Lesson
11-4A
Multiplication as Scaling
How can multiplying by a fraction
change the second factor?
Common
Core
Sue knitted four scarves 4 feet long for her and her
friends Joe, Alan, and June. After a month, they
compared the lengths of their scarves. Some scarves
had stretched and some had shrunk. The results are
shown in the chart.
5.NF.5.b Explaining why
multiplying a given number
by a fraction greater than 1
results in a product greater
than the given number. . .
explaining why multiplying
a given number by a
fraction less than 1 results
in a product smaller than the
given number; and relating
the principle of fraction
(n 3 a)
 ​
  
​ to
 
equivalence _ ​a ​5 _____
b (n 3 b)
the effect of multiplying _ ​a ​
b
by 1. Also 5.NF.5, 5.NF.5.a
Sue
4
Joe
1 ​3 4
1​ __
2
Alan
3 ​3 4
 ​__
June
3 ​3 4
 ​__
4
3
How had the lengths of the scarves changed?
Think of multiplication as scaling or resizing.
Guided Practice
MATHEMATICAL
PRACTICES
Do you know HOW?
Do you UNDERSTAND?
Without multiplying, decide which symbol
belongs in the box: , , or 5.
4. Reason Why does multiplying a
1 ​increase its value?
number by 3​ __
2
1 ​3 2​ __
2 ​ 2​ __
2 ​
1. 3​ __
2
3 ❚ 3
2 ​ 2​ __
4 ​3 2​ __
2. ​ __
❚ 2 ​
5
3
3
5. Reason Does the scaling factor always
have to be the first factor?
3 ​3 __
3 ​
​ 4 ​ 4​ __
3. 4​ __
5 4 ❚ 5
Independent Practice
In 6–11, without multiplying, decide which symbol belongs in the box: , , or 5.
1 ​3 1​ __
2 ​ 1​ __
2 ​
6. 2​ __
2
3 ❚ 3
4 ​ 4​ __
4 ​
​ 35 ​3 4​ __
7. __
5 ❚ 5
2 ​3 __
2 ​
8. 1​ __
​ 5 ​ 1​ __
7 5 ❚ 7
2 ​ 2​ __
2 ​
9. __
​ 13 ​3 2​ __
5 ❚ 5
3 ​3 __
3 ​
10. 3​ __
​ 2 ​ 3​ __
5 2 ❚ 5
1 ​3 2​ __
2 ​ 2​ __
2 ​
11. 4​ __
7 ❚ 7
3
12. Without multiplying, order the following products from least to greatest.
5 ​3 __
1 ​3 __
​ 3 ​ __
​ 34 ​3 __
​ 35 ​ ​ __
​ 3 ​
2 × __
​ 35 ​ 2​ __
4 5
5 5
40
Step 2
Step 1
Draw a picture.
Joe’s scarf stretched.
Perform the multiplication
indicated in the chart
and use a number line
to compare the new
lengths of the scarves.
0
1
2
3
4
5
Step 3
June’s scarf stayed the
same length.
1 ​3 4  4
1​ __
2
__
​ 3 ​3 4 5 4
3
Multiplying a number
by a fraction greater
than 1 results in a
product greater than
the starting number.
Multiplying by a fraction
equal to 1 results in a product
equal to the starting number.
When you multiply by a
fraction equal to 1, you
are finding an equivalent
fraction:
Alan’s scarf shrunk.
6
3 ​3 4  4
​ __
4
Sue’s scarf
Joe’s scarf
Alan’s scarf
Multiplying a number by a fraction
less than 1 results in a product less
than the starting number.
June’s scarf
(4 3 3)
(1 3 3)
4 5 __
​ 41 ​5 __
​ 41 ​3 __
​ 33 ​5 ______
​ 
  
​5 ___
​ 12
   ​
3
MATHEMATICAL
Problem Solving
PRACTICES
13. At a taffy pull, George stretched the taffy to 3 feet. Jose stretched
1 ​times as far as George. Maria stretched it __
​ 23 ​as far. Sally
it 1​ __
3
6 ​as far. Who stretched it the farthest? the least?
stretched it ​ __
6
14. Use Tools Who ran the farthest by the end of the week? Use the
table below that shows the distances in miles.
Holly
Yu
Monday
Tuesday
Wednesday
Thursday
Friday
1
2
3
__
1​   ​
4
1
2
1
1​ __ ​
2
1
4
3
__
2​   ​
4
3
4
1
1​ __ ​
4
1​ __ ​
1​ __ ​
 ​__ ​
2​ __ ​
1
2
1
__
​   ​
2
 ​__ ​
15. Persevere Joni and her friends are stretching rubber bands for an activity
in science class. Joni stretched her rubber band to 24 inches. Manuel stretched
1 ​times as far. Nicole stretched it __
​ 33 ​as far. Molly stretched it __
​ 35 ​as far. Put the
it 2​ __
2
students in order of how far they stretched their rubber bands from least to greatest.
16. Look for Patterns Make up two decimals with an answer close to the given product.
___.___ 3 ___. ___ 5 6.3
17. Reasonableness Put the following products in order from
greatest to least, without multiplying.
​ 47 ​
3 3 __
3 ​3 __
__
​ 1 ​3 __
​ 4 ​ 1​ __
​ 4 ​ __
​ 4 ​3 __
​ 4 ​
2
7
4
7
4
7
Lesson 11-4A
41