Parts of a Circle Using Sectors and Counting Boards

You need to get a packet that contains two congruent circles on cardstock (red and blue), copies of the circles on white paper in case you'd like to make your own circles, and two 3 by 2 counting boards, one with whole numbers and one with fractions, printed on one pieces of cardstock.

Part 1. Getting familiar with the pieces and how their sizes correspond to the numbers on the fraction board, and getting their sizes in degrees.

1. Your first task is to cut out the cardstock circles very carefully, and cut them into sectors as indicated. Do not cut out the counting boards! Take your time, because it is important to cut out the sectors very precisely.

2. We have to find out what fraction of a circle each sector is. You can write your answers on the white paper circles for future reference.

3. To get familiar with your counting boards, put tokens on the fraction board for each sector individually. Notice that there is a sector for each square on the board except for the square with the number 1.

For each circle, put tokens on the fraction board that represent each of the sectors. Now regroup the tokens to see if they add up to one.

Using two boards to compute with fractions

The rules of regrouping on the whole number board and the fraction board are the same.

For example, 2 + 3 = 4 + 1

The same regrouping on the second board
shows that (1/4) + (1/6) = (1/12) + (1/3),
which is far from being obvious.

Similarly, it is obvious that 3 + 1 + 1 + 1 = 6, which corresponds to putting one token on 3 and three tokens on 1. But the same configuration on the fraction board shows that (1/4) + (1/12) + (1/12) + (1/12) = 1/2.

4. Get a protractor. Measure the angle of each sector. You may write your answers on the white paper boards. You can check with the calculator if your measurements are correct.

Part 2. Arithmetic with small fractions in early grades.

1. Addition examples

Get a sector. In how many ways can you make it with the other sectors? You may mix sectors of two colors, red and blue. Each of your solutions corresponds to one addition fact about fractions. Use the board to find the answer, and to show how you regroup the tokens.

One twelfth plus one twelfth plus one twelfth plus one fourth. Now regroup.

One third equals one fourth plus one twelfth

One half equals one third plus one twelfth plus one twelfth

One fourth equals one sixth plus one twelfth

2. Subtraction examples

Choose two configurations (two groups of sectors). What is their difference? Try to answer it first just by looking at it. Do the subtraction on the counting boards.

One half minus (one fourth plus one twelfth) equals one sixth

One third minus one fourth equals one sixth

One sixth plus one sixth minus one twelfth minus one twelfth minus one twelfth equals one twelfth

Exercises such as these can be continued until children become really familiar with the sectors and how they correspond to numbers on the boards.

Part 3. Patterns and tangrams

Some examples

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