Activities

1. True or false?  When multiplying 6 by a whole number, the solution is larger than 6.
   
   
2. True or false?  When multiplying 7 by a fraction, the solution is smaller than 7.
   
   
3. Which is larger, 4 x ½ or 4 x 2?
   
   
4. Which is smaller, 3 x ¼ or 3 x 4?
   
   
5. If Karen wants to make her number larger, does she multiply by a whole number or a fraction?

6. In the expression 6 x ¼, the solution is _______ times the size of 1/4.
   
   
7. In the expression 2/3 x 4/5, the solution is  _______ times the size of 2/3  
   
   
8. Cassie says that when you multiply a number by a/b, the product must be a/b smaller.  Is she correct?
   
   
9. When you multiply a whole number by the fraction b/b, the product is not smaller than either of the numbers being multiplied.  True or false?
   
   
10. In the expression b/c x 8, c is larger than b.   Is the product greater or less than 8?
    
    
11. In the expression c/b x 9, c is larger than b.  Is the product greater or less than 9?
    
    
12. In the expression c/b x 24, c is larger than b.   Circle any correct statement below.
    
    
    1. The product is greater than 24.
    2. The product is less than 24.
    3. The product is c/b times greater than 24.
    4. The product is equal to 24.
       
       
13. Randy has made a sculpture of a dog.  He would like another identical dog sculpture that is double the size.  Should he multiply its dimensions by ½ or 2?
    
    
14. The dimensions of a painting are 4 by 2 ½.  A copy was made with the dimensions 1 by 5/8.  The new painting dimensions are what fraction of the original?
    
    
15. Use the expression 3/7 x 24 to determine which statements below are correct.
    
    
    1. The product is greater than 24.
    2. The product is less than 24.
    3. Three sevenths of the product is 24.
    4. Three sevenths of 24 is the product.

16. Karmen says that when you multiply a number by a/b, the product must be a/b times smaller.  Is he correct? If yes, explain your reasoning.  If not, provide a counterexample.
    
    
17. In the expression c/b x 12, c is larger than b.   Explain how you know the product is greater than
     either factors.
    
    
18. Randy has just finished a painting.  He would like another identical painting that is a/b of the size
     of the original painting.  List the steps Randy needs perform to figure out the size of the new  
     painting. Explain why your method will work.
    
    
19. Lucas multiplies the dimensions of his original painting by a/b and gets larger dimension for his second painting.  Susan multiplies the dimension of her original painting by c/d and gets smaller dimensions for her new painting.  Explain what this tells us about a/b and c/d and what it tells us about the new dimensions with respect to the originals.
    
    
20. When a number is multiplied by a fraction greater than one, the product is greater than both factors.  Explain why this is an accurate statement.
    
    
21. When a number is multiplied by a fraction less than one, the product is less than both the factors.  Explain why this is an accurate statement.
    
    
22. In the expression c/b x 17, c is larger than b.   Choose any correct statement below and justify your reasoning.
    
    
    1. The product is greater than 17.
    2. The product is less than 17.
    3. The product is c/b times greater than 17.
    4. The product is equal to 17.
       
       
23. 14 x a/5 is the expression Bailey is asked to solve and then draw a conclusion.  Her work is below.
    
      a/5 + a/5 + a/5 + a/5 + a/5 + a/5 + a/5 + a/5 + a/5 + a/5 + a/5 + a/5 + a/5 + a/5 = 14a/5  

The conclusion is that when a whole number is multiplied by a fraction, the answer is always greater than either original numbers.

Is Bailey’s work correct?  Is her conclusion correct?  Why or why not?

24. Given the expression 12 x b/9, what values of b will result in a product that is greater than both factors?  Explain why.
    
    
25. Given the expression c/5 x 39, what values of c will result in a product that is less than both factors?  Explain why.

Answer Key/Rubric

1. True
   
   
2. True
   
   
3. 4 x 2
   
   
4. 3 x ¼
   
   
5. Whole number

6. 6
   
   
7. 4/5
   
   
8. Yes
   
   
9. True, multiplying by b/b is the same as multiplying by 1
   
   
10. Less than 8; you are multiplying by a fraction or a number less than 1
    
    
11. Greater than 9; you are multiplying by an improper fraction, or a number greater than 1
    
    
12. a, c
    
    
13. 2
    
    
14. ¼
    
    
15. b, d

16. Incorrect.  12 x 6/3 = 24 and 24 is larger than 12.  Answers will vary.  The only way it will end up smaller is if a<b, but as soon as a is greater than or equal to b, the statement is incorrect.
    
    
17. Answers will vary.  If c is greater than b that means that the fraction is larger than 1.  When multiplying a number by another number that is greater than one the solutions is larger than both numbers.
    
    
18. Answers will vary.  Steps should contain something similar to :  1.  Measure the length and width of the original painting.  2.  Multiply each dimension by the fraction a/b to get the new dimensions. Students must somehow explain multiplying by the fraction is scaling up the dimensions if the fraction is greater than one and scaling down if the fraction is less than one.
    
    
19. Students must indicate that a>b and therefore the fraction is greater than one.  When you multiply any number by a fraction greater than one, the product will be greater than both factors. Lucas will have a larger painting.   And c
20. Answers will vary. When multiplying a number by 1, the product is that number (multiplication identity).  Therefore when multiplying a number by a fraction greater than one, the result must be greater than the original number. 
    
    
21. Answers will vary. When multiplying a number by 1, the product is that number (multiplication identity).  Therefore when multiplying a number by a fraction less than one, the result must be less than the original number.
    
    
22. Students can choose a or c.  Answers will vary.  [Student chooses a]  If c is greater than b that means that the fraction is larger than 1.  When multiplying a number by another number that is greater than one the product is larger than both factors.  [Student chooses c]  Since 17 is being multiplied by c/b, which is greater than one, the product is c/b 17 times, by definition of multiplication.
    
    
23. Her work is correct.  Her conclusion is incorrect.  Answers will vary for the explanation.  If a>5 her conclusion will hold true, but when a<5 (example:  3/5x14 = 8 2/5) the conclusion is false.
    
    
24. b >9.  Explanations will vary.  When multiplying a number by 1, the product is that number (multiplication identity).  Therefore when multiplying a number by a fraction greater than one, the result must be greater than the original number.
    
    
25. c <5.  Explanations will vary.  When multiplying a number by 1, the product is that number (multiplication identity).  Therefore when multiplying a number by a fraction less than one, the result must be less than the original number.