Oakland Terrace Math 6 (A) Student Work

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Math A Units 4 & 5~ Possible Outcomes - Exponential Growth

What Does This Look Like In:

Ms. Lyons's class explored probability and possible outcomes using a Pre-K and Kindergarten website:

www.poissonrouge.com

On one part of the website, students can make combinations of people, choosing from different heads, bodies and legs.

When students go to the Mix and Match page, the computer randomly assigns a head, a body and legs. In the lesson, students first consider how many different combinations the computer could assign if there were 4 heads, 2 bodies and 2 sets of legs (see above).
This student makes a tree diagram to determine how many combinations are possible. Tree diagrams are important for students to conceptualize possible outcomes in probability. Students must than relate the tree diagram to the computation, as this student has done. 4 heads x 2 bodies x 2 legs = 16 combinations.

The actual website has many more combinations. In fact, there are 11 heads, 11 bodies and 11 legs to mix and match. Some students tried to make tree diagrams, but it became clear very quickly that the tree diagrams would be too big and inefficient.

Student work showing possible outcomes: 11 x 11 x 11 = 1331 possible combinations

If students are unclear about the computation for figuring out the number of outcomes, direct them back to the first activity in which there were only 4 heads, 2 bodies and 2 legs. Once they understand that they multiply the number of heads by the number of bodies by the number of legs, they should be able to relate the computation to the larger numbers. This student has shown that there are 1,331 possible outcomes when the computer mixes and matches.

There is not simply one figure on the screen; there are three figures. And although there are 1,331 different combinations that can be made for the first figure, the number of combinations of all three figures is much greater. At this point, if the students are ready, powers should be discussed. The number of outcomes for one figure could be thought of as 11 to the 3rd power. For the whole screen, with 11 heads possible on all 3 figures, 11 bodies possible and 11 legs possible, the calculation becomes much more interesting. Instead of 11 to the 3rd power, the number of possible outcomes for the page is 11 to the 9th power. Using google, in the search line, type in 11 ^ 9 and you will find the answer.

Picture of 3 people, each with different combinations of a head, a body and legs.

11^9 = 2, 357, 947, 691 possible outcomes

Three images of Marilyn Monroe in the style of Andy Warhol. Each face has different colored skin, hair, background, lips and eye shadow.

Students then moved on to the art gallery on the same website:

www.poissonrouge.com/artgallery

Students can change the color of the background, skin, hair, lips and eye shadow on the faces of Marilyn Monroe.

Students had to consider how many different images of Marilyn Monroe are possible. This student determined how many colors each feature could be changed to. The student shows that for one image of Marilyn Monroe, there are 6, 480 possible combinations.

When thinking about exponents, the possible combinations for one image would be expressed as:

Students should consider that there is not simply one image of Marilyn Monroe. On the screens we used, there were actually 15 images of Marilyn Monroe. For 15 images, the possible combinations would be:

When students understand the logic of the tree diagram, it helps them understand the concept of exponential growth, such as with the enormous number of combinations from this seemingly simple design of Marilyn Monroe images.

Indicators:

5.6.1.1 find all possible outcomes of simple experiments using such methods as lists, tree diagrams, area models, and organized lists.
6.6.1.1 read, write, and represent numbers using exponents.

Download a PDF file of this lesson.