4th Grade · 4.OA.A.1 · 4.OA.A.2

Multiplicative comparison in 4th grade

Multiplicative comparison means comparing two amounts by saying how many times as big one is as the other, such as "the dog weighs 7 times as much as the cat". In 4th grade, your child learns to read multiplication this way and to solve word problems that ask "how many times as many". In earlier grades, comparing always meant adding or subtracting.

What the standards say

The official Common Core wording, quoted exactly. The plain version follows.

4.OA.A.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

What it means, in plain words

Reading multiplication as comparison 4.OA.A.1

Until now, your child has read $4 \times 5$ as "4 groups of 5". In 4th grade, the same fact also describes a comparison: $20 = 4 \times 5$ says that 20 is 4 times as many as 5. This is called multiplication as comparison.

Your child also turns a comparison sentence into an equation. "A tree is 5 times as tall as a 3-foot bush" means the tree is $5 \times 3 = 15$ feet tall, so the equation is $15 = 5 \times 3$ . A bar model makes it visible: one short bar for the bush and a long bar made of 5 copies of it for the tree. The clear wording is "times as many" or "times as much". "Times more than" can be read two ways, so it is best avoided.

Your child should be able to:

Read an equation such as $20 = 4 \times 5$ as "20 is 4 times as many as 5"
Say the same comparison the other way: 20 is also 5 times as many as 4
Write a comparison sentence as a multiplication equation
Draw a bar model where one bar is made of copies of the other

Example

What two comparisons does $30 = 6 \times 5$ describe?

30 is 6 times as many as 5.
30 is also 5 times as many as 6.
Both sentences are true, because 6 groups of 5 and 5 groups of 6 both make 30.

Multiplicative comparison word problems 4.OA.A.2

Multiplicative comparison word problems ask for one of three things: the larger amount, the smaller amount or how many times as big one amount is. To find the larger amount, your child multiplies. To find the smaller amount or the number of times, your child divides. The unknown number is written as a letter, such as $n$ .

The hard part is telling two kinds of comparison apart. "How many more?" is answered by subtracting: 20 is 15 more than 5. "How many times as many?" is answered by dividing: 20 is 4 times as many as 5. Multiplicative comparison in 4th grade is the first time a comparison question needs multiplication or division, so your child learns to decide which kind of question it is before using any numbers.

Your child should be able to:

Tell "how many more" questions apart from "how many times as many" questions
Multiply to find the larger amount
Divide to find the smaller amount or how many times as many
Write an equation with a letter for the unknown number

Example

A dog weighs 63 pounds and a cat weighs 9 pounds. How many times as heavy as the cat is the dog?

The question asks "how many times as heavy", so it is a multiplication comparison.
Write it with a letter for the unknown: $63 = n \times 9$ .
$63 \div 9 = 7$ , so the dog is 7 times as heavy as the cat.

Example

A bookcase holds 48 books. That is 6 times as many books as a small shelf holds. How many books does the small shelf hold?

The bookcase holds more, so 48 is the larger amount: $48 = 6 \times s$ .
$48 \div 6 = 8$
The small shelf holds 8 books.

Where children usually get stuck

For "a pumpkin weighs 6 times as much as a 5-pound melon", your child writes $5 + 6 = 11$ pounds.
For three years, comparing has meant adding or subtracting. Nothing in the wording warns the child that this question works differently.
What helps. Draw a short bar for the melon and a long bar made of 6 copies of it for the pumpkin. Ask how much each copy weighs. The pumpkin weighs $6 \times 5 = 30$ pounds.
For "a jacket costs 72 dollars, which is 8 times as much as a hat", your child multiplies $72 \times 8$ .
The child sees "times" and multiplies the two numbers, without checking which amount is the bigger one.
What helps. Ask first which item costs more. The jacket does, and its price is given, so the hat must cost less. Dividing gives $72 \div 8 = 9$ dollars.
Asked "how many more" and then "how many times as many" about 12 apples and 3 apples, your child answers 9 both times.
Both questions sound like comparing, and the child has only one tool for comparing.
What helps. Lay out 12 and 3 small objects. For "how many more", line them up and count the extras (9). For "how many times as many", put the 12 into groups of 3 and count the groups (4).

Try four practice questions

Real 4th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Lila wrote a "times as many as" sentence about the multiplication equation below.
$\displaystyle 3 \times 7 = 21$
Lila's sentence: "3 is 7 times as many as 21."
Select the statement that best describes Lila's sentence.
A
Lila's sentence puts 21 in the multiplier spot (right after "is"), but 21 should be the total at the start.
B
Lila's sentence puts 21 at the end, with the factors flipped from the equation's order.
C
Lila's sentence puts 21 at the end, in the same order as the equation.
D
Lila's sentence correctly matches the equation.
Show the answer and how to solve it
The matching description is the one that says Lila's sentence puts 21 at the end, in the same order as the equation. A correct sentence would be "21 is 3 times as many as 7" or "21 is 7 times as many as 3."
1. 1Look at the equation. The product (the total) is 21. The factors are 3 and 7.
2. 2A correct "times as many as" sentence puts the product (21) first as the total. For $3 \times 7 = 21$ , two correct sentences are: "21 is 3 times as many as 7" and "21 is 7 times as many as 3."
3. 3Now look at Lila's sentence: "3 is 7 times as many as 21." Find where 21 (the product) is placed. It is at the end — not at the start where it belongs.
4. 4If Lila's sentence were true, it would mean $3 = 7 \times 21 = 147$ . But the equation says $3 \times 7 = 21$ . So Lila's sentence does not match the equation. Notice also that the factors 3 and 7 appear in the same left-to-right order as in the equation.
5. 5The matching description is the one that says Lila's sentence puts 21 at the end, in the same order as the equation.
Common mistake. A common mistake is reading the equation left-to-right and writing the sentence in that order — but "times as many as" requires the product (total) at the START, not the end.
The idea underneath. In a "times as many as" sentence, the product (the total) always comes first. For $3 \times 7 = 21$ , the sentence must start with 21 — like "21 is 3 times as many as 7" or "21 is 7 times as many as 3."
2On level
Aisha collected 7 stickers. Marcus collected 28 stickers. Marcus collected how many times as many stickers as Aisha?
Show the answer and how to solve it
Answer. 4
The answer is $4$ . Marcus has $28$ stickers, which is $4$ times as many as Aisha's $7$ stickers, because $28 \div 7 = 4$ .
1. 1Identify the two amounts. Aisha has $7$ stickers (the smaller amount), and Marcus has $28$ stickers (the larger amount).
2. 2To find how many times as many, divide the larger amount by the smaller amount.
3. 3Marcus has $4$ times as many stickers as Aisha, so enter $4$ as the answer.
Common mistake. A common mistake is confusing "times as many" (multiplication and division) with "more than" (addition and subtraction). A student might enter $21$ (the difference $28 - 7 = 21$ ) instead of $4$ . "Times as many" always means dividing the larger amount by the smaller amount.
The idea underneath. 'Times as many' is a multiplicative comparison: the larger amount equals the smaller amount multiplied by the factor. Divide the larger amount by the smaller amount to find the factor.
3On level
Alex is asked to write a multiplication equation that uses $k$ to represent the sentence ' $24$ is $4$ times as many as $k$ .'
$\displaystyle k = 4 \times 24$
Select the statement that best describes Alex's equation.
A
Alex's equation does not represent the sentence; the wrong operation is used.
B
Alex's equation does not represent the sentence; $k$ is in the wrong role.
C
Alex's equation correctly represents the sentence.
D
Alex's equation does not represent the sentence; a different letter from $k$ is used.
Show the answer and how to solve it
Alex's equation puts $k$ alone on one side, making $k$ the result of $4 \times 24$ . But the sentence says $24$ is the product and $k$ is a factor, so the equation does not represent the sentence — $k$ is in the wrong role.
1. 1First, figure out what the sentence means as an equation. ' $24$ is $4$ times as many as $k$ ' means $24$ equals $4$ times $k$ .
2. 2Now look at Alex's equation. It says $k$ equals $4$ times $24$ , which would make $k = 96$ .
3. 3Compare the two. The sentence has $24$ alone on one side as the product, with $k$ as a factor. Alex's equation has $k$ alone on one side as the product, with $24$ as a factor. The roles of $k$ and $24$ are switched.
4. 4Alex's equation does not match the sentence. The correct answer is the choice that says $k$ is in the wrong role.
Common mistake. A common mistake is confusing which value is the product in a 'times as many' sentence. In ' $24$ is $4$ times as many as $k$ ,' $24$ is the product (the bigger total) and $k$ is a factor — so the equation should put $24$ alone on one side and $4 \times k$ on the other.
The idea underneath. In ' $24$ is $4$ times as many as $k$ ,' the number $24$ is the product and $k$ is a factor that gets multiplied by $4$ . The equation puts the product on one side and the multiplication on the other — so $24$ goes with $4 \times k$ , never with $k$ alone.
4Stretch
Diego has 24 marbles. Lila has 8 marbles. Diego has $m$ times as many marbles as Lila. Which equation represents the story?
A
$\displaystyle m \times 8 = {}$
$\displaystyle 24$
B
$\displaystyle m + {}$
$\displaystyle 8 = {}$
$\displaystyle 24$
C
$\displaystyle m \times 24 = {}$
$\displaystyle 8$
D
$\displaystyle m = {}$
$\displaystyle 3$
Show the answer and how to solve it
The equation $m \times 8 = 24$ represents the story. It says: $m$ times Lila's count of marbles ( $8$ ) equals Diego's count, $24$ .
1. 1The story compares Diego's and Lila's marble counts using the phrase 'times as many.' This is a multiplicative comparison, so the equation must use multiplication, not addition. The unknown $m$ is the multiplier — how many times greater Diego's count is than Lila's.
2. 2Diego has $24$ marbles, Lila has $8$ marbles, and Diego has $m$ times as many marbles as Lila. So $m$ multiplied by Lila's count, $8$ , equals Diego's count, $24$ .
3. 3The equation $m \times 8 = 24$ matches the story. (The question only asks which equation represents the comparison — you do not need to solve for $m$ .)
Common mistake. Confusing 'times as many' (multiplication) with 'more than' (addition). Writing $m + 8 = 24$ would mean Diego has $8$ more marbles than Lila, not that Diego's count is $m$ times Lila's count of $8$ .
The idea underneath. In a 'times as many' story, the larger amount equals the multiplier times the smaller amount. The equation always sets up this multiplicative relationship, no matter which quantity is unknown.