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2.5.3: Guess and Check, Work Backward

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Guess and Check, Work Backward

Suppose that you and your brother both play baseball. Last season, you had 12 more hits than 3 times the number of hits that your brother had. If you had 159 hits, could you figure out how many hits your brother had?

More Problem Solving Strategies

This lesson will expand your toolbox of problem-solving strategies to include guess and check and working backward . Let’s begin by reviewing the four-step problem-solving plan:

Step 1: Understand the problem.

Step 2: Devise a plan – Translate.

Step 3: Carry out the plan – Solve.

Step 4: Look – Check and Interpret.

Develop and Use the Strategy: Guess and Check

The strategy for the “guess and check” method is to guess a solution and use that guess in the problem to see if you get the correct answer. If the answer is too big or too small, then make another guess that will get you closer to the goal. You continue guessing until you arrive at the correct solution. The process might sound like a long one; however, the guessing process will often lead you to patterns that you can use to make better guesses along the way.

Let's use the guess and check method to solve the following problem:

Nadia takes a ribbon that is 48 inches long and cuts it in two pieces. One piece is three times as long as the other. How long is each piece?

We need to find two numbers that add to 48. One number is three times the other number.

Guess 5 and 15. The sum is 5+15=20, which is too small.

Guess bigger numbers 6 and 18. The sum is 6+18=24, which is too small.

However, you can see that the previous answer is exactly half of 48.

Multiply 6 and 18 by two.

Our next guess is 12 and 36. The sum is 12+36=48. This is correct.

Develop and Use the Strategy: Work Backward

The “work backward” method works well for problems in which a series of operations is applied to an unknown quantity and you are given the resulting value. The strategy in these problems is to start with the result and apply the operations in reverse order until you find the unknown. Let’s see how this method works by solving the following problem.

Let's solve the following problem by working backwards :

Anne has a certain amount of money in her bank account on Friday morning. During the day she writes a check for $24.50, makes an ATM withdrawal of $80, and deposits a check for $235. At the end of the day, she sees that her balance is $451.25. How much money did she have in the bank at the beginning of the day?

We need to find the money in Anne’s bank account at the beginning of the day on Friday. From the unknown amount, we subtract $24.50 and $80 and we add $235. We end up with $451.25. We need to start with the result and apply the operations in reverse.

Start with $451.25. Subtract $235, add $80, and then add $24.50.

451.25−235+80+24.50=320.75

Anne had $320.75 in her account at the beginning of the day on Friday.

Plan and Compare Alternative Approaches to Solving Problems

Most word problems can be solved in more than one way. Often one method is more straightforward than others. In this section, you will see how different problem-solving approaches compare when solving different kinds of problems.

Now, let's solve the following problem by using the both the guess and check method and the working backward method:

Nadia’s father is 36. He is 16 years older than four times Nadia’s age. How old is Nadia?

This problem can be solved with either of the strategies you learned in this section. Let’s solve the problem using both strategies.

Guess and Check Method:

We need to find Nadia’s age.

We know that her father is 16 years older than four times her age, or 4× (Nadia’s age) + 16.

We know her father is 36 years old.

Work Backward Method:

Nadia’s father is 36 years old.

To get from Nadia’s age to her father’s age, we multiply Nadia’s age by four and add 16.

Working backward means we start with the father’s age, subtract 16, and divide by 4.

Example 2.5.3.1

Earlier, you were told that you had 12 more hits than 3 times the number of hits that your brother had. If you had 159 hits, how many hits did your brother have?

Since we know how many hits you had, we can work backward to determine the number of hits that your brother had.

Because you had 12 more hits than 3 times the number of hits that your brother had, we do the opposite: subtract 12 and divide by 3.

159−12=147

147÷3=49

Your brother had 49 hits.

Example 2.5.3.2

Hana rents a car for a day. Her car rental company charges $50 per day and $0.40 per mile. Peter rents a car from a different company that charges $70 per day and $0.30 per mile. How many miles do they have to drive before Hana and Peter pay the same price for the rental for the same number of miles?

Hana’s total cost is $50 plus $0.40 times the number of miles.

Peter’s total cost is $70 plus $0.30 times the number of miles.

Guess the number of miles and use this guess to calculate Hana’s and Peter’s total cost.

Keep guessing until their total cost is the same.

Guess 50 miles.

Check $50+$0.40(50)=$70 $70+$0.30(50)=$85

Guess 60 miles.

Check $50+$0.40(60)=$74 $70+$0.30(60)=$88

Notice that for an increase of 10 miles, the difference between total costs fell from $15 to $14. To get the difference to zero, we should try increasing the mileage by 140 miles.

Guess 200 miles

Check $50+$0.40(200)=$130 $70+$0.30(200)=$130correct

• Nadia is at home and Peter is at school, which is 6 miles away from home. They start traveling toward each other at the same time. Nadia is walking at 3.5 miles per hour and Peter is skateboarding at 6 miles per hour. When will they meet and how far from home is their meeting place?
• Peter bought several notebooks at Staples for $2.25 each and he bought a few more notebooks at Rite-Aid for $2 each. He spent the same amount of money in both places and he bought 17 notebooks in total. How many notebooks did Peter buy in each store?
• Andrew took a handful of change out of his pocket and noticed that he was holding only dimes and quarters in his hand. He counted that he had 22 coins that amounted to $4. How many quarters and how many dimes does Andrew have?
• Anne wants to put a fence around her rose bed that is one-and-a-half times as long as it is wide. She uses 50 feet of fencing. What are the dimensions of the garden?
• Peter is outside looking at the pigs and chickens in the yard. Nadia is indoors and cannot see the animals. Peter gives her a puzzle. He tells her that he counts 13 heads and 36 feet and asks her how many pigs and how many chickens are in the yard. Help Nadia find the answer.
• Andrew invests $8000 in two types of accounts: a savings account that pays 5.25% interest per year and a more risky account that pays 9% interest per year. At the end of the year, he has $450 in interest from the two accounts. Find the amount of money invested in each account.
• There is a bowl of candy sitting on our kitchen table. This morning Nadia takes one-sixth of the candy. Later that morning Peter takes one-fourth of the candy that’s left. This afternoon, Andrew takes one-fifth of what’s left in the bowl and finally Anne takes one-third of what is left in the bowl. If there are 16 candies left in the bowl at the end of the day, how much candy was there at the beginning of the day?
• Nadia can completely mow the lawn by herself in 30 minutes. Peter can completely mow the lawn by himself in 45 minutes. How long does it take both of them to mow the lawn together?

Mixed Review

• Rewrite √500 as a simplified square root.
• To which number categories does −2/13 belong?
• Simplify 1/2|19−65|−14.
• Which property is being applied? 16+4c+11=(16+11)+4c
• Is {(4,2),(4,−2),(9,3),(9,−3)} a function?
• Write using function notation: y=(1/12)x−5.
• Jordyn spent $36 on four cases of soda. How much was each case?

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.14.

Additional Resources

Activity: Guess and Check, Work Backward Discussion Questions

Practice: Guess and Check, Work Backward

Real World Application: Car Loan

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Working backward to solve problems - maurice ashley.

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Let’s Begin…

Imagine where you want to be someday. Now, how did you get there? Retrograde analysis is a style of problem solving where you work backwards from the endgame you want. It can help you win at chess -- or solve a problem in real life. At TEDYouth 2012, chess grandmaster Maurice Ashley delves into his favorite strategy.

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Maths with David

Problem solving. work backwards.

A good mathematics problem will not have an obvious solution. We must consider what information we have been given and what we already know, and how these things may relate to the problem. If we are systematic in our thinking this will often lead us to the solution.

If we don’t seem to be making progress, we can:

• Read the problem again;
• Reread the information given and focus on key information that could be useful;
• Use what we already now to reduce the scope of the problem;
• Record work done carefully, so it is easy to retrace steps and to verify or change the method.

Examples of working backwards to tackle a problem

First we will read the two examples and have a quick think about them and then we will look at how working backwards can help us with each one:

Angle Problem

The following diagram shows an isosceles triangle and a square drawn on a straight line. Find the size of angle A:

Fraction Shaded Problem

The following diagram is of a square with four semicircles drawn inside. What faction of the square is shaded?

Worked Solutions

We can start by adding in the angles that we know about in the square, marking each of them as 90 degrees. Then we can think about the isosceles triangle and the properties of it that we know about. Using this we can calculate that each of the base angles in this triangle must be 70 degrees. Once we have marked each of these angles on, it should be clearer to see what angle A is.

We don’t have a formula for finding this specific area and we don’t know any measurements, so we will have to use letters to represent some of the unknown lengths.

Let’s start by looking at one semicircle, the radius of which we will call r:

Using the formula for the area of a circle, we know that the area of this semicircle is ½πr 2 .

But we don’t want the whole semicircle. We just want the petals. The “white” shape that we need to exclude has a strange shape, but we can tackle the problem instead by excluding a triangle within the semi-circle, which will leave us with “half-petals” from which we can easily find the area of the petals:

Because we have already named the radius of the semicircle as “r”, we can find the area of this “white” triangle, because it has base length 2r and height r, so its area is ½ x 2r x r = r 2 .

So the two red half petals on the diagram above have a combined area or ½πr 2 – r 2 . There are four times these in the whole diagram, so the total area shaded is 4(½πr 2 – r 2 ).

The area of the rectangle is 2r x 2r = 4r 2 .

So the fraction we want is 4(½πr 2 – r 2 ) / 4r 2 . This simplifies to five ½π – 1, which is about 0.57.

The following questions can be solved in different ways. The worked solutions provided afterwards are based on the ways suggested above.

29 Questions of increasing difficulty

Worked Solutions to Questions

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Fun teaching resources & tips to help you teach math with confidence

Math Strategies: Problem Solving by Working Backwards

As I’ve shared before, there are many different ways to go about solving a math problem, and equipping kids to be successful problem solvers is just as important as teaching computation and algorithms . In my experience, students’ frustration often comes from not knowing where to start. Providing them with strategies enables them to at least get the ideas flowing and hopefully get some things down on paper. As in all areas of life, the hardest part is getting started! Today I want to explain how to teach  problem solving by working backwards .

* Please Note : This post contains affiliate links which help support the work of this site. Read our full disclosure here .*

–>Pssst! Do your kids need help making sense of and solving word problems? You might like this set of editable word problem solving templates ! Use these with any grade level, for any type of word problem :

Solve a Math Problem by Working Backwards: 

Before students can learn to recognize when this is a helpful strategy, they must understand what it means. Working backwards is to start with the final solution and work back one step at a time to get to the beginning.

It may also be helpful for students to understand that this is useful in many aspects of life, not just solving math problems.

To help show your students what this looks like, you might start by thinking about directions. Write out some basic directions from home to school:

• Start: Home
• Turn right on Gray St.
• Turn left on Sycamore Ln.
• Turn left on Rose Dr.
• Turn right on Schoolhouse Rd.
• End: School

Ask students to then use this information to give directions from the school back home . Depending on the age of your students, you may even want to draw a map so they can see clearly that they have to do the opposite as they make their way back home from school. In other words, they need to “undo” each turn to get back, i.e. turn left on Schoolhouse Rd. and then right on Rose Dr. etc.

In math, these are called inverse operations . When using the “work backwards” strategy, each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backwards they will need to subtract. And if they multiply working forwards, they must divide when working backwards.

Once students understand inverse operations , and know that they must start with the solution and work back to the beginning, they will need to learn to recognize the types of problems that require working backwards.

In general, problems that list a series of events or a sequence of steps can be solved by working backwards.

Here’s an example:

Sam’s mom left a plate of cookies on the counter. Sam ate 2 of them, his dad ate 3 of them and they gave 12 to the neighbor. At the end of the day, only 4 cookies were left on the plate. How many cookies did she make altogether?

In this case, we know that the final cookie amount is 4. So if we work backwards to “put back” all the cookies that were taken or eaten, we can figure out what number they started with.

Because cookies are being taken away, that denotes subtraction. Thus, to get back to the original number we have to do the opposite: add . If you take the 4 that are left and add the 12 given to the neighbors, and add the 3 that Dad ate, and then add the 2 that Sam ate, we find that Sam’s mom made 21 cookies .

You may want to give students a few similar problems to let them see when working backwards is useful, and what problems look like that require working backwards to solve.

Have you taught or discussed problem solving by working backwards  with your students? What are some other examples of when this might be useful or necessary?

Don’t miss the other useful articles in this Problem Solving Series:

• Problem Solve by Drawing a Picture
• Problem Solve by Solving an Easier Problem
• Problem Solve with Guess & Check
• Problem Solve by Finding a Pattern
• Problem Solve by Making a List

So glad to have come across this post! Today, word problems were the cause of a homework meltdown. At least tomorrow I’ll have a different strategy to try! #ThoughtfulSpot

I’m so glad to hear that! I hope you found some useful ideas!! Homework meltdowns are never fun!! Best of luck!

This is really a great help! We have just started using this method for some of my sons math problems and it helps loads. Thanks so much for sharing on the Let Kids Be Kids Linkup!

That’s great Erin! I hope this is a helpful method and makes things easier for your son! 🙂

I’ve not used this method before but sounds like a good resource to teach. Thanks for linking #LetKidsBeKids

I hope this proves to be helpful for you!

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Problem Solving: Working Backwards

Teaching problem solving strategies is as important as teaching computation.  i like to call problem solving, creative thinking, as i believe that give the process a more positive spin.  kids are less intimidated by the word than they are a problem solving as that implies that there is a “problem”.   students need to learn to interpret what they are reading and draw out information.  this can be difficult for some students.  i have found that students’ frustration with working on “story problems” has been where to start and what information do you need.   each month i am tacking a different problem solving strategy this month it is working backwards ., working backwards is sometimes the best strategy and one of the most difficult for students to understand.  i was one of those that didn’t understand this strategy because rarely did i read all the information.  i saw numbers, key words and started putting things together but not always correctly.  also being dyslexic it was hard enough reading the information let alone interpreting it and then working backwards and apply inverse operations.  , when introducing the different types of problem solving strategies; looking for patterns, make a list, draw a picture, guess and check, and working backwards, working backwards can sometimes be a fun one to start first.  giving the students a trick about reading through all the directions is unintimidating and proves a point in an amusing way.  , do you remember the problem where you ask students to read all the directions before starting and then the last question says don’t do the work just sign your name at the top  these worksheets are a great way to get students to read all the directions, take in all the information and then work from there.    .

After doing these worksheets apply working backwards to math.  Our first goal in any problem is to read the whole problem.  Is there an answer provided?  Does the question have key words like equal to, amount left is, leaving ___?  Once you have established that there is an answer to the problem we need to start working backwards which also means using the inverse operation .   In mathematics we are also developing the idea of inverse operation, using the opposite operation than the one given. This is a great math way to show how that works.

MANGO Math Kits provide great games and activities that make students enjoy and understand math.  We adhere to NCTM and Common State Standards .

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Working Backwards, Leaping Forwards

In mathematics lessons, you might be given a question and asked to work out the answer, but these problems are a little bit different! To solve them, you'll need to think about the relationships between functions and operations and their inverses, and work backwards in order to leap forward!  

Missing Multipliers   Age 7 to 14 Challenge Level

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Integration Matcher   Age 16 to 18 Challenge Level

Function Pyramids   Age 16 to 18 Challenge Level

Inverse Problems Save the Day   Age 14 to 18 Challenge Level

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Working Backwards Toward Answers

Mathematics, Data Analysis and Probability, Operations (+, -, x, /, etc.), Problem Solving

Students learn to work backwards in order to solve a problem.

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Working Backwards Strategy Problem Solving

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Description

These word problems give upper elementary students a chance to practice the working backward problem solving strategy. * This resource was updated in 2023 and now includes 24 working backward word problems in five different formats.

The formats include:

• Four worksheets with six working backward problems on each
• Full page worksheet with one problem a page and a workspace
• Strip format- multiple on a page to cut into strips to glue into journals (ink saver!)
• Full page format to glue onto chart paper for group work
• Digital format to assign in Google Classroom

Answer keys included for all formats

You may also be interested in the following problem solving resources:

Draw a picture or diagram problem solving (worksheet and group work formats)

Guess and check problem solving (worksheet and group work formats)

Finding the Pattern Problem Solving (worksheet and group work formats)

Multi-step problem solving (six formats + digital included)

Area and Perimeter single and multi-step problem solving (five formats + digital included)

*This resource is not editable

Your download will include a .zip file with all five formats as separate files.

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