Observing Student Reasoning and Proof
 Introduction | Sums of Numbers | Problem Reflection #1 | Products of Numbers | Problem Reflection #2 | Classroom Practice | Observe a Classroom | Your Journal

Remember, at this grade level, we do not expect students to develop formal proofs. We are more interested in fostering their ability to develop convincing arguments to justify their ideas and solutions. Let's look at how upper elementary students might use reasoning to discuss a classmate's conjecture about even sums.

A fourth-grade class has begun working on the Sums of Numbers problem -- an investigation of properties of numbers and relationships between numbers. Robert's group was assigned a number and asked to look for patterns in operations involving that number. The group examined relationships between the sum of the number added to itself (that is, doubled numbers) and the product of the number multiplied by itself (that is, square numbers). In the course of thinking about this investigation, Robert made a conjecture, which the teacher turned over to the whole class for consideration. The teacher also gave the class some additional things to consider, related to Robert's conjectures:

 Robert said that you always get an even sum when you add a number to itself. Is his conjecture always true? 1. Say Robert's conjecture in your own words. 2. Test his conjecture with several numbers. 3. How many numbers must you test to make sure that it is always true? Is it true for numbers that you can't imagine or write because they are very large? 4. Test Robert's conjecture in another way. What do you notice? 5. What is another way to show that it is always true?

Before looking at the students' answers to the questions the teacher posed (below), consider each question on your own, and try to imagine how a student might answer. When you've had a chance to reflect on each question, select "Show Answer" to see the students' responses. As you analyze the student work, ask yourself: How did the teacher's questions help students reason through this problem? What is the value of this task to the students?

Note: In the fourth-grade setting, it is reasonable to apply Robert's conjecture to whole numbers, rather than fractions, decimals, negative numbers, etc. Remember that even and odd numbers are defined as integers (positive whole numbers and their opposites, and 0). Even numbers can be divided evenly by 2 and can be represented as 2n, while odd numbers can be represented as 2n + 1, with n standing for an integer.

 Sample Answer: Robert's conjecture means "Pick any number, like 0, 1, 2, 3, etc. When you add the number to itself, the answer will be even."
 2. Test his conjecture with several numbers. Show Answer
 Sample Answer: We tested 1 + 1 = 2; 4 + 4 = 8; 103 + 103 = 206; 0 + 0 = 0; 1,357 + 1,357 = 2,714; 97,531 + 97,531 = 195,062. We think the sums will always be even numbers.
 3. How many numbers must you test to make sure that it is always true? Is it true for numbers that you can't imagine or write because they are very large? Show Answer
 Sample Answer: You would have to check all numbers, because there might be one number that doesn't work. But it would take forever to check every number.
 4. Test Robert's conjecture in another way. What do you notice? Show Answer
 Sample Answer: When we did even and odd numbers before, we used linking cubes. If a number can be made into two equal sticks that match, it's even. When you add a number twice, no matter what number you use, the two sticks line up because they are the same. So, the sum is even:
 5. What is another way to show that it is always true? Show Answer
 Sample Answer: a. Make an organized list of sums by adding 1 + 1, 2 + 2, 3 + 3, etc.: The pattern starts with 2, an even number. It shows that the sums always go up by 2. That's because you add 1 to each of the two numbers every time you move to the next row. For example, 5 + 5 = 10; 6 + 6 is next and has one more for each number. This proves that any number added to itself will give an even number. b. If you add a number to itself, you double it. If you divide the new number by 2, it divides it evenly because it's a double. You get back to the start: Even numbers can always be divided by 2 evenly because of the doubles. So, Robert is right –– you are always making an even number.

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