Understanding Fraction Representations

Session 2

Amy LeHewElementary Math Facilitator Meeting

October 2012

4/5 OR 5/4

Which is Closer to 1?

Solutions?

4/5 OR 5/4

Last time we looked closely at types of visual fraction models and considered the connection between the task and visual model used to solve it.

Visual Fraction Models

What visual fraction model is used by most students in our schools?

What visual fraction model is used the least?

In CMS

Take a look at the 3rd grade fraction standards

Highlight the phrase “visual fraction model” everywhere it appears.

Underline the phrase “number line” everywhere it appears.

Numbers and Operations with Fractions

Develop understanding of fractions as numbers.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. 2. Understand a fraction as a number on the number line; represent fractions on a number line

diagram. 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.

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

Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.

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

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.

Turn and make a statement to your partner

about something you learned while highlighting/underlining.

Numbers and Operations with Fractions

Support understanding of important properties of fractions◦ Numerical unit◦ Relationship between whole numbers and

fractions◦ Density of rational numbers (infinite # between

any two)◦ A number can be named infinitely many wayshttp://lmr.berkeley.edu/docs/Pt3-Ch13NCTM%20Yearbook07-4.pdf

Number Lines

Most widely used fraction model? Area◦ Partition pizzas, brownies, etc.

Limitations of area models◦ Count pieces without attending to the whole

(don’t distinguish between fractional part of a set from continuous quantity (area)).

Area Models

How can number lines help students understand fraction concepts that are often

obscured by area models?

Most students were successful identifying whole numbers on a variety of number lines.

Of the students who were successful, they counted

Very few were successful in identifying fractions on a number line

“two-sixths” for the point depicted in the figure below: it is “six spaces and that’s two.”

One-third is three pieces and one thing There isn’t another fraction name Don’t know

Most had difficulty generating equivalent names for the same point

1 1

15 or 9

How do you know?

Which is Larger?

“The smaller the denominator, the bigger

the fraction”

Always True? Sometimes True?

Read the classroom scenario on pages 20-22◦ When you finish, reflect silently on the following:

Do student in your building consider fractions as numbers? Do teachers?

What other generalizations do students and teachers make about fractions? Are some of these generalizations helpful?

Classroom Scenario

Consider the following two statements. Declare Always True, Sometimes True

If sometimes true, show an example and a counter example.

1. The larger the denominator, the smaller the fraction.

2. Fractions are always less than one.

3. Finding a common denominator is the only way to compare fractions with different denominators.

Always True? Sometimes True?

Which is larger 5/6 OR 7/8 ?

What’s the Research?

5/6 If the denominator is

smaller, the piece is bigger

5/6 The 1/6

piece is bigger than 1/8

5/6 The smaller the number the

bigger the pieces.

This one (7/8 ) because it has more pieces

7/8 Since 8 is bigger than 6

and 7 is bigger than 5

How do we ensure students make meaning of the numerator and denominator?

How can we make sure students think of fractions as numbers

Using your blank number line sheet and Cuisenaire rods, label the first line in halves

2nd line in Thirds3rd line in Fourths4th line in Sixths5th line in Twelfths

Construct a Number Line

0 1

What number is halfway between zero and one-half?

What number is one-fourth more than one-half?

What number is one-sixth less than one?What number is one-third more than one?What number is halfway between one-twelfth

and three-twelfths?What would you call a number that is halfway

between zero and one-twelfth?

Answer With A Partner

How might this impact student’s understanding of fractions as numbers on a number line? (using the Cuisenaire rods to make a number)

How does a number line diagram help students make meaning of the numerator and denominator?

This activity found on page 23-26 Look at page 27+ for using 2 wholes

From Area to Linear

Time has run out for the tortoise and the hare!

The Hare jumped five-eighths of the way

The tortoise inched along to three-fourths of the way.

Who is winning?

The Tortoise and the Hare

How would constructing a number line diagram to solve this task help students

make meaning of the numerator and denominator?

The Tortoise and the Hare

With fractions? Yea or Na

Open Number Lines

Use < = > to compare the following sum:

(do not add) *You must represent your thinking on a number line diagram.

1/2 + 1/4 ______

1/3 + 1/5

DO NOT ADD!!!

3.NF Closest to 1/2 3.NF Comparing sums of unit fractions 3.NF Find 1 3.NF Find 2/3 3.NF Locating Fractions Greater than One on

the Number Line 3.NF Locating Fractions Less than One on the

Number Line 3.NF Which is Closer to 1?

Illustrative Mathematics Project

Bring 3rd grade unit 7: Finding Fair Shares Bring 4th grade unit 6: Fraction Cards and

Decimal Squares

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