Problem Solving in English

Problem Solving Strategies

There are numerous approaches to solving math problems. 'Model Drawing' is the first one that we have introduced because we feel that it has the greatest impact in building children's confidence in dealing with math problems. Most students enjoy visual effects. Seeing abstract relationships, represented by concrete and colourful images, helps in understanding, leading to the solution of the problem. Our section on Model Drawing is by no means exhaustive but it will open a new doorway for the student who has been struggling with math problems.

Besides the Model-Drawing Approach there are several other strategies, which are necessary for the student to master, to achieve proficiency in math problem solving. In our next section, we introduce the important and most useful ones. These are:

1) Draw a Picture 2) Look for a Pattern 3) Guess and Check

4) Make a Systematic List 5) Logical Reasoning 6) Work Backwards

The student may also come across problems which may need the use of more than one strategy before a solution can be found.

OTHER PROBLEM-SOLVING STRATEGIES

Primary 3 Primary 4 Primary Primary 6

5

Draw a Picture/Find a Pattern


Draw a Picture


Guess & Check

Guess & Check

Guess & Check

Guess & Check

Work Backwards

Work Backwards

Work Backwards

Work Backwards

Systematic Listing

Systematic Listing

Systematic Listing

Systematic Listing

Logical Reasoning

Logical Reasoning

Logical Reasoning

Logical Reasoning

These Pages are FREE for online use.

Example of working backwards

Problem Solving Strategies Working Backwards

Question: Jack walked from Santa Clara to Palo Alto. It took 1 hour 25 minutes to walk from Santa Clara to Los Altos. Then it took 25 minutes to walk from Los Altos to Palo Alto. He arrived in Palo Alto at 2:45 P.M. At what time did he leave Santa Clara?

Strategy:

1) UNDERSTAND:

What do you need to find?

You need to find what the time was when Jack left Santa Clara.

2) PLAN:

How can you solve the problem?

You can work backwards from the time Jack reached Palo Alto. Subtract the time it took to walk from Los Altos to Palo Alto. Then subtract the time it took to walk from Santa Clara to Los Altos.

3) SOLVE:

Start at 2:45. This is the time Jack reached Palo Alto.Subtract 25 minutes. This is the time it took to get from Los Altos to Palo Alto.Time is: 2:20 P.M.

Subtract: 1 hour 25 minutes. This is the time it took to get from Santa Clara to Los Altos..

Jack left Santa Clara at 12:55 P.M.

EXAMPLE OF QUEATIONS OF WORKING BACKWARDS

Name: Linda BachWho is asking: TeacherLevel: Elementary

Question:I am having problems finding examples of problems that require "Working Backwards" used as a strategy for solving. We are required to give a presentation on Monday, October 25, 1999 in our school districts math class. We are trying to become better problem solvers and how to teach problem solving in the elementary classroom. Help! I can't find anything in my web searches. I am mathematically challenged so I am also very terrified of any math above 5th grade level. Thank you for your help.Linda Bach

Hi Linda,

Here is one problem that you can easily modify to include fractions or percentages if those are topics you are dealing with.

Mary has some jelly beans. Joan had 3 times as many as Mary but ate 4 and now she has 5. How many jelly beans does Mary have?

To solve this problem you work backwards. Joan has 5 jellybeans now so she had 9 before she ate 4. This 9 is 3 times what Mary has, so Mary must have 3.

You can introduce another step if you wish.

Mary has some jelly beans. Joan had 3 times as many as Mary but ate 4 and now John has 2 more jelly beans that Joan. John has 7 jelly beans, how many does Mary have?

A second problem that is a little more complex but is fun to do involves two problem solving strategies, one of which is working backwards.

Chris is training Hoppity, her pet rabbit, to climb stairs. It will hop up one or two stairs at a time. If a flight of stairs has ten steps, in how many ways can Hoppity hop up the this flight of stairs?

Ten stairs is quite a few so try an simpler problem with fewer stairs.

If there is only one stair, then there is only one way to climb this stair.

If there are two stairs, then there are two ways to climb the stairs, by taking two steps or one big step (covering two stairs)

If there are three stairs, then the rabbit can take 3 small steps take a big step and then a small one or a small step followed by a big step.

If there are four stairs, then the rabbit can climb the stairs in four small stepstwo small steps and a large step one small step, then a large, then a smallone large step, then two small steps or two large steps

Thus we have

Number of steps 1 2 3 4

Number of ways to climb the steps 1 2 3 5

Now look at the four step problem with a "looking backwards" strategy. Suppose the rabbit has just arrived at the top. It got there either by being on the third step and hopping up one step or it was on the second step and got there by hopping up two steps.

From what we did before there are 3 ways to get to step three and 2 ways to get to step two. Thus, there are 3 + 2 = 5 ways to get to step four.

Similarly for a 5 step flight of stairs there are 5 ways to get to step four and 3 ways to get to step three and thus there are 5 + 3 = 8 ways to get to step five.

So now we have

Number of steps 1 2 3 4 5

Number of ways to climb the steps 1 2 3 5 8

The pattern in the bottom row of the table should now be clear. Each number in this row (after the first two) is the sum of the two previous numbers. Thus it is easy to continue this row.

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10,944,...

Hence there are 89 nine ways for the rabbit to climb ten stairs (and 10,944 ways for it to climb 20 stairs!)

I hope this helps,

Cheers,Penny

Blake’s Topic Bank

Problem Solving

Working Backwards

by Sharon Shapiro

This unit contains:

n Teaching notes

n 3 teaching examples

n 1 BLM

n 18 task cards

n Answers

UNIT P5 n

Problem Solving

Upper Primary© Blake Education—Problem Solving: Working Backwards 1

THE PROBLEM

SOLVING PROCESS

It is important that students follow a logical and

systematic approach to their problem solving. Following

these four steps will enable students to tackle

problems in a structured and meaningful way.

STEP 1: UNDERSTANDING

THE PROBLEM

v Encourage students to read the problem

carefully a number of times until they fully

understand what is wanted.They may need to

discuss the problem with someone else or

rewrite it in their own words.

v Students should ask internal questions such as,

what is the problem asking me to do, what

information is relevant and necessary for solving

the problem.

v They should underline any unfamiliar words and

find out their meanings.

v They should select the information they know

and decide what is unknown or needs to be

discovered.They should see if there is any

unnecessary information.

v A sketch of the problem often helps their

understanding.

STEP 2: STUDENTS SHOULD

DECIDE ON A STRATEGY OR PLAN

Students should decide how they will solve the

problem by thinking about the different strategies

that could be used.They could try to make

predictions, or guesses, about the problem. Often

these guesses result in generalisations which help

to solve problems. Students should be discouraged

from making wild guesses but they should be

encouraged to take risks.They should always think

in terms of how this problem relates to other

problems that they have solved.They should keep

a record of the strategies they have tried so that

they don’t repeat them.

Some possible strategies include:

v Drawing a sketch, graph or table.

v Acting out situations, or using concrete

materials.

v Organising a list.

v Identifying a pattern and extending it.

v Guessing and checking.

v Working backwards.

v Using simpler numbers to solve the problem,

then applying the same methodology to the

real problem.

v Writing a number sentence.

v Using logic and clues.

v Breaking the problem into smaller parts.

STEP 3: SOLVING THE PROBLEM

v Students should write down their ideas as they

work so they don’t forget how they approached

the problem.

v Their approach should be systematic.

v If stuck, students should reread the problem

and rethink their strategies.

v Students should be given the opportunity to

orally demonstrate or explain how they reached

an answer.

STEP 4: REFLECT

v Students should consider if their answer makes

sense and if it has answered what was asked.

v Students should draw and write down their

thinking processes, estimations and approach, as

this gives them time to reflect on their

practices.When they have an answer they

should explain the process to someone else.

v Students should ask themselves ‘what if’ to link

this problem to another.This will take their

exploration to a deeper level and encourage

their use of logical thought processes.

v Students should consider if it is possible to do

the problem in a simpler way.

Problem Solving

Working Backwards

Sharon Shapiro Upper PrimaryThe strategy of working backwards is used to solve

problems that include a number of linked factors or

events, where some of the information has not been

provided, usually at the beginning of the problem.

To solve these problems it is usually necessary to

start with the answer and work methodically

backwards to fill in the missing information.

This strategy is extremely useful in dealing with

a situation or a sequence of events.The events

occur one after the other and each stage, or piece

of information, is affected by what comes next.

Students begin at the end, with the final action,

and work through the process in reverse order

to establish what happened in the original situation.

In order to use the strategy of working

backwards effectively, students will need to

develop the following skills and understanding.

USING THE OPPOSITE OPERATION

WHEN WORKING BACKWARDS

When you are solving a problem by starting at

the end and working backwards, any mathematical

operations you come across will have to be reversed.

This means that if the problem requires you to add

something, then when working backwards you must

subtract it, or if multiplying when working forwards,

you must divide when working backwards.

So if the problem the correct way round is –

÷ 8 x 2 = 14

then backwards it will be –

14 ÷ 2 x 8 = 56

Or

Jack is 35 years younger than Karen. Frank is half

of Jack’s age. Jennifer is 17 years older than Frank.

If Jennifer is 35 years old, how old is Karen?

Jennifer is 35. She is 17 years older than Frank.

So using the opposite operation plus becomes

minus. So Frank is 35 – 17 = 18

Frank is half Jack’s age so the opposite operation

is 18 x 2 = 36

Jack is 35 years younger than Karen so 36 + 35 = 71

Therefore Karen is 71 years old.

STARTING WITH THE ANSWER AND

WORKING BACKWARDS

In a problem where you know the final outcome

but don’t know the starting point, beginning

at the end of the problem and working backwards

is the best way of arriving at a solution.

For example, in a dancing competition all the

contestants started dancing together.After three

minutes half the people were eliminated. During

the next ten minutes half of the remaining were

eliminated.At the 15 minute mark, half again were

eliminated, and at the 20 minute mark, half of

those still remaining were eliminated. In the last

two minutes one more contestant was eliminated

leaving a winner of the competition. How many

dancers were there in the beginning?

You know that there is one winner and that

the number of contestants was halved at certain

intervals. Using this information, it is possible to

work backwards and find out how many dancers

entered the competition.

Start with the winner 1 person dancing

Last 2 minutes 1+1 = 2 dancers

After 20 minutes Double = 4 dancers




32 dancers started.

© Blake Education—Problem Solving: Working Backwards

Teaching Notes Working Backwards

2

?EXAMPLE 1

John is four years younger than Carmel but Jane

is 24 years older than Carmel. If Jane is 35, how

old is John?

Un d e r s t a n d i n g t h e p ro bl em

WHAT DO WE KNOW?

John is four years younger than Carmel.

Jane is 24 years older than Carmel.

Jane is 35 years old.

WHAT DO WE NEED TO FIND OUT?

Questioning:

How old is John?

P l a n n i n g a n d

c ommu n i c a t i n g a s o l u t i o n

Begin with the information you know, Jane’s age,

and work backwards to calculate John’s age.

Jane is 35 years old. She is 24 years older

than Carmel.

So, 35 – 24 years = 11.

Therefore, Carmel is 11 years old.

John is four years younger than Carmel.

so, 11 – 4 = 7

Therefore, John is seven years old.

R e f l e c t i n g a n d ge n e r a l i s i n g

By starting with the known factor of Jane’s age

we were able to work backwards and calculate the

answer.This strategy can be applied to problems

which include a sequence of events where we

know the end result but don’t know the starting

point.You can check you answer by working

forwards through the problem to see if you reach

the correct end point.

E x t e n s i o n

Additional people who are older and younger

can be added to further complicate the problem.

Students can construct their own problems using

the ages of their families or friends.


Teaching Examples Working Backwards

3EXAMPLE 2

Four students in the class weighed themselves.

Carter was 15 kilograms lighter than Adrian. Gary

was twice as heavy as Carter and Jeremy was

seven kilograms heavier than Gary. If Jeremy

weighed 71 kilograms what was Adrian’s weight?


WHAT DO WE KNOW?

There are four students in the class.

Carter was 15 kilograms lighter than Adrian.

Gary was twice as heavy as Carter.

Jeremy was seven kilograms heavier than Gary.

Jeremy weighed 71 kilograms.


Questioning:

What was Adrian’s weight?



Start at the end of the problem with Jeremy’s

weight which is 71 kilograms then work back

through the sequence of factors to calculate

Adrian’s weight.

Begin by working out Gary’s weight. Jeremy is

seven kilograms heavier than Gary, so subtract

the seven kilograms from Jeremy’s weight of 71

kilograms.

71 kg – 7 kg = 64 kg.

Gary is twice as heavy as Carter. Now that we

know Gary’s weight is 64 kilograms, we can

calculate Carter’s weight.

64 ÷ 2 = 32 kg.

Carter’s weight is 32 kilograms.

Adrian’s weight is Carter’s weight + 15 kilograms.

32 kg + 15 kg = 47 kg.

Therefore,Adrian weighs 47 kilograms.


By using a step-by-step process, starting with the

known and following a backwards sequence to find

the missing information, the solution was easy to

calculate.An unmethodical approach could have

resulted in the wrong operations being used or

steps being missed out.Working backwards and

using the opposite operations enabled a systematic

approach, which can also be applied to any

problem of a similar type.

E x t e n s i o n

To vary the problem, students could include more

than four people in the problem, or add extra

operations to the sequence.



4EXAMPLE 3

In a spelling competition all the competitors were

on stage together.After three minutes, a fifth

of the students had made mistakes and were

excluded from the competition. In the next five

minutes half of those remaining were eliminated

by extremely difficult words.Two minutes later

four students were found cheating and were sent

home.After fifteen minutes of the competition half

of the remaining students had made mistakes and

left the stage. In the last few minutes one more

competitor made an unfortunate mistake and one

contestant was left as the winner of the spelling

competition. How many children originally entered

the competition?


WHAT DO WE KNOW?

Students were participating in a spelling

competition.

Students who made mistakes or cheated

were eliminated.

There was one winner at the end of the

competition


Questioning:

How many children originally entered the

competition?



Start at the end and reverse

the process. The winner

A few minutes before the end

there was one more contestant = 2 spellers

Fifteen minutes into the competition,

double the number = 4 spellers

Ten minutes into the competition

add four to number = 8 spellers

Five minutes into the competition

double the number = 16 spellers

Three minutes into the competition a fifth of the

competitors had been eliminated so

16 spellers =

4

–

5

of the total.

There were 20 children entered in the spelling

competition.


By reading the sentences in the problem carefully

one at a time and recording all the known

information it was possible to begin at the end and

by reversing all operations to work backwards to

reach a solution. Don’t forget to work forward

through the problem, once you have the solution,

to check your answer.

E x t e n s i o n

The problem can be made more complicated by

asking students to work with a variety of fractions

or by including more steps.



5BLM Working Backwards

6

H Understanding the problem

What do you know? List the important facts from the problem. Draw a

double line under your starting point.

H What do you need to find out?

What is the problem asking you to do? What are you uncertain about?

Do you understand all aspects of the problem? Is there any unfamiliar

or unclear language?

H Planning and

communicating a solution

Find your starting point. Work

backwards in a logical step-bystep way. How many steps are

required? Is all the information

necessary? Will using objects

to represent the people or places

make the problem easier to solve?

H Reflecting and generalising

Did the strategy work as planned? Is your answer correct? You can

check this by working forward through the problem. Will you be able to

apply this method of problem solving to other similar problems? Could

you have used a different method to solve the problem?

H Extension

How can this strategy be applied to more complicated problems involving

bigger numbers and additional factors?


This page may be reproduced by the original purchaser for non-commercial classroom use.PROBLEM SOLVING TASK CARDS -Working Backwards

Problem 1

When three girls jumped on a weighing

scale together, it read 164 kilograms.

One girl stepped off and the scale moved

down to 104 kilograms. One more girl

jumped off and the scale showed 55

kilograms. What was each girl’s weight?

Problem 2

Arnold baked cupcakes over

the weekend. Each day during

the week he took three cakes

to school to share with his

friends. On Saturday when he

counted there were 18 left.

How many had he baked?

Problem 3

Daniel has lots of pets. He

has four more goldfish than

he has turtles. He has one

less canary than goldfish.

Six of his pets are birds

(canaries and parrots). He

has two parrots. How many

pets does Daniel have?


This page may be reproduced by the original purchaser for non-commercial classroom use.

7

Level

1

Level

1

Level

1

Measurement

Number123

Number123PROBLEM SOLVING TASK CARDS -Working Backwards

Problem 4

Jemima has twice as much money

as Matthew. Jemima has four times

as much money as Sally. Sally has

$3 more than Andrew. If Matthew

has $14, how much money do

Andrew, Sally and Jemima have?

Problem 5

Jack, Terence, Sharon and Alex

are neighbours. Jack is half as

old as Sharon. Sharon is three

years older than Alex. Alex’s and

Sharon’s ages added together

equal 17 years. Terence is eight.

Who is the youngest?

Problem 6

A teacher bought five flags of different

countries, to use in a class activity. She

added them to the flags she already had

in the classroom. She borrowed four

more flags, but two of these weren’t

used. In the end ten flags were used in

the activity. How many flags were there

in the classroom already?



8

Level

1

Level

1

Level

1

Number123

Number123


Problem 7

Six people entered a block building contest. Lynda built her

pile of blocks twice as high as Selwyn’s. Michelle created a

pile that was three times higher than Lynda’s pile. Warren

built his pile one block higher than Michelle’s. Jane’s pile of

blocks was six higher than Warren’s.

If Adrian piled up 27 blocks, which was two blocks higher

than Jane’s pile, how high was each person’s pile? How many

blocks would be needed altogether?

Problem 8

Four girls each caught a fish while at the

beach. Teri’s was double the size of

Jane’s. Jane’s fish was shorter than

Lynn’s by nine centimetres. Lynn’s was

18 centimetres longer than Reina’s who

caught a fish 30 centimetres long. How

long were Teri’s, Jane’s and Lynn’s fish?

Problem 9

Joshua is five years older than

David. Simon is four years older

than Joshua and nine years older

than David. Simon’s and David’s

ages added together equal 15.

How old are the boys?



9

Level

2

Level

2

Level

2

Space

Measurement


Problem 10

At the party Jade ate less than

four jelly beans. Nicole ate twice

as many jelly beans as Jade. Kahlee

had twice as many as Nicole. Chris

had two more than Kahlee. Chris

ate ten jelly beans. How many jelly

beans were eaten at the party?

Problem 11

Mark, Neil, Frances and Patrick entered a

skipping competition. Patrick skipped eight

more times than Mark before his foot

caught on the rope. Mark jumped three more

times than Neal. Neal skipped half as many

times as Frances. Frances skipped eighty

times. How many times did Patrick skip?

Problem 12

When Ariella climbs aboard there are

already some people sitting the bus. At the

next bus stop an additional five people get

on and two people get off. Two stops later

seven people climb on board. All 15 people

get off the bus at the ferry. How many

were on the bus when Ariella climbed on?



10

Level

2

Level

2

Level

2

Number123

Number123


Problem 13

There is an old jar packed full of textas

on the table. There are twice as many

red textas as blue and one more yellow

than red. Eight textas are either yellow

or green. There are three green textas.

How many blue textas are in the jar?

Problem 14

My new shoes arrived in a rectangular

cardboard box. The length of the box

was double the width and the width

was double the height. The length was

28 centimetres. What was the volume

of the shoebox?

Problem 15

Three people went strawberry collecting and picked 65

strawberries between them. At the first plant they each

picked the same amount of strawberries. At the second

plant they each collected three times the amount that

they had collected at the first plant. After picking from

the third plant they had five times the amount they had

after picking strawberries from the first two plants. At

the fourth plant they collected only five strawberries

altogether. How many strawberries did each person

collect at the first bush?



11

Level

3

Level

3

Level

2

Number123

Measurement


Problem 16

Twenty-four ladybirds were sitting at various places

around the garden. One sixth of the ladybirds flew

away to settle in the garden. Half of the remaining

ladybirds sat on a yellow sunflower. Then half of

those on the sunflower flew onto a fence post.

There were no ladybirds on the hedge, but one

fifth of the ladybirds on the sunflower flew onto

the lavender bush. If one ladybird was on the grass,

how many were on the tree trunk?

Problem 17

Heather loves roses. In her rose

garden she has half as many pink

roses as red, and four times as many

red ones as white. There are 36 roses

that are either yellow or white.

Twenty are yellow. How many roses

are in Heather’s garden?

Problem 18

Four children were bouncing balls.

Jeff’s ball bounced eight more times

than Michael’s. Michael’s ball bounced

half as many times as Olivia’s. Olivia’s

bounced eight times. How many times

did Jeff’s ball bounce?



12

Level

3

Level

3

Level

3

Number123

Number123

Number123P ro bl em 1

The weights of the three girls –

3 girls = 164 kg

2 girls = 104 kg

1 girl = 55 kg

2nd girl, 164 – 104 = 60 kg

3rd girl, 104 – 55 = 49 kg

P ro bl em 2

Number of cupcakes baked by Arnold ?

Monday – 3

Tuesday – 3

Wednesday – 3

Thursday – 3

Friday – 3

Saturday 18 cupcakes left

Therefore, working backwards,

18 + (5 x 3) = 33 cupcakes.

P ro bl em 3

6 birds – 2 parrots = 4 canaries

To find the number of goldfish, add 1 to the

number of canaries.

4 + 1 = 5 goldfish

To find the number of turtles, subtract 4 from

the number of goldfish.

5 – 4 = 1 turtle

Therefore, 2 parrots + 4 canaries + 5 goldfish

+ 1 turtle = 12 pets in total.

P ro bl em 4

Matthew $14

Jemima $14 x 2 = $28

Sally $28 ÷ 4 = $7

Andrew $7 – 3 = $4

P ro bl em 5

Terence is 8 years old.

Sharon’s and Alex’s ages total 17 years.

Sharon is 3 years older than Alex.

So, 17 – 3 = 14. 14 ÷ 2 = 7.

Alex is 7.

Sharon is 7 + 3 = 10.

Jack is half Sharons age so, 10 ÷ 2 = 5.

Therefore, Jack is the youngest.

P ro bl em 6

Original number of flags in classroom

+ 5 flags

+ (4 – 2) = 2 flags

= 10 flags

Therefore working backwards,

10 – 2 – 5 = 3 flags in the classroom.

P ro bl em 7

Working backwards,

Jane 27 – 2 = 25

Warren 25 – 6 = 19

Michelle 19 – 1 = 18

Lynda 18 ÷ 3 = 6

Selwyn 6 ÷ 2 = 3

Total number of blocks

= 3 + 6 + 18 + 19 + 25 + 27 = 98

P ro bl em 8

Reina’s fish 30 cm

Lynn’s fish 30 + 18 = 48 cm

Jane’s fish 48 – 9 = 39 cm

Teri’s fish 39 x 2 = 78 cm

P ro bl em 9

David’s and Simon’s ages total 15.

Simon is 9 years older than David.

So 15 – 9 = 6. 6 ÷ 2 = 3.

David’s age is 3.

Simon’s is 3 + 9 = 12.

Joshua is 5 years older than David, so 3 + 5 = 8.



Answers to Task Cards

13P ro bl em 1 0

Working backwards,

Chris, 10

Kahlee, 10 – 2 = 8

Nicole, 8 ÷ 2 = 4

Jane, 4 ÷ 2 = 2

Total jelly beans eaten = 2 + 4 + 8 + 10 = 24.

P ro bl em 1 1

Frances, 80 skips

Neil, 80 ÷ 2 = 40

Mark, 40 + 3 = 43

Patrick, 43 + 8 = 51

P ro bl em 1 2

People on bus at beginning ?

Ariella gets on, + 1

1st stop, + 5 – 2

2nd stop, + 7

Total getting off bus, 15

Working backwards,

15 – 7 = 8

8 + 2 – 5 = 5

5 – 1 = 4

Four people were on the bus in the beginning.

P ro bl em 1 3

Green textas, 3

Yellow and green = 8

8 – 3 = 5 yellow textas

One more yellow than red, 5 – 1 = 4 red

Twice as many red as blue, 4 ÷ 2 = 2 blue textas.

P ro bl em 1 4

Length 28 cm

Length is double the width so,

28 ÷ 2 = 14 cm

Width is double the height so,

14 ÷ 2 = 7 cm

The volume is length x width x height,

28 x 14 x 7 = 2,744 cm2

.

P ro bl em 1 5

Total strawberries 65

Working backwards,

65 – 5 = 60

60 ÷ 5 = 12

12 equals the total number of strawberries picked

off the first and second plants.To work out how

many were picked from the first bush you must

divide by four to get four equal parts, one part

being what was picked off the first plant and three

parts being what was picked off the second plant.

Therefore 12 ÷ 4 = 3. So there were three

strawberries picked off the first plant, one by each

of the three people.

P ro bl em 1 6

Total ladybirds minus

1

–

6

24 ÷ 6 = 4

24 – 4 = 20 ladybirds left in garden

Half on sunflower 20 ÷ 2 = 10

Half flew away onto fence post 10 ÷ 2 = 5

0 on hedge

1

–

5

fly onto lavender bush 5 ÷ 5 = 1

1 on grass

Total accounted for, 4 + 10 + 5 + 1 + 1 = 21

So 24 – 21 = 3 on tree trunk.

P ro bl em 1 7

20 roses are yellow

36 are yellow or white

36 – 20 = 16 white

4 times as many red as white, 16 x 4 = 64 red

half as many pink as red 64 ÷ 2 = 32 pink

32 + 64 + 16 + 20 = 132 roses

P ro bl em 1 8

Olivia’s ball bounced 8 times.

Michael’s ball bounced half as many times as

Olivia’s, so 8 ÷ 2 = 4.

Jeff’s ball bounced 8 more times than Michael’s,

so 4 + 8 = 12.



14

Problem Solving Strategy of Working Backwards

Problem Solving website prepared for

Problem Solving Strategies for City

College course EDUC 6200E, Dr. Moresh

What is Working Backwards in problem solving?

The strategy of working backwards entails starting with the end results and reversing the steps you need to get those results, in order to figure out the answer to the problem.

When do we use this strategy? What are real life examples? There are at least two different types of problems which can best be solved by this strategy:

(1) When the goal is singular and there are a variety of alternative routes to take.situation, the strategy of working backwards allows us to ascertain which of the alternative routes was optimal.

An example of this is when you are trying to figure out the best route to take to get from your house to a store. You would first look at what neighborhood the store is in and trace the optimal route backwards on a map to your home.

(2) When end results are given or known in the problem and you're asked for the initial conditions.

An example of this is when we are trying to figure out how much money we started with at the beginning of the day, if we know how much money we have at the end of the day and all of the transactions we made during the day.

Sample Problem:

Joe forgot to check how much money he began the day with. During the day, he spent $8.00 on breakfast, withdrew $40.00 from the ATM, got his dry cleaning for $12.00, bought 5 shirts for $22.00 a piece (plus 8% sales tax). At the end of the day, he had $100.00, how much did he start the day with?

Solution:

Rather than letting x = the initial amount and creating a long algebraic equation, if we use the working backwards strategy, the problem is more easily solved.

$100 Initial amount

+ 1.08($22*5) = $218.80 (adding back shirt purchase)

+ $12 = $230.80 (adding back dry cleaning)

- $40 = $190.80 (subtracting ATM withdrawal)

+ 8 = $198.80 (adding back breakfast)

Joe began the day with $198.80.

Below are more sample problems:Problem #1

Joe gives Nick and Tom as many peanuts as each already has. Then Nick gives Joe and Tom as many peanuts as each of them then has. Finally, Tom gives Nick and Joe as many peanuts as each has. If at the end each has sixteen peanuts, how many peanuts did each have at the beginning?

Solution:

Working backwards, we can start with the end result and analyze each step:

Joe Nick Tom

Peanuts at end: 16 16 16

Previous round: 8 8 32

(because Tom must have given each half of their 16 peanuts, or 8 peanuts)

Previous round: 4 28 16

(because Nick gave each half of their peanuts)

Peanuts at start: 26 14 8

(because Joe gave each half of their peanuts)

The answer, therefore, is that Joe began with 26 peanuts, Nick began with 14, and Tom with 8.

Problem #2

I have seven coins whose total value is $0.57. What coins do I have? And, how many of each coin do I have?

Solution:

In this problem, we can solve it most easily by working backwards -- instead of starting with zero and adding the different combinations of coins towards a goal of $0.57, we can begin with the end result and work backwards towards $0.00.

Step 1: There must be 2 pennies. (The only other option would be to use seven pennies, but that would use up all the coins prematurely.)

Step 2: Now we need to figure out how to use the 5 remaining coins to make a total of $0.55. Because 5 dimes is less than $0.55, we must use at least one quarter.

Step 3: Now we need to use 4 coins to make up the remaining $0.30. At this point, all the remaining coins must be dimes and nickels, and the only possible combination is to use 2 dimes and 2 nickels.

Problem #3

The grid represents a section of town with only 2-way streets. You want to go from point A to point B without backtracking. How many different routes are possible? (You can only go north or east at each intersection.)

B

A

Solution:

Starting at point B, we work backwards, one intersection at a time.solution table, the number at each intersection denotes the number of possible routes to B from that intersection. Each number is simply the sum of the two numbers north and east of it.

Using this system, we can fill in all the numbers back to Point A, and we see that there are seventy possible routes.

1 1 1 1 B

5

4

3

2

1

15

10

6

3

1

35

20

10

4

1

70 35 15 5 1A

Documents

Problem Solving in English