Unit 8, Lesson 8
Keeping Track of All Possible Outcomes
Grade 7
8.1
Warm-up
Bracelets made by a tribe of Native American jewelers in Arizona and New Mexico are available with several different options. How many different bracelets are possible if each one includes 1 stone, 1 design, and 1 size?
stones
- turquoise
- pink coral
- lapis lazuli
designs
- animal
- floral
sizes
- small
- medium
- large
8.2
Activity
Consider the experiment: Flip a coin, and then roll a number cube.
Elena, Kiran, and Priya each use a different method for finding the sample space of this experiment.
-
Elena carefully writes a list of all the options: Heads 1, Heads 2, Heads 3, Heads 4, Heads 5, Heads 6, Tails 1, Tails 2, Tails 3, Tails 4, Tails 5, Tails 6.
- Kiran makes a table:
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| H | H1 | H2 | H3 | H4 | H5 | H6 |
| T | T1 | T2 | T3 | T4 | T5 | T6 |
-
Priya draws a tree with branches in which each pathway represents a different outcome:
- Compare the three methods. What is the same about each method? What is different? Be prepared to explain why each method produces all the different outcomes without repeating any.
-
Which method do you prefer for this situation?
Pause here so your teacher can review your work.
-
Find the sample space for each of these experiments using any method. Make sure you list every possible outcome without repeating any.
-
Flip a dime, then flip a nickel, and then flip a penny. Record whether each lands heads or tails up.
-
Han’s closet has: a blue shirt, a green shirt, a white shirt, blue pants, khaki pants, and black pants. He must select one shirt and one pair of pants to wear for the day.
- Spin a color, then spin a number.
Circular spinner divided into four equal parts. The first part is red and labeled “R,” the second part is blue and labeled “B,” the third part is green and labeled “G,” and the fourth part is yellow and labeled “Y.” The pointer is in the part labeled “B.”Circular spinner divided into five equal parts. Starting from the top right, and moving clockwise, the first part is labeled 1, the second, 2, the third, 3, the fourth, 4, and the fifth, 5. The pointer is in the part labeled “5.”
- Spin the hour hand on an analog clock, then choose a.m. or p.m.
-
Student Lesson Summary
Sometimes we need a systematic way to count the number of outcomes that are possible in a given situation. For example, suppose there are 3 people (A, B, and C) who want to run for the president of a club and 4 different people (1, 2, 3, and 4) who want to run for vice president of the club. We can use a tree, a table, or an ordered list to count how many different combinations are possible for a president to be paired with a vice president.
With a tree, we can start with a branch for each of the people who want to be president. Then for each possible president, we add a branch for each possible vice president, for a total of possible pairs. We can also start by counting vice presidents first and then adding a branch for each possible president, for a total of possible pairs.
Tree diagram with three branches for the first choice, labeled “A,” “B”, and “C.” Choices “A”, “B”, and “C” each have four branches labeled with a different number from 1 through 4.
Tree diagram with four branches for the first choice, labeled 1, 2, 3, and 4. Choices 1, 2, 3, and 4 each have three branches, labeled with a different letter “A,” “B,” or “C.”
A table can show the same result:
| 1 | 2 | 3 | 4 | |
|---|---|---|---|---|
| A | A1 | A2 | A3 | A4 |
| B | B1 | B2 | B3 | B4 |
| C | C1 | C2 | C3 | C4 |
So does this ordered list:
A1, A2, A3, A4, B1, B2, B3, B4, C1, C2, C3, C4
None