit juice. How much of each kind should be mixed together to get a 4 Liters of a 10% concentration of fruit juice?

40. Ellen can wash her car in 60 minutes. Her older sister Sarah can do the same job in 45 minutes. How long will it take if they wash the car together?

41. A plane flies 500 mi/h. The plane can travel 1100 miles with the wind in the same amount of time as it travels 900 miles against the wind. What is the speed of the wind?

42. A company produces boxes that are 5 feet long, 4 feet wide, and 3 feet high. The company wants to increase each dimension by the same amount so that the new volume is twice as big. How much is the increase in dimension?

43. James invested half of his money in land, a tenth in stock, and a twentieth in saving bonds. Then, he put the remaining 21000 in a CD. How much money did James saved or invested?

44. The motherboards for a desktop computer can be manufactured for 50 dollars each. The development cost is 250000. The first 20 motherboard are samples and will not be sold. How many salable motherboards will yield an average cost of 6325 dollars?

45. How much of a 70% orange juice drink must be mixed with 44 gallons of a 20% orange juice drink to obtain a mixture that is 50% orange juice?

46. A company sells nuts in bulk quantities. When bought in bulk, peanuts sell for $1.20 per pound, almonds for $ 2.20 per pound, and cashews for $3.20 per pound. Suppose a specialty shop wants a mixture of 280 pounds that will cost $2.59 per pound. Find the number of pounds of each type of nut if the sum of the number of pounds of almonds and cashews is three times the number of pounds of peanuts. Round your answers to the nearest pound.

47. A Basketball player has successfully made 36 of his last 48 free throws. Find the number of consecutive free throws the player needs to increase his success rate to 80%.

48. John can wash cars 3 times as fast as his son Erick. Working together, they need to wash 30 cars in 6 hours. How many hours will it takes each of them working alone?

49. In a college, about 36% of student are under 20 years old and 15% are over 40 years old. What is the probability that a student chosen at random is under 20 years old or over 40 years?

50. Twice a number plus the square root of the number is twelve minus the square root of the number.

51. The light intensity, I , of a light bulb varies inversely as the square of the distance from the bulb. A a distance of 3 meters from the bulb, I = 1.5 W/m^2 . What is the light intensity at a distance of 2 meters from the bulb?

52. The lengths of two sides of a triangle are 2 and 6. find the range of values for the possible lengths of the third side.

53. Find three consecutive integers such that one half of their sum is between 15 and 21.

54. After you open a book, you notice that the product of the two page numbers on the facing pages is 650. What are the two page numbers?

55. Suppose you start with a number. You multiply the number by 3, add 7, divide by ½, subtract 5, and then divide by 12. The result is 5. What number did you start with?

56. You have 156 feet of fencing to enclose a rectangular garden. You want the garden to be 5 times as long as it is wide. Find the dimensions of the garden.

57. The amount of water a dripping faucet wastes water varies directly with the amount of time the faucet drips. If the faucet drips 2 cups of water every 6 minutes, find out how long it will take the faucet to drip 10.6465 liters of water.

58. A washer costs 25% more than a dryer. If the store clerk gave a 10% discount for the dryer and a 20% discount for the washer, how much is the washer before the discount if you paid 1900 dollars.

59. Baking a tray of blueberry muffins takes 4 cups of milk and 3 cups of wheat flour. A tray of pumpkin muffins takes 2 cups of milk and 3 cups of wheat flour. A baker has 16 cups of milk and 15 cups of wheat flour. You make 3 dollars profit per tray of blueberry muffins and 2 dollars profit per tray of pumpkin muffins. How many trays of each type of muffins should you make to maximize profit?

60. A company found that -2p + 1000 models the number of TVs sold per month where p can be set as low as 200 or as high as 300. How can the company maximize the revenue?

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These Are the 7 Hardest Math Problems Ever Solved — Good Luck in Advance

That’s the beauty of math: There’s always an answer for everything, even if takes years, decades, or even centuries to find it. So here are seven more brutally difficult math problems that once seemed impossible until mathematicians found a breakthrough.

Still with us?

Grab any map and four crayons. It’s possible to color each state (or country) on the map, following one rule: No states that share a border get the same color.

The fact that any map can be colored with five colors—the Five Color Theorem —was proven in the 19th century. But getting that down to four took until 1976.

Now, the real numbers are larger, but are they the second infinite size? This turned out to be a much harder question, known as The Continuum Hypothesis (CH) .

If CH is true, then |ℝ| is the second infinite size, and no infinite sets are smaller than ℝ, yet larger than ℕ. And if CH is false, then there is at least one size in between.

So what’s the answer? This is where things take a turn.

CH has been proven independent, relative to the baseline axioms of math. It can be true, and no logical contradictions follow, but it can also be false, and no logical contradictions will follow.

Gödel’s and Cohen’s halves of the proof each take a graduate level of Set Theory to approach, so it’s no wonder this unique story has been esoteric outside mathematical circles.

The Ancient Greeks wondered about constructing lines and shapes in various ratios, using the tools of an unmarked compass and straightedge. If someone draws an angle on some paper in front of you, and gives you an unmarked ruler, a basic compass, and a pen, it’s possible for you to draw the line that cuts that angle exactly in half. It’s a quick four steps, nicely illustrated like this , and the Greeks knew it two millennia ago.

So if you ever time-travel to ancient Greece, you can tell them their attempts at the angle trisection problem are futile.

In 2019, mathematicians finally solved a math puzzle that had stumped them for decades. It’s called a Diophantine Equation, and it’s sometimes known as the “summing of three cubes”: Find x, y, and z such that x³+y³+z³=k, for each k from one to 100.

These ten brutally difficult math problems once seemed impossible until mathematicians eventually solved them—even if it took them years, decades, or centuries.

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Choose Your Test

Sat / act prep online guides and tips, the 15 hardest sat math questions ever.

Want to test yourself against the most difficult SAT math questions? Want to know what makes these questions so difficult and how best to solve them? If you're ready to really sink your teeth into the SAT math section and have your sights set on that perfect score, then this is the guide for you.

We've put together what we believe to be the 15 most difficult questions for the current SAT , with strategies and answer explanations for each. These are all hard SAT Math questions from College Board SAT practice tests, which means understanding them is one of the best ways to study for those of you aiming for perfection.

Image: Sonia Sevilla /Wikimedia

Brief Overview of SAT Math

The third and fourth sections of the SAT will always be math sections . The first math subsection (labeled "3") does not allow you to use a calculator, while the second math subsection (labeled as "4") does allow the use of a calculator. Don't worry too much about the no-calculator section, though: if you're not allowed to use a calculator on a question, it means you don't need a calculator to answer it.

Each math subsection is arranged in order of ascending difficulty (where the longer it takes to solve a problem and the fewer people who answer it correctly, the more difficult it is). On each subsection, question 1 will be "easy" and question 15 will be considered "difficult." However, the ascending difficulty resets from easy to hard on the grid-ins.

Hence, multiple choice questions are arranged in increasing difficulty (questions 1 and 2 will be the easiest, questions 14 and 15 will be the hardest), but the difficulty level resets for the grid-in section (meaning questions 16 and 17 will again be "easy" and questions 19 and 20 will be very difficult).

With very few exceptions, then, the most difficult SAT math problems will be clustered at the end of the multiple choice segments or the second half of the grid-in questions. In addition to their placement on the test, though, these questions also share a few other commonalities. In a minute, we'll look at example questions and how to solve them, then analyze them to figure out what these types of questions have in common.

But First: Should You Be Focusing on the Hardest Math Questions Right Now?

If you're just getting started in your study prep (or if you've simply skipped this first, crucial step), definitely stop and take a full practice test to gauge your current scoring level. Check out our guide to all the free SAT practice tests available online and then sit down to take a test all at once.

The absolute best way to assess your current level is to simply take the SAT practice test as if it were real , keeping strict timing and working straight through with only the allowed breaks (we know—probably not your favorite way to spend a Saturday). Once you've got a good idea of your current level and percentile ranking, you can set milestones and goals for your ultimate SAT Math score.

If you're currently scoring in the 200-400 or the 400-600 range on SAT Math, your best bet is first to check out our guide to improving your math score to be consistently at or over a 600 before you start in trying to tackle the most difficult math problems on the test.

If, however, you're already scoring above a 600 on the Math section and want to test your mettle for the real SAT, then definitely proceed to the rest of this guide. If you're aiming for perfect (or close to) , then you'll need to know what the most difficult SAT math questions look like and how to solve them. And luckily, that's exactly what we'll do.

WARNING: Since there are a limited number of official SAT practice tests , you may want to wait to read this article until you've attempted all or most of the first four official practice tests (since most of the questions below were taken from those tests). If you're worried about spoiling those tests, stop reading this guide now; come back and read it when you've completed them.

Now let's get to our list of questions (whoo)!

Image: Niytx /DeviantArt

The 15 Hardest SAT Math Questions

Now that you're sure you should be attempting these questions, let's dive right in! We've curated 15 of the most difficult SAT Math questions for you to try below, along with walkthroughs of how to get the answer (if you're stumped).

No Calculator SAT Math Questions

$$C=5/9(F-32)$$

The equation above shows how temperature $F$, measured in degrees Fahrenheit, relates to a temperature $C$, measured in degrees Celsius. Based on the equation, which of the following must be true?

A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of $5/9$ degree Celsius.
A temperature increase of 1 degree Celsius is equivalent to a temperature increase of 1.8 degrees Fahrenheit.
A temperature increase of $5/9$ degree Fahrenheit is equivalent to a temperature increase of 1 degree Celsius.

A) I only B) II only C) III only D) I and II only

ANSWER EXPLANATION: Think of the equation as an equation for a line

where in this case

$$C= {5}/{9} (F−32)$$

$$C={5}/{9}F −{5}/{9}(32)$$

You can see the slope of the graph is ${5}/{9}$, which means that for an increase of 1 degree Fahrenheit, the increase is ${5}/{9}$ of 1 degree Celsius.

$$C= {5}/{9} (F)$$

$$C= {5}/{9} (1)= {5}/{9}$$

Therefore, statement I is true. This is the equivalent to saying that an increase of 1 degree Celsius is equal to an increase of ${9}/{5}$ degrees Fahrenheit.

$$1= {5}/{9} (F)$$

$$(F)={9}/{5}$$

Since ${9}/{5}$ = 1.8, statement II is true.

The only answer that has both statement I and statement II as true is D , but if you have time and want to be absolutely thorough, you can also check to see if statement III (an increase of ${5}/{9}$ degree Fahrenheit is equal to a temperature increase of 1 degree Celsius) is true:

$$C= {5}/{9} ({5}/{9})$$

$$C= {25} /{81} (\which \is ≠ 1)$$

An increase of $5/9$ degree Fahrenheit leads to an increase of ${25}/{81}$, not 1 degree, Celsius, and so Statement III is not true.

The final answer is D.

The equation ${24x^2 + 25x -47}/{ax-2} = -8x-3-{53/{ax-2}}$ is true for all values of $x≠2/a$, where $a$ is a constant.

What is the value of $a$?

A) -16 B) -3 C) 3 D) 16

ANSWER EXPLANATION: There are two ways to solve this question. The faster way is to multiply each side of the given equation by $ax-2$ (so you can get rid of the fraction). When you multiply each side by $ax-2$, you should have:

$$24x^2 + 25x - 47 = (-8x-3)(ax-2) - 53$$

You should then multiply $(-8x-3)$ and $(ax-2)$ using FOIL.

$$24x^2 + 25x - 47 = -8ax^2 - 3ax +16x + 6 - 53$$

Then, reduce on the right side of the equation

$$24x^2 + 25x - 47 = -8ax^2 - 3ax +16x - 47$$

Since the coefficients of the $x^2$-term have to be equal on both sides of the equation, $−8a = 24$, or $a = −3$.

The other option which is longer and more tedious is to attempt to plug in all of the answer choices for a and see which answer choice makes both sides of the equation equal. Again, this is the longer option, and I do not recommend it for the actual SAT as it will waste too much time.

The final answer is B.

If $3x-y = 12$, what is the value of ${8^x}/{2^y}$?

A) $2^{12}$ B) $4^4$ C) $8^2$ D) The value cannot be determined from the information given.

ANSWER EXPLANATION: One approach is to express

$${8^x}/{2^y}$$

so that the numerator and denominator are expressed with the same base. Since 2 and 8 are both powers of 2, substituting $2^3$ for 8 in the numerator of ${8^x}/{2^y}$ gives

$${(2^3)^x}/{2^y}$$

which can be rewritten

$${2^3x}/{2^y}$$

Since the numerator and denominator of have a common base, this expression can be rewritten as $2^(3x−y)$. In the question, it states that $3x − y = 12$, so one can substitute 12 for the exponent, $3x − y$, which means that

$${8^x}/{2^y}= 2^12$$

The final answer is A.

Points A and B lie on a circle with radius 1, and arc ${AB}↖⌢$ has a length of $π/3$. What fraction of the circumference of the circle is the length of arc ${AB}↖⌢$?

ANSWER EXPLANATION: To figure out the answer to this question, you'll first need to know the formula for finding the circumference of a circle.

The circumference, $C$, of a circle is $C = 2πr$, where $r$ is the radius of the circle. For the given circle with a radius of 1, the circumference is $C = 2(π)(1)$, or $C = 2π$.

To find what fraction of the circumference the length of ${AB}↖⌢$ is, divide the length of the arc by the circumference, which gives $π/3 ÷ 2π$. This division can be represented by $π/3 * {1/2}π = 1/6$.

The fraction $1/6$ can also be rewritten as $0.166$ or $0.167$.

The final answer is $1/6$, $0.166$, or $0.167$.

$${8-i}/{3-2i}$$

If the expression above is rewritten in the form $a+bi$, where $a$ and $b$ are real numbers, what is the value of $a$? (Note: $i=√{-1}$)

ANSWER EXPLANATION: To rewrite ${8-i}/{3-2i}$ in the standard form $a + bi$, you need to multiply the numerator and denominator of ${8-i}/{3-2i}$ by the conjugate, $3 + 2i$. This equals

$$({8-i}/{3-2i})({3+2i}/{3+2i})={24+16i-3+(-i)(2i)}/{(3^2)-(2i)^2}$$

Since $i^2=-1$, this last fraction can be reduced simplified to

$$ {24+16i-3i+2}/{9-(-4)}={26+13i}/{13}$$

which simplifies further to $2 + i$. Therefore, when ${8-i}/{3-2i}$ is rewritten in the standard form a + bi, the value of a is 2.

In triangle $ABC$, the measure of $∠B$ is 90°, $BC=16$, and $AC$=20. Triangle $DEF$ is similar to triangle $ABC$, where vertices $D$, $E$, and $F$ correspond to vertices $A$, $B$, and $C$, respectively, and each side of triangle $DEF$ is $1/3$ the length of the corresponding side of triangle $ABC$. What is the value of $sinF$?

ANSWER EXPLANATION: Triangle ABC is a right triangle with its right angle at B. Therefore, $\ov {AC}$ is the hypotenuse of right triangle ABC, and $\ov {AB}$ and $\ov {BC}$ are the legs of right triangle ABC. According to the Pythagorean theorem,

$$AB =√{20^2-16^2}=√{400-256}=√{144}=12$$

Since triangle DEF is similar to triangle ABC, with vertex F corresponding to vertex C, the measure of $\angle ∠ {F}$ equals the measure of $\angle ∠ {C}$. Therefore, $sin F = sin C$. From the side lengths of triangle ABC,

$$sinF ={\opposite \side}/{\hypotenuse}={AB}/{AC}={12}/{20}={3}/{5}$$

Therefore, $sinF ={3}/{5}$.

The final answer is ${3}/{5}$ or 0.6.

Calculator-Allowed SAT Math Questions

The incomplete table above summarizes the number of left-handed students and right-handed students by gender for the eighth grade students at Keisel Middle School. There are 5 times as many right-handed female students as there are left-handed female students, and there are 9 times as many right-handed male students as there are left-handed male students. if there is a total of 18 left-handed students and 122 right-handed students in the school, which of the following is closest to the probability that a right-handed student selected at random is female? (Note: Assume that none of the eighth-grade students are both right-handed and left-handed.)

A) 0.410 B) 0.357 C) 0.333 D) 0.250

ANSWER EXPLANATION: In order to solve this problem, you should create two equations using two variables ($x$ and $y$) and the information you're given. Let $x$ be the number of left-handed female students and let $y$ be the number of left-handed male students. Using the information given in the problem, the number of right-handed female students will be $5x$ and the number of right-handed male students will be $9y$. Since the total number of left-handed students is 18 and the total number of right-handed students is 122, the system of equations below must be true:

$$x + y = 18$$

$$5x + 9y = 122$$

When you solve this system of equations, you get $x = 10$ and $y = 8$. Thus, 5*10, or 50, of the 122 right-handed students are female. Therefore, the probability that a right-handed student selected at random is female is ${50}/{122}$, which to the nearest thousandth is 0.410.

Questions 8 & 9

Use the following information for both question 7 and question 8.

If shoppers enter a store at an average rate of $r$ shoppers per minute and each stays in the store for average time of $T$ minutes, the average number of shoppers in the store, $N$, at any one time is given by the formula $N=rT$. This relationship is known as Little's law.

The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little's law to estimate that there are 45 shoppers in the store at any time.

Little's law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84 shoppers per hour make a purchase and each of these shoppers spend an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store?

ANSWER EXPLANATION: Since the question states that Little's law can be applied to any single part of the store (for example, just the checkout line), then the average number of shoppers, $N$, in the checkout line at any time is $N = rT$, where $r$ is the number of shoppers entering the checkout line per minute and $T$ is the average number of minutes each shopper spends in the checkout line.

Since 84 shoppers per hour make a purchase, 84 shoppers per hour enter the checkout line. However, this needs to be converted to the number of shoppers per minute (in order to be used with $T = 5$). Since there are 60 minutes in one hour, the rate is ${84 \shoppers \per \hour}/{60 \minutes} = 1.4$ shoppers per minute. Using the given formula with $r = 1.4$ and $T = 5$ yields

$$N = rt = (1.4)(5) = 7$$

Therefore, the average number of shoppers, $N$, in the checkout line at any time during business hours is 7.

The final answer is 7.

The owner of the Good Deals Store opens a new store across town. For the new store, the owner estimates that, during business hours, an average of 90 shoppers per hour enter the store and each of them stays an average of 12 minutes. The average number of shoppers in the new store at any time is what percent less than the average number of shoppers in the original store at any time? (Note: Ignore the percent symbol when entering your answer. For example, if the answer is 42.1%, enter 42.1)

ANSWER EXPLANATION: According to the original information given, the estimated average number of shoppers in the original store at any time (N) is 45. In the question, it states that, in the new store, the manager estimates that an average of 90 shoppers per hour (60 minutes) enter the store, which is equivalent to 1.5 shoppers per minute (r). The manager also estimates that each shopper stays in the store for an average of 12 minutes (T). Thus, by Little's law, there are, on average, $N = rT = (1.5)(12) = 18$ shoppers in the new store at any time. This is

$${45-18}/{45} * 100 = 60$$

percent less than the average number of shoppers in the original store at any time.

The final answer is 60.

Question 10

In the $xy$-plane, the point $(p,r)$ lies on the line with equation $y=x+b$, where $b$ is a constant. The point with coordinates $(2p, 5r)$ lies on the line with equation $y=2x+b$. If $p≠0$, what is the value of $r/p$?

ANSWER EXPLANATION: Since the point $(p,r)$ lies on the line with equation $y=x+b$, the point must satisfy the equation. Substituting $p$ for $x$ and $r$ for $y$ in the equation $y=x+b$ gives $r=p+b$, or $\bi b$ = $\bi r-\bi p$.

Similarly, since the point $(2p,5r)$ lies on the line with the equation $y=2x+b$, the point must satisfy the equation. Substituting $2p$ for $x$ and $5r$ for $y$ in the equation $y=2x+b$ gives:

$5r=2(2p)+b$

$\bi b$ = $\bo 5 \bi r-\bo 4\bi p$.

Next, we can set the two equations equal to $b$ equal to each other and simplify:

$b=r-p=5r-4p$

Finally, to find $r/p$, we need to divide both sides of the equation by $p$ and by $4$:

The correct answer is B , $3/4$.

If you picked choices A and D, you may have incorrectly formed your answer out of the coefficients in the point $(2p, 5r)$. If you picked Choice C, you may have confused $r$ and $p$.

Note that while this is in the calculator section of the SAT, you absolutely do not need your calculator to solve it!

Question 11

A) 261.8 B) 785.4 C) 916.3 D) 1047.2

ANSWER EXPLANATION: The volume of the grain silo can be found by adding the volumes of all the solids of which it is composed (a cylinder and two cones). The silo is made up of a cylinder (with height 10 feet and base radius 5 feet) and two cones (each with height 5 ft and base radius 5 ft). The formulas given at the beginning of the SAT Math section:

Volume of a Cone

$$V={1}/{3}πr^2h$$

Volume of a Cylinder

$$V=πr^2h$$

can be used to determine the total volume of the silo. Since the two cones have identical dimensions, the total volume, in cubic feet, of the silo is given by

$$V_{silo}=π(5^2)(10)+(2)({1}/{3})π(5^2)(5)=({4}/{3})(250)π$$

which is approximately equal to 1,047.2 cubic feet.

Question 12

If $x$ is the average (arithmetic mean) of $m$ and $9$, $y$ is the average of $2m$ and $15$, and $z$ is the average of $3m$ and $18$, what is the average of $x$, $y$, and $z$ in terms of $m$?

A) $m+6$ B) $m+7$ C) $2m+14$ D) $3m + 21$

ANSWER EXPLANATION: Since the average (arithmetic mean) of two numbers is equal to the sum of the two numbers divided by 2, the equations $x={m+9}/{2}$, $y={2m+15}/{2}$, $z={3m+18}/{2}$are true. The average of $x$, $y$, and $z$ is given by ${x + y + z}/{3}$. Substituting the expressions in m for each variable ($x$, $y$, $z$) gives

$$[{m+9}/{2}+{2m+15}/{2}+{3m+18}/{2}]/3$$

This fraction can be simplified to $m + 7$.

Question 13

The function $f(x)=x^3-x^2-x-{11/4}$ is graphed in the $xy$-plane above. If $k$ is a constant such that the equation $f(x)=k$ has three real solutions, which of the following could be the value of $k$?

ANSWER EXPLANATION: The equation $f(x) = k$ gives the solutions to the system of equations

$$y = f(x) = x^3-x^2-x-{11}/{4}$$

A real solution of a system of two equations corresponds to a point of intersection of the graphs of the two equations in the $xy$-plane.

The graph of $y = k$ is a horizontal line that contains the point $(0, k)$ and intersects the graph of the cubic equation three times (since it has three real solutions). Given the graph, the only horizontal line that would intersect the cubic equation three times is the line with the equation $y = −3$, or $f(x) = −3$. Therefore, $k$ is $-3$.

Question 14

$$q={1/2}nv^2$$

The dynamic pressure $q$ generated by a fluid moving with velocity $v$ can be found using the formula above, where $n$ is the constant density of the fluid. An aeronautical engineer users the formula to find the dynamic pressure of a fluid moving with velocity $v$ and the same fluid moving with velocity 1.5$v$. What is the ratio of the dynamic pressure of the faster fluid to the dynamic pressure of the slower fluid?

ANSWER EXPLANATION: To solve this problem, you need to set up to equations with variables. Let $q_1$ be the dynamic pressure of the slower fluid moving with velocity $v_1$, and let $q_2$ be the dynamic pressure of the faster fluid moving with velocity $v_2$. Then

$$v_2 =1.5v_1$$

Given the equation $q = {1}/{2}nv^2$, substituting the dynamic pressure and velocity of the faster fluid gives $q_2 = {1}/{2}n(v_2)^2$. Since $v_2 =1.5v_1$, the expression $1.5v_1$ can be substituted for $v_2$ in this equation, giving $q_2 = {1}/{2}n(1.5v_1)^2$. By squaring $1.5$, you can rewrite the previous equation as

$$q_2 = (2.25)({1}/{2})n(v_1)^2 = (2.25)q_1$$

Therefore, the ratio of the dynamic pressure of the faster fluid is

$${q2}/{q1} = {2.25 q_1}/{q_1}= 2.25$$

The final answer is 2.25 or 9/4.

Question 15

For a polynomial $p(x)$, the value of $p(3)$ is $-2$. Which of the following must be true about $p(x)$?

A) $x-5$ is a factor of $p(x)$. B) $x-2$ is a factor of $p(x)$. C) $x+2$ is a factor of $p(x)$. D) The remainder when $p(x)$ is divided by $x-3$ is $-2$.

ANSWER EXPLANATION: If the polynomial $p(x)$ is divided by a polynomial of the form $x+k$ (which accounts for all of the possible answer choices in this question), the result can be written as

$${p(x)}/{x+k}=q(x)+{r}/{x+k}$$

where $q(x)$ is a polynomial and $r$ is the remainder. Since $x + k$ is a degree-1 polynomial (meaning it only includes $x^1$ and no higher exponents), the remainder is a real number.

Therefore, $p(x)$ can be rewritten as $p(x) = (x + k)q(x) + r$, where $r$ is a real number.

The question states that $p(3) = -2$, so it must be true that

$$-2 = p(3) = (3 + k)q(3) + r$$

Now we can plug in all the possible answers. If the answer is A, B, or C, $r$ will be $0$, while if the answer is D, $r$ will be $-2$.

A. $-2 = p(3) = (3 + (-5))q(3) + 0$ $-2=(3-5)q(3)$ $-2=(-2)q(3)$

This could be true, but only if $q(3)=1$

B. $-2 = p(3) = (3 + (-2))q(3) + 0$ $-2 = (3-2)q(3)$ $-2 = (-1)q(3)$

This could be true, but only if $q(3)=2$

C. $-2 = p(3) = (3 + 2)q(3) + 0$ $-2 = (5)q(3)$

This could be true, but only if $q(3)={-2}/{5}$

D. $-2 = p(3) = (3 + (-3))q(3) + (-2)$ $-2 = (3 - 3)q(3) + (-2)$ $-2 = (0)q(3) + (-2)$

This will always be true no matter what $q(3)$ is.

Of the answer choices, the only one that must be true about $p(x)$ is D, that the remainder when $p(x)$ is divided by $x-3$ is -2.

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You deserve all the naps after running through those questions.

What Do the Hardest SAT Math Questions Have in Common?

It's important to understand what makes these hard questions "hard." By doing so, you'll be able to both understand and solve similar questions when you see them on test day, as well as have a better strategy for identifying and correcting your previous SAT math errors.

In this section, we'll look at what these questions have in common and give examples of each type. Some of the reasons why the hardest math questions are the hardest math questions is because they:

#1: Test Several Mathematical Concepts at Once

Here, we must deal with imaginary numbers and fractions all at once.

Secret to success: Think of what applicable math you could use to solve the problem, do one step at a time, and try each technique until you find one that works!

#2: Involve a Lot of Steps

Remember: the more steps you need to take, the easier to mess up somewhere along the line!

We must solve this problem in steps (doing several averages) to unlock the rest of the answers in a domino effect. This can get confusing, especially if you're stressed or running out of time.

Secret to success: Take it slow, take it step by step, and double-check your work so you don't make mistakes!

#3: Test Concepts That You Have Limited Familiarity With

For example, many students are less familiar with functions than they are with fractions and percentages, so most function questions are considered "high difficulty" problems.

If you don't know your way around functions , this would be a tricky problem.

Secret to success: Review math concepts that you don't have as much familiarity with such as functions . We suggest using our great free SAT Math review guides .

#4: Are Worded in Unusual or Convoluted Ways

It can be difficult to figure out exactly what some questions are asking , much less figure out how to solve them. This is especially true when the question is located at the end of the section, and you are running out of time.

Because this question provides so much information without a diagram, it can be difficult to puzzle through in the limited time allowed.

Secret to success: Take your time, analyze what is being asked of you, and draw a diagram if it's helpful to you.

#5: Use Many Different Variables

With so many different variables in play, it is quite easy to get confused.

Secret to success: Take your time, analyze what is being asked of you, and consider if plugging in numbers is a good strategy to solve the problem (it wouldn't be for the question above, but would be for many other SAT variable questions).

The Take-Aways

The SAT is a marathon and the better prepared you are for it, the better you'll feel on test day. Knowing how to handle the hardest questions the test can throw at you will make taking the real SAT seem a lot less daunting.

If you felt that these questions were easy, make sure not underestimate the effect of adrenaline and fatigue on your ability to solve problems. As you continue to study, always adhere to the proper timing guidelines and try to take full tests whenever possible. This is the best way to recreate the actual testing environment so that you can prepare for the real deal.

If you felt these questions were challenging, be sure to strengthen your math knowledge by checking out our individual math topic guides for the SAT . There, you'll see more detailed explanations of the topics in question as well as more detailed answer breakdowns.

What's Next?

Felt that these questions were harder than you were expecting? Take a look at all the topics covered in the SAT math section and then note which sections were particular difficulty for you. Next, take a gander at our individual math guides to help you shore up any of those weak areas.

Running out of time on the SAT math section? Our guide will help you beat the clock and maximize your score .

Aiming for a perfect score? Check out our guide on how to get a perfect 800 on the SAT math section , written by a perfect-scorer.

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Check out our best-in-class online SAT prep classes . We guarantee your money back if you don't improve your SAT score by 160 points or more.

Our classes are entirely online, and they're taught by SAT experts . If you liked this article, you'll love our classes. Along with expert-led classes, you'll get personalized homework with thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step, custom program to follow so you'll never be confused about what to study next.

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Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology. She is passionate about bringing education and the tools to succeed to students from all backgrounds and walks of life, as she believes open education is one of the great societal equalizers. She has years of tutoring experience and writes creative works in her free time.

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Hard Geometry Problems – Tackling Tough Challenges with Ease

Geometry problems often involve shapes, sizes, positions, and the properties of space. As I delve into the realm of geometry, it’s fascinating to explore the intricate challenges posed by harder problems in this field.

These problems test my understanding of concepts such as congruence, similarity, the Pythagorean Theorem, as well as area and perimeter calculations.

I find that working through these problems hones my analytical skills and enhances my spatial reasoning abilities.

A cluttered desk with scattered textbooks and papers, a calculator, and a ruler. A chalkboard or whiteboard with complex equations and diagrams

Hard geometry problems usually cover a diverse array of topics, from basic triangle rules to complex geometric proofs.

Showcasing mastery in this domain often calls for a blend of creativity and meticulousness, especially as I encounter problems that push the boundaries of my knowledge. For example, proving that two or more shapes are congruent or similar or finding the area of intricate figures can be quite a brain teaser.

The journey through these challenges is not just about finding the right answers; it’s also about appreciating the beauty and precision of geometry itself. Are you ready to join me in unraveling the elegance and complexity of these geometrical puzzles?

Complex Geometry Problems

In my journey through mathematics , I’ve found that complex geometry problems often intimidate students preparing for standardized tests such as the ACT , GRE , and SAT . These problems can involve a variety of geometric figures, from triangles and circles to circular cylinders and squares .

When tackling triangles , I always pay attention to the isosceles and equilateral types. It’s crucial to remember that isosceles triangles have two sides of equal length, and the angles opposite these sides are equal. As for equilateral triangles , all three sides and angles are the same, with each angle measuring $\frac{\pi}{3}$ radians, or $60^\circ$.

Here’s a quick table summarizing triangle rules :

For circles , understanding the terminology is key. The diameter is twice the radius , and a chord that passes through the center of a circle forms a diameter . A semicircle is half of a circle, and when calculating area , I recall the formula $A = \pi r^2$, where $r$ is the radius .

If the problem involves a figure with both a triangle and a semicircle , I check if they share a side or a vertex . This often leads to interesting relationships between angles and sides which are essential to finding a solution .

Although these concepts might seem daunting at first, with practice, solving these types of geometry problems becomes a rewarding and enlightening act.

Solving Geometry Problems

When I tackle geometry problems , I think of them as intricate math puzzles . In my experience, practice is key to becoming proficient.

A student's desk covered in geometry textbooks and papers, with a ruler, compass, and protractor scattered among them

A reliable method I use involves several steps:

Understand the Problem : I carefully read the problem to grasp what’s being asked, especially the value we are trying to find, like the “value of ( x )”.

Draw It Out : I sketch the geometry figure, labeling known measurements and angles, which helps visualize and identify the ASA (Angle-Side-Angle) or other relevant theorems.

Apply Theorems : My familiarity with geometric principles, often refreshed by reading test prep books or resources from Stanford University , comes in handy.

Test Different Approaches : I try different problem-solving techniques, using tools from the Get 800 collection or insights from the MindYourDecisions channel.

Check the Work : I always verify my answers to avoid common mistakes.

Here’s a quick reference I’ve created that might be handy:

The practice doesn’t just involve solving problems from books or test prep books ; it also includes explaining concepts to others, which could be classmates or a teacher .

In my journey of learning and teaching, I’ve found that discussing the process openly in a friendly manner greatly reinforces understanding. As with all things, especially something as logical as geometry, maintaining persistent practice and a can-do attitude is essential in mastering challenging problems.

In grappling with hard geometry problems , I’ve encountered numerous challenges that have pushed my understanding to new heights.

From the intricate relationships between angles to the deep insights required for problem-solving, these problems offer a true test of mathematical skill. Mathematical contests often feature such problems to differentiate between good and exceptional problem solvers.

One thought that stands out is the value of persistence and logic. In facing problems about finding an unknown angle or solving for a particular length, the approach isn’t merely about applying formulas.

It requires a creative combination of geometry principles, sometimes integrating concepts borrowed from other areas of mathematics, such as algebra or trigonometry.

Reflecting on famous problems, like those I’ve come across from various online platforms, it becomes clear just how beautiful and complex geometry can be.

These problems often serve dual purposes, they are not just queries to be answered but lessons that deepen my appreciation and understanding of mathematics. The journey through tough geometry questions is not only about reaching the correct answer but also about appreciating the intricacies of the geometric world.

I encourage my readers to embrace these difficult problems with a sense of adventure. Remember, tackling a geometry problem is more than a test; it’s an opportunity to explore the elegance of mathematics.

Whether you solve a challenging problem on the first try or it takes numerous attempts, each effort enhances your mathematical intuition and prowess.

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120 Math Word Problems To Challenge Students Grades 1 to 8

Written by Marcus Guido

Hey teachers! 👋

Use Prodigy to spark a love for math in your students – including when solving word problems!

Teaching Tools
Subtraction
Multiplication
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You sit at your desk, ready to put a math quiz, test or activity together. The questions flow onto the document until you hit a section for word problems.

A jolt of creativity would help. But it doesn’t come.

Whether you’re a 3rd grade teacher or an 8th grade teacher preparing students for high school, translating math concepts into real world examples can certainly be a challenge.

This resource is your jolt of creativity. It provides examples and templates of math word problems for 1st to 8th grade classes.

There are 120 examples in total.

The list of examples is supplemented by tips to create engaging and challenging math word problems.

120 Math word problems, categorized by skill

Addition word problems.

$A teacher is teaching three students with a whiteboard happily.$

Best for: 1st grade, 2nd grade

1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop. How many shots were there in total?

2. Adding to 20: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store to get 3 more pieces of gum. How many pieces of gum does Adrianna have now?

3. Adding to 100: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store and got 70 pieces of strawberry gum and 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

4. Adding Slightly over 100: The restaurant has 175 normal chairs and 20 chairs for babies. How many chairs does the restaurant have in total?

5. Adding to 1,000: How many cookies did you sell if you sold 320 chocolate cookies and 270 vanilla cookies?

6. Adding to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In June, the hobby store sold 15,498 more trading cards than normal. In total, how many trading cards did the hobby store sell in June?

7. Adding 3 Numbers: Billy had 2 books at home. He went to the library to take out 2 more books. He then bought 1 book. How many books does Billy have now?

8. Adding 3 Numbers to and over 100: Ashley bought a big bag of candy. The bag had 102 blue candies, 100 red candies and 94 green candies. How many candies were there in total?

Subtraction word problems

Best for: 1st grade, second grade

9. Subtracting to 10: There were 3 pizzas in total at the pizza shop. A customer bought 1 pizza. How many pizzas are left?

10. Subtracting to 20: Your friend said she had 11 stickers. When you helped her clean her desk, she only had a total of 10 stickers. How many stickers are missing?

11. Subtracting to 100: Adrianna has 100 pieces of gum to share with her friends. When she went to the park, she shared 10 pieces of strawberry gum. When she left the park, Adrianna shared another 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?

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12. Subtracting Slightly over 100: Your team scored a total of 123 points. 67 points were scored in the first half. How many were scored in the second half?

13. Subtracting to 1,000: Nathan has a big ant farm. He decided to sell some of his ants. He started with 965 ants. He sold 213. How many ants does he have now?

14. Subtracting to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In July, the hobby store sold a total of 20,777 trading cards. How many more trading cards did the hobby store sell in July compared with a normal month?

15. Subtracting 3 Numbers: Charlene had a pack of 35 pencil crayons. She gave 6 to her friend Theresa. She gave 3 to her friend Mandy. How many pencil crayons does Charlene have left?

16. Subtracting 3 Numbers to and over 100: Ashley bought a big bag of candy to share with her friends. In total, there were 296 candies. She gave 105 candies to Marissa. She also gave 86 candies to Kayla. How many candies were left?

Multiplication word problems

A hand holding a pen is doing calculation on a pice of papper

Best for: 2nd grade, 3rd grade

17. Multiplying 1-Digit Integers: Adrianna needs to cut a pan of brownies into pieces. She cuts 6 even columns and 3 even rows into the pan. How many brownies does she have?

18. Multiplying 2-Digit Integers: A movie theatre has 25 rows of seats with 20 seats in each row. How many seats are there in total?

19. Multiplying Integers Ending with 0: A clothing company has 4 different kinds of sweatshirts. Each year, the company makes 60,000 of each kind of sweatshirt. How many sweatshirts does the company make each year?

20. Multiplying 3 Integers: A bricklayer stacks bricks in 2 rows, with 10 bricks in each row. On top of each row, there is a stack of 6 bricks. How many bricks are there in total?

21. Multiplying 4 Integers: Cayley earns $5 an hour by delivering newspapers. She delivers newspapers 3 days each week, for 4 hours at a time. After delivering newspapers for 8 weeks, how much money will Cayley earn?

Division word problems

Best for: 3rd grade, 4th grade, 5th grade

22. Dividing 1-Digit Integers: If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag?

23. Dividing 2-Digit Integers: If you have 80 tickets for the fair and each ride costs 5 tickets, how many rides can you go on?

24. Dividing Numbers Ending with 0: The school has $20,000 to buy new computer equipment. If each piece of equipment costs $50, how many pieces can the school buy in total?

25. Dividing 3 Integers: Melissa buys 2 packs of tennis balls for $12 in total. All together, there are 6 tennis balls. How much does 1 pack of tennis balls cost? How much does 1 tennis ball cost?

26. Interpreting Remainders: An Italian restaurant receives a shipment of 86 veal cutlets. If it takes 3 cutlets to make a dish, how many cutlets will the restaurant have left over after making as many dishes as possible?

Mixed operations word problems

A female teacher is instructing student math on a blackboard

27. Mixing Addition and Subtraction: There are 235 books in a library. On Monday, 123 books are taken out. On Tuesday, 56 books are brought back. How many books are there now?

28. Mixing Multiplication and Division: There is a group of 10 people who are ordering pizza. If each person gets 2 slices and each pizza has 4 slices, how many pizzas should they order?

29. Mixing Multiplication, Addition and Subtraction: Lana has 2 bags with 2 marbles in each bag. Markus has 2 bags with 3 marbles in each bag. How many more marbles does Markus have?

30. Mixing Division, Addition and Subtraction: Lana has 3 bags with the same amount of marbles in them, totaling 12 marbles. Markus has 3 bags with the same amount of marbles in them, totaling 18 marbles. How many more marbles does Markus have in each bag?

Ordering and number sense word problems

31. Counting to Preview Multiplication: There are 2 chalkboards in your classroom. If each chalkboard needs 2 pieces of chalk, how many pieces do you need in total?

32. Counting to Preview Division: There are 3 chalkboards in your classroom. Each chalkboard has 2 pieces of chalk. This means there are 6 pieces of chalk in total. If you take 1 piece of chalk away from each chalkboard, how many will there be in total?

33. Composing Numbers: What number is 6 tens and 10 ones?

34. Guessing Numbers: I have a 7 in the tens place. I have an even number in the ones place. I am lower than 74. What number am I?

35. Finding the Order: In the hockey game, Mitchell scored more points than William but fewer points than Auston. Who scored the most points? Who scored the fewest points?

Fractions word problems

A student is drawing on a notebook, holding a pencil.

Best for: 3rd grade, 4th grade, 5th grade, 6th grade

36. Finding Fractions of a Group: Julia went to 10 houses on her street for Halloween. 5 of the houses gave her a chocolate bar. What fraction of houses on Julia’s street gave her a chocolate bar?

37. Finding Unit Fractions: Heather is painting a portrait of her best friend, Lisa. To make it easier, she divides the portrait into 6 equal parts. What fraction represents each part of the portrait?

38. Adding Fractions with Like Denominators: Noah walks ⅓ of a kilometre to school each day. He also walks ⅓ of a kilometre to get home after school. How many kilometres does he walk in total?

39. Subtracting Fractions with Like Denominators: Last week, Whitney counted the number of juice boxes she had for school lunches. She had ⅗ of a case. This week, it’s down to ⅕ of a case. How much of the case did Whitney drink?

40. Adding Whole Numbers and Fractions with Like Denominators: At lunchtime, an ice cream parlor served 6 ¼ scoops of chocolate ice cream, 5 ¾ scoops of vanilla and 2 ¾ scoops of strawberry. How many scoops of ice cream did the parlor serve in total?

41. Subtracting Whole Numbers and Fractions with Like Denominators: For a party, Jaime had 5 ⅓ bottles of cola for her friends to drink. She drank ⅓ of a bottle herself. Her friends drank 3 ⅓. How many bottles of cola does Jaime have left?

42. Adding Fractions with Unlike Denominators: Kevin completed ½ of an assignment at school. When he was home that evening, he completed ⅚ of another assignment. How many assignments did Kevin complete?

43. Subtracting Fractions with Unlike Denominators: Packing school lunches for her kids, Patty used ⅞ of a package of ham. She also used ½ of a package of turkey. How much more ham than turkey did Patty use?

44. Multiplying Fractions: During gym class on Wednesday, the students ran for ¼ of a kilometre. On Thursday, they ran ½ as many kilometres as on Wednesday. How many kilometres did the students run on Thursday? Write your answer as a fraction.

45. Dividing Fractions: A clothing manufacturer uses ⅕ of a bottle of colour dye to make one pair of pants. The manufacturer used ⅘ of a bottle yesterday. How many pairs of pants did the manufacturer make?

46. Multiplying Fractions with Whole Numbers: Mark drank ⅚ of a carton of milk this week. Frank drank 7 times more milk than Mark. How many cartons of milk did Frank drink? Write your answer as a fraction, or as a whole or mixed number.

Decimals word problems

Best for: 4th grade, 5th grade

47. Adding Decimals: You have 2.6 grams of yogurt in your bowl and you add another spoonful of 1.3 grams. How much yogurt do you have in total?

48. Subtracting Decimals: Gemma had 25.75 grams of frosting to make a cake. She decided to use only 15.5 grams of the frosting. How much frosting does Gemma have left?

49. Multiplying Decimals with Whole Numbers: Marshall walks a total of 0.9 kilometres to and from school each day. After 4 days, how many kilometres will he have walked?

50. Dividing Decimals by Whole Numbers: To make the Leaning Tower of Pisa from spaghetti, Mrs. Robinson bought 2.5 kilograms of spaghetti. Her students were able to make 10 leaning towers in total. How many kilograms of spaghetti does it take to make 1 leaning tower?

51. Mixing Addition and Subtraction of Decimals: Rocco has 1.5 litres of orange soda and 2.25 litres of grape soda in his fridge. Antonio has 1.15 litres of orange soda and 0.62 litres of grape soda. How much more soda does Rocco have than Angelo?

52. Mixing Multiplication and Division of Decimals: 4 days a week, Laura practices martial arts for 1.5 hours. Considering a week is 7 days, what is her average practice time per day each week?

Comparing and sequencing word problems

Four students are sitting together and discussing math questions

Best for: Kindergarten, 1st grade, 2nd grade

53. Comparing 1-Digit Integers: You have 3 apples and your friend has 5 apples. Who has more?

54. Comparing 2-Digit Integers: You have 50 candies and your friend has 75 candies. Who has more?

55. Comparing Different Variables: There are 5 basketballs on the playground. There are 7 footballs on the playground. Are there more basketballs or footballs?

56. Sequencing 1-Digit Integers: Erik has 0 stickers. Every day he gets 1 more sticker. How many days until he gets 3 stickers?

57. Skip-Counting by Odd Numbers: Natalie began at 5. She skip-counted by fives. Could she have said the number 20?

58. Skip-Counting by Even Numbers: Natasha began at 0. She skip-counted by eights. Could she have said the number 36?

59. Sequencing 2-Digit Numbers: Each month, Jeremy adds the same number of cards to his baseball card collection. In January, he had 36. 48 in February. 60 in March. How many baseball cards will Jeremy have in April?

Time word problems

66. Converting Hours into Minutes: Jeremy helped his mom for 1 hour. For how many minutes was he helping her?

69. Adding Time: If you wake up at 7:00 a.m. and it takes you 1 hour and 30 minutes to get ready and walk to school, at what time will you get to school?

70. Subtracting Time: If a train departs at 2:00 p.m. and arrives at 4:00 p.m., how long were passengers on the train for?

71. Finding Start and End Times: Rebecca left her dad’s store to go home at twenty to seven in the evening. Forty minutes later, she was home. What time was it when she arrived home?

Money word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade

60. Adding Money: Thomas and Matthew are saving up money to buy a video game together. Thomas has saved $30. Matthew has saved $35. How much money have they saved up together in total?

61. Subtracting Money: Thomas has $80 saved up. He uses his money to buy a video game. The video game costs $67. How much money does he have left?

62. Multiplying Money: Tim gets $5 for delivering the paper. How much money will he have after delivering the paper 3 times?

63. Dividing Money: Robert spent $184.59 to buy 3 hockey sticks. If each hockey stick was the same price, how much did 1 cost?

64. Adding Money with Decimals: You went to the store and bought gum for $1.25 and a sucker for $0.50. How much was your total?

65. Subtracting Money with Decimals: You went to the store with $5.50. You bought gum for $1.25, a chocolate bar for $1.15 and a sucker for $0.50. How much money do you have left?

67. Applying Proportional Relationships to Money: Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents? Assume the relationship is directly proportional.

68. Applying Percentages to Money: Retta put $100.00 in a bank account that gains 20% interest annually. How much interest will be accumulated in 1 year? And if she makes no withdrawals, how much money will be in the account after 1 year?

Physical measurement word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade

72. Comparing Measurements: Cassandra’s ruler is 22 centimetres long. April’s ruler is 30 centimetres long. How many centimetres longer is April’s ruler?

73. Contextualizing Measurements: Picture a school bus. Which unit of measurement would best describe the length of the bus? Centimetres, metres or kilometres?

74. Adding Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Last year, Micha’s dad used 100 litres of gas. This year, her dad used 90 litres of gas. How much gas did he use in total for the two years?

75. Subtracting Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Over the past two years, Micha’s dad used 200 litres of gas. This year, he used 100 litres of gas. How much gas did he use last year?

A tablet showing an example of Prodigy Math's battle gameplay.

76. Multiplying Volume and Mass: Kiera wants to make sure she has strong bones, so she drinks 2 litres of milk every week. After 3 weeks, how many litres of milk will Kiera drink?

77. Dividing Volume and Mass: Lillian is doing some gardening, so she bought 1 kilogram of soil. She wants to spread the soil evenly between her 2 plants. How much will each plant get?

78. Converting Mass: Inger goes to the grocery store and buys 3 squashes that each weigh 500 grams. How many kilograms of squash did Inger buy?

79. Converting Volume: Shad has a lemonade stand and sold 20 cups of lemonade. Each cup was 500 millilitres. How many litres did Shad sell in total?

80. Converting Length: Stacy and Milda are comparing their heights. Stacy is 1.5 meters tall. Milda is 10 centimetres taller than Stacy. What is Milda’s height in centimetres?

81. Understanding Distance and Direction: A bus leaves the school to take students on a field trip. The bus travels 10 kilometres south, 10 kilometres west, another 5 kilometres south and 15 kilometres north. To return to the school, in which direction does the bus have to travel? How many kilometres must it travel in that direction?

Ratios and percentages word problems

Best for: 4th grade, 5th grade, 6th grade

82. Finding a Missing Number: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. Jenny has 28 trophies. How many does Meredith have?

83. Finding Missing Numbers: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. The difference between the numbers is 12. What are the numbers?

84. Comparing Ratios: The school’s junior band has 10 saxophone players and 20 trumpet players. The school’s senior band has 18 saxophone players and 29 trumpet players. Which band has the higher ratio of trumpet to saxophone players?

85. Determining Percentages: Mary surveyed students in her school to find out what their favourite sports were. Out of 1,200 students, 455 said hockey was their favourite sport. What percentage of students said hockey was their favourite sport?

86. Determining Percent of Change: A decade ago, Oakville’s population was 67,624 people. Now, it is 190% larger. What is Oakville’s current population?

87. Determining Percents of Numbers: At the ice skate rental stand, 60% of 120 skates are for boys. If the rest of the skates are for girls, how many are there?

88. Calculating Averages: For 4 weeks, William volunteered as a helper for swimming classes. The first week, he volunteered for 8 hours. He volunteered for 12 hours in the second week, and another 12 hours in the third week. The fourth week, he volunteered for 9 hours. For how many hours did he volunteer per week, on average?

Probability and data relationships word problems

$Two students are calculating on a whiteboard$

Best for: 4th grade, 5th grade, 6th grade, 7th grade

89. Understanding the Premise of Probability: John wants to know his class’s favourite TV show, so he surveys all of the boys. Will the sample be representative or biased?

90. Understanding Tangible Probability: The faces on a fair number die are labelled 1, 2, 3, 4, 5 and 6. You roll the die 12 times. How many times should you expect to roll a 1?

91. Exploring Complementary Events: The numbers 1 to 50 are in a hat. If the probability of drawing an even number is 25/50, what is the probability of NOT drawing an even number? Express this probability as a fraction.

92. Exploring Experimental Probability: A pizza shop has recently sold 15 pizzas. 5 of those pizzas were pepperoni. Answering with a fraction, what is the experimental probability that he next pizza will be pepperoni?

93. Introducing Data Relationships: Maurita and Felice each take 4 tests. Here are the results of Maurita’s 4 tests: 4, 4, 4, 4. Here are the results for 3 of Felice’s 4 tests: 3, 3, 3. If Maurita’s mean for the 4 tests is 1 point higher than Felice’s, what’s the score of Felice’s 4th test?

94. Introducing Proportional Relationships: Store A is selling 7 pounds of bananas for $7.00. Store B is selling 3 pounds of bananas for $6.00. Which store has the better deal?

95. Writing Equations for Proportional Relationships: Lionel loves soccer, but has trouble motivating himself to practice. So, he incentivizes himself through video games. There is a proportional relationship between the amount of drills Lionel completes, in x , and for how many hours he plays video games, in y . When Lionel completes 10 drills, he plays video games for 30 minutes. Write the equation for the relationship between x and y .

Geometry word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade, 8th grade

96. Introducing Perimeter: The theatre has 4 chairs in a row. There are 5 rows. Using rows as your unit of measurement, what is the perimeter?

97. Introducing Area: The theatre has 4 chairs in a row. There are 5 rows. How many chairs are there in total?

98. Introducing Volume: Aaron wants to know how much candy his container can hold. The container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. What is the container’s volume?

99. Understanding 2D Shapes: Kevin draws a shape with 4 equal sides. What shape did he draw?

100. Finding the Perimeter of 2D Shapes: Mitchell wrote his homework questions on a piece of square paper. Each side of the paper is 8 centimetres. What is the perimeter?

101. Determining the Area of 2D Shapes: A single trading card is 9 centimetres long by 6 centimetres wide. What is its area?

102. Understanding 3D Shapes: Martha draws a shape that has 6 square faces. What shape did she draw?

103. Determining the Surface Area of 3D Shapes: What is the surface area of a cube that has a width of 2cm, height of 2 cm and length of 2 cm?

104. Determining the Volume of 3D Shapes: Aaron’s candy container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. Bruce’s container is 25 centimetres tall, 9 centimetres long and 9 centimetres wide. Find the volume of each container. Based on volume, whose container can hold more candy?

105. Identifying Right-Angled Triangles: A triangle has the following side lengths: 3 cm, 4 cm and 5 cm. Is this triangle a right-angled triangle?

106. Identifying Equilateral Triangles: A triangle has the following side lengths: 4 cm, 4 cm and 4 cm. What kind of triangle is it?

107. Identifying Isosceles Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 5 cm. What kind of triangle is it?

108. Identifying Scalene Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 6 cm. What kind of triangle is it?

109. Finding the Perimeter of Triangles: Luigi built a tent in the shape of an equilateral triangle. The perimeter is 21 metres. What is the length of each of the tent’s sides?

110. Determining the Area of Triangles: What is the area of a triangle with a base of 2 units and a height of 3 units?

111. Applying Pythagorean Theorem: A right triangle has one non-hypotenuse side length of 3 inches and the hypotenuse measures 5 inches. What is the length of the other non-hypotenuse side?

112. Finding a Circle’s Diameter: Jasmin bought a new round backpack. Its area is 370 square centimetres. What is the round backpack’s diameter?

113. Finding a Circle's Area: Captain America’s circular shield has a diameter of 76.2 centimetres. What is the area of his shield?

114. Finding a Circle’s Radius: Skylar lives on a farm, where his dad keeps a circular corn maze. The corn maze has a diameter of 2 kilometres. What is the maze’s radius?

Variables word problems

$A hand is calculating math problem on a blacboard$

Best for: 6th grade, 7th grade, 8th grade

115. Identifying Independent and Dependent Variables: Victoria is baking muffins for her class. The number of muffins she makes is based on how many classmates she has. For this equation, m is the number of muffins and c is the number of classmates. Which variable is independent and which variable is dependent?

116. Writing Variable Expressions for Addition: Last soccer season, Trish scored g goals. Alexa scored 4 more goals than Trish. Write an expression that shows how many goals Alexa scored.

117. Writing Variable Expressions for Subtraction: Elizabeth eats a healthy, balanced breakfast b times a week. Madison sometimes skips breakfast. In total, Madison eats 3 fewer breakfasts a week than Elizabeth. Write an expression that shows how many times a week Madison eats breakfast.

118. Writing Variable Expressions for Multiplication: Last hockey season, Jack scored g goals. Patrik scored twice as many goals than Jack. Write an expression that shows how many goals Patrik scored.

119. Writing Variable Expressions for Division: Amanda has c chocolate bars. She wants to distribute the chocolate bars evenly among 3 friends. Write an expression that shows how many chocolate bars 1 of her friends will receive.

120. Solving Two-Variable Equations: This equation shows how the amount Lucas earns from his after-school job depends on how many hours he works: e = 12h . The variable h represents how many hours he works. The variable e represents how much money he earns. How much money will Lucas earn after working for 6 hours?

How to easily make your own math word problems & word problems worksheets

Two teachers are discussing math with a pen and a notebook

Armed with 120 examples to spark ideas, making your own math word problems can engage your students and ensure alignment with lessons. Do:

Link to Student Interests: By framing your word problems with student interests, you’ll likely grab attention. For example, if most of your class loves American football, a measurement problem could involve the throwing distance of a famous quarterback.
Make Questions Topical: Writing a word problem that reflects current events or issues can engage students by giving them a clear, tangible way to apply their knowledge.
Include Student Names: Naming a question’s characters after your students is an easy way make subject matter relatable, helping them work through the problem.
Be Explicit: Repeating keywords distills the question, helping students focus on the core problem.
Test Reading Comprehension: Flowery word choice and long sentences can hide a question’s key elements. Instead, use concise phrasing and grade-level vocabulary.
Focus on Similar Interests: Framing too many questions with related interests -- such as football and basketball -- can alienate or disengage some students.
Feature Red Herrings: Including unnecessary information introduces another problem-solving element, overwhelming many elementary students.

A key to differentiated instruction , word problems that students can relate to and contextualize will capture interest more than generic and abstract ones.

Final thoughts about math word problems

You’ll likely get the most out of this resource by using the problems as templates, slightly modifying them by applying the above tips. In doing so, they’ll be more relevant to -- and engaging for -- your students.

Regardless, having 120 curriculum-aligned math word problems at your fingertips should help you deliver skill-building challenges and thought-provoking assessments.

The result?

A greater understanding of how your students process content and demonstrate understanding, informing your ongoing teaching approach.

Why Is Math So Hard? + Ways to Make It Easier

Math is often considered one of the most challenging subjects students learn. Between equations and multi-step word problems, it’s easy to simply give up.

But here’s a hot take: Learning math doesn't have to be a frustrating experience—even when that learning happens after you're out of school. With the right strategies and tools, math can be easy… and even enjoyable . So keep reading to learn about why math can be hard, strategies for making it easier, and how the Elevate app can help.

Why is Math So Hard?

Students struggle with math for a variety of reasons—some of them may resonate with you whether you're still a student or not.

One of the most common reasons people struggle with math is that math involves abstract concepts that can be pretty difficult to understand. Unlike other subjects that are more concrete, math deals with numbers, symbols, and equations that can be difficult to grasp. (Plus, if you’re an auditory learner, you might find it challenging to visualize and understand math.)

Another reason for difficulty in math is a potentially weak foundation established in elementary school and middle school math classes. No shade to your past math teachers, but math concepts build upon themselves, so if you haven’t fully grasped the fundamental concepts early on, it might be difficult to keep up with learning more complex topics.

Fear and anxiety can also play a role in the perceived difficulty of math. You might have previously experienced negative feedback or discouragement from teachers or peers after solving math problems, and chances are that’s led to doubt your abilities in the subject.

And finally, cultural and societal attitudes toward math can contribute to its perceived difficulty. Some people view math as a subject that only certain people are capable of excelling in, such as those who are naturally gifted at it. This mindset can discourage you from pursuing math or cause you to feel like a failure if you don't understand it right away.

The Role of Cognitive Processes in Understanding Math

Now. let’s take a closer look inward. How does your brain process math, exactly?

The short answer: Your brain processes mathematical information through a variety of cognitive processes. Let’s break them down:

One important process is logical reasoning , which involves identifying patterns and relationships between numbers and symbols. Logical reasoning helps you make sense of mathematical concepts and solve problems by breaking them down into smaller, more manageable parts.
Another cognitive skill involved in learning math is problem-solving . This skill involves using logic and critical thinking to identify the steps required to solve a particular math problem. Effective problem-solving requires you to understand the underlying concepts related to the problem, as well as the rules and procedures for manipulating numbers and symbols.
In addition to these skills, critical thinking is also important for learning and understanding math. Critical thinking involves evaluating information, identifying assumptions, and making logical deductions based on evidence. In math, critical thinking is essential for analyzing complex problems and developing creative solutions.
Finally, working memory plays a crucial role in math learning and understanding. Meaning, you must be able to hold multiple pieces of information in your mind simultaneously in order to solve problems and understand concepts. Developing your working memory capacity can help you improve your ability to tackle complex math problems. ( Lucky for you, we have some tips to help you do that .)

Overall, understanding how your brain process mathematical information is essential for effective math learning and understanding. By developing skills such as logical reasoning, problem-solving, critical thinking, and working memory capacity, you can improve your ability to succeed in this important subject.

How to Make Math Easier

Now that you know more about your brain processes mathematical information, it’s time to put it to work. But don’t worry, these two practical strategies are sure to make math a little easier—and less daunting:

1. Practice and Consistency is Key

You know what they say: Practice makes perfect. And while it might seem like a no-brainer, practice and consistency are two of the most effective ways to improve math skills . By practicing math regularly, you can develop a better understanding of mathematical concepts, improve your problem-solving skills, build your confidence in your ability to solve problems, retain information, and build on your previous knowledge.

2. Use A Step-by-Step Approach

Another strategy for making math easier is to use a step-by-step approach to solving complex problems. This involves breaking down the problem into smaller parts, identifying the key information—the “why”—that’s needed to solve the problem, and using visual aids or diagrams to help you understand the problem. (Remember when your math teacher used to instruct you to draw dots on numbers’ endpoints to help you practice counting?)

While these two strategies are great starting points, there are many other resources and tools available to aid in learning math and making it a little easier. Math tutors can provide one-on-one support and personalized instruction tailored to your needs. Plus, online resources such as Khan Academy or Mathway provide access to instructional videos, practice problems, and step-by-step solutions for various types of math problems.

Harnessing the Power of the Elevate App to Improve Your Math Skills

To practice building your practical math skills in a fun and low-stakes setting, you can also use certain mobile apps. One popular brain training app that incorporates a wide variety of math games is Elevate.

With more than 40 games spread across math skill groups, reading, writing, speaking, and memory, the Elevate app offers personalized training programs based on your goals—like improving your math skills—and it adapts to your skill level and performance over time. And it makes learning math much more fun than it was in high school!

Elevate's Top Brain Games for Learning Math

The Elevate app includes several brain games designed to enhance and improve your math skills, testing your knowledge in areas like multiplication, percentages, addition, subtraction, and even budgeting. Plus, you can track your progress right in the app.

Subtraction is a great way to strengthen your mental math.
Percentages teaches you how to easily calculate both simple and complex percentages.
Discounting lets you practice rapidly organizing prices from least to most expensive with taxes and discounts.
Averages teaches you how to quickly and accurately calculate the average of a set of numbers.
Tipping teaches you how to rapidly calculate common tip percentages.
Purchasing helps you calculate common spending habits, so you can be more confident in your purchasing decisions.
Measuring teaches you how to use mental math for things like cooking and construction.
Estimation helps you increase the accuracy of your mental calculations.
Division helps you more accurately divide a set of numbers, you can more accurately split bills and other household expenses.
Equivalence teaches you tips for breaking down complex fractions.
Conversion helps you easily convert weights and distances for cooking, travel, and more.
Percentages teaches you how to calculate markups and discounts in your head.

Real Success Stories: Users' Math Improvements with Elevate

Many people who use Elevate have seen big improvements in their math skills, as evidenced by this sampling of the app’s reviews:

“I needed some serious work on my math abilities. Now I’m at the top of my class in math all because of this app. I seriously believe you should try it.”
“I use this almost every day. I believe it keeps my mind sharp. I also feel a sense of accomplishment when I have completed the day's training and I’m seeing progress in math!”
“I have an enormous math phobia that this app is actually helping me deal with. I volunteer once a week at a place where I close out the cash register and add up a bunch of bills and coins. Previously, I’d use my calculator to do the brain work, but now I’m to a point where I can do some of the calculations in my head!”

How to Overcome Math Anxiety

As previously mentioned, math anxiety can be a significant obstacle if you’re looking to improve your math skills. But understanding the causes and effects of math anxiety is essential to overcoming it.

Understand the Causes and Effects of Math Anxiety

Math anxiety can be caused by several factors, including a lack of confidence in math abilities, fear of failure, and negative experiences with math in the past. If you struggle with math, you might know the feeling of overwhelm or frustration by complex problems all too well.

The effects of math anxiety can be severe, including decreased performance on math tests and assignments, as well as a lack of interest in math for more practical reasons, such as tipping or budgeting.

Cultivate a Math-Friendly Mindset

You can try several practical tips to overcome your math anxiety.

Practicing relaxation techniques such as deep breathing or visualization can help calm your nerves and reduce anxiety. Breaking down complex problems into smaller parts can also make math easier to understand and help you feel less overwhelmed.

And if you’re willing to go the extra mile, seeking help from math teachers or tutors is another effective way to overcome your math anxiety. A tutor can provide one-on-one support and instruction tailored to your needs, helping you build up your confidence.

Additionally, reframing any negative attitudes toward math and focusing on its practical applications can help you develop a more positive mindset toward the subject. For example, understanding algebra can help with budgeting and financial planning. Algebraic equations can be used to calculate interest rates, loan payments, and other financial calculations to help you build your dream life. By focusing on these practical applications, you might become more motivated to learn algebra and develop a more positive attitude toward the subject.

Overall, overcoming math anxiety requires both understanding its causes and effects and developing practical strategies for cultivating a positive mindset toward math. By implementing these strategies and seeking support when needed, you can improve your math skills. You’ve got this!

Start Improving Your Math Skills Today

It’s no secret: Math is challenging . But by learning practical strategies and tools and shifting your perspective about math, you can make it easier… and even a little fun. One way to build practical math skills is by downloading the Elevate brain training app on iOS or Android . It has fun games specifically dedicated to improving your practical math skills. Give it a try for free today!

How to Easily Tip Without a Calculator

Many of us are overly reliant on phone calculators. Learn how to calculate a tip with mental math.

Working Memory Vs. Short-Term Memory

Learn the differences between working memory and short-term memory and tips for improving them.

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How do you solve word problems in math?

Master word problems with eight simple steps from a math tutor!

Author Amber Watkins

Published April 2024

Key takeaways
Students who struggle with reading, tend to struggle with understanding and solving word problems. So the best way to solve word problems in math is to become a better reader!
Mastery of word problems relies on your child’s knowledge of keywords for word problems in math and knowing what to do with them.
There are 8 simple steps each child can use to solve word problems- let’s go over these together.

Table of contents

How to solve word problems

Lesson credits

As a tutor who has seen countless math worksheets in almost every grade – I’ll tell you this: every child is going to encounter word problems in math. The key to mastery lies in how you solve them! So then, how do you solve word problems in math?

In this guide, I’ll share eight steps to solving word problems in math.

How to solve word problems in math in 8 steps

Step 1: read the word problem aloud.

For a child to understand a word problem, it needs to be read with accuracy and fluency! That is why, when I tutor children with word problems, I always emphasize the importance of reading properly.

Mastering step 1 looks like this:

Allow your child to read the word problem aloud to you.
Don’t let your child skip over or mispronounce any words.
If necessary, model how to read the word problem, then allow your child to read it again. Only after the word problem is read accurately, should you move on to step 2.

Step 2: Highlight the keywords in the word problem

The keywords for word problems in math indicate what math action should be taken. Teach your child to highlight or underline the keywords in every word problem.

Here are some of the most common keywords in math word problems:

Subtraction words – less than, minus, take away
Addition words – more than, altogether, plus, perimeter
Multiplication words – Each, per person, per item, times, area
Division words – divided by, into
Total words – in all, total, altogether

Let’s practice. Read the following word problem with your child and help them highlight or underline the main keyword, then decide which math action should be taken.

Michael has ten baseball cards. James has four baseball cards less than Michael. How many total baseball cards does James have?

The words “less than” are the keywords and they tell us to use subtraction .

Step 3: Make math symbols above keywords to decode the word problem

As I help students with word problems, I write math symbols and numbers above the keywords. This helps them to understand what the word problem is asking.

Let’s practice. Observe what I write over the keywords in the following word problem and think about how you would create a math sentence using them:

Step 4: Create a math sentence to represent the word problem

Using the previous example, let’s write a math sentence. Looking at the math symbols and numbers written above the word problem, our math sentence should be: 10 – 5 = 5 !

Each time you practice a word problem with your child, highlight keywords and write the math symbols above them. Then have your child create a math sentence to solve.

Step 5: Draw a picture to help illustrate the word problem

Pictures can be very helpful for problems that are more difficult to understand. They also are extremely helpful when the word problem involves calculating time , comparing fractions , or measurements .

Step 6: Always show your work

Help your child get into the habit of always showing their work. As a tutor, I’ve found many reasons why having students show their work is helpful:

By showing their work, they are writing the math steps repeatedly, which aids in memory
If they make any mistakes they can track where they happened
Their teacher can assess how much they understand by reviewing their work
They can participate in class discussions about their work

Step 7: When solving word problems, make sure there is always a word in your answer!

If the word problem asks: How many peaches did Lisa buy? Your child’s answer should be: Lisa bought 10 peaches .

If the word problem asks: How far did Kyle run? Your child’s answer should be: Kyle ran 20 miles .

So how do you solve a word problem in math?

Together we reviewed the eight simple steps to solve word problems. These steps included identifying keywords for word problems in math, drawing pictures, and learning to explain our answers.

Is your child ready to put these new skills to the test? Check out the best math app for some fun math word problem practice.

Parents, sign up for a DoodleMath subscription and see your child become a math wizard!

Amber Watkins

Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring elementary through college level math. "Knowing that my work in math education makes such an impact leaves me with an indescribable feeling of pride and joy!"

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March 12, 2024

The Simplest Math Problem Could Be Unsolvable

The Collatz conjecture has plagued mathematicians for decades—so much so that professors warn their students away from it

By Manon Bischoff

Close up of lightbulb sparkling with teal color outline on black background

Mathematicians have been hoping for a flash of insight to solve the Collatz conjecture.

James Brey/Getty Images

At first glance, the problem seems ridiculously simple. And yet experts have been searching for a solution in vain for decades. According to mathematician Jeffrey Lagarias, number theorist Shizuo Kakutani told him that during the cold war, “for about a month everybody at Yale [University] worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.”

The Collatz conjecture—the vexing puzzle Kakutani described—is one of those supposedly simple problems that people tend to get lost in. For this reason, experienced professors often warn their ambitious students not to get bogged down in it and lose sight of their actual research.

The conjecture itself can be formulated so simply that even primary school students understand it. Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x : if x is odd, you calculate 3 x + 1; otherwise calculate x / 2. Repeat these instructions as many times as possible, and, according to the conjecture, you will always end up with the number 1.

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For example: If you start with 5, you have to calculate 5 x 3 + 1, which results in 16. Because 16 is an even number, you have to halve it, which gives you 8. Then 8 / 2 = 4, which, when divided by 2, is 2—and 2 / 2 = 1. The process of iterative calculation brings you to the end after five steps.

Of course, you can also continue calculating with 1, which gives you 4, then 2 and then 1 again. The calculation rule leads you into an inescapable loop. Therefore 1 is seen as the end point of the procedure.

Bubbles with numbers and arrows show Collatz conjecture sequences

Following iterative calculations, you can begin with any of the numbers above and will ultimately reach 1.

Credit: Keenan Pepper/Public domain via Wikimedia Commons

It’s really fun to go through the iterative calculation rule for different numbers and look at the resulting sequences. If you start with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Or 42: 42 → 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1. No matter which number you start with, you always seem to end up with 1. There are some numbers, such as 27, where it takes quite a long time (27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → ...), but so far the result has always been 1. (Admittedly, you have to be patient with the starting number 27, which requires 111 steps.)

But strangely there is still no mathematical proof that the Collatz conjecture is true. And that absence has mystified mathematicians for years.

The origin of the Collatz conjecture is uncertain, which is why this hypothesis is known by many different names. Experts speak of the Syracuse problem, the Ulam problem, the 3 n + 1 conjecture, the Hasse algorithm or the Kakutani problem.

German mathematician Lothar Collatz became interested in iterative functions during his mathematics studies and investigated them. In the early 1930s he also published specialist articles on the subject , but the explicit calculation rule for the problem named after him was not among them. In the 1950s and 1960s the Collatz conjecture finally gained notoriety when mathematicians Helmut Hasse and Shizuo Kakutani, among others, disseminated it to various universities, including Syracuse University.

Like a siren song, this seemingly simple conjecture captivated the experts. For decades they have been looking for proof that after repeating the Collatz procedure a finite number of times, you end up with 1. The reason for this persistence is not just the simplicity of the problem: the Collatz conjecture is related to other important questions in mathematics. For example, such iterative functions appear in dynamic systems, such as models that describe the orbits of planets. The conjecture is also related to the Riemann conjecture, one of the oldest problems in number theory.

Empirical Evidence for the Collatz Conjecture

In 2019 and 2020 researchers checked all numbers below 2 68 , or about 3 x 10 20 numbers in the sequence, in a collaborative computer science project . All numbers in that set fulfill the Collatz conjecture as initial values. But that doesn’t mean that there isn’t an outlier somewhere. There could be a starting value that, after repeated Collatz procedures, yields ever larger values that eventually rise to infinity. This scenario seems unlikely, however, if the problem is examined statistically.

An odd number n is increased to 3 n + 1 after the first step of the iteration, but the result is inevitably even and is therefore halved in the following step. In half of all cases, the halving produces an odd number, which must therefore be increased to 3 n + 1 again, whereupon an even result is obtained again. If the result of the second step is even again, however, you have to divide the new number by 2 twice in every fourth case. In every eighth case, you must divide it by 2 three times, and so on.

In order to evaluate the long-term behavior of this sequence of numbers , Lagarias calculated the geometric mean from these considerations in 1985 and obtained the following result: ( 3 / 2 ) 1/2 x ( 3 ⁄ 4 ) 1/4 x ( 3 ⁄ 8 ) 1/8 · ... = 3 ⁄ 4 . This shows that the sequence elements shrink by an average factor of 3 ⁄ 4 at each step of the iterative calculation rule. It is therefore extremely unlikely that there is a starting value that grows to infinity as a result of the procedure.

There could be a starting value, however, that ends in a loop that is not 4 → 2 → 1. That loop could include significantly more numbers, such that 1 would never be reached.

Such “nontrivial” loops can be found, for example, if you also allow negative integers for the Collatz conjecture: in this case, the iterative calculation rule can end not only at –2 → –1 → –2 → ... but also at –5 → –14 → –7 → –20 → –10 → –5 → ... or –17 → –50 → ... → –17 →.... If we restrict ourselves to natural numbers, no nontrivial loops are known to date—which does not mean that they do not exist. Experts have now been able to show that such a loop in the Collatz problem, however, would have to consist of at least 186 billion numbers .

A plot lays out the starting number of the Collatz sequence on the x-axis with the total length of the completed sequence on the y-axis

The length of the Collatz sequences for all numbers from 1 to 9,999 varies greatly.

Credit: Cirne/Public domain via Wikimedia Commons

Even if that sounds unlikely, it doesn’t have to be. In mathematics there are many examples where certain laws only break down after many iterations are considered. For instance,the prime number theorem overestimates the number of primes for only about 10 316 numbers. After that point, the prime number set underestimates the actual number of primes.

Something similar could occur with the Collatz conjecture: perhaps there is a huge number hidden deep in the number line that breaks the pattern observed so far.

A Proof for Almost All Numbers

Mathematicians have been searching for a conclusive proof for decades. The greatest progress was made in 2019 by Fields Medalist Terence Tao of the University of California, Los Angeles, when he proved that almost all starting values of natural numbers eventually end up at a value close to 1.

“Almost all” has a precise mathematical meaning: if you randomly select a natural number as a starting value, it has a 100 percent probability of ending up at 1. ( A zero-probability event, however, is not necessarily an impossible one .) That’s “about as close as one can get to the Collatz conjecture without actually solving it,” Tao said in a talk he gave in 2020 . Unfortunately, Tao’s method cannot generalize to all figures because it is based on statistical considerations.

All other approaches have led to a dead end as well. Perhaps that means the Collatz conjecture is wrong. “Maybe we should be spending more energy looking for counterexamples than we’re currently spending,” said mathematician Alex Kontorovich of Rutgers University in a video on the Veritasium YouTube channel .

Perhaps the Collatz conjecture will be determined true or false in the coming years. But there is another possibility: perhaps it truly is a problem that cannot be proven with available mathematical tools. In fact, in 1987 the late mathematician John Horton Conway investigated a generalization of the Collatz conjecture and found that iterative functions have properties that are unprovable. Perhaps this also applies to the Collatz conjecture. As simple as it may seem, it could be doomed to remain unsolved forever.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

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6 Ways to Improve Students’ Math Literacy

Middle and high school math teachers can use these ideas to build students’ reading comprehension and reasoning skills using real tasks like budgeting.

Student working on math problems on a whiteboard

While a lot has changed in math instruction over the years, the idea that students need to be math literate has been constant. Being math literate means much more than calculations. Life events such as buying a home, paying taxes, or even estimating how much you’ll spend on groceries require modeling and reasoning skills.

State and district tests often include problems that are real-world based, and that means that students will need to use reading comprehension, along with math skills, in order to show proficiency. This can be particularly difficult for students with learning disabilities, those who have had interrupted schooling, and/or emergent multilingual learners. It’s imperative that math teachers develop a tool kit to help students decipher the math moves needed for such problems.

Through my dissertation research and my many years of teaching mathematics with great math teachers, I have found simple ways to help students become more math literate. Here are some practical ideas on how teachers can help students become math literate, from the perspective of Algebra 1 teachers from various backgrounds.

6 Ways to Help Students Gain Math Literacy

1. Use sentence frames. Sentence frames are a simple way to help students of all backgrounds learn how to state their answers and ask any questions they have about a word problem. Teachers can post sentence frames on a board or even on students’ desks for easy access. Here are some examples:

“I agree with this answer because ____.”
“I believe the answer is ____ because ____.”
“I showed my work by ____.”
“One strategy that may be helpful is ____ because ____.”

2. Bring back the highlighter. Many Algebra 1 teachers agree that the highlighter is a great way to help emphasize learning in mathematics. The highlighter gives students control of the parts of the problem that they find important. A good suggestion is to demonstrate the use of highlighting key words and have highlighters available for every task and assessment. It also helps students see the patterns in math problems.

3. Speak “algebra.” Students in all math classes need to be speaking math in their classes. It’s important to use the appropriate vocabulary words that pertain to the lesson. This is particularly important as students see formal math language in textbooks and standardized tests. Yes, breaking down the vocabulary for comprehension is a great tactic, but bringing it back to the standard math vocabulary is how we make connections.

4. Use word walls. The word wall was an important part of many math classrooms a few decades ago. They made sure that students saw math words that related to a particular topic being taught. For example, when introducing a polynomial unit, teachers would often put words such as monomial , trinomial , and polynomial on the word wall.

Many Algebra 1 teachers feel that having the visual is most important as students learn about new topics. Students need visual reminders. One suggestion was for students to “own” the word wall by passing out the words in advance and having them hang up each word as it was introduced throughout the unit. Students can make the words artsy and creative in an effort to personalize the resource.

5. Provide foldables or graphic organizers. The use of foldables in the math classroom is a game changer for many students, especially in the post-pandemic era. These low-tech student- or teacher-made “books” constructed out of folded paper provide learners with a handy place to write down the main concepts introduced in a unit. Students have been so used to math technology that there is a need to bring them back to tactile methods.

Providing a foldable to sum up or even begin a unit is a cost-effective method and allows students to use paper and scissors in a creative way to refer to math vocabulary and common word problems. You can find lots of free ideas for foldables online , and there are sites that sell them as well. Some of the designs are very creative and bring a bit of visual art into your math classroom.

6. Have students write relevant word problems. Every time there’s a new curriculum or textbook, word problems get a refresher to connect with the current generation, but there’s no reason why students can’t make up their own. Allow them to write their own word problems, using the context you’re teaching. Not only will students own their own learning, but also they will be able to use critical thinking skills to combine math, vocabulary, and everyday life to further their understanding.

Making the math classroom become a laboratory of reading and math enables students to become owners of the learning process. Students can be math literate, which will allow teachers to facilitate learning processes with all types of word problems, and consequently improve math scores and prepare students for the world of infusing mathematics into their everyday lives.

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Do You Understand the Problem You’re Trying to Solve?

To solve tough problems at work, first ask these questions.

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Problem solving skills are invaluable in any job. But all too often, we jump to find solutions to a problem without taking time to really understand the dilemma we face, according to Thomas Wedell-Wedellsborg , an expert in innovation and the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

In this episode, you’ll learn how to reframe tough problems by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for just one root cause can be misleading.

Key episode topics include: leadership, decision making and problem solving, power and influence, business management.

HBR On Leadership curates the best case studies and conversations with the world’s top business and management experts, to help you unlock the best in those around you. New episodes every week.

Listen to the original HBR IdeaCast episode: The Secret to Better Problem Solving (2016)
Find more episodes of HBR IdeaCast
Discover 100 years of Harvard Business Review articles, case studies, podcasts, and more at HBR.org .

HANNAH BATES: Welcome to HBR on Leadership , case studies and conversations with the world’s top business and management experts, hand-selected to help you unlock the best in those around you.

Problem solving skills are invaluable in any job. But even the most experienced among us can fall into the trap of solving the wrong problem.

Thomas Wedell-Wedellsborg says that all too often, we jump to find solutions to a problem – without taking time to really understand what we’re facing.

He’s an expert in innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

In this episode, you’ll learn how to reframe tough problems, by asking questions that reveal all the factors and assumptions that contribute to the situation. You’ll also learn why searching for one root cause can be misleading. And you’ll learn how to use experimentation and rapid prototyping as problem-solving tools.

This episode originally aired on HBR IdeaCast in December 2016. Here it is.

SARAH GREEN CARMICHAEL: Welcome to the HBR IdeaCast from Harvard Business Review. I’m Sarah Green Carmichael.

Problem solving is popular. People put it on their resumes. Managers believe they excel at it. Companies count it as a key proficiency. We solve customers’ problems.

The problem is we often solve the wrong problems. Albert Einstein and Peter Drucker alike have discussed the difficulty of effective diagnosis. There are great frameworks for getting teams to attack true problems, but they’re often hard to do daily and on the fly. That’s where our guest comes in.

Thomas Wedell-Wedellsborg is a consultant who helps companies and managers reframe their problems so they can come up with an effective solution faster. He asks the question “Are You Solving The Right Problems?” in the January-February 2017 issue of Harvard Business Review. Thomas, thank you so much for coming on the HBR IdeaCast .

THOMAS WEDELL-WEDELLSBORG: Thanks for inviting me.

SARAH GREEN CARMICHAEL: So, I thought maybe we could start by talking about the problem of talking about problem reframing. What is that exactly?

THOMAS WEDELL-WEDELLSBORG: Basically, when people face a problem, they tend to jump into solution mode to rapidly, and very often that means that they don’t really understand, necessarily, the problem they’re trying to solve. And so, reframing is really a– at heart, it’s a method that helps you avoid that by taking a second to go in and ask two questions, basically saying, first of all, wait. What is the problem we’re trying to solve? And then crucially asking, is there a different way to think about what the problem actually is?

SARAH GREEN CARMICHAEL: So, I feel like so often when this comes up in meetings, you know, someone says that, and maybe they throw out the Einstein quote about you spend an hour of problem solving, you spend 55 minutes to find the problem. And then everyone else in the room kind of gets irritated. So, maybe just give us an example of maybe how this would work in practice in a way that would not, sort of, set people’s teeth on edge, like oh, here Sarah goes again, reframing the whole problem instead of just solving it.

THOMAS WEDELL-WEDELLSBORG: I mean, you’re bringing up something that’s, I think is crucial, which is to create legitimacy for the method. So, one of the reasons why I put out the article is to give people a tool to say actually, this thing is still important, and we need to do it. But I think the really critical thing in order to make this work in a meeting is actually to learn how to do it fast, because if you have the idea that you need to spend 30 minutes in a meeting delving deeply into the problem, I mean, that’s going to be uphill for most problems. So, the critical thing here is really to try to make it a practice you can implement very, very rapidly.

There’s an example that I would suggest memorizing. This is the example that I use to explain very rapidly what it is. And it’s basically, I call it the slow elevator problem. You imagine that you are the owner of an office building, and that your tenants are complaining that the elevator’s slow.

Now, if you take that problem framing for granted, you’re going to start thinking creatively around how do we make the elevator faster. Do we install a new motor? Do we have to buy a new lift somewhere?

The thing is, though, if you ask people who actually work with facilities management, well, they’re going to have a different solution for you, which is put up a mirror next to the elevator. That’s what happens is, of course, that people go oh, I’m busy. I’m busy. I’m– oh, a mirror. Oh, that’s beautiful.

And then they forget time. What’s interesting about that example is that the idea with a mirror is actually a solution to a different problem than the one you first proposed. And so, the whole idea here is once you get good at using reframing, you can quickly identify other aspects of the problem that might be much better to try to solve than the original one you found. It’s not necessarily that the first one is wrong. It’s just that there might be better problems out there to attack that we can, means we can do things much faster, cheaper, or better.

SARAH GREEN CARMICHAEL: So, in that example, I can understand how A, it’s probably expensive to make the elevator faster, so it’s much cheaper just to put up a mirror. And B, maybe the real problem people are actually feeling, even though they’re not articulating it right, is like, I hate waiting for the elevator. But if you let them sort of fix their hair or check their teeth, they’re suddenly distracted and don’t notice.

But if you have, this is sort of a pedestrian example, but say you have a roommate or a spouse who doesn’t clean up the kitchen. Facing that problem and not having your elegant solution already there to highlight the contrast between the perceived problem and the real problem, how would you take a problem like that and attack it using this method so that you can see what some of the other options might be?

THOMAS WEDELL-WEDELLSBORG: Right. So, I mean, let’s say it’s you who have that problem. I would go in and say, first of all, what would you say the problem is? Like, if you were to describe your view of the problem, what would that be?

SARAH GREEN CARMICHAEL: I hate cleaning the kitchen, and I want someone else to clean it up.

THOMAS WEDELL-WEDELLSBORG: OK. So, my first observation, you know, that somebody else might not necessarily be your spouse. So, already there, there’s an inbuilt assumption in your question around oh, it has to be my husband who does the cleaning. So, it might actually be worth, already there to say, is that really the only problem you have? That you hate cleaning the kitchen, and you want to avoid it? Or might there be something around, as well, getting a better relationship in terms of how you solve problems in general or establishing a better way to handle small problems when dealing with your spouse?

SARAH GREEN CARMICHAEL: Or maybe, now that I’m thinking that, maybe the problem is that you just can’t find the stuff in the kitchen when you need to find it.

THOMAS WEDELL-WEDELLSBORG: Right, and so that’s an example of a reframing, that actually why is it a problem that the kitchen is not clean? Is it only because you hate the act of cleaning, or does it actually mean that it just takes you a lot longer and gets a lot messier to actually use the kitchen, which is a different problem. The way you describe this problem now, is there anything that’s missing from that description?

SARAH GREEN CARMICHAEL: That is a really good question.

THOMAS WEDELL-WEDELLSBORG: Other, basically asking other factors that we are not talking about right now, and I say those because people tend to, when given a problem, they tend to delve deeper into the detail. What often is missing is actually an element outside of the initial description of the problem that might be really relevant to what’s going on. Like, why does the kitchen get messy in the first place? Is it something about the way you use it or your cooking habits? Is it because the neighbor’s kids, kind of, use it all the time?

There might, very often, there might be issues that you’re not really thinking about when you first describe the problem that actually has a big effect on it.

SARAH GREEN CARMICHAEL: I think at this point it would be helpful to maybe get another business example, and I’m wondering if you could tell us the story of the dog adoption problem.

THOMAS WEDELL-WEDELLSBORG: Yeah. This is a big problem in the US. If you work in the shelter industry, basically because dogs are so popular, more than 3 million dogs every year enter a shelter, and currently only about half of those actually find a new home and get adopted. And so, this is a problem that has persisted. It’s been, like, a structural problem for decades in this space. In the last three years, where people found new ways to address it.

So a woman called Lori Weise who runs a rescue organization in South LA, and she actually went in and challenged the very idea of what we were trying to do. She said, no, no. The problem we’re trying to solve is not about how to get more people to adopt dogs. It is about keeping the dogs with their first family so they never enter the shelter system in the first place.

In 2013, she started what’s called a Shelter Intervention Program that basically works like this. If a family comes and wants to hand over their dog, these are called owner surrenders. It’s about 30% of all dogs that come into a shelter. All they would do is go up and ask, if you could, would you like to keep your animal? And if they said yes, they would try to fix whatever helped them fix the problem, but that made them turn over this.

And sometimes that might be that they moved into a new building. The landlord required a deposit, and they simply didn’t have the money to put down a deposit. Or the dog might need a $10 rabies shot, but they didn’t know how to get access to a vet.

And so, by instigating that program, just in the first year, she took her, basically the amount of dollars they spent per animal they helped went from something like $85 down to around $60. Just an immediate impact, and her program now is being rolled out, is being supported by the ASPCA, which is one of the big animal welfare stations, and it’s being rolled out to various other places.

And I think what really struck me with that example was this was not dependent on having the internet. This was not, oh, we needed to have everybody mobile before we could come up with this. This, conceivably, we could have done 20 years ago. Only, it only happened when somebody, like in this case Lori, went in and actually rethought what the problem they were trying to solve was in the first place.

SARAH GREEN CARMICHAEL: So, what I also think is so interesting about that example is that when you talk about it, it doesn’t sound like the kind of thing that would have been thought of through other kinds of problem solving methods. There wasn’t necessarily an After Action Review or a 5 Whys exercise or a Six Sigma type intervention. I don’t want to throw those other methods under the bus, but how can you get such powerful results with such a very simple way of thinking about something?

THOMAS WEDELL-WEDELLSBORG: That was something that struck me as well. This, in a way, reframing and the idea of the problem diagnosis is important is something we’ve known for a long, long time. And we’ve actually have built some tools to help out. If you worked with us professionally, you are familiar with, like, Six Sigma, TRIZ, and so on. You mentioned 5 Whys. A root cause analysis is another one that a lot of people are familiar with.

Those are our good tools, and they’re definitely better than nothing. But what I notice when I work with the companies applying those was those tools tend to make you dig deeper into the first understanding of the problem we have. If it’s the elevator example, people start asking, well, is that the cable strength, or is the capacity of the elevator? That they kind of get caught by the details.

That, in a way, is a bad way to work on problems because it really assumes that there’s like a, you can almost hear it, a root cause. That you have to dig down and find the one true problem, and everything else was just symptoms. That’s a bad way to think about problems because problems tend to be multicausal.

There tend to be lots of causes or levers you can potentially press to address a problem. And if you think there’s only one, if that’s the right problem, that’s actually a dangerous way. And so I think that’s why, that this is a method I’ve worked with over the last five years, trying to basically refine how to make people better at this, and the key tends to be this thing about shifting out and saying, is there a totally different way of thinking about the problem versus getting too caught up in the mechanistic details of what happens.

SARAH GREEN CARMICHAEL: What about experimentation? Because that’s another method that’s become really popular with the rise of Lean Startup and lots of other innovation methodologies. Why wouldn’t it have worked to, say, experiment with many different types of fixing the dog adoption problem, and then just pick the one that works the best?

THOMAS WEDELL-WEDELLSBORG: You could say in the dog space, that’s what’s been going on. I mean, there is, in this industry and a lot of, it’s largely volunteer driven. People have experimented, and they found different ways of trying to cope. And that has definitely made the problem better. So, I wouldn’t say that experimentation is bad, quite the contrary. Rapid prototyping, quickly putting something out into the world and learning from it, that’s a fantastic way to learn more and to move forward.

My point is, though, that I feel we’ve come to rely too much on that. There’s like, if you look at the start up space, the wisdom is now just to put something quickly into the market, and then if it doesn’t work, pivot and just do more stuff. What reframing really is, I think of it as the cognitive counterpoint to prototyping. So, this is really a way of seeing very quickly, like not just working on the solution, but also working on our understanding of the problem and trying to see is there a different way to think about that.

If you only stick with experimentation, again, you tend to sometimes stay too much in the same space trying minute variations of something instead of taking a step back and saying, wait a minute. What is this telling us about what the real issue is?

SARAH GREEN CARMICHAEL: So, to go back to something that we touched on earlier, when we were talking about the completely hypothetical example of a spouse who does not clean the kitchen–

THOMAS WEDELL-WEDELLSBORG: Completely, completely hypothetical.

SARAH GREEN CARMICHAEL: Yes. For the record, my husband is a great kitchen cleaner.

You started asking me some questions that I could see immediately were helping me rethink that problem. Is that kind of the key, just having a checklist of questions to ask yourself? How do you really start to put this into practice?

THOMAS WEDELL-WEDELLSBORG: I think there are two steps in that. The first one is just to make yourself better at the method. Yes, you should kind of work with a checklist. In the article, I kind of outlined seven practices that you can use to do this.

But importantly, I would say you have to consider that as, basically, a set of training wheels. I think there’s a big, big danger in getting caught in a checklist. This is something I work with.

My co-author Paddy Miller, it’s one of his insights. That if you start giving people a checklist for things like this, they start following it. And that’s actually a problem, because what you really want them to do is start challenging their thinking.

So the way to handle this is to get some practice using it. Do use the checklist initially, but then try to step away from it and try to see if you can organically make– it’s almost a habit of mind. When you run into a colleague in the hallway and she has a problem and you have five minutes, like, delving in and just starting asking some of those questions and using your intuition to say, wait, how is she talking about this problem? And is there a question or two I can ask her about the problem that can help her rethink it?

SARAH GREEN CARMICHAEL: Well, that is also just a very different approach, because I think in that situation, most of us can’t go 30 seconds without jumping in and offering solutions.

THOMAS WEDELL-WEDELLSBORG: Very true. The drive toward solutions is very strong. And to be clear, I mean, there’s nothing wrong with that if the solutions work. So, many problems are just solved by oh, you know, oh, here’s the way to do that. Great.

But this is really a powerful method for those problems where either it’s something we’ve been banging our heads against tons of times without making progress, or when you need to come up with a really creative solution. When you’re facing a competitor with a much bigger budget, and you know, if you solve the same problem later, you’re not going to win. So, that basic idea of taking that approach to problems can often help you move forward in a different way than just like, oh, I have a solution.

I would say there’s also, there’s some interesting psychological stuff going on, right? Where you may have tried this, but if somebody tries to serve up a solution to a problem I have, I’m often resistant towards them. Kind if like, no, no, no, no, no, no. That solution is not going to work in my world. Whereas if you get them to discuss and analyze what the problem really is, you might actually dig something up.

Let’s go back to the kitchen example. One powerful question is just to say, what’s your own part in creating this problem? It’s very often, like, people, they describe problems as if it’s something that’s inflicted upon them from the external world, and they are innocent bystanders in that.

SARAH GREEN CARMICHAEL: Right, or crazy customers with unreasonable demands.

THOMAS WEDELL-WEDELLSBORG: Exactly, right. I don’t think I’ve ever met an agency or consultancy that didn’t, like, gossip about their customers. Oh, my god, they’re horrible. That, you know, classic thing, why don’t they want to take more risk? Well, risk is bad.

It’s their business that’s on the line, not the consultancy’s, right? So, absolutely, that’s one of the things when you step into a different mindset and kind of, wait. Oh yeah, maybe I actually am part of creating this problem in a sense, as well. That tends to open some new doors for you to move forward, in a way, with stuff that you may have been struggling with for years.

SARAH GREEN CARMICHAEL: So, we’ve surfaced a couple of questions that are useful. I’m curious to know, what are some of the other questions that you find yourself asking in these situations, given that you have made this sort of mental habit that you do? What are the questions that people seem to find really useful?

THOMAS WEDELL-WEDELLSBORG: One easy one is just to ask if there are any positive exceptions to the problem. So, was there day where your kitchen was actually spotlessly clean? And then asking, what was different about that day? Like, what happened there that didn’t happen the other days? That can very often point people towards a factor that they hadn’t considered previously.

SARAH GREEN CARMICHAEL: We got take-out.

THOMAS WEDELL-WEDELLSBORG: S,o that is your solution. Take-out from [INAUDIBLE]. That might have other problems.

Another good question, and this is a little bit more high level. It’s actually more making an observation about labeling how that person thinks about the problem. And what I mean with that is, we have problem categories in our head. So, if I say, let’s say that you describe a problem to me and say, well, we have a really great product and are, it’s much better than our previous product, but people aren’t buying it. I think we need to put more marketing dollars into this.

Now you can go in and say, that’s interesting. This sounds like you’re thinking of this as a communications problem. Is there a different way of thinking about that? Because you can almost tell how, when the second you say communications, there are some ideas about how do you solve a communications problem. Typically with more communication.

And what you might do is go in and suggest, well, have you considered that it might be, say, an incentive problem? Are there incentives on behalf of the purchasing manager at your clients that are obstructing you? Might there be incentive issues with your own sales force that makes them want to sell the old product instead of the new one?

So literally, just identifying what type of problem does this person think about, and is there different potential way of thinking about it? Might it be an emotional problem, a timing problem, an expectations management problem? Thinking about what label of what type of problem that person is kind of thinking as it of.

SARAH GREEN CARMICHAEL: That’s really interesting, too, because I think so many of us get requests for advice that we’re really not qualified to give. So, maybe the next time that happens, instead of muddying my way through, I will just ask some of those questions that we talked about instead.

THOMAS WEDELL-WEDELLSBORG: That sounds like a good idea.

SARAH GREEN CARMICHAEL: So, Thomas, this has really helped me reframe the way I think about a couple of problems in my own life, and I’m just wondering. I know you do this professionally, but is there a problem in your life that thinking this way has helped you solve?

THOMAS WEDELL-WEDELLSBORG: I’ve, of course, I’ve been swallowing my own medicine on this, too, and I think I have, well, maybe two different examples, and in one case somebody else did the reframing for me. But in one case, when I was younger, I often kind of struggled a little bit. I mean, this is my teenage years, kind of hanging out with my parents. I thought they were pretty annoying people. That’s not really fair, because they’re quite wonderful, but that’s what life is when you’re a teenager.

And one of the things that struck me, suddenly, and this was kind of the positive exception was, there was actually an evening where we really had a good time, and there wasn’t a conflict. And the core thing was, I wasn’t just seeing them in their old house where I grew up. It was, actually, we were at a restaurant. And it suddenly struck me that so much of the sometimes, kind of, a little bit, you love them but they’re annoying kind of dynamic, is tied to the place, is tied to the setting you are in.

And of course, if– you know, I live abroad now, if I visit my parents and I stay in my old bedroom, you know, my mother comes in and wants to wake me up in the morning. Stuff like that, right? And it just struck me so, so clearly that it’s– when I change this setting, if I go out and have dinner with them at a different place, that the dynamic, just that dynamic disappears.

SARAH GREEN CARMICHAEL: Well, Thomas, this has been really, really helpful. Thank you for talking with me today.

THOMAS WEDELL-WEDELLSBORG: Thank you, Sarah.

HANNAH BATES: That was Thomas Wedell-Wedellsborg in conversation with Sarah Green Carmichael on the HBR IdeaCast. He’s an expert in problem solving and innovation, and he’s the author of the book, What’s Your Problem?: To Solve Your Toughest Problems, Change the Problems You Solve .

We’ll be back next Wednesday with another hand-picked conversation about leadership from the Harvard Business Review. If you found this episode helpful, share it with your friends and colleagues, and follow our show on Apple Podcasts, Spotify, or wherever you get your podcasts. While you’re there, be sure to leave us a review.

We’re a production of Harvard Business Review. If you want more podcasts, articles, case studies, books, and videos like this, find it all at HBR dot org.

This episode was produced by Anne Saini, and me, Hannah Bates. Ian Fox is our editor. Music by Coma Media. Special thanks to Maureen Hoch, Adi Ignatius, Karen Player, Ramsey Khabbaz, Nicole Smith, Anne Bartholomew, and you – our listener.

See you next week.

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ORIGINAL RESEARCH article

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Research on Teaching Strategies and Skills in Different Educational Stages

BIOTECHNOLOGY PROJECT-BASED LEARNING ENCOURAGES LEARNING AND MATHEMATICS APPLICATION Provisionally Accepted

1 Universidad Técnica Particular de Loja, Ecuador

The final, formatted version of the article will be published soon.

Project-based learning (PBL) is a promising approach to enhance mathematics learning concepts in higher education. Here, teachers provide guidance and support to PBL implementation. The objective of this study was to develop PBL-based biotechnological projects as a strategy for mathematics learning. The methodology design was applied to 111 university students from Biochemical, Chemical Engineering and Business Administration careers. Knowledge, skills, perceptions, and engagement were measured through questionnaires, workshops, rubrics, and survey instruments. As a result, the paired comparison between tests, questionnaires and project shows significant differences (P<.001) between the experimental group and the control group. It is concluded that the teaching of mathematics should be oriented to the development of competencies, abilities, and skills that allow students to generate real solutions and broaden their vision of the applicability of their knowledge using new learning strategies. Key words: Mathematical models, Biotechnology, Project based learning.Science, technology, engineering, and mathematics (STEM) education has become a crucial topic both inside and outside of school (Han et al., 2015). Currently, mathematics learning tends to be oriented towards textbooks, and students can only work on math problems based on what the teacher exemplifies; however, if given different contextbased problems, they will have difficulty solving them (Fisher et al., 2020). Likewise, the traditional classroom model does not encourage student's interest in STEM (Sahin, 2009) The research gap is between what students learn at the university and what they really need in the workplace (Holmes et al., 2015). Higher education institutions have been trying to provide students with both (i) hard skills, such as cognitive knowledge and professional skills (Vogler et al., 2018), and (ii) soft skills, such as problem-solving and teamwork (Lennox and Roos 2017). However, these skills are difficult to achieve through traditional learning. One learning that creates an active, collaborative atmosphere, and can increase selfconfidence in students is Project-based learning (PBL) (Cruz et al., 2022; Guo et al., 2022;Markula and Aksela 2022). The PBL method is applied as a teaching model that involves

Keywords: Mathematical Models, Biotechnology, Project Based Learning (PBL), Learning, Learning mathematics activities

Received: 02 Jan 2024; Accepted: 03 Apr 2024.

Copyright: © 2024 Vivanco and Jiménez-Gaona. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Mr. Oscar A. Vivanco, Universidad Técnica Particular de Loja, Loja, Ecuador

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The Accidental Tax Cutter in Chief

President biden says he wants to rake in more money from corporations and high earners. but so far, he has cut more taxes than he’s raised..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Michael Barbaro. This is “The Daily.”

[THEME MUSIC]

Today, in his campaign for re-election, President Biden says that raising taxes is at the heart of his agenda. But as it turns out so far, he’s done the opposite as president. My colleague Jim Tankersley explains.

It’s Wednesday, April 3.

Jim, welcome back. We haven’t seen you since the State of the Union. Always a pleasure.

So, so great to be here. And yeah, I finally recovered from staying up all night with you guys.

Yeah, you don’t even know all night. You stopped and we kept going.

That’s true. I did. I got a robust three hours that night. You’re right.

[LAUGHS]:: So Jim, in your capacity as really the chief economic thinker covering this president, you recently came across something very surprising.

Yeah, it started with a pretty basic question for me. I like to do this crazy thing, Michael, where I like to take candidates’ promises and see if they’ve come true.

It’s a little bit wild, but it’s what I do for fun. And in this case, I wanted to look at a very central promise of President Biden’s campaign in 2020, which he has repeated while in office.

I promise you, I guarantee you we can build back, and build back better with an economy that rewards work, not wealth.

The promise was he was going to raise taxes.

But I tell you what I’m going to do, and I make no apologies for it. I’m going to ask the wealthiest Americans and the biggest corporations of the Fortune 500 companies, 91 making a collective billions of dollars, didn’t pay a single solitary penny in federal tax!

Not taxes on the middle class, not taxes on low income workers, but he was going to raise taxes on corporations and the rich.

But I’m going to make sure they pay their fair share!

He was going to make them pay their fair share. And he leaned into it.

Guess what? You’re going to start paying your fair share. I’m going to ask them to finally begin to pay the fair share. It’s not a punishment. Pay your fair share.

He said it over and over.

Fair share? Translation — it’s back to the failed policies of the 1970s.

Republicans loved this. They repeated it too. They told voters that the president was going to raise taxes.

Joe Biden bragging about raising taxes on corporations. That means less money for those very employers to hire people back.

They talked about all the ways in which corporate tax increases could rebound on workers.

Joe Biden will shut down your economy, raise taxes, wants a $4 trillion tax increase. He’s the only politician I’ve ever seen who said, we will raise your taxes. You’re supposed —

It was a very big part of the economic debate for the campaign.

And I wanted to know, is that true? Has that actually played out in the policy agenda the president has had?

So I asked some economists at the Tax Policy Center in Washington to run an analysis and just say, let’s look at all of the ways Biden has changed the tax code in all of those laws he’s signed, and ask, has he raised taxes as president? And it turns out the answer is he has not raised taxes.

On net, he has cut more taxes than he’s raised.

How much more has he cut taxes than raised them?

So by the math that economists use when they look at budgets, the traditional way of scoring tax changes, he has cut taxes by $600 billion on net.

Hmm. A lot of money, a lot of tax cuts.

It’s a lot of tax cuts. The president has been a net tax cutter.

So Jim, why and how did Biden end up cutting taxes, especially if his stated intent was to raise taxes?

Well, there’s two sides of this equation and two complementary explanations for what’s happened here. The first side is the tax increases that Biden ran on, he’s only done a couple of them. He has trillions of dollars of ideas for how to raise taxes on rich people and corporations. The Treasury Department publishes an entire book full of them every year called “The Green Book.”

But in the actual legislation he signed, there’s only been a couple, really. There was a tax on stock buybacks that companies do and then a new minimum tax for certain multinational corporations that have very low tax rates. Those add up to real money, but they are not, in the grand scheme of Biden’s tax increases, a really large amount of the agenda he’s proposed.

So explanation number one, he just hasn’t been that successful in passing tax increases, and there’s a lot of reasons for that. The biggest one is just the simplest one is that he’s just had a really hard time persuading members of Congress, including Democrats, to back some of his favorite tax increases. He wants to raise the corporate income tax rate, which President Trump cut in his 2017 tax bill. Biden wants to raise it to 28 percent from 21 percent. Congress has not had any appetite to go along with that.

He wanted to get rid of what’s called the carried interest loophole, a long-time white whale of Democratic policy making. But he could not get even 50 Democrats to go along with that. Senator Kyrsten Sinema of Arizona was opposed to it, and so it didn’t get included.

And he did some pieces of legislation on a bipartisan basis. And in those cases, Republicans were just not going to pay for anything by raising taxes, and so he had to take those off the table there. So it’s all added up to just not very much activity in Congress to raise taxes on what Biden wants to do.

Got it. So that’s the side of a ledger where Biden simply fails to increase taxes because he can’t get Congress to increase taxes.

Right. But there’s another side, which is also that Biden has signed into law a decent number of tax cuts.

And that starts from the very beginning. Just a couple of months into his presidency, if you’ll recall, we’re still in the depths of the COVID-19 pandemic, the economy is wobbling after it had started to rebound. Biden proposes what is essentially a stimulus bill.

And he includes some tax cuts in there, a tax cut for families, a child tax credit. And it also includes — you remember those direct checks that people got as part of that bill?

Yes, $1,400. I remember them.

Yes, those were technically tax cuts.

So the stimulus bill starts with that. The next year, he passes this bill that is trying to accelerate manufacturing of things like semiconductors in the United States. That’s the CHIPS Act. And that includes some corporate tax cuts for companies that invest in the kind of manufacturing that Biden wants. This is industrial policy via carrots for corporations. And Biden is handing them out as part of this bill.

So tax cuts there. And then finally the Inflation Reduction Act, which includes the largest climate effort in American history is a bunch of corporate tax cuts at its core, tax cuts for manufacturing of solar panels, tax cuts for people to buy electric vehicles, tax cuts for all sorts of things tied to the transition from fossil fuels to lower emission sources of energy. And those tax cuts add up. They add up for corporations. They add up for individuals. And in the end, that full suite of tax cuts that he’s passed across all of this legislation outweighs the modest tax increases that were also included in the Inflation Reduction Act to reduce its cost.

Got it. So a very big reason why Biden ends up cutting taxes, beyond the fact that he’s not able to raise them through Congress, is that that’s what it took, according to those in his administration, to get American industry and American consumers to change their behavior in line with policy goals such as getting more domestic computer chip manufacturing and getting more people to buy electric vehicles they decided the way to do that was to give people tax breaks, which means he cut their taxes.

Right, people and companies. The president certainly has talked throughout the campaign about wanting to give middle class families a break. But he has also, in the process of crafting policy, really come to rely on tax cuts for people and for corporations as a way of achieving these policy goals. And in many cases, again, this is what he had to do to pass these bills through even Democrats in Congress.

Senator Joe Manchin of West Virginia didn’t want to just send money to companies that were making solar panels. He wanted there to be tax incentives for it. And so that is part of the reason why these were created as tax incentives. And so all of this adds up to more of a tax cutting record than you might have imagined when Biden was on the campaign trail.

I’m curious who really ended up benefiting from these tax cuts. You said they went to people and to corporations, but on the whole, did they end up reaching lower income Americans, middle income Americans, or the rich?

Well, we don’t have a full distributional analysis, which is what you’re asking for, of the entirety of Biden’s tax changes. But what we can say this — particularly the ones that were in that early stimulus bill, the recovery plan, those were very much targeted toward lower income and middle income Americans.

There were income limits on who could get things like the Child Tax Credit. Obviously, the direct payments went to people who were middle class or less. So the analysis of that would suggest that these were tax cuts for lower income people, for middle class people. And on the flip side, what I think we are likely to see with the electric vehicle credit through the Inflation Reduction Act is that while there are some income limits on who can qualify for that credit, that the people who end up claiming that credit tend to be the higher earners among the people who qualify.

Right. Who buys a Tesla, after all? Somebody with a fair amount of money.

Right, exactly. And of course, the corporate tax cuts go to companies, flow through to their shareholders. There’s a huge debate in the academic literature among politicians about how much of that benefit actually ends up going to their workers versus stays with shareholders. But we can broadly say that Joe Biden has done a lot for certain corporations who are trying to advance his manufacturing goals in particular to reduce their tax bills. And that is certainly not in line with the rhetoric you hear him talking about most of the time about making corporations pay their fair share. And the White House acknowledges this. I asked them about it. And they basically said, we think there’s a difference between just cutting the corporate tax rate in a way that helps anybody no matter what they’re doing and what we’re trying to do, which is basically reward corporations for accelerating the energy transition.

Understood. But where does this ultimately leave Biden’s campaign promise to make the tax code fairer and to make sure that the well-off in particular and corporations are paying their, what he calls, fair share?

Well, I think by Biden’s own measurements, by his own ambitions, he would have to agree that he is nowhere close to what he believes would be a fair share for corporations. Because Biden is still running on this. As he enters his re-election campaign, as it really heats up, a rematch with Donald Trump, the president is really leaning into this message of we need to do more. We need to raise more taxes on corporate America. It is time for these companies and for high earners to pay their fair share.

Right. I didn’t get it done in the first term. But if you elect me, I’ll get it done in the second.

Give me another shot, and this time, I promise, will be different.

[MUSIC PLAYING]

We’ll be right back.

So Jim, let’s talk about Biden’s tax raising plans for a theoretical second term and why anyone should have any faith that he could get it done, if there’s a second term, given the experience so far of his first term.

Yeah, well, man, there’s a lot of plans to talk about. I don’t think we can get through all of them, but we can certainly hit the highlights here. So we can start with the couple of things that Biden has been able to do to raise taxes on corporations. He wants to take those and then plus them up.

He’s put this new minimum tax on corporations. It’s a 15 percent minimum tax on certain multinationals. He now wants to raise that to 21 percent.

He wants to take that corporate stock buyback tax which is 1 percent right now, and he’d like to quadruple it to 4 percent. And then he goes after some things large and small. He wants to do new taxes that hit the use of corporate and private jets. He wants to do new taxes on companies that pay large amounts of compensation to their executives.

And then we get to some really big taxes on high earning individuals. So the president has said over and over again, he won’t raise taxes on anyone making less than $400,000 a year. But he’s got a bunch of taxes for the people above that. So he wants to raise the top marginal income tax rate. He wants to take it from 37 percent, which is the level set by President Trump’s 2017 tax law, and bring it back to 39.6 percent, which is what it was before. He also wants to impose what he calls a billionaires tax.

OK. It’s a 25 percent tax on the total value of all of the assets of anyone worth more than $100 million.

OK, wait. I have several questions about this.

First being a fact check, if it’s a billionaires tax, it’s interesting that it’s going after people who have just 100 million.

Yeah, I think most billionaires would be offended at the inclusion of 100 millionaires in that. Yes, totally agree. That is factually inaccurate, the name.

Right. But beyond that, this sounds very much like a wealth tax, which we don’t really have in our system.

Yeah, it’s a sort of wealth tax. The Biden people don’t call it a wealth tax, but it is a tax on something other than income that you report every year to the IRS as having been earned. It goes beyond just, oh, I got interest from my stock holdings or I made money from my job. It’s, oh, the value of my art collection increased last year, and now Biden’s going to tax me on that increase, even if I didn’t sell the art.

That’s a real change, and that reflects the president’s view that people with enough money to buy enormous art collections that appreciate enormously in value should be paying more in taxes.

Right. And of course, a tax like this is extremely perhaps maddeningly hard to actually pull off. It’s hard to get someone to describe their art collection’s value so that you can apply a 25 tax to it. So this might end up being more of a political statement than a practical tax.

Yeah, there’s also questions about whether it’s constitutional. So there’s all sorts of drama around this proposal, but it is certainly, if nothing else, a statement of the president’s intent to make people worth a lot of money pay a lot more in taxes.

OK, so that’s a lot of proposed tax increases, almost all of them focused on those who are rich and corporations. Overall, Jim, what stands out to you about this Biden term two tax increase plan?

I think we could very fairly say that it’s the largest tax increasing plan by a sitting president or a presidential nominee for a party in American history.

He wants to get a lot of money from corporations and people who earn or are worth a lot of money.

But the rub, of course, is it’s hard to see the Congressional math that lets Biden accomplish these tax increases, some of which, like you said, he couldn’t get done the first time. Why would we think he would get them done the second time even if he wins this fall?

Yeah, it would be really difficult. Biden would have to win in November. Democrats would have to take the House of Representatives back from Republicans, which is certainly possible. It’s very closely divided right now. And they’d need to hold at least 50 seats in the Senate. And then those 50 Democrats in the Senate would have to be willing to go along with far more in tax increases than Democrats were last time around.

So if there is a second term, it feels like we should assume it will be very difficult perhaps even quite unlikely he’s going to get to push through a lot of these taxes. Which makes me wonder, Jim, why is Biden running on a tax program that he knows has so little chance of becoming reality and when it’s pretty clear that he’s gotten a lot of stuff done without raising taxes? It turns out that’s not been all that essential to getting infrastructure or climate bills done. So why is he making this so central?

Several reasons. One of them is it’s very important to him rhetorically to talk about fiscal responsibility. Big parts of the Biden agenda, the CHIPS bill, the infrastructure bill, some other legislation, were not actually paid for. The spending and tax cuts were not offset by tax increases.

So they’re going to add to the debt.

Right. So they’re going to add to the debt. Same is true of the stimulus bill. But moving forward, the president has said that he’s going to pay for his agenda and he’s actually going to have some extra tax dollars coming in left over to help pay down future budget deficits. And on paper, it’s the way to pay for Biden’s other big, expansive plans that he hasn’t been able to do but wants to — universal child care, federal paid leave, investing in elder care, just a whole bunch of things that he still wants to do more — housing initiatives.

The president needs money to make a case that he’s being fiscally responsible, and this is the money that would do that.

So that’s one reason. Another reason is the calendar. Biden and his team are looking ahead to the end of 2025, and they know that if he wins another term, he will be in office at a rare moment in Washington, when basically tax policy has to be on the Congressional agenda.

Well, Republicans, when they passed their tax cuts in 2017, set a bunch of them to expire at the end of 2025 in order to lower the cost of the bill.

These are the Trump tax cuts.

The Trump tax cuts. And that includes all the tax cuts for individuals. So now that those are coming due, there’s going to be a fight in Washington over whether to extend them or make them permanent or change them in some way or just let them expire, and Democrats know there’s going to be a huge fight that will reach almost certainly the floor of the House and the Senate. And so Biden wants to be ready.

He wants to be ready with a suite of policy proposals that Democrats can basically pull off the shelf and try to use to put Republicans in a box. Basically say, we would like to keep taxes low or cut them further for low income workers, middle class workers. But we want to pay for that by raising taxes on the rich. You Republicans also want to do nice things for low and middle class workers, but you want to cut taxes for corporations and the rich, and we think that’s a political loser for you.

So Biden is ready with what they think will be a political winner for Democrats in this almost certain floor tax fight at the end of 2025.

And that brings us to the last reason why Biden is doing this, and maybe the most important, which is it’s really good politics.

Just explain that. Why is talking about tax increases, net tax increases, such good politics?

If you talk to Democratic pollsters, if you talk to people inside the White House, outside the White House, political strategists anywhere in Biden’s orbit, they all agree that the public loves the idea of forcing rich people and corporations to pay their, quote, “fair share.” It’s just become a winning and central political argument in Democratic campaigns, the idea that corporations avoid taxes, that rich people avoid taxes, and that Joe Biden is trying to position himself as a champion of the idea that they need to pay more. Those corporations and those rich people need to pay more, and he’s going to make it happen.

You’re describing this as something that is kind of a new political reality. Is that right?

Yeah, it’s evolved over the last decade or so I think. For a long time in Washington, the conventional wisdom was just couldn’t talk about tax increases of any kind. They were poison. There was a whole anti-tax movement that did a really good job of messaging that, and Democratic candidates got very scared of talking about raising taxes even on the very, very rich.

That started to turn over time. But it’s really changed. I think we saw in the 2020 election that the Democratic primary had just enormous amounts of taxes on corporations and the rich funding all sorts of policy proposals — Medicare for all and universal child care and trillions and trillions of dollars — and Democratic candidates like Liz Warren and Bernie Sanders competing to see who could tax corporations and the rich the most.

Biden is a product of that primary. He was one of the most moderate people in that group, but his proposals are really outside of the historical norm for Democratic candidates up until then. And that reflects the fact that pollsters have been doing all this research, finding that the American people, including independents and increasingly numbers of Republicans, just don’t think corporations pay their fair share and are open to the idea they should pay more.

This is really interesting, and it makes me think that what you’re really saying is that there might have been a time when a Democratic nominee like Joe Biden might have reveled in his image as an overall tax cutter. But that is not this moment, and that is not this candidate. He wants to be a tax increaser. He thinks that is where the politics are.

I think that’s exactly right when you think about tax increaser as tax increaser on the rich and on corporations. There’s two ways to be a successful populist politician. One of them is to be like Trump and run around saying you’re going to do enormous tax cuts for everybody, which is a Republican version of populism. Trump, my biggest tax cut in history, I’m going to do another huge, enormous tax cut. It’s going to be so big you won’t believe it.

There might have been a time when Democrats tried to follow that playbook. But Biden’s not doing that. He’s leaning into the other side of populism. He’s telling workers, hey, I’m on your side with these big companies. They’re trying to screw you, and I’m not going to stand for it. And so I’m going to raise their taxes. I’m going to make them pay more so that there’s more money for you, whether that’s more tax cuts or more programs or whatever.

And that is the Democratic version of populism right now, and that’s the one that Joe Biden is running on. And that’s why he’s so happy to talk about raising corporate taxes because it’s a way to tell workers, hey, I’m on your side.

Right. Even if that’s not what he’s done or ever may be able to do.

Yeah. Part of the problem with populism is that you make a lot of promises you can’t keep, and this certainly, in his first term, has been an area where the president has talked a much bigger game than he’s been able to execute. The second term might be different, but that doesn’t really matter for the campaign. What matters is the rhetoric.

Well, Jim, thank you very much. We appreciate it.

Thank you. Always a pleasure.

Here’s what else you need to know today. On Tuesday, Israel confirmed that it had carried out the airstrike that killed seven aid workers delivering food to civilians in Gaza. The attack, which occurred on Monday, struck a convoy run by the World Central Kitchen, a nonprofit group. At the time of the attack, the aid workers were traveling in clearly marked cars that designated them as non-combatants.

Israel’s Prime Minister Benjamin Netanyahu described the attack as unintentional and said that his government deeply regretted the deaths. In its own statement, World Central Kitchen called the strike unforgivable and said that as a result, it would suspend its aid work in Gaza, where millions of people are in dire need of both food and medicine.

Today’s episode was produced by Stella Tan and Mary Wilson with help from Michael Simon Johnson. It was edited by Lisa Chow, contains original music by Dan Powell and Marion Lozano, and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Michael Barbaro. See you tomorrow.

April 4, 2024 • 32:37 Israel’s Deadly Airstrike on the World Central Kitchen
April 3, 2024 • 27:42 The Accidental Tax Cutter in Chief
April 2, 2024 • 29:32 Kids Are Missing School at an Alarming Rate
April 1, 2024 • 36:14 Ronna McDaniel, TV News and the Trump Problem
March 29, 2024 • 48:42 Hamas Took Her, and Still Has Her Husband
March 28, 2024 • 33:40 The Newest Tech Start-Up Billionaire? Donald Trump.
March 27, 2024 • 28:06 Democrats’ Plan to Save the Republican House Speaker
March 26, 2024 • 29:13 The United States vs. the iPhone
March 25, 2024 • 25:59 A Terrorist Attack in Russia
March 24, 2024 • 21:39 The Sunday Read: ‘My Goldendoodle Spent a Week at Some Luxury Dog ‘Hotels.’ I Tagged Along.’
March 22, 2024 • 35:30 Chuck Schumer on His Campaign to Oust Israel’s Leader
March 21, 2024 • 27:18 The Caitlin Clark Phenomenon

Hosted by Michael Barbaro

Featuring Jim Tankersley

Produced by Stella Tan and Mary Wilson

With Michael Simon Johnson

Edited by Lisa Chow

Original music by Dan Powell and Marion Lozano

Engineered by Chris Wood

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In his campaign for re-election, President Biden has said that raising taxes on the wealthy and on big corporations is at the heart of his agenda. But under his watch, overall net taxes have decreased.

Jim Tankersley, who covers economic policy for The Times, explains.

On today’s episode

Jim Tankersley , who covers economic policy at the White House for The New York Times.

President Biden, wearing a blue sweater, speaks into a microphone. In the room behind him, rows of American flags hang from the ceiling.

Background reading

An analysis prepared for The New York Times estimates that the tax changes President Biden has ushered into law will amount to a net cut of about $600 billion over four years.

“Does anybody here think the tax code’s fair?” For Mr. Biden, tax policy has been at the center of his efforts to make the economy more equitable.

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The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

Jim Tankersley writes about economic policy at the White House and how it affects the country and the world. He has covered the topic for more than a dozen years in Washington, with a focus on the middle class. More about Jim Tankersley

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6 Tools to Help You Solve Difficult Math Problems
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The 10 Hardest Math Problems That Were Ever Solved
In 2019, mathematicians finally solved one of the hardest math problems —one that had stumped them for decades. It's called a Diophantine Equation, and it's sometimes known as the "summing ...
Quadratic Equations: Very Difficult Problems with Solutions
Problem 6 sent by Κυριάκος There is a two-digit number whose digits are the same, and has got the following property: When squared, it produces a four-digit number, whose first two digits are the same and equal to the original's minus one, and whose last two digits are the same and equal to the half of the original's.
100 Hard Word Problems in Algebra
Let x represent the number of weeks and y represent the total amount of money saved. Determine in how many weeks Jenna and Khalil will have the same amount of money. 17. The sum of three consecutive terms of a geometric sequence is 104 and their product is 13824.find the terms. 18.
Hard Math Problems That'll Make Your Head Spin
Answer: 1. Many of these hard math problems don't require doing difficult math as much as they require identifying relationships between groups of numbers. In this case, don't look at the ...
Free Math Worksheets
Khan Academy's 100,000+ free practice questions give instant feedback, don't need to be graded, and don't require a printer. Math Worksheets. Khan Academy. Math worksheets take forever to hunt down across the internet. Khan Academy is your one-stop-shop for practice from arithmetic to calculus. Math worksheets can vary in quality from ...
These Are the 7 Hardest Math Problems Ever Solved
These Are the 7 Hardest Math Problems Ever Solved — Good Luck in Advance. In 2019, mathematicians finally solved a math puzzle that had stumped them for decades. It's called a Diophantine Equation, and it's sometimes known as the "summing of three cubes": Find x, y, and z such that x³+y³+z³=k, for each k from one to 100.
The 15 Hardest SAT Math Questions Ever
B) x − 2 is a factor of p ( x). C) x + 2 is a factor of p ( x). D) The remainder when p ( x) is divided by x − 3 is − 2. ANSWER EXPLANATION: If the polynomial p ( x) is divided by a polynomial of the form x + k (which accounts for all of the possible answer choices in this question), the result can be written as.
How to Work Through Hard Math Problems
Strategies for Difficult Math Problems — and Beyond. Here are a few strategies for dealing with hard problems, and the frustration that comes with them: Do something. Yeah, the problem is hard. Yeah, you have no idea what to do to solve it. At some point you have to stop staring and start trying stuff. Most of it won't work.
Hard Algebra Questions
Steps. Example. Write down the inequality. ( x - 5 > 10 ) Perform the same operation on both sides. ( x - 5 + 5 > 10 + 5 ) Simplify to find the solution. ( x > 15 ) Remember, if I multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign flips.
Solving equations & inequalities
Why we do the same thing to both sides: Variable on both sides. Intro to equations with variables on both sides. Equations with variables on both sides: 20-7x=6x-6. Equation with variables on both sides: fractions. Equation with the variable in the denominator.
7 of the hardest problems in mathematics that have been solved
7. The goat problem. An illustration of the goat grazing problem. Mnchnstnr. Problem: The goat problem is a much more recently solved mathematical problem. It involves calculating the grazing area ...
Hard Geometry Problems
I encourage my readers to embrace these difficult problems with a sense of adventure. Remember, tackling a geometry problem is more than a test; it's an opportunity to explore the elegance of mathematics. Whether you solve a challenging problem on the first try or it takes numerous attempts, each effort enhances your mathematical intuition ...
Wolfram Problem Generator: Unlimited AI-generated Practice Problems
Wolfram for Education. Wolfram Demonstrations. Mathematica. MathWorld. Online practice problems with answers for students and teachers. Pick a topic and start practicing, or print a worksheet for study sessions or quizzes.
How to Solve Math Problems Faster: 15 Techniques to Show Students
So, students would solve 952 by designating 25 as the last two digits. They would then multiply 9 x 10 to receive 90. Putting these numbers together, the result is 9,025. Just like that, a hard problem becomes easy multiplication for many students. 12. Calculating Percentages
It Will Be Hard
The challenging homework you're faced with requires complex and difficult thought. It is not designed to be completed all at once, in one evening, or even in one day. Our most successful students start the homework soon after the related class. This allows sufficient time to think through problems, take breaks, and ask questions.
Hardest GCSE Maths Questions
However hard or easy you find maths, practice is vital to improve. At Save My Exams we have resources - including practice papers - to help with your Maths revision, to test yourself, and to improve in GCSE Maths. Exam paper questions organised by topic and difficulty. Our worksheets cover all topics from GCSE, IGCSE and A Level courses.
Wolfram Problem Generator: Online Practice Questions & Answers
Only Wolfram Problem Generator directly integrates the popular and powerful Step-by-step Solutions from Wolfram|Alpha. You can use a single hint to get unstuck, or explore the entire math problem from beginning to end. Online practice problems for math, including arithmetic, algebra, calculus, linear algebra, number theory, and statistics.
120 Math Word Problems To Challenge Students Grades 1 to 8
Subtraction word problems. Best for:1st grade, second grade 9. Subtracting to 10: There were 3 pizzas in total at the pizza shop.A customer bought 1 pizza. How many pizzas are left? 10. Subtracting to 20: Your friend said she had 11 stickers.When you helped her clean her desk, she only had a total of 10 stickers.
Why Is Math So Hard? + Ways to Make It Easier
But don't worry, these two practical strategies are sure to make math a little easier—and less daunting: 1. Practice and Consistency is Key. You know what they say: Practice makes perfect. And while it might seem like a no-brainer, practice and consistency are two of the most effective ways to improve math skills.
Maths Question Generator
Difficulty. Min: Max: Fully customisable Maths Question Generator. Create up to 9 different groups of randomly generated questions, each testing a specific topic and level of difficulty.
Mathway
Free math problem solver answers your algebra homework questions with step-by-step explanations. Mathway. Visit Mathway on the web. Start 7-day free trial on the app. Start 7-day free trial on the app. Download free on Amazon. Download free in Windows Store. get Go. Algebra. Basic Math. Pre-Algebra. Algebra. Trigonometry. Precalculus.
How Do You Solve Word Problems in Math?
Step 2: Highlight the keywords in the word problem. The keywords for word problems in math indicate what math action should be taken. Teach your child to highlight or underline the keywords in every word problem. Here are some of the most common keywords in math word problems: Subtraction words- less than, minus, take away.
Step-by-Step Math Problem Solver
QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students. The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and ...
The Simplest Math Problem Could Be Unsolvable
Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x: if x is odd, you calculate 3 x + 1; otherwise calculate x ...
Improving Students' Math Literacy in Middle and High School
6 Ways to Help Students Gain Math Literacy. 1. Use sentence frames. Sentence frames are a simple way to help students of all backgrounds learn how to state their answers and ask any questions they have about a word problem. Teachers can post sentence frames on a board or even on students' desks for easy access.
Do You Understand the Problem You're Trying to Solve?
To solve tough problems at work, first ask these questions. Problem solving skills are invaluable in any job. But all too often, we jump to find solutions to a problem without taking time to ...
Frontiers
Project-based learning (PBL) is a promising approach to enhance mathematics learning concepts in higher education. Here, teachers provide guidance and support to PBL implementation. The objective of this study was to develop PBL-based biotechnological projects as a strategy for mathematics learning. The methodology design was applied to 111 university students from Biochemical, Chemical ...
The Accidental Tax Cutter in Chief
2. Hosted by Michael Barbaro. Featuring Jim Tankersley. Produced by Stella Tan and Mary Wilson. With Michael Simon Johnson. Edited by Lisa Chow. Original music by Dan Powell and Marion Lozano ...