Modeling the Mean 6th Grade Mathematics 5E Lesson Plan:
Objectives:
SWBAT
- Understand the mean as a number that “evens out” or “balances” a distribution.
- Find the mean of a set of data both mathematically and by using a model.
Engage (10 minutes): Begin the lesson by talking about the U.S. Census, which is a survey that counts the number of people living in the United States and looks at key characteristics about these people, such as household size. The census focuses on counting the people who live in households rather than asking questions like, “How many people are in your family?” Ask students why the census asks how many people are in your household rather than how many people are in your family. I also brought in facts from the 2010 Census and had students guess how many people were found to be living in the U.S. (308 million) as well as the most populated (California at 37 million) and least populated state (Wyoming at 563,00).
Explore 1 ( 10 minutes): Group students in pairs and pass out Getting Ready 3.1 as well as different colored cubes to students. Allow students about 10 minutes to work in pairs to create different colored stacks of cubes to represent each household and to answer questions 1, 2, and 3 which ask students to order the stacks from shortest to tallest, find the median, mode, and to “even out” the stacks to find the mean.
Explain 1 (10 minutes): Discuss with the class how they found the mean or 4 by “evening out” the cubes by moving cubes from taller stacks to shorter stacks.
Discussion points include:
- When you moved the cubes to even out the stacks, the stacks for households with fewer than four people were made taller, and the stacks for households with more than four people were made shorter.
- You moved two cubes from each of the two stacks with six cubes. So you decreased the stack heights by a total of four cubes. (Write 2+2 =4 on the board.)
- You added two cubes to the stack with two cubes, and you added one cube to each of the two stacks with three cubes. So you increased the stack heights by a total of four cubes. (Write 2+1+1 on the board).
- Ask students what they notice about amount of increase and decrease. (They are the same.). Explain how mean is also known as average, even though some households are larger and some are smaller than the mean, the average or mean number of people per household is 4.
Explore 2 (15 minutes): Pass out Problem 3.1 to students and give them time to work in pairs. The problem includes another sample data set of household size for a group of students and asks students to determine the total number of people in all the students households combined, find and compare the mean of this data set to the mean in Getting Ready, and try to develop other ways to find the mean besides using cubes.
Explain 2 (15 minutes): Ask different groups to share explain their solutions to the questions on Problem 3.1 handout. Make sure students justify their strategies by explaining why it works. Most students will use the strategy of evening out cubes to find the mean but some students may come up with other strategies such as finding a balance point in the distribution using a line plot or the standard algorithm of adding up all of the people in each household and dividing that total by the number of households. If the standard algorithm does not come up as a strategy, probe students through the following questions:
- What was the total number of people in all households combined? (24)
- How did you find the total number of people in all households combined? (adding them all up- write 6+4+3+4+3+4=24 on the board).
- How many students were surveyed? (6 by counting number of stacks)
- How do the total number of households, the total number of students surveyed, and the mean relate? (24/6 = 4).
- Would this strategy also work for Getting Ready 3.1? ( Have students check.)
Elaborate: If time permits have students investigate other sets of data that have larger numbers of data values. Have students do the same set of explorations they did for Problem 3.1, but with data sets that involve eight households, each having a mean of four people.
Evaluate (10 minutes): Pass out index cards to each student and put a sample data set on the board with about 6 different data values. Ask students to find the mean on their index card both mathematically and by drawing a model on their index card. Collect the index cards as students leave class.