Thought for the day: Consistently showing students that you are interested in their ideas is generally more motivating than praise!

Concepts: halves (and possibly thirds) as names; dependence of halves and thirds on the size of whole; fractions as relative to a whole; many ways of showing the same fraction of a whole

Beginning

I notice six rectangles, four large ones all the same size and two small ones both the same size.
I notice that all of the rectangles have a brown shape inside them.
I wonder which shape has the most brown in it.
I wonder if some of the shapes have the same amount of brown in them.
I notice that the brown parts of some of the rectangles look like half the rectangle.

Exploring

I wonder if it would help to split the rectangles into smaller parts.
I notice that the large rectangles could have four small squares going across and two squares going down.
I notice that the small rectangles could have three small squares going across and two squares going down.
I notice that sometimes I can tell how big the brown part is by counting the squares inside of it.
I notice that I could fit two of the brown shapes into the whole rectangle in some of the rectangles. (This means the brown shape is half of the rectangle.)
I wonder how many ways I can make brown parts that are half of a rectangle.
I notice that when the brown part is half, then so is the white part.
I notice that when the rectangle is bigger, you need a larger brown part to make half.
(Suggestions: Help students draw grids on top of the rectangles (4 by 2 for the large rectangles and 3 by 2 for the small rectangles works well. If this is too hard, draw pictures on grid paper for them, but only after they have had a chance to think about them with a grid.)

Creating

As they notice and wonder, students may create new ideas and drawings that flow out of things that they have noticed and wondered about. For example, they may:

Create more drawings that show different ways to make half (and explain why they show half).
Create drawings that show thirds. (If they don’t know what this means, ask them to guess first. Help them understand that three of the brown shapes would fit into the whole rectangle.
Create two drawings where a third in one picture has more brown than half in the other picture. Explain what makes this happen.
Create stories about halves and thirds (or even about drawings that they make).

Reflecting and Extending

I notice two brown parts that are same size, but one of them is half and the other is not (pictures 3 and 4).
I notice that two things can both be half even when they are different sizes (see pictures 1 and 3).
I notice that I have to look at both the brown part and the whole rectangle to see if the brown part is half.
I wonder
if there is a name for how much the brown parts are when they are not half of the rectangle (pictures 4 and 6).
I wonder if two things that are half of the same rectangle can have different shapes.
I wonder if the brown parts have to be connected in order to make half.

Notes

Suppose that the pictures are numbered from left to right and top to bottom like this:

1 2
3 4
5 6

Students can make many interesting comparisons:

Pictures 1, 2, 3, and 5 all show half.
Pictures 1, 2, 5, and 6 have the same amount of brown. (The grid may help them see this, especially for 1 , 2, and 6.)
Picture 5 is challenging to understand with the grid, because the brown part covers parts of grid squares.
The brown parts in pictures 3 and 4 are the same size, but one of them is half (3) and the other is not.
The brown part of 6 is the same size as in pictures 1, 2, and 5, but it is not half, while 1, 2, and 5 are.
Some students may be able to recognize that the brown part of 4 is less than half, and the brown part of 6 is more than half.
Some students may be able to predict meanings and draw pictures for a fourth, a fifth, etc.

Encourage students to draw creative pictures of one half. The two halves do not need to look the same. It is also okay for the brown (or white!) parts to be made of disconnected parts. A few students may even be able to use partial grid squares and/or slanted line segments in their drawings.

Notice that students are beginning to think about area informally with this prompt (when they try to decide which pictures “have the most brown”). They even touch on the idea of using small “unit” squares to help them make some of the comparisons.

Do not press very young students to use traditional fraction notation yet. They need to develop other fraction concepts first.