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.

• How do you know that you have all factors of 24? Here's how three different teachers reasoned about this problem:

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.

 Question: Why is this statement true: In order to find all the factors of a number, you only need to check the whole numbers up to the square root of that number? Use the illustration above to explain your answer. How would you extend this type of reasoning to any number n? Show Answer
 Sample Answer: As you see in the illustration, the square root of 24 is between 4 (the square root of 16) and 5 (the square root of 25), so you only need to check up to to find all the factors. The next factor larger than is 6, which you've already recorded as a factor in the pair (4, 6). So, you can now be sure that you've found all the factors of 24. By definition of square roots, n will factor as . Again, think of finding other factors from this "middle point." If you change the first number to be something larger than , the second factor must get smaller to make the product stay constant at n. So, you're guaranteed that when you factor n into a product of exactly two numbers, at least one of the two will be less than or equal to . This is true of any pair of two factors for any number n.

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