Activities

1. ½ ÷ 1
   
   
2. 1 ÷ 1/3
   
   
3. Write the equation for the following:    1/4 goes into 1 four times
   
   
4. a/b ÷ 1 = ?
   
   
5. List the steps for dividing two fractions.

6. Solve 1/3 ÷ 8.
   
   
7. Solve 12/6 ÷ ½ .
   
   
8. What is the solution to 1/12 ÷ 9?
   1. ¼
   2. 1/108
   3. ¾
   4. 2/3
      
      
9. 18/3 ÷ 1/2 = ?
   
   
10. Write the solution to 25 ÷ 1/3.
    
    
11. List three solutions to ¼ ÷ 5. 
    
    
12. Is the solution to 18 ÷ 1/9 greater than one or less than one?   Show work to justify your answer.
    
    
13. Is the solution to 1/9 ÷ 18 greater than one or less than one?  Show work to justify your answer.
    
    
14. If a, b, and c are whole numbers, is the solution a whole number or a fraction?  4ab ÷ 1/2c
    
    
15. True or false, 24 ÷ 1/6 =  96?  Show all calculations to justify your answer.

16. Compare the solution to 14 ÷ 1/5 and 1/5 ÷ 14.  List one difference that does not include the actual answer to both expressions.  Explain your reasoning.

17. Without solving the expressions below, explain how you know the solution to one of the expressions is much larger than the solution to the other expression.    18/3 ÷ ½       ½ ÷  18/3

18. Draw a model to show 3 ÷ 1/4.  Explain the connection of the model to the calculations for the solution to the expression.

19. Draw a model to show 1/3 ÷ 4.  Explain the connection of the model to the calculations for the solution to the expression.

20. Flint and Dorsey have created a model for the expression 1/3 ÷ 4.  Who is right?  Explain why?
    
    Flint                                                                                               Dorsey
    
    

Flint’s answer                                                                         Dorsey’s answer

(1 of the 4 shaded blocks represents the answer)             (4 shaded triangles represent the answer)

21. Write an expression for the following situation, solve it and explain your reasoning.  There are 12 pans of brownies.  A serving is 1/9 of a pan.  How many people will receive a brownie?
    
    
22. One large pan of brownies has been cut into 9 brownies.  Clara has 5 friends coming over and she wants everyone to have the same amount of the brownie pan.  What fraction of the large brownie will each person get?  How many brownies will each person receive? The answer to both of these questions represents the same amount of brownie.  Explain why.
    
    
23. John and Sophie wrote their answer to the following problem.  Who is correct and why? 
    A large pizza is cut into 12 pieces and there are 9 pizzas.  A serving is 1/6 of a pizza.  How many people can be served?  Explain your answer.
    
    John:  1 ÷1/6 = 6.   6 x 9 = 54    54 people can be served. Since a serving is 1/6 of a pizza, then 6 people can be served per pizza and there are 9 pizzas so 6 times 9 is 54.
    
    Sophie:  12÷ 1/6 = 72      72 x 9 = 648   So 648 people can be served.  A pizza has 12 slices and a serving is 1/6 of a pizza, then 12 ÷ 1/6 = 72.  Since there are 9 pizzas and 72 servings per pizza, nine times seventy two is 648.
    
    
24. Write an expression for the following situation, solve it, and explain your reasoning.  There is 1/3 of a cake left on the counter.  How much of the cake will 9 people receive?
    
    
25. Expression #1:  2 ÷ 1/b             Expression #2:  1/d ÷ 47
    Is it possible for the solution to expression one to be larger than expression two?  Explain.        

Answer Key/Rubric

1. 1/2
   
   
2. 3
   
   
3. 1 ÷ ¼ = 4
   
   
4. a/b
   
   
5. Students must somehow indicate that the second fraction is inverted and then numerators are multiplied, denominators are multiplied, and the solutions are placed in their respective locations within the solution fraction.

6. 1/24
   
   
7. 4
   
   
8. b
   
   
9. 12
   
   
10. 75
    
    
11. 1/20, 2/40, 3/60
    
    
12. Greater than one (answer is 162).  There are multiple ways for students to show their calculations – one could be to invert the second fraction and multiplying across.
    
    
13. Less than one (answer is 1/162).  There are multiple ways for students to show their calculations.
    
    
14. Whole number because when you invert the second fraction and multiply, you get 8abc for the answer.
    
    
15. False.  There are multiple ways for students to show their calculations, but the answer should be 144.

16. The solution to the first expression is a whole number whereas the solution to the second expression is a fraction.  Explanations may vary.  An example:  The first expression is asking how many 1/5’s are in 14 and for every whole there are five 1/5’s so 5 times 14 is a whole number.  The second expression is asking how many 14’s are there in 1/5 and 14 does not go into 1/5 so the answer is a fraction.
    
    
17. In the first expression 18/3’s is equivalent to 6.  The solution to the first expression is a whole number whereas the solution to the second expression is a fraction.  Explanations may vary.  An example:  The first expression is asking how many 1/2’s are in 6 and for every whole there are two 1/2’s so 2 times 6 is a whole number.  The second expression is asking how many 6’s are there in 1/2 and 6 does not go into 1/2 so the answer is a fraction.
    
    
18. 
    

Answers will vary.  An Example:   draw a model that shows 3 wholes divided into fourths.  Counting the number of fourths in 3 gives a solutions of 12 because the expression is asking how many fourths are in three.

19. Answers will vary.  An example:   draw a model that shows one whole which has been divided into thirds (1/3 is shaded) and then the same 1/3 is divided into four. One of the fourths of a 1/3 is equal to 1/12 of the whole.

20. Answers will vary.  Dorsey and Flint both have the correct model.  Flint interprets his correctly and Dorsey interprets hers incorrectly. The whole has been divided into thirds (1/3 is shaded) and then the same third is divided into 4 equal parts. One of the four parts represents the answer because it is being divided by four (Flint’s answer).  Dorsey follows the same procedure except counts all four parts which would represent 1/3 of the whole.
    
    
21. 12 ÷ 1/9.  108 people will receive a brownie. Answers will vary.  If a serving is 1/9 of a pan then there are 9 brownies in each pan (9/9 is equal to one whole).  If there are 9 brownie servings in one pan and there are 12 pans, 12 x 9 = 108.
    
    
22. 1/6 (6 people total)   1 1/2. Answers will vary.  The first question refers to the pan as being the whole.  The second question refers to each brownie as being the whole.
    
    
23. John is correct, 54 people can be served.   Since a serving is 1/6 of a pizza, then 6 people can be served per pizza regardless of how the pizza is actually cut.  The 12 slices are not relevant to the problem because we are not asked for the serving size, which is 2 slices. If six people are served with one pizza then multiply that by the number of pizzas.
    
    
24. 1/3 ÷ 9.  Each person will get 1/27 of the cake.  Answers will vary.  If you have a third and divide that third into 9 parts and then do this to the other thirds even though they are gone, one serving is one piece out of 27.
    
    
25. Yes it is possible.  Answers will vary.  Not only is it possible but it must be larger.  The solution to the first expression is a whole number whereas the solution to the second expression is a fraction.  The first expression is asking how many 1/b’s are in 2 and for every whole there are b 1/b’s so b times 2 is a whole number.  The second expression is asking how many 47’s are there in 1/b and 47 does not go into 1/b (a fraction less than one) so the answer is a fraction