challenges and opportunites of common core: fraction...

Challenges and Opportunites of Common Core: Fraction

Teaching and Learning

Tad Watanabe Kennesaw State University

science.kennesaw.edu/~twatanab

Stages of Fraction Teaching & Learning in CCSS

•  Laying foundations: Grades 1 ~ 2

•  Understanding fractions as numbers: Grades 3 ~ 4

•  Fraction arithmetic: Grades 4 ~ 6

Laying Foundations

•  1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

Laying Foundations

•  2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Some Key Ideas

•  Equal partitioning •  Describing shares •  Exploring the relationship between

the size of shares and the number of shares

Equal Partitioning

Challenge – How do students know that shares are

equal?

Equal Partitioning

Challenge – It’s about quantities, not shapes.

Reasoning Empirical More abstract

Both and

are half of the same sized share. Therefore they are equal.

Describing Shares

Singapore Textbook, Standard Edition, 2B, p. 62

Describing Shares

•  CCSS does not specify the use of fraction notations in Grades 1 & 2.

•  Gunderson & Gunderson (1957) – At the beginning of fraction instruction,

we should write out the fraction words rather than use standard notation: 3-fourths, instead of ¾.

Number and Size of Shares


Number and Size of Shares

Challenge – Fractions as

relationships and fractions as quantities

– Both pizzas are divided into halves and quarters.

– Which half is bigger?

– Which is bigger, half or qurater?

Fractions as Numbers •  Grade 3 Overview

Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

Fractions as Numbers

Opportunity: Unitary view of fractions •  3.NF.1

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.

Unitary View of Fractions

Tokyo Shoseki (2006), 3B, p. 57

Unitary View of Fractions


Fractions on the number line

•  3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Fractions on the number line

Challenge – 1 as a whole – 1 as a point on the number line vs. 1

as the distance from 0, or the length of a unit interval

Bridging to number line


Fractions as Numbers •  3.NF.3

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

•  4.NF.1 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 themseselves are the same size. Use this principle to recognize and generate equivalent fractions.

Equivalent Fractions


Equivalent Fractions Challenge (Grade 4)

If we divide each thrid into 2 equal parts…

Therefore, .

But, why “x” when we “divide”?

Comparing Fractions

•  3.NF.3.d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Comparing Fractions as Numbers

•  Whole = 1 •  Key understandings – The numerator tells us how many units

there are. – The denominator tell us the size of the

unit in terms of the number of equal parts 1 must be partitioned.

– The size of fractions are determined by considering BOTH.

Important Misconception

•  Given two proper fractions, the one with the smaller difference between the numerator and the denominator is larger.

Example 2/3 > 2/5 because 3-2 < 5-2

Complement


Comparing Fractions (Gr. 4)

•  4.NF.2 … or by comparing to a benchmark fraction such as ½. …


Whole Numbers and Fractions

•  3.NF.3.c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form of 3 = 3/1; recognize that 6/1 = 6; located 4/4 and 1 at the same point of a number line diagram.

Grade 3 footnote Grade 3 expectations in this domaing (NF) are limited to fractions with denominators 2, 3, 4, 6, and 8.

1 as denominator •  Perhaps it may be better to

introduce it in the context of quotient meaning of fraction in Grade 5, i.e., 5 = 5 ÷ 1 = 5/1.

Tokyo Shoseki (2006), 5B, pp. 51 - 52

… For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts.

Grade 3 Overview

Improper fractions, mixed numbers, whole numbers

Tokyo Shoseki (2006), 4B, pp. 40 & 43

Addition/Subtraction

Opportunity – Unitary View of Fractions


Tokyo Shoseki (2006), 3B, pp. 62 - 63


Tokyo Shoseki (2006), 5A, p. 44

Addition/Subtraction •  You can only add/subtract numbers

when they are referring to the same unit. – Grade 1: addition/subtraction of

multiples of 10 Examples: 30 + 40; 80 – 60

– Grade 2 – 5: solving problems involving measurement

– Grade 5: decimal number addition/subtraction Example: 0.3 + 0.4; 0.8 – 0.6

Multiplication/Division

Opportunity – Clear distinction between multiplying

fractions by whole numbers (Grade 4) and multiplying by fractions (Grade 5)

Meaning of Multiplication

•  3.OA.1 Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 roups of 7 objects each.

(# of groups) x (group size) = product

Multiplication

•  Grade 4: multiply a fraction by a whole number 5 x ¾, but not ¾ x 5

•  Grade 5: multiplying by a fraction ¾ x 5 as well as ¾ x ½

Meaning of Multiplication •  5 x ¾ means 5 groups of ¾ in each

group. •  Does not require an adjustment in

the meaning of multiplication. •  5 groups of 3 ¼-units, or 5x3 ¼-

units. 5 x ¾ = 5 x (3 x ¼) = (5 x 3) x ¼

Meaning of Multiplication •  ¾ x (###) means ¾ times as

much of (###) – 5.NF.5 •  4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.

Meaning of Multiplication Challenge •  4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.

So, can both 5 x ¾ and ¾ x 5 be interpreted as 5 times as much of ¾?

Multiplying by Fractions

•  5.NF.4.a Interpret product (a/b) x q as a parts of partition of q into b equal parts; equivalently, as the result of a sequence of operations a x q ÷ b.


Challenge (2/3) x 4 – 2/3 of 4 – partition 4 into 3 equal parts –  take 2 of the equal parts

(2/3) x 4 = 2 x (4 ÷ 3)


Challenge (2/3) x (4/5) – 2/3 of 4/5 – partition 4/5 into 3 equal parts –  take 2 of the equal parts

(2/3) x (4/5) = 2 x {(4/5) ÷ 3}


Challenge (2/3) x (4/5) – 2/3 of 4/5 – partition 4/5 into 3 equal parts –  take 2 of the equal parts

(2/3) x (4/5) = 2 x {(4/5) ÷ 3} 5.NF.7: Apply and extend previous

understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

Division of Fractions

•  Dividing a whole number by a unit fraction – How many unit fractions are in so

many wholes? (measurement division)

•  Dividing a fraction by a whole number – How much is in each group if a

fraction is partitioned into so many parts? (fair sharing division)

Division of Fractions •  4/5 ÷ 2 = ? – If 4 1/5-units are partitioned into 2

groups, how much is in each group? – 4 ÷ 2 = 2 1/5-units in each group. – 4/5 ÷ 2 = (4 ÷ 2)/5

Division of Fractions •  3/5 ÷ 2 = ? – If 3 1/5-units are partitioned into 2

groups, how much is in each group? – 3 ÷ 2 = not a whole number!

•  3/5 = 6/10 – If 6 1/10-units are partitioned into 2

grous, how much is in each group? – 6 ÷ 2 = 3 1/10-units in each group – 3/5 ÷ 2 = 6/10 ÷ 2 = (6 ÷ 2)/10 = 3/10

Division of Fractions •  In general,

a/b ÷ n = a/(b × n)

Multiplication and Division

Opportunity: double number line (2/3) x (4/5) –  2/3 of 4/5 –  partition 4/5 into 3 equal parts –  take 2 of the equal parts


2/5 ÷ 3/4 = ?


•  Mathematical practice: repeated reasoning 2/5 ÷ 3/4 = ? – How much is for 1/4? – Take 4 of that amount to find how

much is for 1.


•  Mathematical practice: repeated reasoning 2/5 ÷ 3/4 = ?

Multiplicatin & Division

Opportunity: Developing a unified understanding of multiplication, division, and proportion.

•  4/5 x 2/3

•  2/5 ÷ 3/4

Unit Approach

•  With 6 gallons of gasoline, Mike’s car can travel 250 miles. How far can he travel with 4 gallons of gasoline?

Finde per-one quantity – partitive division

challenges and opportunites of common core: fraction...

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