What is a Fraction? … be flexible, use appropriate tools strategically

What if we use technology to visualize new concepts and interact with math to investigate and learn? What if we pair a process learning progression with a content learning progression?

By the end of this lesson, we want every learner to be able to say:

I can explain and illustrate that a fraction a/b is the quantity formed by a parts of size 1/b, and I can represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0.

AND

I can apply mathematical flexibility to show what I know using more than one method.

We have completed Jo Boaler’s two courses – How to Learn Math: For Students, and How to Learn Math: For Teachers and Parents.  As a team we are working on our math flexibility with math learners of all ages.  We challenge ourselves to offer more visuals and additional pathways for success. How might we leverage appropriate tools and use them strategically?

Enter: Building Concepts lessons from Texas Instruments.  Kristi Story (@kstorysquared) used What is a Fraction? to review and assess what is already known with our 6th graders.

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To offer a glimpse of the learning experience, a copy of my raw notes from this lesson are below.

Kristi starts with The Number of the Day to chalk talk a number talk.

It is obvious that our students have an understanding of fractions, decimals and percents.  Kristi encourages students to and modeled making connections between different representations of 2 1/5, the number of the day.  Many students answered aloud and enthusiastically moved to the board to draw or write a different representation.  By using the chalk talk method, this number talk encouraged number flexibility and creativity and the number talk offered all learners the opportunity to expand their understanding and fluency.

2+1:5

Kristi launches the TI-Nspire software and the lesson What is a Fraction? and encourages our students to explore and investigate what the software will do and interpret the results.  This led to a side conversation about 1.5/3 and complex fractions.

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Kristi introduces the vocabulary of unit fraction. Interesting discussion and another chance for mathematical flexibility happens when students are asked to describe/illustrate what happens when the value of the denominator increases.  How does the number of equal parts in the interval from 0 to 1 change? What happens to the length of those parts?

Students clearly possess background knowledge of fractions, and Kristi challenges them to become more flexible in representing fractions.  Note: Many students are drawing circles to represent fractions.  In addition, we want them to draw number lines  and rectangles.

The discussion transitions to compare 3/5 to 7/5. Student answers included

3/5 is 3 copies of 1/5.
3/5 is a little more than 1/2
3/5 is 60% of the way between 0 and 1
3/5 is 2/5 back from 1
7/5 is 2/5 more than 1
7/5 is 3/5 less than 2
Both are 2/5 away from 1 but in different directions.

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Kristi and students use Think-Pair-Share to describe how they decided to explain their answer to the question Is 11/8 closer to 1 or 2? Kristi asks everyone improve their answer based on partner feedback. Kristi asks for volunteers to read their partner’s idea.

From me to Kristi:

I thought today was great! I love how you facilitated a discussion encouraging all learners to talk about math. My notes are attached.  Thank you for your willingness to pilot this software with our students.  I was glad to hear that you have enjoyed this start with fractions.

From Kristi:

Thank you for all the feedback. As I said yesterday, it was exciting to present fractions in a way that I think will make a difference in their understanding of fractions. I’m looking forward to continuing this series.

What if we use technology to visualize new concepts and interact with math to investigate and learn?


#LL2LU for What is a Fraction?

Level 4:
I can decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation, and I can justify decompositions by using a visual fraction model.

Level 3:
I can explain and illustrate that a fraction a/b is the quantity formed by a parts of size 1/b, and I can represent a fraction a/b on a number line diagram by marking off a lengths 1/from 0.

Level 2:
I can represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts.

Level 1:
I can explain and illustrate that a fraction 1/b is the quantity formed by 1 part when a whole is partitioned into b equal parts.

I can compare fractions by reasoning about their size.

Level 3:

#LL2LU for Mathematical Flexibility

Level 4:
I can analyze different pathways to success, find connections between pathways and add new strategies to my thinking.

Level 3:
I can apply mathematical flexibility to show what I know using more than one method.

Level 2:
I can show my work to document one successful  method.

Level 1:
I can find and state a correct solution.


What if we pair a process learning progression with a content learning progression?

 

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