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
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
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:
- 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?
- 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 repetition, 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
If the class did not come to consensus 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.
Marcy Cook. "IDEAS: Combinations." The Arithmetic Teacher. 36, 1 (September 1988) 31-36.