Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Exploring Reasoning and Proof
|Try It Yourself: Factors of 24 | Extending the Activity | Problem Reflection | Your Journal|
After you've finished exploring the activity, answer the following extension questions.
Teacher A: I tried all the numbers from 1 to 24. First I tried a width of 1, then 2, then 3, and so on. Then my list said 1, 24, 2, 12, 3, 8, 4, 6. Then I tried 6, 7, and 8. I noticed that 6 x 4 works and so does 8 x 3, but those factors were already on my list; they are just rotated versions of rectangles that I already created. Just to be safe, I thought in my mind about 9, 10, etc., up to 24. None of them made a rectangle that had a new factor.
Teacher B: I have 1, 24, 2, 12, 4, 6, 8, and 3. I kept making one dimension half as long every time. I made a 1 by 24, then a 2 by 12, then a 4 by 6, then an 8 by 3. I could see the pattern: While cutting one dimension in half, the other one doubles. When I got to a width of 3, half of it was 1 1/2, but that's not called a factor because it's not a whole number. I also thought of breaking 24 into three, four, five, or six equal parts, but I didn't find any new factors, and I knew that a higher number of parts wouldn't work, because when I try a number such as 8, its partner factor had to be a smaller number, 3, and that's already on my list.
Teacher C: I worked in order. I made a 1 by 24, then a 2 by 12, then a 3 by 8. Actually, I didn't really need to make rectangles like a 4 by 6 because I know that 4 times 6 is 24. I stopped and thought about it. Twenty-four is a little less than 5 times 5, which is 25. No whole number times 5 equals 24. Fractions aren't allowed because we're finding factors. There isn't going to be a new factor after 5. If there were an undiscovered factor greater than 5, it would need a partner factor that is smaller than 5, but all the smaller factors were already systematically found.
Teacher C's systematic method is leading toward future development of a general convincing argument that you do not need to test factors beyond the next whole number equal to or after the square root of the number.
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