Table of Contents

Unit 3: Reasoning and Explaining (MP2 and MP3)

3.2.4 Levels of Explanation: Level 3

Multiple Representations with Emerging Pattern(s)

In the third level of explanations, students begin to see patterns across multiple representations. Being able to identify and understand these patterns supports learners as they begin to abstract and generalize; in this case, finding the method to decide whether any number is odd or even.

In the example below, Brandon, Chris, and Nathan are presenting their definition of odd and even, and refer to their tests to support their definition. As you saw in the previous video, they built their cube trains for numbers 1 through 30 and have made a representation of their trains (see photo below).

Students’ graph that represents multi-link cube trains of 1 through 30.
The note reads: “This represents the model of multi links we built.”

Brandon: An even number is a number that ends in 0, 2, 4, 6 or 8. It can be split in half equally. If someone counts by twos, they will land on an even number.

Chris: An odd number is a number that ends in 1, 3, 5, 7 or 9. It cannot be split equally, because there would be a remainder. If someone counts by twos, they would not make it to an odd number.

Nathan: 2610 ends in 0. If someone counts by twos, that person would get to 2610. It can be split in half equally. If you add 1305 and 1305, you would get 2610. So 2610 is even.

Chris: 187 ends in 7. If someone counts by twos, they won’t get to that number. It cannot be split in half equally. 187 divided by 2 is 93 R-1. 93 has a remainder, so 187 is odd.

Brandon: 294 ends in 4. If someone counts by twos, they would get to this number. It can be split in half equally. 294 divided by 2 is 147, so 294 is even. This (representation above) represents the model of the multi-links that we built. Then the information we got from the graph, this graph. We recorded it onto this paper to help us understand which numbers were odd and even.

Teacher: Questions?

Tristin: Remember when you said it was 1, 3, 5, 7 or 9 on the back it’s odd? How did you guys know that?

Chris: How did we know if it’s …..

Tristin: How did you figure out that if you put one of those numbers on the back, it would be odd?

Teacher: Good question, Tristin.

Chris: He’s asking us how did we know that? We would know, because we tested it out by dividing, like division. We just picked any random number by putting an odd number in the one’s place. We divided it and ended up with a remainder.


Time to View

The video below shows Brandon, Chris, and Nathan working across strategies.

Download Transcript

Time to Reflect

In your Metacognitive Journal, reflect upon the evidence that the students used in their explanations to argue for their claim.



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Argument is central to thought and the construction of knowledge (e.g., Kuhn, 1992). The significance of argument to conceptual understanding in mathematics is related to the development of students' thinking and reasoning that occurs during the acts of challenge and justification."

Wood, 1999