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Thought for the day: Once students have discussed a prompt like this thoroughly, encourage them to create and compare their own images!

Concepts: equivalent fractions (possibly area)

Beginning

I notice that both rectangles are the same size, and they have the same amount colored in.
I notice four vertical rectangles inside of each big rectangle.
I notice that the lower rectangle is also split into (3) horizontal bars.
I notice that the lower rectangle has more total parts (12).
I wonder if these pictures are about fractions.
I wonder what real-world things these pictures could be about.

Exploring

I notice that the pictures can represent 1/3 and 4/12 (because the parts within each picture are the same size).
I wonder if fractions of the same size always follow a pattern where the numerator and denominator are multiplied by the same number.
I wonder why this happens.
I wonder if I can draw a different kind of picture to show that 1/3 and 4/12 are the same size.
I wonder how I could draw pictures to show other fractions that are the same size as 1/3.

Creating

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

Create a story to match this prompt. (The story should be about 1/3 and 4/12 being the same amount.)
Create other rectangle diagrams to show other equivalent fractions for 1/3.
Create other kinds of diagrams that show multiple equivalent fractions for 1/3.
Create diagrams for other equivalent fractions.

As students create new drawings, they should look for patterns in the diagrams and use them to predict patterns in the ways they can write equivalent fractions.

Reflecting and Extending

I notice that there are many (an infinite number of) ways to represent every fraction.
I notice that the numerators and denominators of equivalent fractions show skip-counting patterns.

The numerators count by 2, and the denominators count by 3.

I wonder if there is a fast way to calculate equivalent fractions.
I wonder if there a fast way to find the simplest equivalent fraction.
I notice that finding equivalent fractions involves multiplying or dividing the numerator and denominator by the same number.

Notes

This image shows an example of visualizing equivalent fractions. It is important for children to see a variety of ways to represent this concept. Circle diagrams with "pie-shaped" pieces tend to be overused (and are often more difficult to draw accurately). Number lines tend to be under-utilized. Once students have discussed the given image in depth, it is always a good idea for them to draw their own diagrams showing both the same relationship (between 1/3 and 4/12) and new relationships. Encourage variety and creativity in the images! In addition, you can ask kids to create real-world situations and stories for the concepts. Where in the world might they see that 1/3 and 4/12 stand for the same amount?

images like these are just one step away from adding and subtracting fractions. Advanced or adventurous learners sometimes spontaneously begin to think about this. If so, encourage them, but don't teach them rules yet. If not, be patient, and give them more time to play around with visualziing equivalent fractions first!
 

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