Modeling & Representation: Fractions and Decimals Grade 3-6

Warm-Up Using all seven shapes in each plastic bag, create a perfect square.

Warm-Up Hint: One half of the square is the two large rectangles.

Warm-Up What fraction of the whole is the square? ?

Objectives Participants will be able to: Identify common pitfalls in the way fractions and decimals are traditionally taught, and ways to avoid them Understand the progression of knowledge and skills students need to be proficient in fractions and decimals from 3 rd -6 th grade. Identify and practice strategies that help students build a conceptual understanding of fractions in order to master standards

Where do you start? At your tables, discuss what prior knowledge or experience a child might have with fractions? EQUAL SHARING: How would you equally divide 4 candy bars amongst 6 people?

Equal Sharing Example

Equal Sharing “Equal Sharing problems allow your students to learn fractions using what they already understand as a foundation. The evolution of children’s strategies for Equal Sharing problems follows a predictable pattern...it is the basis for children’s developing understanding of the multiplicative relationship between the numerator and the denominator.”

What does this represent?

Redefining the “whole” Instead of asking “How many pieces is the brownie cut into?” ask “How many of these parts fit in the WHOLE?”

Beyond Pizzas and Brownies

Story Problems

True or False? A fraction is a part of a whole.

Progression of Clusters: Number and Operations-Fractions (Domain) 3 RD GRADE4 TH GRADE 5 TH GRADE 6 TH GRADE Develop understanding of fractions as numbers: -number line -fraction equivalence Extend understanding of fraction equivalence and ordering -compare fractions with different num/den Build fractions from unit fractions by applying understanding of operations on whole numbers Understand decimal notation for fractions, and compare decimal fractions -With denominators 10 and 100 Use equivalent fractions as a strategy to add and subtract fractions -with unlike denominators -using models and benchmark fractions Apply and extend previous understandings of multiplication and division to multiply and divide fractions Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Fraction Equivalence

The Importance of the Number Line

Progression of Clusters: Number and Operations-Fractions (Domain) 3 RD GRADE4 TH GRADE 5 TH GRADE 6 TH GRADE Develop understanding of fractions as numbers: -number line -fraction equivalence Extend understanding of fraction equivalence and ordering -compare fractions with different num/den Build fractions from unit fractions by applying understanding of operations on whole numbers Understand decimal notation for fractions, and compare decimal fractions -With denominators 10 and 100 Use equivalent fractions as a strategy to add and subtract fractions -with unlike denominators -using models and benchmark fractions Apply and extend previous understandings of multiplication and division to multiply and divide fractions Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Comparing Fractions with Different Denominators “Recognize that comparisons are valid only when the two fractions refer to the same whole.” Fraction Bars/Strips Number Line Placement

Comparing Fractions with Different Denominators

Progression of Clusters: Number and Operations-Fractions (Domain) 3 RD GRADE4 TH GRADE 5 TH GRADE 6 TH GRADE Develop understanding of fractions as numbers: -number line -fraction equivalence Extend understanding of fraction equivalence and ordering -compare fractions with different num/den Build fractions from unit fractions by applying understanding of operations on whole numbers Understand decimal notation for fractions, and compare decimal fractions -With denominators 10 and 100 Use equivalent fractions as a strategy to add and subtract fractions -with unlike denominators -using models and benchmark fractions Apply and extend previous understandings of multiplication and division to multiply and divide fractions Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Math Operations: Fractions and Whole Numbers

Repeated Addition on its way to Grouping and Combining Strategy Direct Modeling

Math Operations: Adding Fractions

Math Operations: Fractions and Whole Numbers “Multiple Groups problems help children reinforce and extend their understanding of fractions in terms of mathematical relationships, which is foundational to understanding equivalence and to operating on fractions-adding, subtracting, multiplying and dividing them.”

Math Operations: Dividing Fractions

Math Operations: Adding Fractions with Unlike Denominators THEY DECOMPOSED! TINA TONY

Multiplying Fractions

Modeling Strategies: Area Models Answer: ¼

Dividing Fractions

Modeling Strategies: Area Models

Progression of Clusters: Number and Operations-Fractions (Domain) 3 RD GRADE4 TH GRADE 5 TH GRADE 6 TH GRADE Develop understanding of fractions as numbers: -number line -fraction equivalence Extend understanding of fraction equivalence and ordering -compare fractions with different num/den Build fractions from unit fractions by applying understanding of operations on whole numbers Understand decimal notation for fractions, and compare decimal fractions -With denominators 10 and 100 Use equivalent fractions as a strategy to add and subtract fractions -with unlike denominators -using models and benchmark fractions Apply and extend previous understandings of multiplication and division to multiply and divide fractions Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Decimals: Begin with Powers of Ten Start out by explaining to your students that fractions with a denominator of 10 could also be written with a decimal point. Seven tenths = =.7 Four and 2 tenths = 4 = 4.2

Decimals: Begin with Powers of Ten

= ?

Decimals: Begin with Powers of Ten 0.3. x 0.4 = ?

Decimals: Students’ Realizations As students started using and discussing decimal notation, they realized that the relationship between the place values to the right of the decimal point is the same as the relationships between the place values to the left of the decimal point. “There are 10 hundreds in a thousand, then you go 1 [place] over and there are 10 tens in a hundred, then you go 1 more over and there are 10 ones in 10, and you can even go 1 more over still and there are 10 tenths in 1. It keeps going and going and going.”

Decimals: Number Line

Power My Learning

CLOSING List 3 important facts you learned List 2 strategies you are excited to try List 1 question you still have