Scott's 6th grade math students engaged in a "number talk," whereby students were asked to multiply a fraction by a mixed number: 1 1/3 by 2/3. Students shared their responses to the whole-class. Students then discussed the two approaches in small groups. This video shows one group of students working through different approaches to solving this problem.
Focus Practices: Reasoning (MP2) and Sensemaking (MP1)
During a number talk prior to the clip, Scott asked two students, Diana and Nico, to use mathematical reasoning (MP2) to explain their approaches to the class:
Approach 1: Diana multiplied the fractions and then added the 1 from the mixed number to get 1 2/9: 1 1/3 * 2/3 = 1 2/9.
Approach 2: Nico first converted the mixed number 1 1/3 into an improper fraction 4/3. He then multiplied the improper fraction by 2/3 to get 8/9: 1 1/3 * 2/3 = 4/3 * 2/3 = 8/9.
Scott asked the class to discuss and evaluate the two approaches in small groups.
1. In what ways was Nora’s approach different from Nico’s approach?
2. What strategies did Nora use to communicate her approach to her group?
3. What other math practices are evident in this group?
Understanding Diana’s Approach
When evaluating Diana’s approach, Nora recognized her error in that she “multiplied the fraction and then just moved the 1”. Olivia demonstrates MP3 (construct viable arguments and critique the reasoning of others) at the very beginning of the clip as she is trying to make sense of Diana's strategy when she asks, "How can you take away the whole number?" and Noah follows up with another question, "but why would the whole number be..." He gets interrupted but clearly both students are exchanging ideas about Diana's strategy.
In her attempt to explain why Diana’s response was incorrect, she proposed her own approach. First, she multiplied the fraction of both terms (1/3 * 2/3). Next, she multiplied the whole number with the fraction (1 * 2/3). Finally, to add the fractions with unlike denominators (2/9 + 2/3), Nora converted the second fraction (2/3) into an equivalent fraction (6/9) before adding the fractions (2/9 + 6/9). Nora is using the distributive property when she decomposes the mixed number into two parts and multiplies the factor, 2/3, by each of the separate parts.
Using multiple approaches to solve problems
Nora recognized that multiple approaches could be used to solve the problem. Olivia and Nora appeared to be using parallel approaches to solve the problem. They recognized the validity of both approaches because they resulted in the same answer (“Yeah, I got the same thing.”). Although we could not hear all of Olivia's reasoning, elements of her approach seemed consistent with Nick's approach, which involved converting the mixed fraction into an improper fraction.
Attempts to clearly explain her approach to others
In response to Victoria’s confusion, Nora elaborated upon her reasoning both verbally and using the symbolic representations in her notebook.
For example, she clarified the procedure used to convert whole numbers into equivalent fractions ("1 is 1/1") to facilitate the multiplication of 1 and 2/3. To facilitate the addition of fractions with different denominators, she noted the need to multiply the numerator and denominator of 2/3 by 3. Nora appears to have been successful in helping Victoria understand her approach.
Highlighting general characteristics of approach
Olivia sought to check her understanding of Nora's approach. Nora identified an error in what entities were multiplied together. In her explanation, Nora used a more conceptual approach that leveraged academic vocabulary (e.g. "whole number" and "fraction" (MP6)) and general rules that could be applied to a range of similar situations. Olivia considered generalizability by questioning whether this approach could be used to solve a range of different problems ("I wonder if you can do that for all of them?") and Nora used her prior experiences to affirm the approach’s viability (“That's how I do all of them") (MP8).
Watch the video from Scott's perspective as he reflects on his students' use of the mathematical practices to explain their ideas to each other.