Number : Real Numbers

Task

Determine if the following statements are sometimes, always, or never true.

When you add two numbers together, you get the same result regardless of the order they are added. For example, 3 + 2 = 5 and 2 + 3 = 5.
When you increase both the numerator and denominator of a fraction by the same amount, the result will always be greater than the original fraction. For example, start with \(\frac{2}{3}\). Now add 5 to the numerator and denominator \(\frac{2 + 5}{3 + 5} = \frac{7}{8} \), which is greater than \(\frac{2}{3}\).
The product of two numbers will always be larger than either of the original numbers. For example, \(3 \times 11=33\). The result, 33, is larger than both 3 and 11.

Sample teacher prompts to support student thinking:

You showed me that it works for these numbers. What about other numbers?
You have tested whole numbers. What about other types of numbers we have studied so far?
Why did you choose these examples?
How do you know that the new fraction is larger than the original fraction? Show me.
How did you compare the two fractions to determine which is larger?
Where would these numbers be on a number line?
How are you convinced that this works for all types of fractions? What strategy could you use to verify that it works for both proper and improper fractions?
Have you tested fractions less than \(\frac{1}{2}\)? Do you think it would still work for these fractions? How could you prove it to me?
How can you describe the group of fractions for which the statement is true and the group for which it is not?
What is special about the fractions for which the statement is (or is not) true? What characteristics do these fractions have that make them true?
Have you tried testing the statement with 0 or 1 as one of the numbers? What do you think might happen with these numbers? What is unique about these numbers that might make them important to test?
Have you tried testing the statement with positive and negative numbers? Do negative numbers follow the same pattern as positive numbers? What would happen if you had a combination of negative and positive numbers?

What specific expectations from the Number strand are most relevant to address in this type of task?
How do tasks like this help students deepen their understanding of number sense and operations?
At what points in the instructional sequence or during a lesson is this type of activity most appropriate to encourage students to reflect?