elementary problem solving in math with answer

K-5 Math Centers

K-5 math ideas, 3rd grade math, need help organizing your k-5 math block, math problem solving 101.

Have you ever given your students a money word problem where someone buys an item from a store, but your students come up with an answer where the person that bought the item ends up with more money than he or she came in with?

Word problem solving is one of those things that many of our children struggle with.  When used effectively, questioning and dramatization can be powerful tools for our students to use when solving these types of problems.

I came up with this approach after co-teaching a lesson with a 3rd grade teachers.  Her kids were having extreme difficulty comprehending a word problem she presented.  So we devised a lesson that would help students better understand problem solving.

The approach we took included the use of several literacy skills, like reading comprehension and writing. First, we started the lesson with a “think aloud” modeled by the teacher.We read and displayed the problem below but excluded ALL of the numbers. See the images below:

The purpose of reading the problem without the numbers is to get the students to understand what is actually happening in the problem.  Typically some students focus solely on keywords when solving word problems, but I do not advise using this approach exclusively.  With math problems, the context of the problem and actions in the problem determine how the child should go about solving it.

Read the Problem Without Numbers & Ask Questions:

After reading the problem (without numbers) to the students, I asked the following questions:

• Can you describe what is happening in your own words?
• What is the main idea of the problem?
• How could you act this out?

Make a Plan & Ask Questions:

After the students articulated what was happening in the problem, we made a plan to solve the problem.  I used the following guiding questions:

• Sample Answers include- We know that Kai has some goldfish. Kai donated or gave away some of the goldfish.
• Sample Answers include – We need to know how many goldfish Kai has.  We also need to know how many he gave anyway.  We also need to know how many bowls there are.
• Sample Answers include-   We need to find out how many fish belong in each bowl.

The class discussed the answers to the questions above. As we discussed the questions above the responses were written out on a problem solving template.

As part of this process, we clarified student understanding of the problem and determined what we needed to find and do to solve the problem.  Next, we walked the students through the process of showing their work using pictures.  Lastly, we checked our answers by writing an equation that matched the pictures to finally solve the problem.

Team Work Counts

After going through the process with the class, we decided to split the students into small groups of 3 and 4 to solve a math problem together.  The groups were expected to use the same process that we used to solve the problem.  It took a while but check out one of the final products below.

Benefits to Using this Process:

• Students understood what the problem is asking them to do
• Students are required to think and communicate as a team
• Students avoid making errors that can come with only using keywords
• Students are required to record their math reasoning using the problem solving template
• After using this process a couple of times, students get used to explaining and justifying their answers
• You become the facilitator of the learning by asking more questions, thereby making students independent thinkers

Things to Consider Include:

• This process in NOT quick.  It requires TIME.  You should not rush the process and expect to have it completed in 20 – 30 minutes in one day.
• This process is not a one time lesson.  Students may not get it the first time.  It should be seen a routine that can be used when solving word problems.

Be sure to let me know how this process works in your classroom in the comments below.

• Read more about: K-5 Math Ideas

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Problem Solving

Problem Solving Strategies

Think back to the first problem in this chapter, the ABC Problem . What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills.  He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities).  He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. [1]

 In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

• First, you have to understand the problem.
• After understanding, then make a plan.
• Carry out the plan.
• Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!). If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Problem 2 (Payback)

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem?

This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers). Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

Problem 3 (Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64… It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem). Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns). Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table.
• Can you explain and justify any of the patterns you see? How can you be sure they will continue?
• What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

Problem 4 (Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions). When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons ↵

Mathematics for Elementary Teachers Copyright © 2018 by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

Be Prepared

y = 12 − 3 x 2 y = 12 − 3 x 2

− 1 2 − 1 2

0, undefined

−5 , −5 , 5 −5 , −5 , 5

x + 1 4 x + 1 4

y = 12 − 2 x −3 y = 12 − 2 x −3

x = 10 3 x = 10 3

− 2 5 x + 6 − 2 5 x + 6

x > 5 x > 5

x is less than 5.

A: ( 5 , 1 ) ( 5 , 1 )  B: ( −2 , 4 ) ( −2 , 4 )  C: ( −5 , −1 ) ( −5 , −1 )  D: ( 3 , −2 ) ( 3 , −2 )  E: ( 0 , −5 ) ( 0 , −5 )  F: ( 4 , 0 ) ( 4 , 0 )

A: ( 4 , 2 ) ( 4 , 2 )  B: ( −2 , 3 ) ( −2 , 3 )  C: ( −4 , −4 ) ( −4 , −4 )  D: ( 3 , −5 ) ( 3 , −5 )  E: ( −3 , 0 ) ( −3 , 0 )  F: ( 0 , 2 ) ( 0 , 2 )

Answers will vary.

ⓐ yes, yes  ⓑ yes, yes

ⓐ no, no  ⓑ yes, yes

x - intercept: ( 2 , 0 ) ( 2 , 0 ) ; y - intercept: ( 0 , −2 ) ( 0 , −2 )

x - intercept: ( 3 , 0 ) ( 3 , 0 ) , y - intercept: ( 0 , 2 ) ( 0 , 2 )

x - intercept: ( 4 , 0 ) ( 4 , 0 ) , y - intercept: ( 0 , 12 ) ( 0 , 12 )

x - intercept: ( 8 , 0 ) ( 8 , 0 ) , y - intercept: ( 0 , 2 ) ( 0 , 2 )

x - intercept: ( 4 , 0 ) ( 4 , 0 ) , y - intercept: ( 0 , −3 ) ( 0 , −3 )

x - intercept: ( 4 , 0 ) ( 4 , 0 ) , y - intercept: ( 0 , −2 ) ( 0 , −2 )

− 2 3 − 2 3

− 4 3 − 4 3

− 3 5 − 3 5

− 1 36 − 1 36

− 1 48 − 1 48

slope m = 2 3 m = 2 3 and y -intercept ( 0 , −1 ) ( 0 , −1 )

slope m = 1 2 m = 1 2 and y -intercept ( 0 , 3 ) ( 0 , 3 )

2 5 ; ( 0 , −1 ) 2 5 ; ( 0 , −1 )

− 4 3 ; ( 0 , 1 ) − 4 3 ; ( 0 , 1 )

− 1 4 ; ( 0 , 2 ) − 1 4 ; ( 0 , 2 )

− 3 2 ; ( 0 , 6 ) − 3 2 ; ( 0 , 6 )

ⓐ intercepts  ⓑ horizontal line  ⓒ slope–intercept  ⓓ vertical line

ⓐ vertical line  ⓑ slope–intercept  ⓒ horizontal line  ⓓ intercepts

• ⓐ 50 inches
• ⓑ 66 inches
• ⓒ The slope, 2, means that the height, h , increases by 2 inches when the shoe size, s , increases by 1. The h -intercept means that when the shoe size is 0, the height is 50 inches.
• ⓐ 40 degrees
• ⓑ 65 degrees
• ⓒ The slope, 1 4 1 4 , means that the temperature Fahrenheit ( F ) increases 1 degree when the number of chirps, n , increases by 4. The T -intercept means that when the number of chirps is 0, the temperature is 40 ° 40 ° .
• ⓒ The slope, 0.5, means that the weekly cost, C , increases by $0.50 when the number of miles driven, n, increases by 1. The C -intercept means that when the number of miles driven is 0, the weekly cost is $60
• ⓒ The slope, 1.8, means that the weekly cost, C, increases by $1.80 when the number of invitations, n , increases by 1.80. The C -intercept means that when the number of invitations is 0, the weekly cost is $35.;

not parallel; same line

perpendicular

not perpendicular

y = 2 5 x + 4 y = 2 5 x + 4

y = − x − 3 y = − x − 3

y = 3 5 x + 1 y = 3 5 x + 1

y = 4 3 x − 5 y = 4 3 x − 5

y = 5 6 x − 2 y = 5 6 x − 2

y = 2 3 x − 4 y = 2 3 x − 4

y = − 2 5 x − 1 y = − 2 5 x − 1

y = − 3 4 x − 4 y = − 3 4 x − 4

y = 8 y = 8

y = 4 y = 4

y = 5 2 x − 13 2 y = 5 2 x − 13 2

y = − 2 5 x + 22 5 y = − 2 5 x + 22 5

y = 1 3 x − 10 3 y = 1 3 x − 10 3

y = − 2 5 x − 23 5 y = − 2 5 x − 23 5

x = 5 x = 5

x = −4 x = −4

y = 3 x − 10 y = 3 x − 10

y = 1 2 x + 1 y = 1 2 x + 1

y = − 1 3 x + 10 3 y = − 1 3 x + 10 3

y = −2 x + 16 y = −2 x + 16

y = −5 y = −5

y = −1 y = −1

x = −5 x = −5

ⓐ yes  ⓑ yes  ⓒ yes  ⓓ yes  ⓔ no

ⓐ yes  ⓑ yes  ⓒ no  ⓓ no  ⓔ yes

y ≥ −2 x + 3 y ≥ −2 x + 3

y ≤ 1 2 x − 4 y ≤ 1 2 x − 4

x − 4 y ≤ 8 x − 4 y ≤ 8

3 x − y ≤ 6 3 x − y ≤ 6

Section 4.1 Exercises

A: ( −4 , 1 ) ( −4 , 1 )  B: ( −3 , −4 ) ( −3 , −4 )  C: ( 1 , −3 ) ( 1 , −3 )  D: ( 4 , 3 ) ( 4 , 3 )

A: ( 0 , −2 ) ( 0 , −2 )  B: ( −2 , 0 ) ( −2 , 0 )  C: ( 0 , 5 ) ( 0 , 5 )  D: ( 5 , 0 ) ( 5 , 0 )

ⓑ Age and weight are only positive.

Section 4.2 Exercises

ⓐ yes; yes  ⓑ no; no  ⓒ yes; yes  ⓓ yes; yes

ⓐ yes; yes  ⓑ yes; yes  ⓒ yes; yes  ⓓ no; no

$722, $850, $978

Section 4.3 Exercises

( 3 , 0 ) , ( 0 , 3 ) ( 3 , 0 ) , ( 0 , 3 )

( 5 , 0 ) , ( 0 , −5 ) ( 5 , 0 ) , ( 0 , −5 )

( −2 , 0 ) , ( 0 , −2 ) ( −2 , 0 ) , ( 0 , −2 )

( −1 , 0 ) , ( 0 , 1 ) ( −1 , 0 ) , ( 0 , 1 )

( 6 , 0 ) , ( 0 , 3 ) ( 6 , 0 ) , ( 0 , 3 )

( 0 , 0 ) ( 0 , 0 )

( 4 , 0 ) , ( 0 , 4 ) ( 4 , 0 ) , ( 0 , 4 )

( −3 , 0 ) , ( 0 , 3 ) ( −3 , 0 ) , ( 0 , 3 )

( 8 , 0 ) , ( 0 , 4 ) ( 8 , 0 ) , ( 0 , 4 )

( 2 , 0 ) , ( 0 , 6 ) ( 2 , 0 ) , ( 0 , 6 )

( 12 , 0 ) , ( 0 , −4 ) ( 12 , 0 ) , ( 0 , −4 )

( 2 , 0 ) , ( 0 , −8 ) ( 2 , 0 ) , ( 0 , −8 )

( 5 , 0 ) , ( 0 , 2 ) ( 5 , 0 ) , ( 0 , 2 )

( 4 , 0 ) , ( 0 , −6 ) ( 4 , 0 ) , ( 0 , −6 )

( −3 , 0 ) , ( 0 , 1 ) ( −3 , 0 ) , ( 0 , 1 )

( −10 , 0 ) , ( 0 , 2 ) ( −10 , 0 ) , ( 0 , 2 )

ⓐ ( 0 , 1000 ) , ( 15 , 0 ) ( 0 , 1000 ) , ( 15 , 0 ) ⓑ At ( 0 , 1000 ) ( 0 , 1000 ) , he has been gone 0 hours and has 1000 miles left. At ( 15 , 0 ) ( 15 , 0 ) , he has been gone 15 hours and has 0 miles left to go.

Section 4.4 Exercises

−3 2 = − 3 2 −3 2 = − 3 2

− 3 4 − 3 4

− 1 3 − 1 3

− 5 2 − 5 2

− 8 7 − 8 7

ⓐ 1 3 1 3   ⓑ 4 12 pitch or 4-in-12 pitch

3 50 3 50 ; rise = 3, run = 50

ⓐ 288 inches (24 feet)  ⓑ Models will vary.

When the slope is a positive number the line goes up from left to right. When the slope is a negative number the line goes down from left to right.

A vertical line has 0 run and since division by 0 is undefined the slope is undefined.

Section 4.5 Exercises

slope m = 4 m = 4 and y -intercept ( 0 , −2 ) ( 0 , −2 )

slope m = −3 m = −3 and y -intercept ( 0 , 1 ) ( 0 , 1 )

slope m = − 2 5 m = − 2 5 and y -intercept ( 0 , 3 ) ( 0 , 3 )

−9 ; ( 0 , 7 ) −9 ; ( 0 , 7 )

4 ; ( 0 , −10 ) 4 ; ( 0 , −10 )

−4 ; ( 0 , 8 ) −4 ; ( 0 , 8 )

− 8 3 ; ( 0 , 4 ) − 8 3 ; ( 0 , 4 )

7 3 ; ( 0 , −3 ) 7 3 ; ( 0 , −3 )

horizontal line

vertical line

slope–intercept

• ⓒ The slope, 2.54, means that Randy’s payment, P , increases by $2.54 when the number of units of water he used, w, increases by 1. The P –intercept means that if the number units of water Randy used was 0, the payment would be $28.
• ⓒ The slope, 0.32, means that the cost, C , increases by $0.32 when the number of miles driven, m, increases by 1. The C -intercept means that if Janelle drives 0 miles one day, the cost would be $15.
• ⓒ The slope, 0.09, means that Patel’s salary, S , increases by $0.09 for every $1 increase in his sales. The S -intercept means that when his sales are $0, his salary is $750.
• ⓒ The slope, 42, means that the cost, C , increases by $42 for when the number of guests increases by 1. The C -intercept means that when the number of guests is 0, the cost would be $750.

not parallel

• ⓐ For every increase of one degree Fahrenheit, the number of chirps increases by four.
• ⓑ There would be −160 −160 chirps when the Fahrenheit temperature is 0 ° 0 ° . (Notice that this does not make sense; this model cannot be used for all possible temperatures.)

Section 4.6 Exercises

y = 4 x + 1 y = 4 x + 1

y = 8 x − 6 y = 8 x − 6

y = − x + 7 y = − x + 7

y = −3 x − 1 y = −3 x − 1

y = 1 5 x − 5 y = 1 5 x − 5

y = − 2 3 x − 3 y = − 2 3 x − 3

y = 2 y = 2

y = −4 x y = −4 x

y = −2 x + 4 y = −2 x + 4

y = 3 4 x + 2 y = 3 4 x + 2

y = − 3 2 x − 1 y = − 3 2 x − 1

y = 6 y = 6

y = 3 8 x − 1 y = 3 8 x − 1

y = 5 6 x + 2 y = 5 6 x + 2

y = − 3 5 x + 1 y = − 3 5 x + 1

y = − 1 3 x − 11 y = − 1 3 x − 11

y = −7 y = −7

y = − 5 2 x − 22 y = − 5 2 x − 22

y = −4 x − 11 y = −4 x − 11

y = −8 y = −8

y = −4 x + 13 y = −4 x + 13

y = x + 5 y = x + 5

y = − 1 3 x − 14 3 y = − 1 3 x − 14 3

y = 7 x + 22 y = 7 x + 22

y = − 6 7 x + 4 7 y = − 6 7 x + 4 7

y = 1 5 x − 2 y = 1 5 x − 2

x = 4 x = 4

x = −2 x = −2

y = −3 y = −3

y = 4 x y = 4 x

y = 1 2 x + 3 2 y = 1 2 x + 3 2

y = 5 y = 5

y = 3 x − 1 y = 3 x − 1

y = −3 x + 3 y = −3 x + 3

y = 2 x − 6 y = 2 x − 6

y = − 2 3 x + 5 y = − 2 3 x + 5

x = −3 x = −3

y = −4 y = −4

y = x y = x

y = − 3 4 x − 1 4 y = − 3 4 x − 1 4

y = 5 4 x y = 5 4 x

y = 1 y = 1

y = 2 x + 4 y = 2 x + 4

y = 3 4 x y = 3 4 x

y = 1.2 x + 5.2 y = 1.2 x + 5.2

Section 4.7 Exercises

ⓐ yes  ⓑ yes  ⓒ no  ⓓ yes  ⓔ yes

ⓐ yes  ⓑ no  ⓒ yes  ⓓ yes  ⓔ yes

ⓐ no  ⓑ no  ⓒ no  ⓓ yes  ⓔ yes

y < 2 x − 4 y < 2 x − 4

y ≤ − 1 3 x − 2 y ≤ − 1 3 x − 2

x + y ≥ 3 x + y ≥ 3

x + 2 y ≤ −2 x + 2 y ≤ −2

2 x − y < 4 2 x − y < 4

4 x − 3 y > 12 4 x − 3 y > 12

• ⓑ Answers will vary.

Review Exercises

ⓐ ( 2 , 0 ) ( 2 , 0 )   ⓑ ( 0 , −5 ) ( 0 , −5 )   ⓒ ( −4.0 ) ( −4.0 )   ⓓ ( 0 , 3 ) ( 0 , 3 )

( 6 , 0 ) , ( 0 , 4 ) ( 6 , 0 ) , ( 0 , 4 )

slope m = − 2 3 m = − 2 3 and y -intercept ( 0 , 4 ) ( 0 , 4 )

5 3 ; ( 0 , −6 ) 5 3 ; ( 0 , −6 )

4 5 ; ( 0 , − 8 5 ) 4 5 ; ( 0 , − 8 5 )

plotting points

ⓐ −$250  ⓑ $450  ⓒ The slope, 35, means that Marjorie’s weekly profit, P , increases by $35 for each additional student lesson she teaches. The P –intercept means that when the number of lessons is 0, Marjorie loses $250.  ⓓ

y = −5 x − 3 y = −5 x − 3

y = −2 x y = −2 x

y = −3 x + 5 y = −3 x + 5

y = 3 5 x y = 3 5 x

y = −2 x − 5 y = −2 x − 5

y = 1 2 x − 5 2 y = 1 2 x − 5 2

y = − 2 5 x + 8 y = − 2 5 x + 8

y = 3 y = 3

y = − 3 2 x − 6 y = − 3 2 x − 6

ⓐ yes  ⓑ no  ⓒ yes  ⓓ yes  ⓔ no

y ≥ 2 3 x − 3 y ≥ 2 3 x − 3

x − 2 y ≥ 6 x − 2 y ≥ 6

Practice Test

ⓐ yes  ⓑ yes  ⓒ no

( 3 , 0 ) , ( 0 , −4 ) ( 3 , 0 ) , ( 0 , −4 )

y = − 3 4 x − 2 y = − 3 4 x − 2

y = 1 2 x − 4 y = 1 2 x − 4

y = − 4 5 x − 5 y = − 4 5 x − 5

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Mathematics for Elementary Teachers

(18 reviews)

Michelle Manes, Honolulu, HI

Copyright Year: 2017

Publisher: University of Hawaii Manoa

Language: English

Formats Available

Conditions of use.

Learn more about reviews.

Reviewed by Kevin Voogt, Assistant Professor, Grace College on 4/20/23

There seem to be subjects missing that are typical of the common core mathematics for elementary teachers texts (e.g., Ratios/Proportions, clear Partitive/Measurement division ideas, percentages, certain ideas in Geometry, Measurement). read more

Comprehensiveness rating: 4 see less

There seem to be subjects missing that are typical of the common core mathematics for elementary teachers texts (e.g., Ratios/Proportions, clear Partitive/Measurement division ideas, percentages, certain ideas in Geometry, Measurement).

Content Accuracy rating: 5

I did not find mathematical errors in the text during my review.

Relevance/Longevity rating: 3

I think there is need for quite a few updates to the text in regards to what is covered in elementary mathematics through the common core. The topics listed in my review of the Comprehensiveness above are just a start. I also see a need to add more activities to each section where prospective elementary teachers could do more exploration of the mathematics rather than what seems to be a more traditional approach of having the text explain it followed by problem sets alone.

Clarity rating: 5

The wording was quite clear and had nice explanations throughout.

Consistency rating: 5

It seems consistent throughout - with recurrent use of the same technical terms as needed.

Modularity rating: 4

There were a few issues with being able to assign the texts at different points within the course, as is the case for many math texts, in that many of the sections rely heavily on prior knowledge. If reorganization were to occur, there would be some need to re-structure how certain sections are taught.

Organization/Structure/Flow rating: 4

The text lacks much of the wonderful mathematical connections that could be made between ideas. While some connections are made, they seem a little outdated at times. I also think it would make more sense to have the properties of operations within their corresponding sections on operations rather than after all 4 operations are introduced.

Interface rating: 5

I did not see any issues with the interface. It was pretty user-friendly.

Grammatical Errors rating: 5

I did not notice any errors during my review.

Cultural Relevance rating: 5

I did not see anything insensitive or offensive in the text.

The text is just a small sampling of the many methods that could be used in teaching these mathematical ideas. I would have liked to see more activities for elementary teachers built into the lessons in each chapter as a means for learning and exploring ideas to facilitate more discussion as this text is used. There also are so many more connections that could be made between mathematical ideas that were lost a bit, especially with the general organization. On the whole, it is a nice resource and I could see it as useful for students studying for their certification exams to get some perspectives on the mathematical ideas they might encounter.

Reviewed by Sandra Zirkes, Teaching Professor, Bowling Green State University on 4/14/23

The text covers whole numbers, fractions, decimals, and operations well, and it provides some topics in geometry and algebraic thinking. However, the topics of ratio, proportion, and percent, as well as a more thorough coverage of geometry and... read more

The text covers whole numbers, fractions, decimals, and operations well, and it provides some topics in geometry and algebraic thinking. However, the topics of ratio, proportion, and percent, as well as a more thorough coverage of geometry and measurement are missing.

All information in the text is mathematically accurate and the writing and diagrams are error-free.

Relevance/Longevity rating: 4

While all of the information in the text is accurate and thought-provoking, some specific approaches are outdated with respect to the current standards and pedagogy. Approaching the concept of place value through the "Dots and Boxes" method, without reference to base ten blocks that are overwhelmingly used in the elementary math classroom, limits the coverage of this important topic. Similarly, approaching fractions using the "pies and kids" scenario is not consistent with the standards which emphasize the understanding of all fractions as iterations of unit fractions.

The text is written using clear and understandable prose that is both mathematically accurate and accessible to college level pre-service teachers.

The text has a clear organization and focus and uses consistent approaches and terminology throughout.

Much of the text is easily divisible into smaller subsections for student use. With respect to reorganization and realignment for a particular course, while some topics are revisited at appropriate points in the text, if those original topics were not covered in the course, revisiting the topic may not provide enough basis for the new topic. For example, the understanding of decimals is highly reliant on a student's understanding of the Dots and Boxes approach to place value earlier in the text.

Organization/Structure/Flow rating: 5

The topics in the text are organized in a logical way that is consistent with the structure of a typical mathematics education course.

Interface rating: 4

Navigating the text itself was seamless and intuitive. However, the videos that I viewed had poor visual quality and there was no audio.

The text is well written with no grammatical errors.

There is no apparent cultural insensitivity in the text.

This text has a problem solving focus and emphasizes deep thinking and reasoning about mathematics. Its approaches are clear and understandable. While its approaches are mathematically correct and thought-provoking, it is missing some key topics such as ratio, proportion, percent, and a more thorough coverage of geometry and measurement, as well as some standards-based approaches such as base ten blocks and understanding fractions as iterations of unit fractions.

Reviewed by Fred Coon, Assistant Professor, Anderson University on 2/16/23

The text covers all major points to help develop future teachers. read more

Comprehensiveness rating: 5 see less

The text covers all major points to help develop future teachers.

Text appears to be accurate.

The content is consist with concepts that elementary teachers should know. The methods are small in diversity.

Topics where well explained.

Text appears to use understandable and consistent terms.

Modularity rating: 5

Units appear to be mostly independent and can be used as stand alone units.

The topics are presented in a manner that build on each other but can be rearrange if desired.

Interface was useful and aided in navigating text.

I found no errors.

The text has no culturally insensitive or offensive items that I noticed.

I would like to have seen more diversity in methods discussed.

Reviewed by Perpetual Opoku Agyemang, Professor of Mathematics, Holyoke Community College on 6/17/21

The content in this text is built to help its readers, especially, pre-service elementary education majors learn to think like a mathematician in some very specific ways. The content addresses the subject framework in a complete yet concise... read more

The content in this text is built to help its readers, especially, pre-service elementary education majors learn to think like a mathematician in some very specific ways. The content addresses the subject framework in a complete yet concise manner. Although it does not provide an effective index/or glossary, LCD was not extensively tackled using factor tree, multiples or tables to express it, I still give props to the author since there are a lot of pictorial examples and a question bank for most of the various concepts. Furthermore, Dots and Boxes game on chapter 1 was very engaging and fun.

This text is very accurate and informative using a variety of felicitous examples to suit a diverse student population.

Conventional concepts are presented in a current and applied manner which allows for easier association with similar organized and retained information. This text could use some updated fraction problems and examples involving mixed numbers. Some of the YouTube videos have no sound at all.

Clarity rating: 4

Content material was presented in an easy to understand prose. Introduction of concepts and new terms were usually done by association or relevant previous knowledge. Some of the concepts like Multiplying Fractions, have YouTube videos embedded in the introductions.

Terminologies and framework are consistent throughout the text. The use of different notations were consistent throughout the various chapters and subunits.

This text has easily divisible content as stand alone subunits. However, numbering these chapters and subunits would have gone a long way to help its readers.

The topics in this text are organized from basic to complex concepts in a logical, clear fashion.

This text has an awesome interface (Online, PDF and XML). Moreover, it is untainted by distractions that may confuse its reader. Hyperlinks should have been included in the content.

I did not spot any grammatical errors in this text.

This content material contains no recognizable cultural insensitivity. It could use more examples involving modern affairs that are inclusive of diverse backgrounds.

I truly love the concise format of this text and how many different examples it uses to explain the concepts. The Geometry of Arts and Science and Tangrams were so informative with fun activities. It's easy to tell when one example ends and another begins, although index/or glossary and a system of links from the table of contents would be greatly appreciated. I did not see Points on a Coordinate Plane. Additionally, the number of exercises per section is too small. Of course this can be remedied by adding more. As with any textbook, the reader will need to supplement certain sections and clarify particular terms and concepts to best fit their situation. Pre-service elementary education majors could transition to this book fairly easily and successfully teach K-6 students in the United States in alignment with current Common Core Math Standards.

Reviewed by April Slack, Math Instructor, Aiken Technical College on 5/13/21

This text covers elementary mathematics strands including place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. Measurement and Data and Statistics strands are not included in this particular text. ... read more

This text covers elementary mathematics strands including place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. Measurement and Data and Statistics strands are not included in this particular text. The last chapter supplies the audience with problem-based learning approaches that include some measurement, but not in the detail of previous chapters of the book. It does incorporate problem solving strategies and pedagogical techniques teachers may use in the classroom. Examples with solutions and clarifying notes are provided throughout the text. The text does address Common Core Standards as well as the eight mathematical process standards. The textbook also provide teachers with a conceptual understanding of elementary mathematics along with appropriate mathematical terminology. The text does not offer an index or glossary.

The mathematics content provided in this text is accurate and provides thorough examples of teaching elementary mathematics for pre-service teachers. I found the text to build conceptual understanding and procedural fluency rather than just focus on basic algorithms to solve math problems. This is especially important for pre-service teachers, as they need to truly understand the "why" behind the math tricks that are often taught in early grades. The embedded links throughout the text are all in working order, as well.

The problem-solving approach to mathematics is especially relevant for elementary pre-service teachers; the intended audience. The book does expand beyond elementary mathematics, however, this is deemed extremely useful for all levels of mathematics teachers. Knowing the mathematical concepts beyond elementary strands allows teachers to know where there students are going and the mathematical purpose of content standards at each grade level. Many of the pedagogical techniques presented in the text are aligned with current research and instructional strategies for the elementary classroom.

This text provides explanations and defines mathematical terminology and has accessible prose. Beginning with the problem solving chapter before the specific content strands allows teachers to apply and consider strategies throughout the text. Often times, textbooks save problem solving for the end, but this text addresses strategies upfront and spirals nicely throughout the text. Some of the examples and visual representations are intended for an audience with mathematical background knowledge and strengths. A pre-service teacher may need help with content review prior to understanding the selection of particular problems highlighted in the text.

The text is well-organized and consistent with terminology throughout. The text is also consistent with provided examples that are used by mathematics teachers in everyday classrooms. There are multiple examples throughout each of the content chapters for pre-service teachers to reference and use in their own experiences.

This book is an easy read and may be easily broken up for weekly reading assignments and reflections. It seems as if mathematics teachers had a hand in writing this book. Bulleted and numbered lists are used throughout the text. The text also presents examples in clear, colored blocks. Visual models are clear and concise.

The book is well-organized with headings, subheadings, and the use of italics and boldface make this book extremely student friendly. The topics and content presented in this text are clear and in a logical order. Bulleted and numbered lists are reader friendly and easily understood. I found having the problem solving chapter appear first in the text stresses the importance and relevance of helping students become natural problem solvers. Often times texts and even worksheets save problem solving until the end, which poses a problem with students in the classroom.

This book is very easily navigated. The contents tab and drop down menu allows for the reader to quickly navigate to particular chapters and specific content. The previous and next buttons located at the bottom of the text allows readers to toggle between chapters very quickly. All embedded links work as they should and visual models are clear and understandable. There are no distractors present when trying to navigate the text. There is no index / glossary offered with this text.

The text is free from grammatical errors.

Cultural Relevance rating: 4

This text is not culturally offensive in any way. The final chapter of the text is dedicated to problem based learning and is centered around Voyaging on Hōkūle`a. The text provides embedded links to culturally relevant videos and models that help illustrate the cultural practices of Polynesians.

This textbook has a solid foundation and is well-organized for it's intended audience, the elementary mathematics pre-service teacher. This text will help build conceptual understanding of mathematics that will lead to procedural fluency for teachers. The text also provides clear examples of instructional strategies to be used in today's classrooms. Methods courses for pre-service teachers will find this text extremely useful and easy to incorporate in elementary mathematics methods instruction.

Reviewed by Kane Jessen, Math Instructor, Community College of Aurora on 8/13/20

This textbook is intended to cover the mathematics topics necessary to prepare pre-service elementary education majors to successfully teach K-6 students in the United States in alignment with current Common Core Math Standards. The textbook is a... read more

This textbook is intended to cover the mathematics topics necessary to prepare pre-service elementary education majors to successfully teach K-6 students in the United States in alignment with current Common Core Math Standards. The textbook is a mostly comprehensive collection of K-6 Common Core elementary math topics ranging from non-numerical problem solving through summative PBL assessments incorporating algebra, geometry and authentic problem solving. However, several topics related to K-6 CCSS Standards are not covered or minimally covered. CCSS topics with minimal coverage include set theory, logic, integers, probability, graphing and data analysis. At the beginning of the book, there is an effective and accessible table of contents with links included. However, sections and subsections are labeled only with names and page numbers. The text does not contain an index, glossary or appendices. Chapter summaries and links to previous concepts/problems are not included but would support student learning if included. More visuals and historical explorations would increase comprehensiveness.

Content was found to be accurate, error-free and unbiased.

Relevance/Longevity rating: 5

The language and examples of this text are written with a constructivist and meta-pedagogical voice that is both academic and accessible. The author immediately addresses the importance of CCSS and consistently utilizes the “Exploding Dots” curriculum. The “Exploding Dots” curriculum is a brave and differentiated approach to holistically teaching multi-base mathematics to K-12 students. “Exploding Dots” has been a core focus of K-12 Global Math Project and was pioneered by James Tanton . As future teachers, students can expect to teach “Exploding Dots” or similar CCSS curriculum sometime during their teaching career.

The language of the text is well-written, accessible and clear. Some sections and examples could be expanded for clarity/depth. Prior definitions/review concepts are not consistently linked.

The text is internally consistent in terms of its own terminology, framework and graphics. The “Exploding Dots” infusion helps maintain continuity throughout the text but is not present in all modules.

This text follows the common sequence that many “Mathematics for Elementary Teachers” textbooks commonly follow. The text is organized into eight modules. The text initially builds upon itself without being overly self-referential. The text’s sections, subsections, definitions, axioms and problem banks are all well delineated but lack sections/subsection numbers/identifiers and links to previous concepts/definitions

This textbook has a solid flow and follows a common sequence shared by most for profit “Mathematics for Elementary Teachers” texts. The text is well organized and builds upon itself.

Minimal issues involving interface were observed. Observed interface issues include, one broken video link and unnumbered sections. Definitions and review topics are not linked or referenced with page numbers/sections, however, this creates minimal usability issues. The text contains adequate procedural visuals and also cultural and historical visuals that enhance the student learning experience.

This text is largely free from grammatical errors. Grammatical errors that were observed were minor and non-persistent.

The text is not culturally insensitive or offensive in any way. It consistently uses examples that are inclusive of a variety of races, ethnicities, and backgrounds. Textbook examples often include references to Hawaiin culture. These references are easily understandable and could be readily adapted for students in other places. In an effort to increase relevance, further additions to the text could be made to provide a more equitable and historical focus on women, minorities and problem based learning cross-sectional explorations similar to the Hōkūleʻa section.

This textbook has a solid structure and great flow, I thoroughly enjoyed reviewing this textbook. I am genuinely excited to incorporate Michelle Manes ‘Mathematics for Elementary Teachers’ into my upcoming semester’s curriculum. With subsequent editions and revisions, this textbook will become a wonderful text for students majoring in primary education, especially those who are either lacking in basic math skills or math confidence.

Reviewed by Reina Ojiri, Assistant Professor, Leeward Community College on 7/27/20

The book begins with a reference to the Common Core State Standards (CCSS) for Mathematics and the eight “Mathematical Practices". Though not all states have adopted and/or are currently using the Common Core Standards, with its incorporation at... read more

Comprehensiveness rating: 2 see less

The book begins with a reference to the Common Core State Standards (CCSS) for Mathematics and the eight “Mathematical Practices". Though not all states have adopted and/or are currently using the Common Core Standards, with its incorporation at the beginning of the text I initially thought that the Common Core standards would be revisited consistently throughout the text.

Though the "Think Pair Share" sections are great additions for discussion to the book they do not include common misconceptions or tips for instructors to use to help guide these discussion prompts. The focus on just one type of discussion "Think Pair Share" also does not give future teachers a broader experience with different cooperative learning strategies in the classroom. There are many strategies in addition to “Think Pair Share” that are also great and seeing the same strategy over and over did not provide variation or keep me engaged as I read through the text.

There are a few key concepts that are not included in the text including Measurement & Data and Statistics & Probability.

The text also does not include an effective index and/or glossary. I have found that students do use the index and/or glossary that is typically in the back of the book to help them find information in the text quickly.

Content Accuracy rating: 3

The content is error-free however some of the images included on the PDF version are blurry and hard to read. There does not seem to be consistency between the different readable versions of the text.

There also seems to be a bias for the dots and boxes strategy throughout the text and the content lacks current practices of teaching concepts.

Just like any text, this textbook needs to be updated to match current best practices and research in math education. Since this text is Attribution-ShareAlike which allows “others to remix, adapt, and build upon your work even for commercial purposes, as long as they credit you and license their new creations under the identical terms” it does seem that updates and instructor/course-specific content will be relatively easy to implement as needed.

Clarity rating: 3

This text is written in a way unique way that makes it easier for students to read through and follow. It is very student friendly however might not be as useful as an instructor text since the instructor needs to fill-in-the-blanks on their own.

Consistency rating: 3

The text is written with consistent terminology however the framework for each chapter is not consistent. Some chapters include Explorations and additional sections while others end consistently with a problem bank.

Modularity rating: 3

The text is divided into smaller reading sections however the titles of each section are not easily recognized by students. Though I imagine the titles were meant to be creative for each section, having something more straight forward to make it easier for students to navigate is more important than creativity especially for future teachers who might be teaching these concepts for the first time.

Organization/Structure/Flow rating: 3

It would be good to organize the material consistently throughout the text (e.g.each section should end with a problem bank). The variation in the different sections can be confusing to both the instructor and student when trying to find something in the text.

I also noticed that the online version does not include page numbers while the PDF version does. This is not helpful when referring students to particular sections of the book. The PDF version also has many completely blank pages. I am not sure if this was meant to be on purpose (for printing purposes) but these pages can be very distracting to the reader.

Interface rating: 2

Navigation throughout the text is fine however, there are noticeable differences between the online and PDF versions of the text. The images in the PDF versions are noticeably blurry and lower quality than those in the online version. In some instances, it seems as though images were screenshot and copied and pasted which could account for the image quality.

Some images, in particular, should not have been included at all and are unreadable, for example, the Hokulea on page 441.

I did not notice grammatical errors.

The connection to the Hawaiian culture was a nice touch.

I would use this text as a reference but would not adopt this book as the main text for my class.

Reviewed by Thomas Starmack, Professor, Bloomsburg University of Pennsylvania on 3/26/20

The book is somewhat dated and does not include current research based best practices like concrete, representational, then abstract. Like most authors, they make assumptions that students have the ability to understand abstract and start the... read more

The book is somewhat dated and does not include current research based best practices like concrete, representational, then abstract. Like most authors, they make assumptions that students have the ability to understand abstract and start the lesson there, which is contradictory to how the brain works and what current research says about effective math instruction and learning.

I agree the content is accurate, but in many areas the learner must have a very strong understanding of mathematical concepts, structures, and applications. There lacks current best practice and current NCTM recommendations to approaching the teaching of mathematical content.

Relevance/Longevity rating: 1

Although mathematical concepts at the elementary level remain the same, the approach to engaging students in learning and the methods of instruction have evolved greatly. The book lacks many of the newer approaches and is outdated. The arrangement of the concepts is okay. I would recommend that the big ideas of teaching math are in the beginning and providing an overview of what is mathematics and best approaches to teaching/learning mathematics. Then scaffold the specific concepts. Fractions is one of the most complex and abstract, and this book starts there as a first topic.

Once again, the book is okay in terms of math learning but dated on best practice approaches. The book does not use jargon per say, but does not provide the best approaches for students to learn how to effectively teach mathematics.

Consistency rating: 4

Yes the book is consistent throughout.

The text is divisible, just not relevant to today nor provides current approaches. The order of the content is not in line with a methods of teaching course I would follow.

Organization/Structure/Flow rating: 2

I think the topics are clear but dated and not in the order as described above.

The text provides a variety of interfaces, none of which are confusing for the student who has a very strong math background. The text does mislead students to think starting with abstract is how to instruct elementary students, which is contradictory to brain research and current best practices.

Grammatical Errors rating: 4

I did not notice any grammar errors.

Cultural Relevance rating: 3

I think the text is culturally appropriate. Not certain about the final chapter as it focuses on one population. Having a chapter or theme woven throughout the text that provides students with a stronger understanding that although mathematics is a universal language, there are cultural differences to teaching and learning as evidenced in the 1999 TIMSS report.

The text is outdated. The text is an okay resource but I would not be able to use as the main guide for learning in a college level methods of teaching elementary mathematics course.

Reviewed by Jamie Price, Assistant Professor, East Tennessee State University on 3/20/20

This book introduces the reader to the standards for mathematical practice (SMP) from the Common Core standards in the introduction. I appreciated this as these standards cover all grades and are a unifying theme of the Common Core standards, yet... read more

This book introduces the reader to the standards for mathematical practice (SMP) from the Common Core standards in the introduction. I appreciated this as these standards cover all grades and are a unifying theme of the Common Core standards, yet many times overlooked. In addition, many states, including mine, that are not following Common Core directly have adopted the SMPs. The book does not cover two of the mathematical strands, namely measurement and statistics/data. Among the strands that are covered, however, the author does a thorough job of explaining the content, using a unified theme throughout, such as dots and boxes introduced in place value that appear again in number operations. I particularly liked the final chapter of the book and its connection to Hawaiian culture. The author could easily incorporate ideas related to teaching and learning measurement into this chapter in order to make the book more comprehensive.

The content was very accurate. I did not come across any mathematical errors or biases. The author did a good job of incorporating "think, pair, share" elements throughout each chapter as a model for future teachers. To further guide future teachers, I would have liked to see the author include information in each chapter about common misconceptions students have when learning the related material and ideas on how to address those misconceptions. In my experience, I find that pre-service teachers are unaware of these misconceptions and it is helpful to make them aware of them so that they can anticipate them in their own classrooms.

The content presented in this book is up-to-date and will remain relevant for a long time. Due to the fact that this book focuses more on content rather than methods, I do not foresee a need for many updates moving forward.

The book is written in a very clear and concise way that is approachable to future and current elementary teachers. The author presents key words in bold throughout the book to draw attention to them. I liked the way that the author included videos as well as written explanations of ideas, such as in the Number and Operations chapter, section titled Addition: Dots and Boxes. The author explains, in words, how to use this method to add multi-digit numbers and follows the written example with a video explanation. This helps to reach a variety of learners and learning styles. The author also addresses common "jargon" associated with particular mathematical concepts, such as proper and improper fractions (section titled What is a Fraction?), and discusses how this jargon can be misleading for students.

Each chapter in the book includes an introduction, multiple opportunities for think-pair-share discussions, and several problem sets to practice. I appreciated the consistency in the Dots and Boxes method introduced in the Place Value chapter and then carried into the Number and Operations chapter.

The book uses a modular approach to present the material. Each module contains numerous sections that help to break up the content into smaller chunks so that the content does not seem overwhelming. The modules are set up in an order that makes sense for the mathematics, but a reader could begin reading at any module and still make sense of the content.

The organization of the topics makes sense according to the mathematics presented and is logical.

I did not find anything distracting or confusing in relation to the interface of the text. The book was easy to navigate, with a clearly defined table of contents. I was able to easily click through the various modules and sections within each module. The book uses figures well to provide engagement to the reader as well as to further clarify content. The use of videos embedded within the modules helps to strengthen understanding of the content. It did take me a minute to find the navigation link that allowed me to move to the next section in a module (right arrow at bottom right corner of the page), but once I found it I was able to navigate seamlessly to each subsequent section.

I did not find any grammatical errors in the text.

In my opinion, this was one of the biggest strengths of this text. The author did a nice job of incorporating Hawaiian culture into the text. For example, the author includes an image in the Place Value chapter (Number Systems section) that references the use of tally marks on a sign at Hanakapiai Beach. In addition, a full chapter was devoted to Voyaging on Hōkūle`a. I particularly liked how the author connected this idea to beginning teaching of elementary mathematics and encouraged future teachers to think about ways to see mathematics outside of traditional mathematical settings.

I am glad that I came across this resource. I primarily teach math methods courses for elementary pre-service teachers, but I found many aspects of this text that I can incorporate into my classes to help students think more deeply about the mathematics that they will teach. I appreciated the author's attempt to challenge students in their thinking about elementary mathematics. Initially, I was surprised to find that there was no "answer key" provided for the many problem sets that were included throughout the text. After reading the quote presented on the introductory page to the Problem Solving chapter, I realized that this may have been an intentional decision made by the author to encourage readers to go beyond "a trail someone else has laid." I find that many pre-service elementary teachers want to "just know the answer" when it comes to mathematics; a no answer key approach will encourage discussion and justification, two elements important to ensuring equity in the teaching and learning of mathematics.

Reviewed by Shay Kidd, Assistant Professor- Mathematics Education, University of Montana - Western on 12/30/19

The content that elementary teachers need to have that is not covered in this book is graphing, probability, statistics, exponents, visual displays of data. The coverage of operations is very specific in the examples and does not cover the wide... read more

The content that elementary teachers need to have that is not covered in this book is graphing, probability, statistics, exponents, visual displays of data. The coverage of operations is very specific in the examples and does not cover the wide range that should be presented in this type of text.

Content Accuracy rating: 4

While the core topics presented are correct, the number of problems that are provided without any solutions is alarming. The majority of problems that are provided are meant for the reader to perform but do not provide any type of answer key for checking the work. In this way, the book seems to assume the reader to have a solid knowledge of the topics already and this book discusses a few different approaches to these topics.

The specific content presented is up-to-date and usable.

The book's prose seem to be more of a teaching guide than a textbook. This is nice for the conversational aspect that a reader may want in their learning, but should be explained more or possibly a change of title for the book. Something more like "Exploring the concepts of Elementary Mathematics" would provide a more reading friendly approach the book offers.

The author has a consistent voice of teaching and presenting the material.

Modularity rating: 2

The break-up of the text with boxes is difficult to follow the purpose of each box. While some of the box styles are clear, such as the think, pair, share or problem boxes, others seem to break up the line of discussion. A problem box may be discussed more directly immediately following the box and the presentation of the problem. Most of the problem boxes are not discussed again in the main text. This cased issues for wanting to read with a specific purpose. When the reader wants to understand a problem more, there is generally not more discussion, but unclear about when that would be provided or not. Other times boxes were used without any "box type" provided and these were just to break up the flow of the text.

Place value was a major topic to start the book and had good coverage, then operations and fractions were discussed, then a return to place value with decimals. It would seem that a connection of place value and decimals would work better to follow the other place value discussion.

Interface rating: 3

There are several pages that have large blank parts or are totally blank. This may be due to the PDF version that I chose. When I did use the internet-connected version, there seems to be a dependence on youtube to help do some of the teaching.

There are a few minor issues that would be resolved with a good proofread.

The book does seem to be written with the Hawaiian culture in mind. This may be difficult for other cultures to connect to or understand but does not present any insensitivities.

The book's title suggests a full discussion of the topics that elementary education pre-service teachers would need to know and teach, but this book is very lacking in the topics required for this. I selected this book to review because I teach classes that would use the textbook, but I would not use this textbook as is. There are a few topics that I plan to add to my own instruction, but the book as a whole needs additional help to be able to stand alone. This really appears to be a teaching guide based on the constant think-pair-share setup. This also is a specific teaching and method that seems to require the students to already have much of the content mastered. It does not teach all the content that is required to the level of the discussion had.

Reviewed by Ryan Nivens, Associate Professor, East Tennessee State University on 10/25/19

The book covers all the expected mathematical strands except for measurement and data/stats. There are some obvious connections to the strands of mathematical practice from the Common Core standards. While the abstract specifically lists MP1, MP2,... read more

The book covers all the expected mathematical strands except for measurement and data/stats. There are some obvious connections to the strands of mathematical practice from the Common Core standards. While the abstract specifically lists MP1, MP2, and MP3, the introduction clearly lists all 8. The chapter "Voyaging on Hōkūle`a" contains activities that will require use of measurement and units, but there is no explanation on how measurement topics should be taught or approached. However, this chapter does provide a good project-based learning set of materials, and is an exceptional resource for navigation. The book also includes a chapter on Problem Solving, which is important for those students who must complete the EdTPA and address the 3rd subject specific emphasis area. All embedded links to Youtube videos or Vimeo videos are working and play within the textbook pages.

I find the mathematics to be entirely accurate. There are many teaching strategies, such as "think pair share" that are found throughout the chapters. This is particularly helpful for future teachers.

This book should last a very long time in terms of relevance.

This book is very clear, with mathematical words in bold and proper definitions provided. The text also addresses common math classroom jargon. For an excellent example of this, see the heading "What is a Fraction" in the chapter on Fractions. Toward the bottom is a sub-heading "Jargon: Improper Fractions" that has students consider the usefulness of proper and improper fractions.

This book is consistently laid out, with multiple examples, problems to try, and diagrams to support the transfer of information.

This book is entirely modular. You can pick it up, and easily start in any chapter and not be lost. The heading, subheading, use of italics and boldface make it easy to locate information. As a mathematics education book, this is quite nice.

A mathematician wrote this, the layout is logical without question.

The book is extremely easy to navigate, with a logical structure to the table of contents that you can easily click through. A drop down menu in the upper left corner allows you to view the outline of the book while still viewing a page, and you can collapse/expand chapters within the menu.

The many figures that are present throughout the textbook are perfectly displayed and fit the reading material.

There is nothing I find distracting in the layout and interface.

I could not find any errors.

An entire chapter is dedicated to Voyaging on Hōkūle`a, with exceptional videos and diagrams to illustrate the cultural practices of the early Polynesians.

I was excited to find this book in the Open Educational Resources library. As a professor who frequently teaches methods courses in mathematics for elementary teachers, I feel that this book may be a terrific book to use to replace previous texts that I've adopted. I would like to see a chapter on Measurement to make the Voyaging on Hōkūle`a chapter more useful. It is obvious from the first page you open to that this book was well planned and thought out. I'm impressed.

Reviewed by Monica Rose Gilmore, Graduate Student, CU Boulder on 7/1/19

This textbook goes into depth about different mathematical concepts that are important for elementary school teachers to understand in teaching mathematics. However, the text is missing a focus on statistics and probability, which are key areas of... read more

Comprehensiveness rating: 3 see less

This textbook goes into depth about different mathematical concepts that are important for elementary school teachers to understand in teaching mathematics. However, the text is missing a focus on statistics and probability, which are key areas of focus in elementary math classrooms. The text is also missing an index or glossary but does define new terms as they are introduced.

The content, mathematical diagrams and depictions are accurate and error-free. Each chapter also accurately shows various ways to understand mathematical concepts. However, the diagrams are geared towards an audience that already has some understanding of advanced mathematics.

The content is organized in a way that necessary updates would be straightforward to implement. More specifically, much of the content reflects current mathematical practices and activities endorsed by up-to-date research in mathematics education.

The text is written in accessible prose and provides context for jargon and technical terminology. Additionally, the text clearly separates different terms for different strategies and concepts. For example, in the Problem Solving Strategy section, the interface is divided into different strategies for the reader to explore. This is helpful in keeping new concepts and strategies organized for the reader.

The text is written with consistent terminology. More specifically, the text consistently gives examples of what concepts are called by mathematicians and teachers. This is helpful for pre-service teachers that might be teaching mathematical concepts and strategies for the first time.

The text is easily divided into smaller reading sections. These sections include not only explanations of mathematical concepts, but also theorems, activities and diagrams which can be referenced by the teacher at any point. Also, the text gives teachers ideas for activities and additional problems to try with students.

Though the topics in the texts are presented in a logical, clear fashion, it might be beneficial for pre-service or elementary teachers to see how to specifically scaffold the different concepts within those topics for elementary students at different grade levels. Additionally, the text could also demonstrate how students typically confuse topics so teachers and pre-service teachers are prepared to navigate new concepts for the class.

The interface is easy to navigate since the content clearly outlines chapters and the topics within them. Sections such as notation and vocabulary, think pair shares and theorems are clearly outlined, organized and conceptually scaffolded. However, it might be helpful to have an index so the reader does not have to click within each topic to find the concept they are exploring.

This text is free from grammatical errors.

This text is not culturally insensitive or offensive and includes examples from the Hawaiian culture. Though the text is mainly made up of mathematical explanations, there are a variety of people's names in different problems that could be attributed to a variety of cultures. Additionally, the text reflects Polya's advice (1945) to try adapt the problem until it makes sense. Though the text includes mainly mathematical explanations, it does call for adapting problems which could potentially be applied to a variety of students of different backgrounds.

Reviewed by Glenna Gustafson, Professor, Radford University on 5/22/19

This is book is fairly comprehensive and I feel could be used by most foundational courses in elementary mathematics. The structure and writing provide a good foundation for students learning the "why" behind the mathematics and becoming... read more

This is book is fairly comprehensive and I feel could be used by most foundational courses in elementary mathematics. The structure and writing provide a good foundation for students learning the "why" behind the mathematics and becoming mathematical thinkers. There were some areas that could possibly use more development. In geometry for example there was no discussion of perimeter, area, and volume. Estimation, measurement of weight, time, and probability also appears to be missing. The text is well organized and written so that the chapters do not have to be completed in the order in which they are presented. While there is not index or glossary, the author uses colored text boxes to explain specific content or terms.

The content of the text is accurate and represented in a variety of formats to support learning, Not only does it provide solutions to problems, but also the mathematical thinking behind those solutions.

The text is very relevant for K-6 elementary pre-service teachers. It would be beneficial to know the specific grade levels that the author considers as "elementary" since this does vary by location. The content is "standard" for most elementary math courses and would not need to be updated often and the consistent layout and formation would make changes easy to make.

The text is written in a conversational tone. The simplicity and straight-forwardness of the text should appeal to those students that have sometimes been overwhelmed by writing in more traditional math texts.

The text is organized consistently from chapter to chapter. The table of contents and chunking of content in the chapters is logical and clear, Each chapter includes graphics as well as sections for: Think-Pair-Share; Definitions; Theorems (when appropriate); and, Problems. This consistent structure makes navigation easy.

The table of contents and chunking of content in the chapters is logical and clear. This also makes it easy to not necessary to move sequentially through the text, but to have the option of reviewing or using only needed topics. Subtitles and graphic captioning are appropriate for the content.

The text is easy to read and the organization within each chapters makes navigation easy.

This text is easy to navigate. The inclusion of graphics, charts, photos, and videos support learning. There are several pages where graphics in the Geometry chapter are skewed in the PDF version, but this does not seem to be a problem in the online version, Not all of the video links work within the PDF version.

There were no obvious grammatical errors. Several of the errors that were found were typos and/or word omissions.

The text is culturally inclusive. One thing that should be noted is that it seems male names are over-represented in the Problem sections. A reference to Hawaiian culture and life is evident. The Hōkūle`a voyage found in the last chapter is a good example of problem based learning and the integration of math with other subject areas.

This would be a wonderful text to use as a supplement or compliment to an elementary math methods course. It is not as overwhelming as other math texts, and would provide pre-service teachers with a good foundational review of math concepts, including vocabulary and some pedagogy.

Reviewed by Karise Mace, Mathematics Instructor, Kuztown University on 5/16/19, updated 11/9/20

This book is fairly comprehensive for a one-semester course, although it does not include much detail about several topics. The section on number systems barely touches on Roman numerals and only mentions Mayan and Babylonian counting systems.... read more

This book is fairly comprehensive for a one-semester course, although it does not include much detail about several topics. The section on number systems barely touches on Roman numerals and only mentions Mayan and Babylonian counting systems. The sections on addition, subtraction, and division would be more robust if the author included other algorithms for these operations. The chapter on Geometry does not address perimeter, area, surface area and volume. The book does not include an index or glossary.

While the book is not error free, it is unbiased. Most of the errors seem to be typographical and/or related to web links or LaTeX. In the section on number systems, the author incorrectly explains how one million would be represented using Roman numerals and incorrectly claims that the Mayans did not use a symbol for zero. Further, the Mayan number system was not a true vigesimal system, as the text indicates.

This text uses a constructivist approach to help students build their understanding of the mathematics included in the book. It is well organized and written so that the chapters do not have to be completed in the order in which they are presented. Because of this, the text should be easy to update. When concepts that are presented earlier in the text are used in later chapters, the author includes a brief but thorough review that would allow students to understand the later chapter even if they had not read and completed the problems in the earlier chapter. The "dots and boxes" approach is timely, as it uses the idea of the "exploding dots" that are part of the Global Math Project (https://www.globalmathproject.org).

The textbook is clearly written and enjoyable to read...even for the math-phobic student. The tone is conversational and is even funny at times. The author defines important mathematical terminology in a way that is both mathematically accurate and accessible to students. The chapter on problem solving is fantastic and really gives students insight into how to think and problem solve like a mathematician. The pies per child model for fractions is not the most effective model for helping students understand fractions and this part of the text would be improved if the author replaced this type of modeling with pattern block modeling.

Overall, the text is consistent in its chapter structure and terminology use. However, there is inconsistent notation when using "dots and boxes."

The text is well-organized but can be reorganized in order to suit an instructor's preference. However, it would be best to complete the chapter on problem solving first, as it sets the stage for the rest of the book. Most of the chapters are structured more like an activity book with lots of great problems and thought provoking questions that will help students think deeply about the mathematical concepts being presented. With the exception of the chapter on problem solving, there is not a whole lot of text for students to read.

Although the topics presented could be reorganized to meet student needs, the order in which they are presented is logical and clear.

With only a few exceptions, the images it the text are clear. In the section titled "Careful Use of Language in Mathematics: =" some of the scale images need to be modified so that the items on the scale appear to actually sit on the pans. The same issue occurs in the section titled "Structural and Procedural Algebra." Some of the images in the sections titled "Platonic Solids" and "Symmetry" spill off of the page. The image that appears on page 89 and then again on page 144 would be more clear if a different font was used to label the line segments.

No grammatical errors were noted. However, there were a few typographical errors that could cause confusion for students as on page 219.

The text was culturally sensitive and nothing offensive was noted. As the focus of the text is purely mathematical, there are not many cultural references at all, unless they are references to historical cultures. The author does use names for hypothetical students that are diverse and represent a variety of ethnicities. The last chapter is an integrated unit that focuses on the Hawaiian culture. Unfortunately, the links and web addresses in this chapter do not work and/or are no longer active.

The book includes three sections at the end of the problem solving chapter in which the author articulately explains the language that mathematicians use to succinctly and precisely explain their problem solving and solutions. These sections will help students who may not think of themselves as mathematicians learn to think like mathematicians. So many mathematics textbooks are full of exercises but no true problems. On the other hand, this text is full of wonderful problem solving and critical thinking problems that are embedded in the sections as well as in the problem banks. The author also includes many "Think/Pair/Share" exercises and questions that will facilitate mathematical thinking and conversation among students. The constructivist approach used by the author will help students build deep understanding about the mathematics covered in the text. While there is some room for revision and improvement, this is a very good text to use with elementary education majors, and I definitely plan to use this book the next time I teach them.

Reviewed by Desley Plaisance, Associate Professor, Nicholls State University on 4/29/19

This textbook seems to be appropriate for the first course typically taught for elementary teachers which usually includes topics of problem solving, place value, number and operations. Most books are able to be used for a second course which... read more

This textbook seems to be appropriate for the first course typically taught for elementary teachers which usually includes topics of problem solving, place value, number and operations. Most books are able to be used for a second course which focuses on geometry. This book could not be used for the second course.

Content seems to be accurate.

Topics are somewhat static for a course like this, so the textbook will not become obsolete within a short period of time.

Appears to be clear.

The flow from topic to topic is consistent in presentation.

Divided into clear sections.

Topics are presented logically and in a similar order to most books of this type.

Easy to navigate with clear images and other items such as tables.

Book is written in simple language and appears to be free of grammatical errors.

Appears to be culturally diverse.

This book could definitely be used for a first course of elementary math for teachers with the teacher providing resources. As with many open books, the print and layout is very simple without cluttering pages with unnecessary items.

Reviewed by Lisa Cooper, Assistant Professor, LSUS on 4/26/19

This text covers many concepts appropriately; however, a few concepts are missing, such as; data analysis and statistics. For more than ten years, data-driven instruction has been a major focus in education along with many other uses. This text... read more

This text covers many concepts appropriately; however, a few concepts are missing, such as; data analysis and statistics. For more than ten years, data-driven instruction has been a major focus in education along with many other uses. This text has a table of contents but not an index and/or glossary; however, does define words in chapters when needed.

The content is well organized and accurate. Multiple representations and diverse examples are provided throughout the text which supports an unbiased approach to those entering elementary education.

The text is quite relevant to the classroom today, incorporating such resources as YouTube, varied strategies to promote differentiated instruction, scaffolding between concepts, and problem-solving opportunities. Some states may find issues with Common Core standards being addressed; however, mathematical practices could be interchanged with the "standards."

The text is written free from educational jargon; it is straightforward and easy to understand.

The text is consistent in its structure; color is not distracting, problems, strategies, diagrams, charts, and definitions are provided throughout.

The text is appealing with the page layout; it's not too busy or distracting. Colors are attractive and text is broken down into appropriate amounts.

The text has a well-organized flow with the layout of each topic/chapter.

The text has charts, pictures, diagrams, real-world examples throughout; several different versions of the text are offered too.

No grammatical errors were observed in my review of the text.

The text provides a variety of backgrounds, races, and ethnicities while providing learning experiences and pedagogical approaches to support student engagement and learning.

Reviewed by Demetrice Smith-Mutegi, Instructor/Coordinator, Marian University on 3/6/19

This text covers place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. However, elementary teachers are expected to also know and understand statistics and probability. This text does not address... read more

This text covers place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. However, elementary teachers are expected to also know and understand statistics and probability. This text does not address this mathematical concept.

The text makes non-traditional, yet, accurate representations of mathematical concepts. In some sections, different solutions are presented and explained. This eliminates bias and provides a diverse representation of ideas when solving math problems.

The text is representative of common core problem-solving standards, however, it does require mathematical knowledge beyond elementary school. The problem-solving nature of the text is very relevant to elementary pre-service and in-service teachers (the audience for the book).

The text language is clear and accessible. There is a section on terminology, which is very helpful. Additional diagrams would help to improve the clarity in some cases.

I was expecting to see videos embedded throughout, after seeing them in the first section. It would be great to have a consistent format throughout the text, however, I understand that it is not always feasible to do so. There were other obvious and clear patterns presented, color-coded sections (think/pair/share), problems, examples.

The chapters and subchapters can be easily accessed, breaking the material into smaller sections.

The topics were presented in a logical, clear fashion, however, not all of the chapters would end with a problem bank. In some cases, there were additional sections after the problem bank. It would be great if each section included key objectives or goals of the section.

The text comes in pdf, XML, and an online web version. The search feature on the online version was a valuable addition.

I did not observe any obvious grammatical errors.

Cultural awareness was very obvious in this text. While it was more relevant to Hawaiian culture, it also included cultural awareness of other cultures and backgrounds.

Overall, this text assumes that the student has successfully completed mathematics through basic calculus. There should be more support in this area, as some elementary math students are not prepared to complete problems with this focus.

This a great "discussion" text.

Reviewed by Kandy Noles Stevens, Assistant Professor of Education, Southwest Minnesota State University on 12/28/18

This text covers the areas applicable to elementary mathematics extremely well (with the exception of omitting probability and data analysis) and provides graphically visual boxes within the text to define terms and instructional strategies. ... read more

This text covers the areas applicable to elementary mathematics extremely well (with the exception of omitting probability and data analysis) and provides graphically visual boxes within the text to define terms and instructional strategies. Additionally, the text provides thinking routines that support understanding more than just the concept, but also, the how's and why's of conceptual understanding.

The content is accurate and organized in a way to supports student learning for those training to become elementary teachers of mathematics.

The content of the book is relevant to today's elementary classroom in that it provides future elementary educators with the content knowledge, but also pedagogical approaches that would support student learning. Additionally, the text is organized in a way that is consistent and provides scaffolding support for those who might struggle with any one of the concepts. For Minnesota standards the only item of note is that there is not a section devoted to probability or data analysis, but the latter is touched upon in other chapters. There are three mentions of the Common Core standards in the text. Minnesota is not an adopter of the CC mathematics standards, but the references to the CCSS are in regards to the practices of mathematics and not on standards specifically.

While a great text for training future math teachers, this book does not read as a "typical" mathematics textbook. Students who have struggled in the past with mathematics might find the authors' writing style to be approachable and accessible for all levels of mathematics competence and confidence.

The text is consistent in its terminology and the structure of the framework is uniform throughout, relying on supporting student learning through exercises, think-pair-share activities, and continuous dialog and reflection.

A majority of the chapters begin with a section that introduces the strand of elementary mathematics covered. Not all chapters have this introduction which may pose challenging to interrupt the mathematical progression of some established courses.

The text is very well organized and has an easy-to-read format and flow.

The text is graphically rich with succinct advanced organizers, diagrams, and photos to support learning.

The text is written with professional level writing and is free of grammatical errors.

A variety of races, ethnicities, and backgrounds are present in the exercises used to support student learning throughout. The end of the text involves a Hōkūle`a voyage as a part of a problem-based learning (integrated curriculum) experience. This was something that really made this text stand out in that it gave future elementary teachers an example of using mathematical concepts in authentic (and exciting) learning experiences. This Polynesian voyage would provide many students with an introduction to life culturally different from their own.

I have been a STEM educator for more than two decades and I come from a long line of mathematics educators. While wrapping up my reading of this text, I happened to have my father (a 46 year veteran mathematics educator) here visiting. I shared the text with him and several times I heard him utter, "I like the way this problem is set up". We both found the book to be very knowledgeable for mathematical conceptual understandings, but even more so for introducing ideas for instructional strategies and classroom discourse to help future teachers become equipped with speaking the "language of mathematics" to guide their future students.

Table of Contents

I. Problem Solving

• Introduction
• Problem or Exercise?
• Problem Solving Strategies
• Beware of Patterns!
• Problem Bank
• Careful Use of Language in Mathematics
• Explaning Your Work
• The Last Step

II. Place Value

• Dots and Boxes
• Other Rules
• Binary Numbers
• Other Bases
• Number Systems
• Even Numbers
• Exploration

III. Number and Operations

• Addition: Dots and Boxes
• Subtration: Dots and Boxes
• Multiplication: Dots and Boxes
• Division: Dots and Boxes
• Number Line Model
• Area Model for Multiplication
• Properties of Operations
• Division Explorations

IV. Fractions

• What is a Fraction?
• The Key Fraction Rule
• Adding and Subtracting Fractions
• What is a Fraction? Revisited
• Multiplying Fractions
• Dividing Fractions: Meaning
• Dividing Fractions: Invert and Multiply
• Dividing Fractions: Problems
• Fractions involving zero
• Egyptian Fractions
• Algebra Connections
• What is a Fraction? Part 3

V. Patterns and Algebraic Thinking

• Borders on a Square
• Careful Use of Language in Mathematics: =
• Growing Patterns
• Matching Game
• Structural and Procedural Algebra

VI. Place Value and Decimals

• Review of Dots & Boxes Model
• Division and Decimals
• More x -mals
• Terminating or Repeating?
• Operations on Decimals
• Orders of Magnitude

VII. Geometry

• Triangles and Quadrilaterals
• Platonic Solids
• Painted Cubes
• Geometry in Art and Science

VIII. Voyaging on Hokule?a

• Worldwide Voyage

Ancillary Material

About the book.

This book will help you to understand elementary mathematics more deeply, gain facility with creating and using mathematical notation, develop a habit of looking for reasons and creating mathematical explanations, and become more comfortable exploring unfamiliar mathematical situations. The primary goal of this book is to help you learn to think like a mathematician in some very specific ways. You will: • Make sense of problems and persevere in solving them. You will develop and demonstrate this skill by working on difficult problems, making incremental progress, and revising solutions to problems as you learn more. • Reason abstractly and quantitatively. You will demonstrate this skill by learning to represent situations using mathematical notation (abstraction) as well as creating and testing examples (making situations more concrete). • Construct viable arguments and critique the reasoning of others. You will be expected to create both written and verbal explanations for your solutions to problems. The most important questions in this class are “Why?” and “How do you know you're right?” Practice asking these questions of yourself, of your professor, and of your fellow students. Throughout the book, you will learn how to learn mathematics on you own by reading, working on problems, and making sense of new ideas on your own and in collaboration with other students in the class.

About the Contributors

Michelle Manes, Associate Professor, Department of Mathematics, University of Hawaii

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25 Fun Math Problems For Elementary And Middle School (From Easy To Very Hard!)

Fun math problems and brainteasers are loved by mathematicians; they provide an opportunity to apply mathematical knowledge, logic, and problem-solving skills all at once.

In this article, we’ve compiled 25 fun math problems and brainteasers covering various topics and question types. They’re aimed at students in upper elementary (3rd-5th grade) and middle school (6th grade, 7th grade, and 8th grade). We’ve categorized them as:

How should teachers use these math problems?

Teachers could make use of these math problem solving questions in a number of ways. They can:

• incorporate the questions into a relevant math lesson.
• set tasks at the beginning of lessons.
• break up or extend a math worksheet.
• keep students thinking mathematically after the main lesson has finished.

Some are based on real life or historical math problems, and some include ‘bonus’ math questions to help extend the problem-solving fun! As you read through these problems, think about how you could adjust them to be relevant to your students and their grade level or to practice different math skills. 

These math problems can also be used as introductory puzzles for math games such as those introduced at the following links:

• Math games for grade 4
• Middle school math games

25 Math Problems Worksheet

Want the fun and challenging questions from this blog wrapped up in a downloadable question and answer format? Get this 25 math problems and answers for your elementary and middle school students.

Math word problems

1. home on time – easy .

Type: Elapsed time, Number, Addition

A movie theater screening starts at 2:35 pm. The movie lasts for 2 hours, 32 minutes after 23 minutes of previews. It takes 20 minutes to get home from the movie theater. What time should you tell your family that you’ll be home?

Answer: 5:50 pm

2. A nugget of truth – mixed

Type: Multiplication Facts, Multiplication, Multiples, Factors, Problem-Solving 

Chicken nuggets come in boxes of 6, 9 or 20, so you can’t order 7 chicken nuggets. How many other impossible quantities can you find (not including fractions or decimals)?

Answer: 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, or 43

There is actually a theorem which can be used to prove that every integer quantity greater than 43 can be ordered.

3. A pet problem – mixed

Type: Number, Problem Solving, Forming and Solving Equations, Simultaneous Equations, Algebra

Eight of my pets aren’t dogs, five aren’t rabbits, and seven aren’t cats. How many pets do I have?

Answer: 10 pets (5 rabbits, 3 cats, 2 dogs)

Looking for more word problems, solutions and explanations? Read our article on word problems for elementary school.

4. the price of things – mixed.

Type: lateral thinking problem

A mouse costs $10, a bee costs $15, and a spider costs $20. Based on this, how much does a duck cost?

Answer: $5 ($2.50 per leg)

Math puzzles

5. a dicey math challenge – easy.

Type: Place value, number, addition, problem-solving

Roll three 6 sided dice to generate three place value digits. What’s the biggest number you can make out of these digits? What’s the smallest number you can make?

Add these two numbers together. What do you get?

Answer: If the digits are the same, the maximum is 666 and the minimum is 111. Then, if you add the numbers together, 666 + 111 = 777. If the digits are different, the maximum is 654 and the minimum is 456. Then, if you add the numbers together, 654 + 456 = 1,110.

Bonus: Who got a different result? Why?

6. PIN problem solving – mixed

Type: Logic, problem solving, reasoning

I’ve forgotten my PIN. Six incorrect attempts locks my account: I’ve used five! Two digits are displayed after each unsuccessful attempt: “2, 0” means 2 digits from that guess are in the PIN, but 0 are in the right place. No two digits in my PIN are the same.

What should my sixth attempt be?

Answer: 6347

7. So many birds – mixed

Type: Triangular Numbers, Sequences, Number, Problem Solving

On the first day of Christmas my true love gave me one gift. On the second day they gave me another pair of gifts plus a copy of what they gave me on day one. On day 3, they gave me three new gifts, plus another copy of everything they’d already given me. If they keep this up, how many gifts will I have after twelve days?

Answer: 364

Bonus: This could be calculated as 1 + (1 + 2) + (1 + 2 + 3) + … but is there an easier way? What percentage of my gifts do I receive on each day?

8. I 8 sum math questions – mixed

Type: Number, Place Value, Addition, Problem Solving, Reasoning

Using only addition and the digit 8, can you make 1,000? You can put 8s together to make 88, for example.

Answer: 888 + 88 + 8 + 8 + 8 = 1,000 Bonus: Which other digits allow you to get 1,000 in this way?

Fraction problems

4 friends entered a math quiz. One answered \frac {1}{5} of the math questions, one answered \frac {1}{10} , one answered \frac{1}{4} , and the other answered \frac{4}{25} . What percentage of the questions did they answer altogether?

Answer: 71%

10. Ancient problem solving – easy

Type: Fractions, Reasoning, Problem Solving

Ancient Egyptians only used unit fractions (like \frac {1}{2} , \frac{1}{3} or \frac{1}{4} . For \frac {2}{3} they’d write \frac{1}{3} + \frac{1}{3} . How might they write \frac{5}{8} ?

Answer: 

\frac {1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} is correct. So is \frac {1}{2} + \frac{1}{8} (They are both still unit fractions even though they have different denominators.)

Bonus: Which solution is better? Why? Can you find any more? What if subtractions are allowed?

Learn more about unit fractions here .

11. everybody wants a pizza the action – hard.

An infinite number of mathematicians buy pizza. The first wants \frac{1}{2} . The second wants \frac{1}{4} pizza. The third & fourth want \frac{1}{8} and \frac{1}{16} each, and so on. How many pizzas should they order?

Answer: 1 Each successive mathematician wants a slice that is exactly half of what is left:

12. Shade it black – hard

Type: Fractions, Reasoning, Problem Solving What fraction of this image is shaded black?

Answer:  

Look at the L-shaped part made up of two white and one black squares: 

\frac{1}{3} of this part is shaded. Zoom in on the top-right quarter of the image, which looks exactly the same as the whole image, and use the same reasoning to find what fraction of its L-shaped portion is shaded. Imagine zooming in to do the same thing again and again…

Multiplication and division problems

13. giving is receiving – easy.

Type: Number, Reasoning, Problem Solving

5 people give each other a present. How many presents are given altogether?

14. Sharing is caring – mixed

I have 20 candies. If I share them equally with my friends, there are 2 left over. If one more person joins us, there are 6 candies left. How many friends am I with?

Answer: 6 people altogether (so 5 friends!)

15. Multiplication facts secrets – mixed

Type: Area, 2D Shape, Rectangles

Here are 77 letters:

B Y H R C G N E O E A A H G C U R P U T S A S H H S B O R E O P E E M E E L A T P E F A D P H L TU T I E E O H L E N R Y T I I A G B M T N F C G E I I G

How many different rectangular grids could you arrange all 77 letters into?

Can you reveal the secret message?

Answer: Four: 1 × 77, 77 × 1, 11 × 7 and 7 × 11. If the letters are arranged into one of these, a message appears, reading down each column starting from the top left.

Bonus: Can you find any more integers with the same number of factors as 77? What do you notice about these factors (think about prime numbers)? Can you use this system to hide your own messages?

16. Laugh it up – hard

Type: Multiples, Least Common Multiple, Multiplication Facts, Division, Time

One friend jumps every \frac{1}{3} of a minute. Another jumps every 31 seconds. When will they jump together? Answer: After 620 seconds

Geometry problems

17. pictures of matchstick triangles – easy.

Type: 2D Shapes, Equilateral Triangles, Problem Solving, Reasoning

Look at the matchsticks arranged below. How many equilateral triangles are there?

Answer: 13 (9 small, 3 medium, 1 large)

Bonus: What if the biggest triangle only had two matchsticks on each side? What if it had four?

18. Dissecting squares – mixed

Type: Reasoning, Problem Solving

What’s the smallest number of straight lines you could draw on this grid such that each square has a line going through it?

19. Make it right – mixed

Type: Pythagorean theorem

This triangle does not agree with Pythagorean theorem. 

Adding, subtracting, multiplying or dividing each of the side lengths by the same whole number can fix it. What is the number?

Answer: 3 

The new side lengths are 3, 4 and 5 and  32 + 42 = 52.

20. A most regular math question – hard

Type: Polygons, 2D Shapes, tessellation, reasoning, problem-solving, patterns

What is the regular polygon with the largest number of sides that will self-tessellate?

Answer: Hexagon.

Regular polygons tessellate if one interior angle is a factor of 360 ° . The interior angle of a hexagon is 120°. This is the largest factor less than 180°.

Problem-solving questions

21. pleased to meet you – easy.

Type: Number Problem, Reasoning, Problem Solving

5 people meet; each shakes everyone else’s hand once. How many handshakes take place?

Person A shakes 4 people’s hands. Person B has already shaken Person A’s hand, so only needs to shake 3 more, and so on.

Bonus: How many handshakes would there be if you did this with your class?

22. All relative – easy

Type: Number, Reasoning, Problem-Solving

When I was twelve my brother was half my age. I’m 40 now, so how old is he?

23. It’s about time – mixed

Type: Time, Reasoning, Problem-Solving

When is “8 + 10 = 6” true?

Answer: When you’re telling the time (8am + 10 hours = 6pm)

24. More than a match – mixed

Type: Reasoning, Problem-Solving, Roman Numerals, Numerical Notation

Here are three matches:

How can you add two more matches, but get eight? Answer: Put the extra two matches in a V shape to make 8 in Roman Numerals:

25. Leonhard’s graph – hard

Type: Reasoning, Problem-Solving, Logic

Leonhard’s town has seven bridges as shown below. Can you find a route around the town that crosses every bridge exactly once?

Answer: No!

This is a classic real life historical math problem solved by mathematician Leonhard Euler (rhymes with “boiler”). The city was Konigsberg in Prussia (now Kaliningrad, Russia). Not being able to find a solution is different from proving that there aren’t any! Euler managed to do this in 1736, practically inventing graph theory in the process.

Math puzzles are everywhere!

Many of these 25 math problems are rooted in real life, from everyday occurrences to historical events. Others are just questions that might arise if you say “what if…?”. The point is that although there are many lists of such problem-solving math questions that you can make use of, with a little bit of experience and inspiration you could create your own on almost any topic – and so could your students. 

For a kick-starter on creating your own math problems, read our article on middle school math problem solving .

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by primary school teacher Tom Briggs and has since been revised and adapted for US schools by math curriculum specialist and former elementary math teacher Katie Keeton.

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Think back to the first problem in this chapter, the ABC Problem . What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills.  He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities).  He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. [1]

 In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

• Understand the problem.
• Devise a plan.
• Carry out the plan.
• Looking back.

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:

• What if the picture was different?
• What if the numbers were simpler?
• What if I just made up some numbers?

You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.

This brings us to the most important problem solving strategy of all:

Problem Solving Strategy 2 (Try Something!). If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.

And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.

Problem 2 (Payback)

Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?

Think/Pair/Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem?

This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.

Problem Solving Strategy 3 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?

Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?

After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.

Problem Solving Strategy 4 (Make Up Numbers). Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!

You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.

Watch the solution only after you tried this strategy for yourself.

If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!

Problem 3 (Squares on a Chess Board)

How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64… It’s a lot bigger!)

Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!

Think / Pair / Share

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?

It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”

Problem Solving Strategy 5 (Try a Simpler Problem). Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?

Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).

Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.

For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:

Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!

For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.

Problem Solving Strategy 8 (Look for and Explain Patterns). Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.

If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:

• Describe all of the patterns you see in the table.
• Can you explain and justify any of the patterns you see? How can you be sure they will continue?
• What calculation would you do to find the total number of squares on a 100 × 100 chess board?

(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)

Problem 4 (Broken Clock)

This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)

Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?

After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?

Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.

In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:

• What is the sum of all the numbers on the clock’s face?
• Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
• How do I know when I am done? When should I stop looking?

Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.

Problem Solving Strategy 10 (Check Your Assumptions). When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?

In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:

Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.

• Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons ↵

Mathematics for Elementary Teachers Copyright © 2018 by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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1.2: Problem or Exercise?

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• Michelle Manes
• University of Hawaii

The main activity of mathematics is solving problems. However, what most people experience in most mathematics classrooms is practice exercises. An exercise is different from a problem.

In a problem , you probably don’t know at first how to approach solving it. You don’t know what mathematical ideas might be used in the solution. Part of solving a problem is understanding what is being asked, and knowing what a solution should look like. Problems often involve false starts, making mistakes, and lots of scratch paper!

In an exercise , you are often practicing a skill. You may have seen a teacher demonstrate a technique, or you may have read a worked example in the book. You then practice on very similar assignments, with the goal of mastering that skill.

What is a problem for some people may be an exercise for other people who have more background knowledge! For a young student just learning addition, this might be a problem:

\[\textit{Fill in the blank to make a true statement} \: \_\_\_ + 4 = 7 \ldotp \nonumber \]

But for you, that is an exercise!

Both problems and exercises are important in mathematics learning. But we should never forget that the ultimate goal is to develop more and better skills (through exercises) so that we can solve harder and more interesting problems.

Learning math is a bit like learning to play a sport. You can practice a lot of skills:

• hitting hundreds of forehands in tennis so that you can place them in a particular spot in the court,
• breaking down strokes into the component pieces in swimming so that each part of the stroke is more efficient,
• keeping control of the ball while making quick turns in soccer,
• shooting free throws in basketball,
• catching high fly balls in baseball,

But the point of the sport is to play the game. You practice the skills so that you are better at playing the game. In mathematics, solving problems is playing the game!

On Your Own

For each question below, decide if it is a problem or an exercise . (You do not need to solve the problems! Just decide which category it fits for you.) After you have labeled each one, compare your answers with a partner.

• This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers.(Note: Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15. )

Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2).

• A soccer coach began the year with a $500 budget. By the end of December, the coach spent $450. How much money in the budget was not spent?
• What is the product of 4,500 and 27?
• Arrange the digits 1–6 into a “difference triangle” where each number in the row below is the difference of the two numbers above it.
• Simplify the following expression: $$\frac{2 + 2(5^{3} - 4^{2})^{5} - 2^{2}}{2(5^{3} - 4^{2})} \ldotp$$
• What is the sum of \(\frac{5}{2}\) and \(\frac{3}{13}\)?
• You have eight coins and a balance scale. The coins look alike, but one of them is a counterfeit. The counterfeit coin is lighter than the others. You may only use the balance scale two times. How can you find the counterfeit coin?

• How many squares, of any possible size, are on a standard 8 × 8 chess board?
• What number is 3 more than half of 20?
• Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1’s are separated by one digit, the 2’s are separated by two digits, the 3’s by three digits, and the 4’s by four digits.

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Adapting Math Word Problems for ELLs

To make word problems less confusing, especially for English language learners, change the language, not the math. Here are some ideas.

All students have a right to rigorous and challenging math classes, and word problems are a ubiquitous part of elementary and middle school math. Complex language structures or overly challenging vocabulary, however, can sometimes create barriers for students that impede access to a rigorous and challenging math curriculum. This is particularly true for English language learners (ELLs).

As teachers, we strive to cultivate mathematical reasoning and help students apply math to real-world contexts. When designing instruction for our English language learners, we have to ensure that they are afforded access to rich math tasks but also attend to the unique challenges of students working to acquire an understanding of the language. Fortunately, by attending to our vocabulary choices and sentence structure, we can adapt word problems and ensure that all students have access to rich mathematical content.

Certain linguistic features commonly found in middle school math classes are especially  problematic. Passive voice, complex sentences, and long noun phrases or clauses can be very difficult for all learners, but especially multilingual students developing English proficiency. Unfamiliar vocabulary, novel context, and poorly worded or vague questions can also create barriers to understanding. Small changes that simplify language, however, can significantly improve accessibility and ensure that more students can tackle rich math tasks.

Adapting the Math Language

Use the active voice: The passive voice can obscure what is actually happening in a word problem. Use the active voice to show people engaging with the world. For instance, rather than “The ball was thrown by the girl,” revise the sentence structure to “The girl threw the ball.”

Separate complex sentences: Break up long, convoluted, and meandering sentences to express key ideas. Consider the difference between “A hot dog costs $3.75 and a side salad costs $1.65. If a group of 5 students ordered 6 hot dogs and 4 side salads, and they left an 18% tip, how much did they pay in total, including the tip?” and the revised problem, “A group of friends ordered 6 cheeseburgers at $6.50 each and 4 side salads at $1.65 each. They left an 18% tip on the total bill. How much did they pay in total, including the tip?”

Both versions require the same mathematical understanding, but the language of the second is clearer and more accessible.

Simplify verb tense: Lean toward simple present tense. “The maintenance crew repairs the AC unit” rather than “has been repairing.”

Center people in the problem: Humanize problems with people rather than impersonal subjects. “85% of parents supported the schedule,” not “85% of the votes supported....”

Use familiar vocabulary: Swap challenging terminology for more recognizable vocabulary. “The school is hosting a fundraiser by selling concessions during the basketball tournament. If they sold 322 hamburgers at $3 each and 211 hot dogs at $2 each, what was the total revenue from the concession stand sales?”

Here’s a suggested alternative: “The school wants to raise money by selling food at a basketball game. They sold 322 hamburgers for $3 each and 211 hot dogs for $2 each. How much did the school make from selling the food?” Of course, some students will require additional supports, such as pictures and labels for key vocabulary found in word problems.

Shorten clauses: Trim unnecessary clauses. Instead of “The math tutor, who has taught for 10 years, helps students,” use “The math tutor helps students. She taught for 10 years.”

Replace obscure questions: Be sure to look for vague questions that distract from the math and substitute clear, direct questions. Change “What was the resulting amount after the chef used 16½ cups of milk?” to “The chef used 16½ cups of milk to make ice cream. Calculate how much ice cream the chef made yesterday.” Is something missing here?

Consider the big idea: Notice that in the previous example, students do not have enough information to solve the problem. When adapting math word problems for English language learners, revise the construction of your questions to clarify the task at hand, but also be mindful to simultaneously help students to think like mathematicians. To paraphrase what math education innovator Dan Meyer notes in his TED Talk on math instruction , real-world problems do not contain a simple list of all the required information.

As you adapt math instruction for English language learners, be sure to design rich experiences and help them to develop a mathematical mindset. What additional information do I need to solve this problem? What can I do to find the missing information? English language learners need accessible English, but they also need experiences that help them develop habits of inquiry, problem-solving, and self-efficacy.

The key is to adapt language without watering down rich mathematical thinking and problem-solving. Be sure to maintain high expectations while providing appropriate linguistic support. With slight modifications to ensure comprehensible and accessible language, your English language learners can tackle the same meaningful math as their peers.

Equity in math education means meeting each student where they are and helping them reach meaningful goals. Adjusting language is one path toward creating a math community that works for everyone.

Remember, context matters: Real-world contexts allow students to see math as a meaningful tool, rather than an abstract set of rules. However, take care not to introduce obscure, unfamiliar contexts that overwhelm ELLs with new vocabulary. Similarly, jumping between many different contexts in short succession can impede understanding.

When selecting contexts for word problems and examples, opt for familiar situations from students’ everyday lives that clearly illuminate the mathematical concepts. Additionally, aim to consistently revisit and reinforce the same contexts when teaching specific concepts, math models, or problem types. Repeated exposure across similar situations allows ELLs to digest both the linguistic and mathematical nuances. As comfort builds, you can broaden into new contexts, always taking care to explain unfamiliar vocabulary or scenarios that are essential to the problem.

The goal is to have students see math as meaningful while preventing contexts from distracting from the essential mathematical reasoning. Familiar, consistent contextualization keeps the focus on math concepts and problem-solving strategies.

The Power of Mathematical Models and Manipulatives

In addition to thoughtful verbal and written language adaptations, mathematical models and manipulatives provide critical visual and tactile scaffolds that support deeper understanding and reasoning for English language learners. Charts, ratio tables, coordinate planes, fraction models, graphs, algebra tiles , base-ten blocks, and more make concepts concrete while mitigating vocabulary barriers.

Leveraging models and manipulatives moves learning toward mathematical action. Students demonstrate conceptual connections nonverbally, allowing alternative pathways to develop understandings. All students access deeper thinking as teachers elevate mathematical visualization alongside precision in academic language.

When planning for math instruction and adapting for our multilingual learners, I’ve found these resources to be particularly helpful:

• Teaching Math to Multilingual Students, Grades K–8: Positioning English Learners for Success ,
• “ Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching ,” and
• Math Workshop: Five Steps to Implementing Guided Math, Learning Stations, Reflection, and More .

Ultimately, we want students to develop a deep conceptual understanding of mathematics and to grow their English language proficiency. Slight adaptations to language, the use of familiar real-world contexts, and deliberately incorporating mathematical models and manipulatives can help students to access the math curriculum and to acquire English.

We’d like to know—what strategies have you successfully used to help improve math accessibility for ELL students? Please comment and share.

3 Ways to Strengthen Math Instruction

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Students’ math scores have plummeted, national assessments show , and educators are working hard to turn math outcomes around.

But it’s a challenge, made harder by factors like math anxiety , students’ feelings of deep ambivalence about how math is taught, and learning gaps that were exacerbated by the pandemic’s disruption of schools.

This week, three educators offered solutions on how districts can turn around poor math scores in a conversation moderated by Peter DeWitt, an opinion blogger for Education Week.

Here are three takeaways from the discussion. For more, watch the recording on demand .

1. Intervention is key

Research shows that early math skills are a key predictor of later academic success.

“Children who know more do better, and math is cumulative—so if you don’t grasp some of the earlier concepts, math gets increasingly harder,” said Nancy Jordan, a professor of education at the University of Delaware.

For example, many students struggle with the concept of fractions, she said. Her research has found that by 6th grade, some students still don’t really understand what a fraction is, which makes it harder for them to master more advanced concepts, like adding or subtracting fractions with unlike denominators.

At that point, though, teachers don’t always have the time in class to re-teach those basic or fundamental concepts, she said, which is why targeted intervention is so important.

Still, Jordan’s research revealed that in some middle schools, intervention time is not a priority: “If there’s an assembly, or if there is a special event or whatever, it takes place during intervention time,” she said. “Or ... the children might sit on computers, and they’re not getting any really explicit instruction.”

2. ‘Gamify’ math class

Students today need new modes of instruction that meet them where they are, said Gerilyn Williams, a math teacher at Pinelands Regional Junior High School in Little Egg Harbor Township, N.J.

“Most of them learn through things like TikTok or YouTube videos,” she said. “They like to play games, they like to interact. So how can I bring those same attributes into my lesson?”

Part of her solution is gamifying instruction. Williams avoids worksheets. Instead, she provides opportunities for students to practice skills that incorporate elements of game design.

That includes digital tools, which provide students with the instant feedback they crave, she said.

But not all the games are digital. Williams’ students sometimes play “trashketball,” a game in which they work in teams to answer math questions. If they get the question right, they can crumble the piece of paper and throw it into a trash can from across the room.

“The kids love this,” she said.

Williams also incorporates game-based vocabulary into her instruction, drawing on terms from video games.

For example, “instead of calling them quizzes and tests, I call them boss battles,” she said. “It’s less frightening. It reduces that math anxiety, and it makes them more engaging.

“We normalize things like failure, because when they play video games, think about what they’re doing,” Williams continued. “They fail—they try again and again and again and again until they achieve success.”

3. Strengthen teacher expertise

To turn around math outcomes, districts need to invest in teacher professional development and curriculum support, said Chaunté Garrett, the CEO of ELLE Education, which partners with schools and districts to support student learning.

“You’re not going to be able to replace the value of a well-supported and well-equipped mathematics teacher,” she said. “We also want to make sure that that teacher has a math curriculum that’s grounded in the standards and conceptually based.”

Students will develop more critical thinking skills and better understand math concepts if teachers are able to relate instruction to real life, Garrett said—so that “kids have relationships that they can pull on, and math has some type of meaning and context to them outside of just numbers and procedures.”

It’s important for math curriculum to be both culturally responsive and relevant, she added. And teachers might need training on how to offer opportunities for students to analyze and solve real-world problems.

“So often, [in math problems], we want to go back to soccer and basketball and all of those things that we lived through, and it’s not that [current students] don’t enjoy those, but our students live social media—they literally live it,” Garrett said. “Those are the things that have to live out in classrooms right now, and if we’re not doing those things, we are doing a disservice.”

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