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
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