working backwards problem solving year 5

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 .

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–>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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KS2 Maths (Working Backwards)

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

16 January 2019

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SATs preparation for 2018 . These topic-focused SATs questions have fully-worked solutions suitable for whiteboard display or sending home to parents. Click 👉 tes.com/…/sats-revision… for similar-style question compilations on the other SATs topics. <hr> This particular compilation is from the NUMBER/ALGEBRA strands and contains questions in which students are given the final answer and have to work backwards to obtain the starting number. <hr> I have designed this compilation to be printed as an A4 or A5 booklet which is in the style of the actual SATs papers and is convenient for use in class or as homework. It can even be given to individual students if a parent is asking for ‘some more work’!

KEY POINTS:

• I have provided full answers, with comments and working where helpful.
• I have maintained the style of the actual SATs questions so that students can become comfortable with the way that SATs questions are presented.
• Most of the questions are from actual SATs papers, but I have also added questions so that this resource matches the requirements the current curriculum better than the older resources that are still in common use (note that many of the older resources of this type contain questions on topics which are no longer examined).
• I have spent a lot of time arranging the questions so that there is a general increase in difficulty as students work through them, and so that they fit on the pages better – this means less wasted space and significant paper-saving when printing 😃 <hr> 👍If you like this resource, then please rate it and/or leave a comment💬. If the rate-resource button on this page doesn’t work, then go to your ratings page by clicking 👉 www.tes.com/…/rate-resources…

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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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18 Word Problems For Year 5: Develop Their Problem Solving Skills Across Single and Mixed KS2 Topics

Emma Johnson

Word problems for Year 5 are a great way to assess pupils’ number fluency. By the time primary school children reach Upper Key Stage 2, they will be building on their knowledge and understanding of the number system and place value, working with larger integers. An increased understanding of the connections between the different maths concepts is crucial, as children will be expected to tackle more complex, multi-step problems.

It is important children are regularly provided with the opportunity to solve a range of word problems and Year 5 maths worksheets with reasoning and problem solving incorporated alongside fluency, into all lessons.

To help you with this, we have put together a collection of 18 word problems aimed at Year 5 maths pupils.

Place value 

Addition and subtraction , multiplication and division , fractions, decimals and percentages, measurement , why are word problems important in year 5 maths, how to teach problem solving in year 5, addition word problems for year 5, subtraction word problems for year 5,  multiplication word problems for year 5, division word problems for year 5, fraction and decimal word problems in year 5, mixed four operation word problems, word problem resources.

All Kinds of Word Problems

Download this free pack of multiplication word problems to help your Year 5 class grow their problem solving skills

Year 5 maths word problems in the national curriculum

The National Curriculum states that by Year 5, pupils should develop their ability to solve a wider range of problems, including increasingly complex properties of numbers and arithmetic, and problems demanding efficient written and mental methods of calculation. Children also begin to use the language of algebra for solving a range of problems.

By Year 5, pupils should be starting to gain confidence in solving more complex two-step and multi-step word problems, covering topics across the Year 5 curriculum.

Solve number and practical problems involving ordering and comparing numbers to at least 1,00,000; counting forwards or backwards in steps of powers of 10; interpreting negative numbers in context and rounding to the nearest 10,100, 1000, 10,000 and 100,000.

Solve addition and subtraction word problems and a combination of these, including understanding the meaning of the equals sign.

Solve problems involving multiplication and division, including using knowledge of factors, multiples, squares and cubes and scaling by simple fractions.

Solve problems involving numbers up to 3 decimal places and problems which require knowing percentage and decimal equivalents.

Solve time word problems and those involving converting between units of time and problems involving measure (for example, length, mass, volume and money word problems ) using decimal notation and scaling.

Solve comparison, sum and difference problems, using information presented in a line graph.

Word problems are an important element of the Year 5 curriculum. By this stage, children need to be building confidence in approaching a range of one, two and multi-step word problems . They require children to be creative and apply the skills they have learnt to a range of real-life situations. 

Children need to be taught the skills for successfully approaching and tackling word problems. Reading the question carefully and identifying the key information needed to tackle the problem, is the first step, followed by identifying which calculations are required for solving it and deciding whether it will be helpful to draw a picture/ visual representation to understand and answer the question. 

Using mental maths skills to round and estimate an answer is also very helpful for children to establish whether their final answer is realistic. Children need to also be able to calculate the inverse, to be able to check their answer, once the problem has been completed.

See also: Mental maths year 5

Here is an example:

A transport museum has 1243 visitors on Monday morning and another 1387 visitors in the afternoon. On Tuesday 736 fewer visitors go to the museum than who visited on Monday.

How many visitors were there altogether on Tuesday?

How to solve:

What do you already know?

• The number of visitors on Monday morning and Monday afternoon are given separately. They need to be added together to give the total number of visitors for Monday.
• ‘Fewer’ means I will need to subtract the number of fewer visitors on Tuesday from the total number of visitors on Monday.
• Column addition and subtraction will be needed to solve this question.

How can this be drawn/represented pictorially?

We can draw a bar model to represent this problem:

• To calculate the total number of visitors on Monday, we need to add 1243 and 1387 together. 1243 + 1387 = 2630
• The number of fewer visitors on Tuesday needs to be subtracted from Monday’s total: 2630 – 736 = 1894.
• The total number of visitors on Tuesday was 1894.

In Year 5, addition word problems can involve whole numbers with over 4 digits and decimal numbers. Children should be able to round number to check the accuracy and begin to solve two step number problems 

Addition question 1

Gemma picks two cards from the cards below and adds them together.  She is able to make three different totals. What will they be?

2365     6281      9782

Answer (2 marks) : 12,147   16,063  8646

Addition question 2

Ahmed adds two of these numbers mentally.

357         280          294          232

In his calculation, he exchanges twice to create one ten and one hundred. Write Ahmed’s calculation and work out the total.

Answer (1 mark) : 357 + 294 = 651

Addition question 3

Change one digit in the calculation below, so that the answer is a multiple of 10.

Answer (1 mark) : 723 + 347 = 1070

Subtraction word problems in Year 5 require pupils to be confident subtracting numbers over 4 digits and problems involving decimal numbers. Pupils need to be able to round numbers to check accuracy and to use subtraction when solving mixed word problems. 

Subtraction question 1

A coach is travelling 4924 km across the USA It has 2476 km to go. How many kilometres has the coach already travelled?

Answer (1 mark) : 2448 km

Subtraction question 2

A primary school printed 7283 maths worksheets in the Summer term. 2156 were for Key Stage 1 pupils. How many were printed for Key Stage 2?

Answer (1 mark): 5127 worksheets

Subtraction question 3

 A clothing company made £57,605 profit in 2021 and £73,403 in 2022.  How much more profit did the company make in 2022 than in 2021?

Answer (1 mark): £15,798

In year 5, multiplication word problems include problems involving times tables and multiplying whole numbers up to 4-digits by 1 or 2-digit numbers. Pupils also need to be able to combine multiplication with other operations, in order to solve two-step word problems. 

Multiplication question 1

In this diagram, the numbers in the circles are multiplied together to make the answer in the square between them.

Complete the missing numbers.

Answer (1 mark)

Multiplication question 2

Mrs Jones was printing the end of year maths test. Each test had 18 pages and 89 pupils were sitting the test. Mrs Jones also needed to print out 12 copies for the teachers and Teaching Assistants who were helping to run the test. 

How many pieces of paper did Mrs Jones need to put in the photocopier, to make sure she had enough for all the tests?

Multiplication question 3

A school is booking a trip to Alton Towers. Tickets cost £22 per pupil. There are 120 children in each year group and all the children from 3 year groups will be going.

What will be the total price for all the tickets?

Answer (2 marks): £7920

Third Space Learning often ties word problems into our online one-to-one tutoring. Each lesson is personalised to the needs of the individual student, growing maths knowledge and problem solving skills.

In Year 5, division word problems can involve whole numbers up to 4 digits being divided by 1-digit numbers. Pupils need to understand how to answer word problems, when the answer involves a remainder.

Division question 1

Tom has 96 cubes and makes 12 equal towers. Masie has 63 cubes and makes 9 equal towers. Whose towers are tallest and by how many cubes?

Answer (2 marks): Tom

Tom’s tower has more cubes. His towers have 1 more cube than Maise’s towers.

96 ÷ 12 = 8

Division question 2

A cake factory has made cakes to deliver to a large event. 265 cakes have been baked. How many boxes of 8 cakes can be delivered to the event?

Answer (2 marks) : 33 boxes

Division question 3

Lily collected 1256 stickers. She shared them between her 8 friends. How many stickers did each friend get

Answer (1 mark): 157 stickers

In Year 5, fractions word problems and decimals can include questions involving ordering, addition and subtraction of fractions. They can also involve converting between fractions and decimals. 

Fraction and decimal question 1

Isobel collected 24 conkers. She gave \frac{1}{8} of the conkers to her brother. How many conkers did she have left?

Answer (1 mark): 21 conkers left

\frac{1}{8} of 24 = 3

24 – 4 = 21

Fraction and decimal question 2

Ahmed counted out 32 sweets. He gave \frac{1}{4} of the sweets to his brother and \frac{3}{8} of the sweets to his friend. How many sweets did he have left?

Answer (2 marks): 12 sweets

\frac{1}{4} of 32 = 8

\frac{3}{8} of 32 = 12

He gave away 20 sweets, so had 12 left for himself.

Fraction and decimal question 3

Two friends shared some pizzas, 1 ate 1 \frac{1}{2} pizzas, whilst the other ate \frac{5}{8} of a pizza. How much did they eat altogether?

Answer (1 mark): 2 \frac{1}{8} of pizza

1 \frac{1}{2} = \frac{12}{8}

\frac {12}{8} + \frac{5}{8} = \frac{17}{8} = 2 \frac{1}{8} pizzas

Problems with mixed operations, or ‘multi-step’ word problems, require two or more operations to solve them. A range of concepts can be covered within mixed problems, including the four operations, fractions, decimals and measures. These are worth more marks than some of the more straightforward, one-step problems.

Mixed operation question 1

At the cake sale, Sam buys 6 cookies and a cupcake . He pays £2.85 altogether. Naya buys 2 cookies and pays 90p altogether.

How much does the cupcake cost?

Answer (2 marks): 60p for one cupcake

Mixed operation question 2

Large biscuit tin – 48 biscuits (picture of boxes here) Small biscuit tin – 30 biscuits Ben bought 2 large tins of biscuits and 3 small tins.

How many biscuits did he buy altogether?

Answer (2 marks): 186 biscuits

2 x 48 = 96

3 x 30 = 90

96 + 90 = 186 biscuits

Mixed operation question 3

The owner of a bookshop bought a box of 15 books for £150.

He sold the books individually for £12 each. 

How much profit did he make?

Answer (2 marks)

Third Space Learning offers a wide array of maths and word problems resources for other year groups such as word problems for year 6 , word problems for year 3 and word problems for year 4 . Our word problem collection covers all four operations and other specific maths topics such as ratio word problems and percentage word problems .

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Working Backwards at KS1

Starting from the end sounds rather strange, but in fact can be a very useful problem-solving skill.  These activities lend themselves to being tackled in this way.

This collection is one of our Primary Curriculum collections - tasks that are grouped by topic.

Tables Teaser

How will you work out which numbers have been used to create this multiplication square?

What Was in the Box?

This big box adds something to any number that goes into it. If you know the numbers that come out, what addition might be going on in the box?

Number Lines

A resource to try once children are familiar with number lines, and they have begun to use them for addition. It could be a good way to talk about subtraction. Leah and Tom each have a number line. Can you work out where their counters will land?

Secret Number

Annie and Ben are playing a game with a calculator. What was Annie's secret number?

Starfish Spotting

How many starfish could there be on the beach, and how many children, if I can see 28 arms?

The Tall Tower

As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells?

Stop the Clock

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

Growing Garlic

Ben and his mum are planting garlic. Can you find out how many cloves of garlic they might have had?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Doing and Undoing

An investigation looking at doing and undoing mathematical operations focusing on doubling, halving, adding and subtracting.

Junior Frogs

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?