Katie’s 7^{th} grade students participated in a two-part card sorting activity. First, she asked students to sort cards with fractions in any way that made sense to them (MP1). Next, Katie asked her students to add decimal cards to their grouped fraction cards. She used this activity to assess students’ initial ideas about the relationship between fractions and decimals. Some fractions had 10 or 100 in their denominators to help students make sense of the relationships between fractions and decimals. On the second day, she built on students’ ideas to develop a model to convert between fractions and decimals.

**Focus Practices**: Structure (MP7), Repeated Reasoning (MP8), and Reasoning (MP2)

How would you help students use the ideas raised during this activity to reach a consensus understanding about the relationships between fractions and decimals?

## Overview of practices shown here

MP7 (Structure) is featured in this activity because the students examined the structures found in the fractions and decimals to order and group them. MP8 (Repeated reasoning) and MP2 (explain reasoning) are used in service of understanding the underlying structure in decimals and fractions. Students used mathematical reasoning to justify how they grouped the cards (MP2). It is important to notice that the *teacher* did not evaluate the students’ responses because she wanted her *students* to reach conclusions on their own (MP2). On the second day, students used a graphic organizer to develop their ideas on how to calculate equivalent fractions and decimals. Based on their repeated calculations, students generated a rule for determining equivalency between decimals and fractions.

**Practices in Context**

**Group 1**

The first group of students reasoned abstractly and quantitatively to justify their 4 groups (MP2) and discuss the patterns and relationships they noticed (MP7): 1) 1/2 and 5/10 because they were equal, 2) all fractions with 100 as denominator, 3) all fractions with denominators with 10 or above, and 4) all fractions with single denominators. The teacher asked the students to document their reasoning on paper.

**Group 2**

The second group ordered the fractions from least to greatest. After pushing them to explain what they meant by least to greatest, students shared that they ordered first by denominator and then by numerator. In doing so, Katie prompted students to identify the structure of the mathematical pattern they used to order the fractions (MP7).

**Group 3**

The third group indicated that 4/5 and 8/10 were equivalent. When pushed to explain their answer, they indicated that they could “multiply both of them into 100” to equal 80/100, which implied that the equivalent fractions would have a common denominator. The students indicated that they would multiply the numerator and denominator by 20 to get 80/100. In doing so, the students applied properties of operations to determine equivalent fractions (MP2). The teacher also prompted students to consider the mathematical structure when asked what to multiply both the numerator and denominator by the same number to create equivalent fractions (MP7). She did the same for 8/10, except she told them that they would be multiplying both numerator and denominator by 10. Thus, students converted 8/10 to 80/100 using the same reasoning used to convert 4/5 to 80/100 (MP8).

**Group 4**

Students justified why 35/50, 70/100, and 0.70 were in the same group. They explained the relationship they saw between the decimals in the hundredths place and the fractions that had 100 in the numerator. This group used properties of fractions and decimals and their understanding of place value to group equivalent cards together (MP2). These students were on the verge of making use of regularity in repeated reasoning by generalizing the same method to determine equivalent fractions. They used their knowledge of place value to determine that the decimal of 0.35, pronounced "thirty-five hundredths," is equivalent to the fraction 35/100. Similarly, they used the same understanding of decimal place value to determine that forty hundredths (0.40) is the same as 40/100. Students used the definition of place value to evaluate the reasonableness of their results (MP8).

**Day 2**

Katie asked students to consider ways to convert fractions to decimals. After repeating this process several times, she asked them to write a rule for converting all fractions to decimals. Katie prompted students to identify the mathematical structure of the task in order to identify the most effective solution path (MP7). Students noticed repeated calculations and looked for a general method of determining how to convert a fraction into a decimal (MP8).

Watch Katie explain her design of the task and the ways in which she setup collaborative small groups in her class.

This “funnel” demonstrates how a group of teachers began to think about how the frameworks for math and science are related. The poster symbolizes a pathway towards a solution. In this case, the solution is Katie’s overall goal of getting students to understand how to convert fractions to decimals. She introduced the idea with an open-ended card sort activity to guide students towards this understanding. On the poster, the wide part of the funnel represented where students shared what they know about their current understanding of relationships between fractions and decimals. The teacher used this knowledge to funnel their ideas towards the big idea that they were required to learn. The constraints around the narrow part of the funnel represent the scaffolding the teacher provided during the second day to guide students towards an algorithm for converting fractions to decimals. She provided a safe environment for students to discuss how and why they thought their algorithms made sense. Katie’s role was to facilitate students’ discussions asking questions and allowing room for mistakes so students have opportunities to explore their ideas and make sense of what they are learning.

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