my homework lesson 7 problem solving work backwards

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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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Primary 3 Working Backwards & Its Method

Math heuristics for problem solving, primary 3 working backwards & its method, what is working backwards in math.

The scenario occurs when the quantity data is insufficient to work from the beginning . Working Backwards is a problem-solving strategy in which you start with the end goal and work backward to figure out the steps needed to get there. In other words, instead of starting from the beginning and moving forward, you start from the end and move backward. This strategy is commonly used in math problems that ask you to find a starting value or figure out what happened before a given situation.

How to Solve Math Questions with Working Backwords Method?

Let's take a look at this primary 3 word problem example:.

Watch the tutorial for free!

Sarah had some pens. She bought 34 pens. She then threw away 29 pens as they were spoilt. In the end, she had 64 pens. How many pens did Sarah have at first?

Identify the Concept

We know this is a Working Backwards question as…

Workings Explained

Always remember when we work backwards, everything will be reversed. Example the 2nd sentence – “She bought 34 pens”. We know when we buy things, we will have more. We need to add. However, when we work backwards, instead of adding, we need to subtract.

• We will start drawing the model from the end by drawing a box and label it “End”. Put the end amount “64” in the box.
• Draw arrow to point to the left, draw another box. On top of the arrow write “+29” as “Sarah threw away 29 pens”. Instead of subtract, we need to add. In the box, write “93” (64+29=93).
• Draw another arrow to point to the left, draw another box. On top of the arrow, write “-34” as Sarah bought 34 pens. Instead of adding, we need to subtract. In the box, write “59” (93-34=59). Label the box “At First” or “Before”.

Sarah had 59 pens at first.

We know this is a Working Backwards question as we were not told the number of pens Sarah had but were asked for the number of pens she had at first.

See other problem-solving strategies and methods

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Working backward to solve problems - maurice ashley.

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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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McGraw Hill My Math Grade 4 Chapter 9 Lesson 5 Answer Key Problem-Solving Investigation: Work Backward

All the solutions provided in  McGraw Hill My Math Grade 4 Answer Key PDF Chapter 9 Lesson 5 Problem-Solving Investigation: Work Backward will give you a clear idea of the concepts.

McGraw-Hill My Math Grade 4 Answer Key Chapter 9 Lesson 5 Problem-Solving Investigation: Work Backward

2. Plan I will work backward to solve the problem.

4. Check Does your answer make sense? Explain. Answer: Fraction of the bag Nathan used for the cake = \(\frac{1}{4}\) My answer makes sense because the values are matching and completes the problem.

Explanation: Nathan used some flour for a cake recipe. Quantity of the bag of flour for a bread recipe he used = \(\frac{1}{4}\) Quantity of the bag of flour left = \(\frac{2}{4}\) Fraction of the bag Nathan used for the cake = Quantity of the bag of flour left – Quantity of the bag of flour for a bread recipe he used = \(\frac{2}{4}\) – \(\frac{1}{4}\) = [(2 – 1) ÷ 4] = \(\frac{1}{4}\) Check: Quantity of the bag of flour for a bread recipe he used + Fraction of the bag Nathan used for the cake + Quantity of the bag of flour left = \(\frac{1}{4}\) + \(\frac{1}{4}\) + \(\frac{2}{4}\) = [(1 + 1) ÷ 4] + \(\frac{2}{4}\) = \(\frac{2}{4}\) + \(\frac{2}{4}\) = [(2 + 2) ÷ 4] = \(\frac{4}{4}\) = 1.

4. Check Does your answer make sense? Explain. Answer: Fraction of the pizza Lily eat = \(\frac{1}{6}\). My answer makes sense because the values are matching and completes the problem.

Explanation: Quantity of of the pizza Noah ate = \(\frac{2}{6}\) Quantity of of the pizza Kaylee ate = \(\frac{1}{6}\) Quantity of of the pizza left = \(\frac{2}{6}\) Fraction of the pizza Lily eat = (Quantity of of the pizza Noah ate + Quantity of of the pizza Kaylee ate) – Quantity of of the pizza left = (\(\frac{2}{6}\) + \(\frac{1}{6}\)) – \(\frac{2}{6}\) = [(2 + 1) ÷ 6] – \(\frac{2}{6}\) = \(\frac{3}{6}\) – \(\frac{2}{6}\) = [(3 – 2) ÷ 6] = \(\frac{1}{6}\) Check: Quantity of of the pizza Noah ate + Quantity of of the pizza Kaylee ate + Fraction of the pizza Lily eat + Quantity of of the pizza left = \(\frac{2}{6}\) + \(\frac{1}{6}\) + \(\frac{2}{6}\) + \(\frac{1}{6}\) = [(2 + 1) ÷ 6] + [(2 + 1) ÷ 6] = \(\frac{3}{6}\) + \(\frac{3}{6}\) = [(3 + 3) ÷ 6] = \(\frac{6}{6}\) = 1.

Apply the Strategy Solve each problem by working backward. Question 1. Mathematical PRACTICE Make Sense of Problems Chloe did some of her homework before dinner She did \(\frac{2}{6}\) of her homework after dinner. She has \(\frac{1}{6}\) of her homework left. What fraction of her homework did Chloe do before her dinner? Write in simplest form. Answer: Fraction of her homework Chloe do before her dinner = \(\frac{1}{6}\)

Explanation: Portion of of her homework after dinner she did = \(\frac{2}{6}\) Portion of of her homework she has left = \(\frac{1}{6}\) Fraction of her homework Chloe do before her dinner = Portion of of her homework after dinner she did – Portion of of her homework she has left = \(\frac{2}{6}\) – \(\frac{1}{6}\) = [(2 – 1) ÷ 6] = \(\frac{1}{6}\)

Question 2. There were 12 goals scored during the game. Team A scored \(\frac{8}{12}\) of the goals. Team B scored 2 goals during the first half of the game. What fraction of the goals did Team B score during the second half of the game? Write in simplest form. Answer: Fraction of the goals did Team B score during the second half of the game = \(\frac{28}{3}\)

Explanation: Number of goals scored during the game = 12. Number of goals scored by Team A = \(\frac{8}{12}\) Number of goals scored by Team B during the first half of the game = 2. Fraction of the goals did Team B score during the second half of the game = Number of goals scored during the game  – Number of goals scored by Team A – Number of goals scored by Team B during the first half of the game) = 12 – (\(\frac{8}{12}\) + 2) = 12 – [(8 + 24) ÷ 12] = 12 – (32 ÷ 12) = [(12 × 12) – 32] ÷ 12 = (144 – 32) ÷ 12 = 112 ÷ 12 or \(\frac{112}{12}\) ÷ \(\frac{4}{4}\) = \(\frac{28}{3}\)

Explanation: Number of reptiles the pet store has = 12. Number of turtles = \(\frac{5}{12}\) Number of snakes = \(\frac{2}{12}\) Fraction of the reptiles is lizards = Number of reptiles the pet store has – (Number of turtles + Number of snakes) = 12 – ( \(\frac{5}{12}\) + \(\frac{2}{12}\)) = 12 – [(5 + 2) ÷ 12] = 12 – \(\frac{7}{12}\) = {[(12 × 12) – 7] ÷ 12} = [144 – 7) ÷ 12 = 137 ÷ 12 or \(\frac{137}{12}\)

Review the Strategies Use any strategy to solve each problem.

• Work backward.
• Use logical reasoning.
• Look for a pattern.
• Make a model.

Question 4. There are 16 books on a shelf. Four-sixteenths of the books are about animals. Two are adventure. The rest are mystery. How many are mystery books? Answer: Number of books are mystery = \(\frac{55}{4}\)

Explanation: Number of books on a shelf = 16. Number of books are about animals = Four-sixteenths or \(\frac{4}{16}\) Number of books are adventure = 2. Number of books are mystery = Number of books on a shelf – (Number of books are about animals + Number of books are adventure) = 16 – (\(\frac{4}{16}\) + 2) = 16 – [4 + (2 × 16) ÷ 16] = 16 – [(4 + 32) ÷ 16] = 16 – (36 ÷ 16) = {[(16 × 16) – 36] ÷ 16} = (256 – 36) ÷ 16 = 220 ÷ 16 or \(\frac{220}{16}\) ÷ \(\frac{4}{4}\) = \(\frac{55}{4}\)

Question 5. There are 10 pieces of chalk. Two-tenths of the chalk is pink. One piece is blue. The rest are white. How many pieces of chalk are white? Answer: Number of pieces of chalk are white = \(\frac{44}{5}\)

Explanation: Number of pieces of chalk = 10. Number of pieces of chalk are pink = Two-tenths or \(\frac{2}{10}\) Number of pieces of chalk are blue = 1. Number of pieces of chalk are white = Number of pieces of chalk – (Number of pieces of chalk are pink + Number of pieces of chalk are blue) = 10 – (\(\frac{2}{10}\) + 1) = 10 – {[(2 + (1 × 10)] ÷ 10} = 10 – [(2 + 10) ÷ 10] = 10 – (12 ÷ 10) = {[(10 × 10) – 12] ÷ 10} = [(100 – 12) ÷ 10] = (88 ÷ 10) ÷ \(\frac{2}{2}\) = \(\frac{44}{5}\)

Question 6. Giselle played with some friends on Monday. She played with 2 times as many friends on Wednesday. This was 4 more than on Friday. On Friday, she played with 4 friends. How many did she play with on Monday? Answer: Number of friends she played on Monday = 4.

Explanation: Number of times as many friends she played on Wednesday = 2. This was 4 more than on Friday. => Number of friends she played on Friday = 4. Number of friends she play with on Wednesday = 4 × Number of friends she played on Friday = 4 × 2 = 8. Number of friends she played on Monday = Number of friends she play with on Wednesday – Number of friends she played on Friday = 8 – 4 = 4.

Explanation: Number of buttons for each costume she needs = 3 . Number of costumes = 22. Number of buttons she needs for 22 costumes = Number of buttons for each costume she needs × Number of costumes = 22 × 3 = 66. Number of buttons she needs for 21 costumes = Number of buttons for each costume she needs × Number of costumes = 21 × 3 = 63.

McGraw Hill My Math Grade 4 Chapter 9 Lesson 5 My Homework Answer Key

Problem Solving Solve each problem by working backward. Question 1. Marla, Jamie, and Sarah have each led their book club’s monthly meeting. Jamie has led \(\frac{2}{6}\) of the meetings, and Sarah has led \(\frac{1}{6}\) of the meetings. What fraction of the meetings has Marla led? Answer: Fraction of the meetings has Marla led = \(\frac{1}{6}\)

Explanation: Number of books Jamie has led of the meetings = \(\frac{2}{6}\) Number of books Sarah has led of the meetings = \(\frac{1}{6}\) Fraction of the meetings has Marla led = Number of books Jamie has led of the meetings – Number of books Sarah has led of the meetings = \(\frac{2}{6}\) – \(\frac{1}{6}\) = [(2 – 1) ÷ 6] = \(\frac{1}{6}\)

Question 2. Suzanne dropped her penny jar. She found some of the pennies, but some are still missing. She found \(\frac{6}{10}\) of the pennies on the rug. She found \(\frac{3}{10}\) of the pennies on the couch. What fraction of pennies is still missing? Answer: Fraction of pennies is still missing = \(\frac{3}{10}\)

Explanation: Number of the pennies on the rug she found = \(\frac{6}{10}\) Number of the pennies on the couch she found = \(\frac{3}{10}\) Fraction of pennies is still missing = Number of the pennies on the rug she found – Number of the pennies on the couch she found = \(\frac{6}{10}\) – \(\frac{3}{10}\) = [(6 – 3) ÷ 10] = \(\frac{3}{10}\)

Question 3. Mathematical PRACTICE Use Number Sense Noah spent some of his allowance on Monday. He spent \(\frac{1}{6}\) of his allowance on Tuesday and \(\frac{3}{6}\) of it on Friday. Noah has none of his allowance money left. What fraction of his allowance did he spend on Monday? Answer: Amount of his allowance Noah spent on Monday = \(\frac{1}{3}\)

Explanation: Amount of his allowance Noah spent on Tuesday = \(\frac{1}{6}\) Amount of his allowance Noah spent on Friday = \(\frac{3}{6}\) Amount of his allowance Noah spent on Monday = Amount of his allowance Noah spent on Friday – Amount of his allowance Noah spent on Tuesday = \(\frac{3}{6}\) – \(\frac{1}{6}\) = [(3 – 1) ÷ 6] = \(\frac{2}{6}\) ÷ \(\frac{2}{2}\) = \(\frac{1}{3}\)

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