## Overview and Objective

In this lesson, students explore a visual representation of the mean of a data set. Students use virtual manipulatives to find the mean of a data set and add a value to a data set so the set has a given mean.

## Polypads for This Lesson

To assign these to your classes in Mathigon, save a copy to your Mathigon account. All tiles expect the squares are locked on these Polypads. Click here to learn about unlocking tiles in order to change these Polypads after saving a copy to your account.

## Warm-Up

Share Mean 1 with students and invite them to engage with the Polypad as they like to answer the question. After a few minutes of work time, share some student work with the class. Ask some students to share their strategies with the class. Hold off from mentioning the word mean or connecting this process to finding the mean of a data set. That will come later in the lesson.

Click here to learn how to share Polypads with students and how to view their work.

Most students likely moved around the tiles to create equal stacks of 15.

### Approach #1

Perhaps some students used the strategies shown in these two videos below. If so, invite students to share their approaches with the class. If not, share these videos and ask student if they are able to describe what is happening in these videos. Click here to learn more about merging and splitting number tiles.

### Approach #2

### Approach #3

In approach #2, merging all the number tiles is a visual representation of finding the sum of the numbers. Splitting the merged tiles into 5 equal groups shows dividing the sum by 5.

At this point, ask students what the "15" is telling them about the data set of 14, 17, 7, 21, and 16. Depending on their background knowledge, they may know this as the **average **or **mean **of a data set. Some may know the standard process for finding the mean as finding the sum of the numbers and dividing by the number of data points. Discuss as a class how all three of these approaches find the mean of a data set.

## Main Activity

Share Mean 2, Mean 3, Mean 4, and Mean 5 with students. Some notes:

**Mean 3**: The answer is 7.5. The number tiles cannot be split in half. Students may ask for help here. Encourage them to find ways on their own to represent this on the canvas. The question blank is set to accept 7.5 as a correct answer. 7 1/2 or 15/2 will be marked as incorrect, even though it is correct. If students enter a fraction, encourage them to represent the number differently.

**Mean 4: **Students use four dodecahedron dice to create their own example. This video shows an example.

**Mean 5: **Here, students are asked to create an additional value in a data set so the new data set has a specific mean. Not all students may reach this question in this lesson. This question can be used as an extension question as needed. Here is two ways to use the Polypad tools to explore this question.

**Approach #1**

**Approach #2**

## Closure

Share some student work with the class. Invite students to share which approaches they found most useful when answering these questions. To close the lesson, ask students to find the mean of **67, 32, 81, and 60**. Consider having students work on this in pairs, as an exit ticket for your review, or as part of a whole class discussion. As an exit-ticket, you'll be able to see where students are along the continuum of concrete representations and abstract reasoning.

## Support and Extension

For students ready for additional extension in this lesson, consider asking questions like the following: **Create a set of 4 different numbers that have a mean of 32.75. Show work to convince a classmate that the mean of your 4 numbers is 32.75**.

For students needing additional support with these ideas, continue to use small numbers that can easily be represented with number tiles. If these students continue to just arbitrarily move tiles around the Polypad into equal stacks, revisit the ideas in Approach #2 from the Warm-Up with them. Focusing on this approach may help them move to the idea of finding the sum of the numbers and dividing by the number of numbers.