Illuminations: Combinations

Combinations

Ice Cream Cones

Students are encouraged to discover all the combinations for ice cream cones. They use problem-solving skills (including elimination and collection of organized data) to draw their conclusions. Students also discuss whether or not order matters mathematically for the given problem.

Learning Objectives
 
Students will:
  • Predict the number of two-color combinations that can be made from eight colors.
  • Determine the number of combinations that can be made.
  • Generalize the number of possible combinations that can be made.
  • Determine whether or not order matters mathematically.

Materials
 
Ice-Cream Cones Activity Sheet
Eight crayons for each student: brown for chocolate, white for vanilla, red for strawberry, orange for orange sherbet, green for lime, yellow for lemon, purple for grape, and black to make chocolate chips

Instructional Plan

Begin the class by reviewing the previous lesson, Shorts and Shirts. Ask students to determine the number of possible outfits given 5 colors of shirts and 6 colors of shorts. Students should respond by saying there are 30 outfits possible. Tell students that today they will be coloring ice cream cones to determine the total number of combinations possible.

Distribute the Ice-Cream Cones activity sheet to each student.

Ice Cream Cones Activity Sheet Ice Cream Cones Activity Sheet

Each student will also need the following 8 colors of crayons: brown (chocolate), white (vanilla), red (strawberry), orange (orange sherbet), green (lime), yellow (lemon), purple (grape), and black (chocolate chips.)

Before starting the activity, read the scenario on the activity sheet. Individually, students should predict the number of different two-topping ice cream cones that are possible. Predictions may include any of the following:

  • 16 (8 colors × 2 scoops)
  • 64 (8 colors for the first scoop × 8 colors for the second scoop)
  • 56 (8 colors for the first scoop × 7 colors for the second scoop, because students may focus on the word "different")
  • and so on.

 

Students should work individually to color and count their ice cream cones. In groups of two or three, discuss the results: both the types of ice cream cones created and the number.

Two issues should arise in group discussions:

  1. Does order matter on the ice cream cone? For example, is an ice cream cone with vanilla on top and lime on the bottom different from one with lime on top and vanilla on the bottom?

  2. Can the same flavor be used for both scoops?

As students discuss these questions in their groups, you may choose to let students decide on their own how to proceed. This will, of course, lead to different answers. Another option is to let students discuss these issues and make a class decision before proceeding; that way, all students are attempting to solve the same problem.

Depending on the assumptions that students make regarding order and repetitition, the answers will vary:

  • Order Matters
    • Flavor May Not Be Repeated: The first scoop can be any of the eight flavors, and the second scoop can be any of the seven flavors not used in the first scoop. Therefore, there are 8 × 7 = 56 possible two‑scoop combinations.
    • Flavor May Be Repeated: Either scoop can be any of the eight flavors, so there are 8 × 8 = 64 possible two‑scoop combinations.

  • Order Does Not Matter
    • Flavor May Not Be Repeated: This case is best solved by making an organized list, tree diagram, or table. An organized list is shown below:

        C-V C-Li C-Le C-G C-O C-S C-CC
        V-Li V-Le V-G V-O V-S V-CC  
        Li-Le Li-G Li-O Li-S Li-CC    
        Le-G Le-O Le-S Le-CC      
        G-O G-S G-CC        
        O-S O-CC          
        S-CC            
      C = chocolate, V = vanilla, Li = lime, Le = lemon,
      G = grape, O = orange, S = strawberry, CC = chocolate chip

      Note that in this case, there are 28 possible two-scoop combinations.

    • Flavor May Be Repeated: In addition to the 28 combinations listed above, there are an additional eight possibilities, one for each case when both scoops are the same flavor. Consequently, there are 28 + 8 = 36 combinations in this case.

If the class did not come to concensus prior to completing the activity sheet, then a rich discussion can occur. Ask students who found 28 ice cream cones to share with other members of the class the reasoning behind their answer. [8 × 7 ÷ 2 = 28.]

Students who arrived at 56 combinations should also be allowed to share their reasoning, namely that order matters, but the same flavor cannot be used for both scoops. [8 × 7 = 56.] Ask them to state why they consider vanilla‑lime to be different from lime‑vanilla.

Students who found either 36 or 64 combinations allowed for repetition of flavors, which is also a reasonable assumption.

Though students may find four different answers to the original question, be sure to point out that none of the answers are wrong. It is important that students think about why their answers may be different, and you should push them to articulate why they got a different answer than another student.

Questions for Students
 

Why was more than one correct answer possible for this problem?

[Student responses should include an understanding of the different ways of interpreting the problem posed to them.]

How did your answer compare to other students' answers?

[Student responses will depend upon their answers.]

How is today's lesson similar to the shirts and shorts lesson? How is it different?

[Students may suggest that in both lessons, we used an organized method of identifying all of the possible combinations, and not just finding a total count.]

Did you use the counting principle in today's lesson? If so, how?

Extensions
 
  1. Pose the following questions to students: "What if you allow either one or two scoops? How is the solution different?"

NCTM Standards and Expectations
 
Data Analysis & Probability 3-5
  1. Predict the probability of outcomes of simple experiments and test the predictions.

References
 
  • Marcy Cook. "IDEAS: Combinations." The Arithmetic Teacher. 36, 1 (September 1988) 31-36.
  
1 period   

NCTM Resources

Number Stories of Long Ago

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