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Thought for the day: Some images are easy to modify in order to suit the needs of your students!

Concepts: representing fractions; area


Examples of noticing and wondering

I notice that three of the rectangles have 18 squares and one of them has 12 squares.
I notice that that there are 6 green squares and 6 blue squares filled, but only 4 orange squares filled in.
I notice that it is hard to count the number of purple squares, because they are not all whole squares (but it looks close to 6).
I notice that if I cut the purple triangle into parts, I can arrange it into 6 squares.
I notice that I can fit 3 of the green rectangles and 3 of the blue rectangles into the whole rectangle.

I wonder why the number of squares filled in is not always the same.
I wonder if the rectangle with the orange squares has fewer squares filled in because the rectangle is smaller.
I wonder if I can do the same thing with the orange and purple shapes if I cut them and rearrange them.
I wonder if this image is about fractions.
I wonder how many other ways I can show 1/3.



This type of image appears in an extended activity called "Building Fractions" on the Deep Math Projects page of this site.

The image above shows some different ways to represent the fraction 1/3, though it will probably take students a lot of time and conversation to realize this. After they have done some open-ended noticing and wondering for a while, you can ask them to compare and contrast the four pictures. The eventual goal may be to find things that all four pictures have in common. You can easily change some of the pictures in order to suit the needs your group of students.

The blue and green pictures may suggest the idea counting the squares, because they both clearly have the same number (6). It is harder to count squares in the purple triangle. Students may think of other strategies such as cutting it apart and rearranging it or seeing it as half of a 3-by-4 rectangle. The blue, green, and purple shapes all have the same number of squares: 6.

In order to figure out what the smaller rectangle has in common with the other three, it may help to compare the colored part to the whole. For the green and the blue shapes, it is easy to see that three copies of it fit into the whole. Also, you can see that three copies of the orange shape fit into its whole if you cut one of the three pieces into two parts. The purple triangle is harder, but you can also cut and rearrange its pieces so that three copies of it fit into its rectangle. 

You might follow up with this image by asking students to create and discuss their own pictures that represent 1/3 in creative ways.

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