Thought for the day: Noticing and wondering do not have to happen in any particular order. Allow students' ideas to flow naturally.

Concepts: average (mean); properties of addition (possibly division)



I notice that the color of each segment matches the numeral.
I notice that the numbers could stand for the lengths of the segments.
I notice that the total length is 15 units.
I notice that in the top picture, the colored segments are different lengths, but in the bottom picture, they are the same.
I notice that both pictures have 3 parts.
I wonder what kind of real-world thing this picture could be about.

I notice that the picture shows the sum, 7 + 2 + 6, rearranged into 3 equal parts.
I notice that I could make the second picture by moving 2 units from ‘7’ segment and 1 unit from the ‘6’ segment to the ‘2’ segment.
I wonder if I could draw pictures like this for other sums.
I wonder if the number of parts always has to stay the same for both lines.
I wonder if I could figure out the "equal parts" length without drawing a picture.
I notice that this picture is about visualizing means (averages).
I wonder if this picture could help me find other combinations of three numbers that have the same mean.
I wonder if I could use pictures like this to help me invent strategies to divide numbers.



As they notice and wonder, students may create their own diagrams like this prompt. Many of their ideas will flow from things that they have wondered about. For example, they may

Create stories to fit the prompt.
Create other diagrams with sums of 15.
Create other diagrams having other sums.
Create diagrams in which the sums have more (or fewer) than 3 parts.
Use number line diagrams to create new strategies for calculating means.
Create other types of diagrams (besides number lines) that show the concept of “mean.”

Reflecting and Extending

I notice that the mean is about equal sharing.
I wonder what would happen if the equal parts were not whole numbers.
I notice that the mean is kind of a “typical” number for a list.
I notice that the mean of two numbers is always exactly halfway between them.
I wonder what is the fastest way to find the mean when there are a lot of numbers.


Students often learn to think of the mean or average as a series of steps: "add the numbers and find divide by the number of numbers." The image above shows a meaning for this process. You are simply collecting everything together and then sharing it out equally.  You may use the prompt either to introduce the concept of the mean, or if students are already familiar with how to calculate it, to help them understand it conceptually.

To develop the concept further, you can ask students to (1) think of real-world situations for the image (2) talk about how 5 in the example is a "typical" size of the set of numbers 7, 2, and 6, (3) draw other pictures segments to represent means of others sets of numbers, or (4) draw other kinds of pictures (not necessarily segments) to illustrate combining things into a total and then sharing them out equally.

For example, suppose that each person in a group of kids had some cookies. If you listed the number of cookies that each child had, the mean of the numbers in that list would be the number of cookies each child would have if they shared the cookies equally among them.

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