Lesson: Solving Word Problems

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Lesson 3: Solving Word Problems

Instructional Objectives

• Solve higher-order word problems involving percentage change.

• Relate the concepts of percentage of percentage and fraction of fraction

Warm-Up Activity

This activity prepares pupils to solve word problems by finding percentage change.

• Have pupils review the strategy of finding of percentage change by drawing a model to solve the following problem.

The usual price of a novel was $25. It was sold at $20.

a) Find the discount on the price of the book. ($5)

b) Find the percentage discount on the usual price. (20%)

Solving word problems involving percentage (Pupil’s Book 6A page 122)

Before you learn ...

• Read the scenario aloud. Have pupils draw a model to find the percentage increase in Gwen’s height.

• Invite volunteers to share their models with the class.

• Explain to pupils that they will learn to solve more complex word problems involving percentage change in this lesson.

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(Pupil’s Book 6A pages 122 and 123)

Pupils learn to use a comparison model to solve word problems on percentage change given the original and final quantities.

• Direct pupils’ attention to 1 on page 122 of Pupil’s Book 6A.

• Read and explain the context of the problem. Use the four-step problem-solving method to guide pupils through the process.

• Step 1 Ask: What have we gathered from the problem? (We know the amount Shaun collected on the first day and the second day of the funfair. We also know that the total amount he had collected over 3 days.) How can we find the amount he collected on the third day? (We can subtract the amounts collected on the first and second days from the total.

$2538 − $925 − $728 = $885)

• Ask: What is the difference in the amount of money collected from the first day to the second day?

($925 − $728 = $197) What is the difference in the amount of money collected from the second day to the third day? ($885 − $728 = $157)

Digital Element

Launch Learn from the . Have pupils interact with the Digital Element to enhance their understanding of the concept.

Best Practice

Review the four-step problem-solving method:

Step 1 What have I gathered from the problem?

Step 2 How can I solve it?

Step 3 What do I need to find?

Step 4 How can I check my answer?

For the questions in LEARN and TRY throughout this lesson, have pupils practise drawing the models on their own before demonstrating how to do so.

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b

represent the amount of money collected on each of the 3 days? (comparison model) Why? (We are comparing the difference between the amount collected on the first day and the second day, and also between the second day and the third day.) Guide pupils to draw and label the model as shown in Pupil’s Book 6A.

• Step 3 Ask: What do we need to find? (We need to find the amount of money Shaun collected on the third day, the difference in the amount of money collected from the first day to the second day, and the difference in the amount of money collected from the second day to the third day.) Have pupils indicate the parts of the model that they need to find.

• Remind pupils the strategy to find the percentage change.

Change

Percentage change = ×100

Original quantity

• Guide pupils to identify the amount collected on the first day in as the original quantity, since they need to compare the amount collected on the second day with respect to the amount collected on the first day. Highlight to pupils that they need to find the percentage decrease.

• Guide pupils to identify the amount collected on the second day in as the original quantity, since they need to compare the amount collected on the third day with respect to the amount collected on the second day. Highlight to pupils that they need to find the percentage increase.

• Have pupils find the corresponding percentage changes. Remind pupils to correct their answers to 1 decimal place.

• Step 4 Ask: How can we check our answer? (We can work backwards.) Point out that when they use the percentage changes correct to 1 decimal place, the difference in amounts will not be exact.

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(Pupil’s Book 6A page 124)

1 requires pupils to use a comparison model to find percentage changes given the original and final quantities.

• Remind pupils to use the four-step problem-solving method to help them.

• Have pupils identify the whole (100%) first for each part of the problem.

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4 1 3 a d

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Pupils learn to use models to solve word problems involving sequential percentage changes.

• Step 1 Ask: What have we gathered from the problem?

(We know that 30% of a strip of leather was cut, resulting in 175 cm of leather strip left. We also know that 25% of the 175-cm strip was cut.)

• Step 2 Ask: How can we solve it? (We can draw a model.) Ask: What kind of model should we draw to represent the original length and the remaining length after making the belt and after making the bracelet?

(comparison model or part-whole model) Why? (We are comparing the length of a leather strip to its length before it was cut.) Guide pupils to draw and label the model as shown in Pupil’s Book 6A.

• Step 3 Ask: What do we need to find? (We need to find the original length of the strip of leather and the its length after making the belt and the bracelet.) Lead pupils to see that the original length of the leather strip represents the original quantity (100%) in , as 30% of the original length was cut. Ask: What percentage of the original length of leather strip is left after cutting 30%? (100% − 30% = 70%) Guide pupils to see that 175 cm represents 70% of the original length. Have pupils use the unitary method to find 100% or the original length of the leather strip.

• Lead pupils to see that the length of the leather strip after making the belt represents the original quantity (100%) in , as 25% of that length was cut. Guide pupils to see that 175 cm represents 100%. Ask: What

percentage of the remaining strip of leather after making the belt is left after cutting the 25%? (100% − 25% = 75%) Have pupils find the final quantity (75%) as the length of remaining strip of leather in the end.

• Step 4 Have pupils work backwards to check their answers for each part.

Use the before-after concept to illustrate percentage increase or decrease. (Pupil’s Book 6A page 126)

This activity provides pupils with opportunities to draw before-after models to illustrate percentage changes.

• Have pupils work in pairs.

Have one pupil in each pair read the word problem and draw a model for the problem.

Have the pupil use the model to solve the problem.

Have the partner in each pair check the answer in using a calculator.

Have pupils switch roles and repeat to with the problems in to .

• Have volunteers share their models and answers with the class.

Differentiated Instruction

Struggling Learners

Pupils may have difficulty seeing how the whole in is different from the whole in . Explain to pupils that they are making different comparisons in both the parts of the problem.

Provide scaffolding to draw models for each part of the problem. Explain to pupils that they have to note the percentage of which quantity is being used. Encourage pupils to highlight the lines to correctly identify the whole (100%) for each part of the problem. For example in 2 , He cut 30% of the strip of leather to make a belt…He then cut off 25% of the remaining strip of leather…

Have pupils label the 100% for every comparison made.

Best Practice

Have pupils in each pair explain how they draw each part of their model to their partners.

Encourage the partners to ask clarification questions.

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2 requires pupils to use models to solve a word problem involving sequential percentage changes.

• Have pupils identify the whole (100%) first for each part of the problem.

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Pupils learn to use models to solve word problems involving fractions and percentages.

• Step 1 Ask: What have we gathered from the problem? (We know that Kamal’s monthly salary last year was

$2000 and that Andrew’s monthly salary last year was

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5 of Kamal’s monthly salary. We also know that Andrew’s monthly salary increased by 20% this year.)

• Step 2 Ask: How can we solve it? (We can draw a model.) Ask: What kind of model should we draw to represent the relationship between their monthly salaries last year? (comparison model) Why? (We are comparing Andrew’s monthly salary last year to Kamal’s monthly salary last year.) Guide pupils to draw and label the model as shown in Pupil’s Book 6A. Say: Andrew’s monthly salary was

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5 of Kamal’s monthly salary. We can represent Andrew’s monthly salary last year with 4 units and Kamal’s monthly salary with 5 units. Guide pupils to draw the model to compare Andrew’s monthly salary this year to his monthly salary last year as shown in Pupil’s Book 6A.

• Step 3 Ask: What do we need to find? (We need to find the increase in Andrew’s monthly salary from last year to this year.) What was Andrew’s monthly salary last year? (

4× $2000 = $1600

5 ) Have pupils indicate the part of the model that they need to find. Lead pupils to see that since they are comparing Andrew’s monthly salary last year with his monthly salary this year, they can take his monthly salary last year as the original quantity (100%). Ask:

What is the percentage increase in his monthly salary? (20% of his monthly salary last year) Lead pupils to represent Andrew’s monthly salary last year as 100% and use unitary method to find 20% of it, in order to get the increase in his monthly salary from last year to this year.

• Step 4 Ask: Have pupils work backwards to check their answers by finding Andrew and Kamal’s monthly salaries last year, and comparing them.

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3 requires pupils to use models to solve a word problem involving fractions and percentages.

• Have pupils identify the whole (100%) for the problem.

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Pupils learn to use models to solve word problems involving percentage of a percentage.

• Step 1 Ask: What have we gathered from the problem? (We know that there are three categories of seats in a theatre, A, B and C. We also know that 75% of the seats are Category A seats.) Ask: What percentage of the total seats are Category B and Category C seats? (100% − 75% = 25%) What percentage of the remaining seats are Category B seats? (60%)

• Step 2 Ask: How can we solve it? (We can draw a model.) Ask: What kind of model should we draw to represent the different categories of seats? (part-whole model) Why? (We are given the percentages of the different parts of the whole.) Guide pupils to draw the model as shown in Pupil’s Book 6A. Have pupils show the given information in the model.

• Step 3 Ask: What do we need to find? (We need to find the percentage of seats in the theatre that are Category B seats.) What else do we need to find? (We need to find the total number of seats in the theatre, given that there are 50 Category C seats.)

• Lead pupils to find the percentage of the total seats in the theatre that are Category B seats from the model. Write on the board: 60% of 25% =

60

100 × 25% = 15%. Highlight to pupils that 60% =

60 100.

• Ask: How can we find the percentage of total seats that are Category C seats? (percentage of the total seats that are Category B and Category C seats − the percentage of the total that are Category B seats) Have pupils find the percentage of total seats that are Category C seats. (25% − 15% = 10%)

• Lead pupils to see that 10% of the total seats is equal to 50. Have pupils use unitary method to find 100%, which represents the number of seats in the theatre altogether.

Make connections between the concepts of ‘percentage of percentage’

and ‘fraction of fraction’. (Pupil’s Book 6A page 131)

This activity provides pupils with opportunities to use models to relate the concept of ‘percentage of a percentage’ to ‘fraction of a fraction’.

• Have pupils work in pairs.

Have one pupil in each pair read the word problem and draw a model to solve it. Have pupils see that the whole can be interpreted in terms of percentage or units.

Have the partner in each pair read the other problem and draw a model to solve it.

Have pupils compare their models and answers in and Invite volunteers to share with the class what they have found out.

Have pupils switch roles and repeat to with the problems in and .

• Have volunteers share their models and answers with the class.

• Conclude the activity by summarising that a percentage of a percentage is the same as a fraction of a fraction.

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3 requires pupils to use models to solve a word problem involving fractions and percentages.

• Have pupils identify the whole (100%) for the problem.

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Let’s Practise

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This practice reinforces the use of four-step problem-solving method to solve word problems involving percentage changes. Encourage pupils to draw models to represent the information in the problems.

to require pupils to to solve word problems involving percentage changes.

Common Error

Some pupils may have difficulty solving word problems involving the before-after concept with different

scenarios, especially when the whole (100%) differs from one pair of comparisons to another pair of comparisons.

Have pupils practise drawing various types of models in order to visualise the relationships of the pieces of information given in the problem. Have pupils find the whole (100%) for each comparison made.

On Your Own

Pupils practise solving higher-order word problems involving percentage changes in Practice 3, pages 101 to 108 of Workbook 6A.

These pages (with the answers) are shown on pages 73 to 80 of this chapter.

Use Practice 3 on pages 93 to 100 of Homework 6A for more practice.