Bake Sale

Task

1. Lindy is having a bake sale. She has 48 chocolate chip cookies to put in bags. How many bags can she fill if she puts the same number in each bag and uses them all? Find all the possibilities. Explain your reasoning.
2. Lindy has 64 vanilla wafer cookies to put in bags. How many bags can she fill if she puts the same number in each bag and uses them all? Find all the possibilities. Explain your reasoning.
3. How many bags can Lindy fill if she puts the chocolate chip cookies and the vanilla wafers in the same bags? She plans to use all the cookies and wants to include an equal number of chocolate chip cookies and an equal number of vanilla wafers in each bag. Explain your reasoning.
4. What is the largest number of bags she can make with an equal number of chocolate chip cookies and an equal number of vanilla wafers in each bag (assuming she uses them all)? Explain your reasoning.

IM Commentary

This problem uses the same numbers and asks essentially the same mathematical questions as "6.NS Factors and Common Factors," but requires students to apply the concepts of factors and common factors in a context. A version of this task could be adapted into a teaching task to help motivate the need for the concept of a common factor.

Solution

1. 1, 2, 3, 6, 8, 12, 16, 24, 48.
   The number of bags she can make are shown in the table:

   Number of CC cookies in each bag	1	2	3	4	6	8	12	16	24	48
   Number of bags	48	24	16	12	8	6	4	3	2	1

   Since the number of cookies that goes in a bag is a factor of the total number of cookies, all the factors of 48 are listed in the first row of the table. Each column corresponds to a situation where the 48 cookies are divided equally among some bags, and the product of the numbers in each column is 48. The number of bags is also a factor of the total number of cookies.
2. 1, 2, 4, 8, 16, 32, 64.
   

   Number of vanilla wafers in each bag	1	2	4	8	16	32	64
   Number of bags	64	32	16	8	4	2	1

   Since the number of cookies that goes in a bag is a factor of the total number of cookies, all the factors of 64 are listed in the first row of the table. Each column corresponds to a situation where the 64 cookies are divided equally among some bags, and the product of the numbers in each column is 64. The number of bags is also a factor of the total number of cookies.
3. 1, 2, 4, 8, 16
   Since the number of bags must be a factor of both 48 and 64, the common factors of 48 and 64 represent all the possibilities for the number of bags she can make. The table below shows these possibilities along with the number of each kind of cookie that would go in each bag.

   Number of bags	1	2	4	8	16
   Number of CC cookies in each bag	48	24	12	6	3
   Number of vanilla wafers in each bag	64	32	16	8	4

4. Looking at the table from the third part of this solution, we see that the largest number of bags she can make corresponds to the largest common factor, which is 16 in this case. If there are 16 bags, there will be 3 chocolate chip cookies and 4 vanilla wafers in each.