Bake Cookies Day (Dec. 18) Meeting (Multiple Topics)

Transcription

1 Bake Cookies Day (Dec. 18) Meeting (Multiple Topics) Topic This meeting s problems cover a variety of topics. However, there is a heavy emphasis on proportional reasoning. Materials Needed Copies of the Bake Cookies Day problem set (Problems and answers can be viewed here, but a more student-friendly version in larger font is available for download from on the MCP Members Only page of the Club Program section.) Calculators Variety of cookies and small bags optional Meeting Plan On Bake Cookies Day we obviously need to eat some cookies even if we don t bake any! This is a great day to bring in a variety of little cookies and enough small bags to have one small bag per student. Let s assume you have 18 kids at your club meeting, and you brought in chocolate chip cookies and oatmeal cookies. Start by telling them that you have 36 chocolate chip cookies and 27 oatmeal cookies. (1) If you want to make as many bags as possible, use all of the cookies and have the same assortment of whole cookies in each bag, how many of each type of cookie should go in each bag? 4 chocolate chip & 3 oatmeal (2) What if you want to put as many whole cookies as possible into each bag while using all of the cookies and having the same assortment in each bag? How many of each type of cookie would go in each bag? 12 chocolate chip & 9 oatmeal (3) Finally, for the club members at the meeting, if everyone is to get the same amount of cookies in his or her bag, how many of each type of cookie should each club member receive? Using whole cookies: 2 chocolate chip & 1 oatmeal, but go ahead and give everyone another half of an oatmeal cookie! Once students are enjoying their cookies, you can set them to work on these dozen delicious cookie problems. Abby, Brooke, Carter and Travis are each providing cookies for a school bake sale. Let s take a look at their process. Making the cookies 1. One of Brooke s recipes calls for five tablespoons of flour for every 2 ounces of butter. How many tablespoons of flour are needed if two pounds of butter are used? There are 16 ounces of butter in one pound School Handbook, Warm-Up Travis wanted to create a unique cookie frosting color for his cookies. He mixed together a sample that was 12 teaspoons of red frosting, 2.5 teaspoons of yellow frosting and 0.5 teaspoons of blue frosting. He then mixed a main batch of frosting using 30 teaspoons of yellow frosting and enough red and blue frosting so as to maintain the original ratio. How many total teaspoons of frosting did he use when making both the sample and the main batch? School Handbook, Warm-Up 9-5 (modified) Packaging the cookies 3. Carter s cookies are each 4 inches in diameter, and he can fit 12 of them perfectly on the bottom of his container, as shown in this top view. What are the dimensions, in inches, of the bottom of his container? 4. Abby also has cookies that are each 4 inches in diameter, and she can fit 11 of them perfectly on the bottom of her container, as shown in this top view. What are the dimensions, in inches, of the bottom of her container? Express any non-integer values as a decimal to the nearest hundredth MATHCOUNTS Club Resource Guide 35 Club Resource Guide.pdf 35 8/18/08 11:24:16 AM

2 5. Brooke initially made 60 sugar cookies, 80 chocolate chip cookies and 100 peanut butter cookies. She plans to make packages of cookies that each contain an identical assortment of whole cookies. How many cookies are in a package, assuming that she makes as many packages as possible and uses all of the cookies she made? School Handbook, Warm-Up A 75-cookie bag contains 30 oatmeal cookies, 30 chocolate chip cookies and 15 peanut butter cookies. If this same ratio of the three types of cookies is used for a 100-cookie bag, how many peanut butter cookies should be in the bag? School Handbook, Workout 3-7 (modified) 7. Travis contributed 9 dozen cookies to the bake sale. The cookies will be put into bags of 6 cookies and bags of 8 cookies. If all 9 dozen cookies will be put into bags, what is the greatest number of 8-cookie bags that can be made? 8. Still using the scenario above, if there must be more 8-cookie bags than 6-cookie bags and all 9 dozen cookies must be used, what is the smallest number of 8-cookie bags that can be made? Selling the cookies 9. Carter plans to sell a package of two cookies for $0.50. At this rate, for how much would he sell 3 dozen of these two-cookie packages? 10. Travis plans to sell individual cookies for 30 cents each, but will have a buy 2, get 1 free deal available. According to this deal, what is the average cost per cookie, in cents, for a customer who buys 2 and gets 1 free? 11. Abby received one 30-cookie order. This represents 2.5% of the total number of cookies in all of her orders. What is the total number of cookies in all of her orders? School Handbook, Workout 2-6 (modified) 12. Twelve eighth-graders can buy and eat 50 dozen cookies in 1 day. At the same rate, how many dozens of cookies will 20 eighth-graders buy and eat in 3 days? School Handbook, Warm-Up 14-9 Answers: 80 tablespoons; 195 teaspoons; 12 by 16 ; by 16 ; 12 cookies; 20 peanut butter cookies; 12 8-cookie bags; 9 8-cookie bags; $18; 20 cents; 1200 cookies; 250 dozen Possible Next Steps If your school has a kitchen that students can use, actually baking some cookies might be a fun activity for the group... even if there is no math involved. If your club wants to purchase club T-shirts or other MATHCOUNTS items, a cookie bake sale might be the way to go, and this activity can be used to kickoff the project MATHCOUNTS Club Resource Guide Club Resource Guide.pdf 36 8/18/08 11:24:16 AM

3 Bake Cookies Day Meeting Problem Set Abby, Brooke, Carter and Travis are each providing cookies for a school bake sale. Let s take a look at their process. Making the cookies 1. One of Brooke s recipes calls for five tablespoons of flour for every 2 ounces of butter. How many tablespoons of flour are needed if two pounds of butter are used? There are 16 ounces of butter in one pound School Handbook, Warm-Up Travis wanted to create a unique cookie frosting color for his cookies. He mixed together a sample that was 12 teaspoons of red frosting, 2.5 teaspoons of yellow frosting and 0.5 teaspoons of blue frosting. He then mixed a main batch of frosting using 30 teaspoons of yellow frosting and enough red and blue frosting so as to maintain the original ratio. How many total teaspoons of frosting did he use when making both the sample and the main batch? School Handbook, Warm-Up 9-5 (modified) Packaging the cookies 3. Carter s cookies are each 4 inches in diameter, and he can fit 12 of them perfectly on the bottom of his container, as shown in this top view. What are the dimensions, in inches, of the bottom of his container? 4. Abby also has cookies that are each 4 inches in diameter, and she can fit 11 of them perfectly on the bottom of her container, as shown in this top view. What are the dimensions, in inches, of the bottom of her container? Express any non-integer values as a decimal to the nearest hundredth. 5. Brooke initially made 60 sugar cookies, 80 chocolate chip cookies and 100 peanut butter cookies. She plans to make packages of cookies that each contain an identical assortment of whole cookies. How many cookies are in a package, assuming that she makes as many packages as possible and uses all of the cookies she made? School Handbook, Warm-Up A 75-cookie bag contains 30 oatmeal cookies, 30 chocolate chip cookies and 15 peanut butter cookies. If this same ratio of the three types of cookies is used for a 100- cookie bag, how many peanut butter cookies should be in the bag? School Handbook, Workout 3-7 (modified) Copyright MATHCOUNTS, Inc MATHCOUNTS Club Resource Guide Problem Set

4 7. Travis contributed 9 dozen cookies to the bake sale. The cookies will be put into bags of 6 cookies and bags of 8 cookies. If all 9 dozen cookies will be put into bags, what is the greatest number of 8-cookie bags that can be made? 8. Still using the scenario above, if there must be more 8-cookie bags than 6-cookie bags and all 9 dozen cookies must be used, what is the smallest number of 8-cookie bags that can be made? Selling the cookies 9. Carter plans to sell a package of two cookies for $0.50. At this rate, for how much would he sell 3 dozen of these two-cookie packages? 10. Travis plans to sell individual cookies for 30 cents each, but will have a buy 2, get 1 free deal available. According to this deal, what is the average cost per cookie, in cents, for a customer who buys 2 and gets 1 free? 11. Abby received one 30-cookie order. This represents 2.5% of the total number of cookies in all of her orders. What is the total number of cookies in all of her orders? School Handbook, Workout 2-6 (modified) 12. Twelve eighth-graders can buy and eat 50 dozen cookies in 1 day. At the same rate, how many dozens of cookies will 20 eighth-graders buy and eat in 3 days? School Handbook, Warm-Up 14-9 **Answers to these problems are on page 36 of the Club Resource Guide.** Copyright MATHCOUNTS, Inc MATHCOUNTS Club Resource Guide Problem Set

5 Bake Cookies Day Solutions ( MCP Club Resource Guide) Problem 1. If there are 16 ounces in 1 pound and we have 2 pounds of butter, then we have 32 ounces of butter. Needing 5 tablespoons of flour for every 2 ounces of butter leads to the proportion 5/2 = x/32. Multiplying the first fraction by 16/16 we get 5/2 = 80/32, so we need 80 tablespoons of flour. Problem 2. The sample batch called for 12 red, 2.5 yellow and 0.5 blue. Multiplying everything by 2, this is equivalent to the ratio 24 red, 5 yellow and 1 blue. The main batch will use 30 yellow. This means we just multiply this previous ratio by 6 and get a main batch of 144 red, 30 yellow and 6 blue. Totaling the sample and main batch amounts give us = 195 teaspoons. Problem 3. The short side of the container s base is equal to 6 radii, and the long side of the container s base is equal to 8 radii. This means the container s base is 12 inches by 16 inches. Problem 4. Similar to the last problem, the longer side is 16 inches. However, the shorter side is shorter than 12 inches. We can see that an equilateral triangle can be drawn, and its altitude will be 4 3. The short side of the container s base is equal to a radius, this triangle s height and another radius. This is = inches, to the nearest hundredth. The dimensions of the container s base are inches by 16 inches. Problem 5. If she makes as many packages as possible with the same ratio of cookies in each package, then the number of packages is the greatest common factor or 60, 80 and 100. This is 20. If there are 20 packages made, then there are 3 sugar cookies, 4 chocolate chip cookies and 5 peanut butter cookies in each package. This is a total of = 12 cookies. Problem 6. The ratio of 30:30:15 can be reduced to 2:2:1. If this ratio must be used to total 100 cookies, we then can set up the equation 2x + 2x + 1x = 100, where 1x represents the number of peanut butter cookies. This reduces to 5x = 100 and x = 20. Therefore, there are 20 peanut butter cookies. Problem 7. Nine dozen cookies is 9 12 = 108 cookies. If there will be 6-cookie bags (small bags) and 8-cookie bags (large bags), we know 6x + 8y = 108 and we need y to be as large as possible. If we only used large bags, the 108 cookies would fill = 13 large bags with 4 cookies left over. These 4 cookies obviously don t fill a small bag. However, if we only fill 12 large bags, then there are 12 cookies left which exactly fill 2 small bags. Therefore, 12 8-cookie (large) bags can be made. Problem 8. The least common multiple of 8 and 6 is 24. Therefore, if we take apart three large bags (24 cookies), these will make four small bags (24 cookies). From the previous solution we know 12 large bags and 2 small bags use all of the cookies. From here we can go to 12 3 = 9 large bags and = 6 small bags. Then we can go to 9 3 = 6 large bags and = 10 small bags. Remember, though, that we need more large bags than small bags, so the smallest number of large bags that meets these criteria is 9 8-cookie (large) bags. Problem 9. If he has 3 dozen of these two-cookie packages, then he has 36 packages. These will sell for 36 $0.50 = $ Copyright MATHCOUNTS, Inc MATHCOUNTS Club Resource Guide Solution Set

6 Problem 10. A customer who buys 2 and then gets 1 free is getting 3 cookies for 60 cents. This is an average of 60 3 = 20 cents per cookie. Problem 11. Letting x = total number of cookies sold, we can solve this problem with the ratio 30/2.5 = x/100. Using cross-products we get 2.5x = Dividing both sides by 2.5 gives us x = Problem 12. We re told 12 kids can eat 50 dozen cookies in 1 day. In 3 days, these same 12 kids will eat 50 3 = 150 dozen. (Notice that not all three values change just two of them.) So now we have 12 kids, 150 dozen, 3 days. However, we want 20 kids. What would 2 kids eat? (This is a sixth of the number of kids we currently have.) We can see 2 kids would eat 150/6 = 25 dozen in 3 days. Now let s go to 10 times the number of kids these 20 kids would eat = 250 dozen cookies in 3 days. Copyright MATHCOUNTS, Inc MATHCOUNTS Club Resource Guide Solution Set