Multiplication as Scaling

Key Concepts

• After this lesson, students will be able to:
• Use multiplication to scale or resize something.
• Interpret multiplication as scaling (resizing), by comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
• Multiply by a fraction equivalent to 1 ,the value is unchanged.
• Multiply by a fraction less than 1, the value becomes smaller.
• Multiply by a fraction greater than 1, the value becomes bigger.

What is the Multiplication a scaling? 

1. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. 

2. Explaining why multiplying a given number by  

A fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); 

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 equivalence a/b = nXa/nXb to the effect of multiplying a/b by 1. 

Scaling

You might think that multiplying makes a number get bigger. But sometimes, multiplying can make a number get smaller, or even stay the same! This process is called scaling. 

Scaling with whole numbers  

Lets try with the number 10. Look at each example below 

First factor X second factor= Product 

10 X 1/2 =  5             The product is less than 10. 

10 X  1  = 10              The product equals 10. 

10 X 1 1/2 = 15          The product is greater than 10 

Go back and look at the second factor in each problem. Is that factor greater than, less than, or equal to 1? 

10 X 1/2 = 5                    

1/2 is less than 1 

10 X 1 = 10                 1 equals 1 

10 X 1 1/2 =15            1 1/2 is greater than 1 

You can use this pattern to predict whether a product will be  greater than, less than,  or equal to the first factor. 

• If you multiply a factor by a number less than 1, the product will be less than that first factor.  
• If you multiply a factor by a number equal to 1,  the product will be equal to that first factor.  
• If you multiply a factor by a number greater than 1,  the product will be greater than that first factor. 

Example 1:  

Solve & Share  

Without multiplying, circle the problem in each set with the greatest product and underline the problem with the least product. Solve this problem any way you choose.? 

Solution: 

When we multiply a fraction equivalent to 1, the value is unchanged . 

When we multiply by a fraction less than 1, the value becomes smaller. 

When we multiply by a fraction greater than 1, the value becomes bigger. 

Example 2:

Sue knitted scarves that are 4 feet long for herself and her friends Joe and Alan. 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. How had the lengths of Joe’s and Alan’s scarves changed? 

Alan’s scarf shrank .  

3/4 x 4 < 4                                             

3/4 is proper fraction .Proper fraction always less than 1. 

When we multiplying a number by a fraction less than 1 , 

the value becomes Smaller. 

Joe’s scarf stretched . 

1 1/2 x 4 < 4 

1 1/2 = 3/2 is improper fraction . Improper fraction is always greater than 1 

Multiplying a number by a fraction greater than 1 results in a product greater than the starting number. 

i.e 

1. Why does multiplying a number by 3 ½ increase its value? 

Solution : 

Whenever you multiply a number greater than 1, you will get an increase in value. 

3 1/2 >1 

3 1/2 X 2 = 7/2 x 2 = 7 

i.e  7 > 3 1/2

2. Which of the following are less than 8? 

1. 8 × 9/10 
2. 8 × 7/6
3. 8 × 3/5 

Solution:     

1. 8 × 9/10                  9/10 <1            8 × 9/10 < 8 

2. 8 × 7/6                   7/6 >1              8 × 7/6 >8 

3. 8 × 3/5                    3/5 <1              8 × 3/5 < 8 

∴ 8 × 9/10 and 8 × 3/5 are less than 8 . 

3. Without multiplying decide which symbol is used in the box: 

1. 3 1/2 × 2 3/4 £ 3 3/4 

2. 2 1/2 × 4 1/4 £ 2 3/4 

3. 5 1/2 × 3 3/4 £ 4 1/2 

Solution : 

Find the product of whole numbers when comparing all mixed fractions. 

1. 3 x 2 = 6        So 3 1/2 × 2 3/4  > 3 3/4 

2. 2 x 4 = 8              So 2 1/2 × 4 1/4  >  2 3/4 

3. 5 x 3 = 15          So 5 1/2 × 3 3/4  > 4 1/2 

4. Without multiplying order the following from least to greatest: 

1. 2 × 3/5 
2. 2 3/4 × 3 1/2  
3. 4 3/4× 2 1/4  
4. 3/5 × 2/5
5. 5/5 × 2 1/2  

Solution: 

1. 2 × 3/5            3/5 <1        2 × 3/5 < 1  

2. 2 x 3 = 6               2 3/4 × 3 1/2 is approximately 6 >1 

3. 4 X 2 = 8               4 3/4 × 2 1/4    is approximately 8>1` 

4. 3/5 × 2/5              2/5 < 1       3/5 × 2/5 < 1 

5. 5/5 × 2 1/2           5/5 =1        5/5 × 2 1/2  = 2 1/2 >1 

So,   

4. 3/5 × 2/5

1. 2 × 3/5    

5. 5/5 × 2 1/2       

2. 2 3/4 × 3 1/2      

3. 4 3/4 × 2 1/4

5. At a taffy pull, George stretched the taffy to 3 feet. Jose stretched it 1 1/3 if times as far as George. Maria stretched it 2/3 as far as George. Sally stretched it 6/6 as far. Who stretched it the farthest? the least? 

Solution: 

George stretched the taffy = 3 feet 

Jose stretched the taffy = 1 1/3 of 3 = 1

1/3  x 3 > 3   ( ∵ 1 1/3 is greater than one )                                

Maria stretched the taffy = 2/3 of 3 = 2/3 x  3 < 3  (∵2/3 is less than one ) 

Sally stretched the taffy = 6/6 of 3 =

6/6 x 3 = 3       ( ∵3/3 is equal to 1 ) 

So,                   

Jose stretched the taffy farthest Maria stretched the taffy the least 

6. Who ran farther by the end of the week? How much farther? Use the table below that shows the distances in miles.? 

Total distance covered by Holly = 1 1/2 + 1/2 + 2 1/4 + 3/4 + 1 1/2

= 1 1/2 + 1/2 + 1 1/2 + 2 1/4 + 3/4

= 3 1/2 +3 

∴Total distance covered by Holly = 6 1/2 miles. 

Total distance covered by Yu = 1 3/4 +1 1/2 + 2 3/4 + 1 1/4 + 1/2

= 1 3/4 + 2 3/4 + 1 1/4 + 1 1/2 + 1/2

= 5 3/4 +2 

∴Total distance covered by Yu = 7 3/4 miles. 

Yu ran farther by the end of the week.  

Exercise

Decide which symbol belongs in the box : <,>, or =.

• 7 7/4
• 8 2/3
• 8

Without multiplying, decide which symbol belongs in the box: <,>, or =.

• 7 1/3 x 2 1/4 7 1/3
• 11 3/4 x 2/2 11 3/4

Without multiplying, order the following products from least to greatest.

• 5/7 x 2 7/9
• 5/7 x 3/4
• 5/7 x 11 1/10
• 5/7 x 5/5

Larry is making fruit salad. For each bowl of fruit salad, she needs 2323 cup of
strawberries. How many cups of strawberries will she use if she makes 21 bowls of
fruit salad?

Concept Map

What have we learned

• Use multiplication to scale or resize something.
• Interpret multiplication as scaling (resizing), by comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
• Multiply by a fraction equivalent to 1, the value is unchanged.
• Multiply by a fraction less than 1, the value becomes smaller.
• Multiply by a fraction greater than 1, the value becomes bigger.