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

Concepts: representing fractions; area (possibly equivalent fractions, adding fractions)

Beginning

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 and 6 blue squares filled, but only 4 brown squares filled.
I notice that it is hard to count the number of purple squares (but it looks close to 6).
I notice that if I put two of the purple triangles together, it fills twelve squares.
I wonder if all four pictures have something in common.

Exploring

I wonder why the number of squares filled in is not always the same.
I wonder if the bottom rectangle has fewer squares filled in because it is smaller.
I notice that I can fit 3 of the green rectangles and 3 of the blue rectangles into the whole rectangle.
I wonder if I can fit three of the orange and purple shapes into their rectangles if I rearrange them.
I wonder if this image is about fractions.
I notice that all four pictures show the fraction 1/3.
I wonder how many other ways I can show 1/3.
I wonder if the shaded parts need to be connected when I show a picture of a fraction.

Creating

As they notice and wonder, students may create their own pictures to represent fractions by choosing different shadings or different rectangles, or by using shapes other than rectangles as wholes. Many of their ideas will flow from things that they have wondered about. For example, they may

Draw more pictures showing 1/3 for a 3 by 8 rectangle. Try for variety and creativity!
Draw pictures showing 1/3 for rectangles of other sizes, including rectangles whose areas are not multiples of 3.
Draw pictures of 1/3 using wholes having other shapes such as triangles, circles, or trapezoids.
Draw pictures of other fractions.
Think of real-world situations that your pictures could represent.
Draw pictures showing more than one fraction inside a single whole.

Reflecting and Extending

I wonder if I could draw a picture of 1/3 for a rectangle that has 8 squares (or some other number that is not a multiple of 3).
I notice that there are many more ways to show fractions than I had realized.
I notice that a single picture shows many fractions. (For example, the top picture can represent 1/3, 6/18, or many other fractions.)
I wonder if I could use pictures like these to add or subtract fractions.

Notes

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.