Introduction

Every multiplication problem can be considered a statement that compares the sizes of the products and factors.

For example, let's consider the following multiplication problem:

{\color{red}{2}} \times {\color{blue}{4}} = 8

We can interpret this equation as a comparison statement in two possible ways:

  • Firstly, this equation tells us that " 8 is {\color{red}{2}} times larger than {\color{blue}{4}}. " This is because we need to add {\color{red}{2}} copies of {\color{blue}{4}} to make 8{:} \underbrace{{\color{blue}{4}} + {\color{blue}{4}}}_{{\color{red}{2}} \,\textrm{times}} = 8

  • Secondly, this equation tells us that " 8 is {\color{blue}{4}} times larger than {\color{red}{2}}. " This is because we need to add {\color{blue}{4}} copies of {\color{red}{2}} to make 8{:} \underbrace{{\color{red}{2}} + {\color{red}{2}}+{\color{red}{2}}+{\color{red}{2}}}_{{\color{blue}{4}} \,\textrm{times}} = 8

Every multiplication problem can be expressed as two separate comparison statements.

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Example: Multiplication Problems as Comparison Statements

According to the equation 7 \times 9 = 63, how many times larger is 63 than 9?

EXPLANATION

The given equation compares the size of the product (63) to its factors (7 and 9). It can be interpreted as a comparison in the following two ways:

  • "The number 63 is 7 times larger than 9. "

  • "The number 63 is 9 times larger than 7. "

Therefore, the answer is 7.

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Practice: Multiplication Problems as Comparison Statements

According to the equation $8 \times 7 = 56,$ what number is $8$ times larger than $7?$

a
 $8$
b
 $7$
c
 $15$
d
 $1$
e
 $56$
Practice: Multiplication Problems as Comparison Statements

According to the equation $4 \times 9 = 36,$ how many times larger is $36$ than $4?$

a
 $9$
b
 $18$
c
 $36$
d
 $12$
e
 $4$
Example: Filling In Missing Spaces in a Comparison Statement

Consider the following multiplication problem:

5 \times 4 = 20

From left to right, what numbers could be inserted into the following to make it a true statement?

\qquad "The number \textrm{___} is 5 times larger than \textrm{___}. "

EXPLANATION

The given equation compares the size of the product (20) to its factors (4 and 5). It can be interpreted as a comparison in the following two ways:

  • "The number 20 is 5 times larger than 4. "

  • "The number 20 is 4 times larger than 5. "

Therefore, from left to right, the missing numbers are 20 and 4.

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Practice: Filling In Missing Spaces in a Comparison Statement

Consider the following multiplication problem:

\[ 3 \times 4 = 12 \]

From left to right, what numbers could be inserted into the following to make it a true statement?

$\qquad$ "The number $\textrm{___}$ is $4$ times larger than $\textrm{___}.$"

a
 $3$ and $12$
b
 $12$ and $4$
c
 $7$ and $3$
d
 $12$ and $3$
e
 $15$ and $7$
Practice: Filling In Missing Spaces in a Comparison Statement

Consider the multiplication problem

\[ 9 \times 8 = 72. \]

From left to right, what numbers could be inserted into the following to make it a true statement?

$\qquad$ "The number $\textrm{___}$ is $\textrm{___}$ times larger than $\textrm{___}.$"

a
 $72, 8,$ and $7$
b
 $8, 72,$ and $9$
c
 $9, 72,$ and $8$
d
 $8, 9,$ and $72$
e
 $72, 8,$ and $9$
Additive vs. Multiplicative Comparison

It's important to recognize the difference between additive and multiplicative comparisons.

Let's consider the following addition problem:

{\color{red}{7}} + {\color{blue}{2}} = 9

We can represent this addition problem as a comparison in the following two ways:

  • Firstly, this equation tells us that " 9 is {\color{blue}{2}} larger than {\color{red}{7}}. " This is because we need to add an additional {\color{blue}{2}} to {\color{red}{7}} to make 9.

  • Secondly, this equation tells us that " 9 is {\color{red}{7}} larger than {\color{blue}{2}}. " This is because we need to add an additional {\color{red}{7}} to {\color{blue}{2}} to make 9.

Notice that the word "times" is missing from our comparison statements. As a general rule, the word "times" is only found in multiplicative comparison statements.

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Example: Distinguishing Between Multiplicative and Additive Comparison

Which of the following statements is represented by the equation 7 \times 2 = 14?

  1. The number 14 is 2 larger than 7.
  2. The number 14 is 2 times larger than 7.
  3. The number 7 is 2 times larger than 14.
EXPLANATION

The given equation compares the size of the product (14) to its factors (2 and 7). It can be interpreted as a comparison in the following two ways:

  • "The number 14 is 7 times larger than 2. "

  • "The number 14 is 2 times larger than 7. "

Therefore, from the given options, the correct answer is "The number 14 is 2 times larger than 7. "

Watch Out! The word "times" is important. The following answer is not correct.

\qquad The number 14 is 2 larger than 7.

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Practice: Distinguishing Between Multiplicative and Additive Comparison

Which of the following statements is represented by the equation $3 \times 12 = 36?$

a
 The number $36$ is $12$ larger than $3.$
b
 The number $3$ is $12$ times larger than $36.$
c
 The number $36$ is $12$ times larger than $3.$
d
 The number $12$ is $36$ larger than $3.$
e
 The number $12$ is $36$ times larger than $3.$
Practice: Distinguishing Between Multiplicative and Additive Comparison

Which of the following statements is represented by the equation $4 \times 16 = 64?$

a
 The number $16$ is $4$ times larger than $64$
b
 The number $16$ is $4$ larger than $64$
c
 The number $64$ is $4$ larger than $16$
d
 The number $4$ is $16$ times larger than $64$
e
 The number $64$ is $4$ times larger than $16$
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