Thought for the day: Noticing and wondering do not have to happen in any particular order. Allow students' ideas to flow naturally. As their thoughts develop, begin guiding them toward learning goals.

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


Examples of noticing and wondering

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 the picture shows the sum, 7 + 2 + 6, rearranged into 3 equal parts.

I wonder if I could draw pictures like this for other numbers.
I wonder if I could figure out the "equal parts" length without drawing a picture.
I wonder if this picture is about visualizing the mean (average).
I wonder if the 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.



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 image either to introduce the concept of the mean, or if students are already familiar with the computation, 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) explain 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.

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