Model Drawing Strategy

In this article, the lesson objectives are: 

• Drawing models to help us solve word problems involving increase in the subject(s)

Model drawing strategy is a useful skill which is used to understand complex word problems visually.

Question 1: 

Timmy had 128 fewer stickers than Chloe. After Chloe bought another 132 stickers, she had thrice as many stickers as Timmy. How many stickers did Chloe have at first?

Solution:

Difference between Chloe and Timmy’s stickers in the end

\(\begin{align}​​ &= 128 + 132\\[2ex] &= 260 \end{align}\)

\(\begin{align}​​ 2 \text{ units} &= 260 \\[2ex] 1 \text{ units} &= 260 \div 2\\[2ex] &= 130 \end{align}\)

Number of stickers Chloe had at first

\(\begin{align}​​ &= 1 \text{ unit} + 128\\[2ex] &= 130 + 128\\[2ex] &= 258 \end{align}\)

Answer:

258 stickers

Question 2: 

Phoebe had 812 more stamps than Rachel at first. After Phoebe bought 804 more stamps, she had 5 times as many stamps as Rachel.

1. How many stamps did Rachel have? 
2. How many stamps would Rachel need to buy in order to have twice as many stamps as Phoebe in the end?

Solution:

A.

Difference between Phoebe and Rachel’s stamps in the end

\(\begin{align}​​ &= 812 + 804\\[2ex] &= 1616 \end{align}\)

\(\begin{align}​​ 4 \text{ units} &= 1616 \\[2ex] 1 \text{ units} &= 1616 \div 4\\[2ex] &= 404 \end{align}\)

Number of stamps Rachel had

\(\begin{align}​​ &=1 \text{ units} \\[2ex] &= 404 \end{align}\)

Answer:

404 stamps

B.

In order for Rachel to have twice as many stamps as Phoebe in the end, 

Number of units Rachel must have in the end

\(\begin{align}​​ &=2 \times 5 \text{ units} \\[2ex] &= 10 \text{ units} \end{align}\)

Number of stamps Rachel needs to buy

\(\begin{align}​​ &= 10 \text{ units }  – 1 \text{ units} \\[2ex] &= 9 \text{ units} \\[2ex] &= 9 \times 404\\[2ex] &= 3636 \end{align}\)

Answer:

3636 stamps

Question 3: 

Julian and Ken had some stickers at first. After Julian bought 24 more stickers, Ken's stickers became 4 times that of Julian's stickers. The total number of stickers Julian and Ken had in the end was 405. How many stickers did Julian have at first?

Solution:

\(\begin{align}​​ 5 \text{ units} &= 405 \\[2ex] 1 \text{ units} &= 405 \div 5\\[2ex] &= 81 \end{align}\)

Number of stickers Julian had at first

\(\begin{align}​​ &= 1 \text{ units } – 24\\[2ex] &= 81 – 24\\[2ex] &= 57 \end{align}\)

Answer:

57 stickers 

Question 4: 

Maddy had \$24 more than Han at first. After Han received \$284, Han had thrice as much as Maddy. Find the total amount of money they had in the end.

Solution: 

Difference between Maddy and Han’s money in the end

\(\begin{align}​​ &= $284 – $24\\[2ex] &= $230 \end{align}\)

\(\begin{align}​​ 2 \text{ units} &= $230 \\[2ex] 1 \text{ units} &= $230 \div 2\\[2ex] &= $115 \end{align}\)

Total amount of money they had in the end

\(\begin{align}​​ &= 3 \text{ units} + 1 \text{ unit} \\[2ex] &= 4 \text{ units} \\[2ex] &= 4 \times $115\\[2ex] &= $460 \end{align}\)

Answer:

$460 

Question 5: 

Jason had \$230 less than Doris. After Jason received \$1500 from his mother, he had 3 times as much money as Doris. How much did both of them have in total at first?

Solution: 

Difference between Jason’s and Doris’s money in the end

\(\begin{align}​​ &= $1500 \;–\; $230\\[2ex] &= $1270 \end{align}\)

\(\begin{align}​​ 2 \text{ units} &= $1270 \\[2ex] 1 \text{ units} &= $1270 \div 2\\[2ex] &= $635 \end{align}\)

Total number of stickers both of them had at first

\(\begin{align}​​ &= 1 \text{ unit} + 1 \text{ unit} \;–\; $230 \\[2ex] &= $635 \;+\; $635 \;–\; $230 \\[2ex] &= $1040 \end{align}\)

Answer:

$1040