Tag Archives: fractions

Fractions with unlike denominators – a lesson plan

Vicki graciously allowed me to teach our 5th graders again today.  Kerry and Marsha gave their time to observe and offer me feedback.

Learning Progression 

Level 4:
I can show what I know in numbers and pictures.

Level 3:  
I can use equivalent fractions to add and subtract fractions.

Level 2:
I can use visual models to add and subtract fractions.

Level 1:
I can decompose fractions into the sum or difference of two fractions.

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We used the Navigator to collect responses from all learners prior to going over each task.

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Think of the questions and the peer-to-peer discussions.  There are as many students answering 4/8 as 14/15.  Can you describe the thinking that might yield an answer of 4/8?

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How might we work toward making thinking visible? Why is peer-to-peer discourse so important? What if we practice flexibility to show what we know more than one way?

Learning Progressions – Zooming out and in

Do we take the time to zoom out as well as zoom in? Are we aware of what is essential to learn in the course that precedes ours or the course after?

Zooming out:

From Common Core State Standards: Numbers and Operations__Fractions:

Grade 6

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Apply and extend previous understandings of numbers to the system of rational numbers.

Grade 5

Use equivalent fractions as a strategy to add and subtract fractions.

Apply and extend previous understandings of multiplication and division

Grade 4 

Extend understanding of fraction equivalence and ordering.

Build fractions from unit fractions

Understand decimal notation for fractions and compare decimal fractions

Grade 3 

Develop understanding of fractions as numbers

Zooming in:

#LL2LU draft for Develop understanding of fractions as numbers.

Grade 6 – Level 4:
I can understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q).

Grade 6 – Level 3:
I can interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions by using visual fraction models and equations to represent the problem.

Grade 6 – Level 2:
I can understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.

Grade 6 – Level 1:
I can extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.


Grade 5 – Level 4:
I can interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions by using visual fraction models and equations to represent the problem.

Grade 5 – Level 3:
I can add and subtract fractions with unlike denominators, including mixed numbers by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

Grade 5 – Level 2:
I can apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Grade 5 – Level 1:
I can interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).


Grade 4 – Level 4:
I can add and subtract fractions with unlike denominators, including mixed numbers by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

Grade 4 – Level 3:
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.

Grade 4 – Level 2:
I can add and subtract mixed numbers with like denominators by replacing each mixed number with an equivalent fraction.

Grade 4 – Level 1:
I can understand a multiple of a/b as a multiple of 1/b, and I can use this understanding to multiply a fraction by a whole number.


Grade 3 – 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.

Grade 3 – Level 3:
I can explain equivalence of fractions in special cases, and I can compare fractions by reasoning about their size using a visual fraction model.

Grade 3 – Level 2:
I can compare two fractions with the same numerator or the same denominator by reasoning about their size.

Grade 3 – Level 1:
I can express whole numbers as fractions, and I can explain why fractions that are equivalent to whole numbers.

What if we vertically align and share learning progressions based on both the zoomed out view and the zoomed in view?

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?