## Reflection: Developing a Conceptual Understanding Math Maps (Part 3) - Section 1: Mathematical Evidence

To help students develop a conceptual understanding about percent, I choose to initially focus on working with finding percentages when the whole was already 100.  Then, once students became comfortable with this concept, I pushed their thinking to find percents when a fraction could easily be converted into 100.

However, in many situations, trying to use equivalent fractions to find a percentage is not the most effective or efficient way.  Therefore, it was important for me to also teach students to use division to find a decimal quotient, then use the decimal to determine the percentage.  This strategy also serves as a great review.

All students found percentages (for fractional parts when the denominator was 20, 50, or 100) using both strategies.

To provide students with a more rigorous challenge, after they completed the group 3 paper, they were provided with an opportunity to find percentages when the total distances included many different numbers (ex: 13, 30, 45, 10, 283 etc).

This challenge handout provides students with a rigorous challenge because for each example, students should choose the best approach for solving (either division or equivalent fractions).

Throughout the independent practice, I check in with students as they complete this challenge portion to conference on their approach.  Through this conferences, I can determine various levels of math proficiency based on the students approach.

Ex 1:  One student who I conferenced with told me that she used the division strategy to solve all of the problems on the sheet.  When I asked her why she choose to use the division strategy for 6/10 she said because that is the best way.  Through this experience I learned that this student is memorizing a strategy.  She needs more support in using the strategies interchangeably.

Ex 2: When I met with a second student that was working on challenge sheet told me that she used the same approach. "I used division to make each of these a decimal".  When I asked her to also look at 6/10 again, she recognized,  "It would be much easier to use equivalent fractions for this one, because 10 goes into 100. I should not have used division for 9/100 either.  That is pretty much already done for me."  This student is demonstrating more proficiency than the other student because with prompting, she was able to realize that different strategies are more appropriate for different examples.

Ex 3:  A third student explained, "I used different strategies for different problems, some where easier to divide because I couldn't make the denominator into 100, for others I used equivalent fractions. When I got to 30/45 I actually used a few strategies.  First, I knew that 15/45 was 33.3%  so I thought, I could just double that and get 66.6%.  I wanted to check my work to see if this was true, so I used the division strategy too.  I got .666 so I knew that I could use my first approach."  This student is demonstrating mathematical proficiency.  She is thinking flexibly about each problem and using repeated reasoning.

I make conference notes throughout the class.  At the group share, I use each of these students' approaches to demonstrate the continuum of progress toward proficiency.

Developing a Conceptual Understanding: Provide Additional Challenges

# Math Maps (Part 3)

Unit 6: Bringing It All Together
Lesson 8 of 10

## Big Idea: Students use mathematical statements to write encouraging statements for a service learning project.

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### Julie Kelley

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