fractions & rations across the common core€¦ · 3rd grade ccss • understand a fraction 1/b as...

Fractions & Rations Across the Common Core

Pamela Weber Harrisfacebook.com/numeracy

@[email protected]

Book signing 2:00 pm

Heinemann Booth

Monday, April 22, 13

mailto:[email protected]:[email protected]

Math is figure-out-able!

[email protected], April 22, 13


Counting StrategiesAdditive Thinking

Multiplicative ThinkingProportional Reasoning

Algebraic Reasoning



ProblemStrings



ProblemStrings

2 x 274 x 278 x 2710 x 279 x 275 x 27

100 x 2799 x 27324/27405/27



Graph it



What kind of relation?

• Open: packs sticks.tns

• Enter the values in lists• Enter the equation in

• What questions couldyou ask?

f (x)1 =



Fractions & Ratios

Counting StrategiesAdditive Thinking

Multiplicative ThinkingProportional Reasoning

Algebraic Reasoning


Interpretations of Rational Numbers

Part-whole (4/5 “4 parts out of 5 equal parts”)Measurement (4/5 “four one-fifth units”)Operator (4/5 “four-fifths of something”)

Quotient (4/5 “4 ÷5”)Ratio (4/5 “the ratio of 4 to 5”)

Susan Lamon Teaching Fractions and Ratios for Understanding


3rd Grade CCSS• Understand a fraction 1/b as the quantity formed by 1

part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

• 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. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

• Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.


Unit Fractions

1a


Unit Fractions

1a

3/4 is three 1/4’s


4th Grade CCSS• Explain why a fraction a/b is equivalent to a fraction (n x

a)/(n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

• Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

• Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.


4th Grade CCSS

• Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4).

• Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x (a/b) = (n x a)/b.)


5th Grade CCSS

• Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b)

• Write a fraction as the quotient of its numerator and denominator (a/b = a ÷ b)

• Solve story problems involving division of whole numbers with fraction or mixed number quotients (e.g., 3 ÷ 4 = 3⁄4)


270 ÷18



Consider the Multiplication and

Division Algorithms



How to Build Multiplicative & Proportional Reasoning at the Same Time?

• Good, rich contexts - See Cathy Fosnot, Bridges from the Math Learning Center, Connected Math, Math in Context, Discovering Algebra

• Mini-lesson: Problem Strings



ProblemStrings



ProblemStrings

2 x 3.24 x 3.28 x 3.210 x 3.29 x 3.25 x 3.215 x 3.2100 x 3.299 x 3.244.8/3.2323.2/3.2313.6/3.2480/[email protected]



6th Grade CCSS

• Understand ratio concepts and use ratio reasoning to solve problems.

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


7th Grade CCSS

• Compute unit rates associated with ratios of fractions

• Recognize and represent proportional relationships between quantities.

• Use proportional relationships to solve multistep ratio and percent problems


8th Grade CCSSUnderstand the connections between proportional relationships, lines, and linear equations.

• Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

• Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.


ProblemStrings

CC PB248109653

40045045618

[email protected]



Fraction Operations


What is 1/4 on a clock?What is 1/3 on a clock?

What is 1/4 + 1/3?


0.26 x 24



ProblemStrings



ProblemStrings

25 x 361/4 x 360.25 x 360.26 x 360.24 x 360.25 x 24 0.75 x 240.76 x 24



Math is Figure-out-able

• Look to most efficient strategies and build proportional reasoning at the same time

• Teachers build own numeracy• It’s about relationships:- Among numbers to solve problems- Between teachers and students to build

young mathematicians



Fractions & Rations Across the Common Core

Pamela Weber Harrisfacebook.com/numeracy

@[email protected]

Book signing 2:00 pm

Heinemann Booth #


fractions & rations across the common core€¦ · 3rd grade ccss • understand a fraction 1/b as...

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