Illuminations: Combinations

Combinations

Shorts and Shirts

Students are encouraged to discover all of the possible combinations for outfits consisting of shirts and shorts. Students apply problem-solving skills (including elimination and collection of organized data) to draw their conclusions.

Learning Objectives
 
Students will:
  • estimate the number of possibilities that can be made from a given number of factors
  • compute the number of possibilities
  • generalize the number of possible combinations that can be made

Materials
 
Colored crayons (red, green, yellow, blue, orange, brown, black, and purple)
Shorts and Shirts Activity Sheet

Instructional Plan

Begin the class by introducing the term combinations. Ask students to explain what a combination is. Guide students to a mathematical definition of combinations. Next, ask students to brainstorm different types of combinations. Students may suggest some of the following:

  • ice cream cones
  • menu combinations
  • outfits
  • safe or padlock combinations

 

In today's lesson, students will explore combinations involving shirt and short outfits. Distribute the Shorts and Shirts activity sheet to each student. Students will be coloring outfits to determine the total number of combinations possible.

Shorts and Shirts Activity Sheet Shorts and Shirts Activity Sheet

Give each student eight crayons in the following colors: red, green, yellow, blue, orange, brown, black, and purple.

Review the problem by reading it with the students:

Each shirt must be a solid color, either yellow, orange, blue or red. Each pair of shorts must be a solid color, either brown, black, green, or purple. How many different outfits can be made? No two outfits should be the same.

Guide students to predict how many different outfits can be colored. They should record their predictions on the activity sheet. (Note: Some students may predict 8 outfits by adding 4 shorts and 4 shirts. Other students may predict 16 outfits by multiplying 4 shorts and 4 shirts.)

Allow enough time for students to color their combinations on the activity sheet. Place students in pairs to compare their results.

As a class, discuss the results. Students should have the following 16 correct combinations:

Yellow shirt, Brown shorts
Yellow shirt, Black shorts
Yellow shirt, Green shorts
Yellow shirt, Purple shorts
Orange shirt, Brown shorts
Orange shirt, Black shorts
Orange shirt, Green shorts
Orange shirt, Purple shorts
Blue shirt, Brown shorts
Blue shirt, Black shorts
Blue shirt, Green shorts
Blue shirt, Purple shorts
Red shirt, Brown shorts
Red shirt, Black shorts
Red shirt, Green shorts
Red shirt, Purple shorts

Ask students to compare their predictions to the correct total number of combinations. Students who made the correct predictions could share their reasoning with the class.

In addition to the organized list shown above, students may also make a table or a tree diagram to solve this problem.

Ask students to think about a general rule or pattern for determining the total number of combinations. Students should see that they could have multiplied the number of shirts possible (4) by the total number of shorts possible (4) to get a correct total number of 16 combinations. As appropriate, tell students that this is an example of the counting principle. You may wish to give additional examples of the counting principle.

You may also ask students to use the Bobbie Bear tool. Students can pick an outfit for Bobbie Bear and customize the outfit similar to the one on the activity sheet.

Bobbie Bear Bobbie Bear

Questions for Students
 

How did your prediction compare to your actual answer? How do you explain this?

[Student responses will vary depending upon their predictions and how close they are to the actual answer. Students who made incorrect predictions should be able to now explain how to find the actual number.]

Which method would be more efficient for finding the total number of outfits: multiplying, drawing a tree diagram, or making a table?

[Students should respond by saying multiplying, because that will be the quickest way to determine a count.]

Which method would be more useful for identifying the different combinations (outfits) possible: multiplying, drawing a tree diagram, or making a table?

[Students should respond by saying either a tree diagram or making a table, because these two methods list the possibilities (whereas multiplying simply gives you a count).]

In your own words, what is the counting principle?

Extensions
 
  1. What if six colors are used for shirts?
    [There would be twenty-four outfits.]
  2. What if eight colors are used for shorts and eight for shirts?
    [Solution: Sixty-four outfits could be colored.]

NCTM Standards and Expectations
 
Data Analysis & Probability 3-5
  1. Propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions or predictions.
  2. 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

Navigating through Data Analysis and Probability in Grades 3‑5

 Activities
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