Modeling & Representation: Fractions and Decimals Grade 3-6.

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?

?

ObjectivesParticipants 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 3rd-6th 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?

• Jose said is represents .

• Amanda said it represent .

• Tanya said it represents 2.

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 BrowniesOn the index card provided, draw a representation of

Story ProblemsWhat story problems could you create to represent …

– in distance (PE class)?– as a ratio?– as a measurement?– as a statistic?

True or False? A fraction is a part

of a whole.

Progression of Clusters:Number and Operations-Fractions (Domain)

3RD GRADE 4TH GRADE 5TH GRADE 6TH 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

The Importance of the Number Line

Progression of Clusters:Number and Operations-Fractions (Domain)

3RD GRADE 4TH GRADE 5TH GRADE 6TH 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)

3RD GRADE 4TH GRADE 5TH GRADE 6TH 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

At your tables, complete these problem using modeling strategies:

The zookeeper has 4 cups of frog food. His frogs eat cup of food each day. How long can he feed the frogs before the foods runs out?

Math Operations: Fractions and Whole Numbers

Repeated Addition on its way to Grouping and Combining Strategy

Direct Modeling

Math Operations: Adding FractionsIt takes of a yard of material to make a pillow. How

many yards of material would it take to make 15 pillows?

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 FractionsNina has 10 ½ yards of fabric to make pillows. If each pillow takes of a yard of material, how many pillows can Nina make before she runs

out of fabric?

Math Operations: Adding Fractions with Unlike Denominators

Tina and Tony painted pictures this afternoon. Tina used a jar of blue paint for her picture. Tony used of the same size jar of blue paint for his picture. How much blue paint did Tina

and Tony use altogether for their paintings?

THEY DECOMPOSED!

TINA TONY

Multiplying Fractions

Modeling Strategies: Area Models

Using an area model, illustrate x ?

Answer: ¼

Dividing Fractions

Modeling Strategies: Area Models

Let’s try it on our own. How would you represent ÷ ?

Progression of Clusters:Number and Operations-Fractions (Domain)

3RD GRADE 4TH GRADE 5TH GRADE 6TH 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 TenStart 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 TenHow might you model .6 and .60 ( and ) using area models?

Decimals: Begin with Powers of Ten1.28 + 1.67 = ?

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

Decimals: Students’ RealizationsAs 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