To study for the tests you should:

Review all the topics we have studied so far. They are listed at Course Topics.

The test will consist of problems that are similar to some of the following problems:

1) This problem refers to the shapes that come in standard pattern tile sets used in schools. If you can fill a shape completely with green triangles, thick blue rhombuses, red trapezoids, and yellow hexagons, then is it also possible to fill this same shape with orange squares and/or thin white rhombuses in addition to the other shapes above? If so, give an example. If not, explain briefly why not.

2) Give three different mathematical topics that you could discuss with pattern tiles. In each case, say how you could use the pattern tiles. Your answer will be scored not only for its accuracy, but also for the quality of the uses you suggest.

3) Problems like those in class activities 1A, 1B, and 1C, the excercises for section 1.1, and the problems for section 1.1.

4) Problems like those in class activities 1F and 1G, the exercises for section 1.3, and the problems for section 1.3.

5) In order to understand fraction multiplication deeply, we must be able to work simultaneously with different wholes. Using the example 2/3 x 2/5, explain why this is so. Discuss what difficulties a child might encounter in making sense of 2/3 x 2/5, and discuss how you might help the child progress.

6) In oder to understand fraction multiplication deeply, we must understand how to divide a whole number of objects into equal parts, such as dividing 8 cookies equally among 3 people. Explain where dividing a whole number of objects into equal parts is needed in understanding 2/3 x 3/5.

7) In addition to understanding the meaning of fractions, list two other important ideas that a child would need to develop in order to understand fraction multiplication deeply. Explain why the ideas you list are important for fraction multiplication.

8) Problems like those on the handout of 1/23.

9) Discuss why we need to develop an understanding of multiplication that goes beyond seeing it as repeated addition.

10) Describe an activity for young children (Pre-K - 2) that might help develop their readiness for understanding multiplication.

11) Write a problem or describe an activity for grades 3 - 5 that might help children in these grades develop their understanding of multiplication.

12) Write an estimation problem or describe an estimation activty for elementary school children (any grade) that might also help children develop their understanding of other mathematical concepts. What other concepts would your problem or activity help develop? In each case, explain why.

13) In the Singapore materials, how is the commutative property of multiplication introduced?

14) In my daughter's 3rd grade math book, the commutative property of multiplication is explained by using a number line to show that 5 + 5 + 5 is equal to 3 + 3 + 3 + 3 + 3. Give two reasons why this is a poor explanation.

15) Discuss how the Singapore materials introduce area.

16) How do the Singapore materials explain the volume formula for a rectangular prism (they use the term cuboid instead of rectangular prism)?

17) Problems like those in class activities 2A - 2E and 2H, 2I, the exercise of section 2.1, and problem 1 of section 2.1.

18) The class activities of section 2.2. The exercises of section 2.2.