Presented by:Jenny Ray, Mathematics SpecialistKentucky Department of Education

Northern KY Cooperative for Educational Services

Jenny C. Ray Math Specialist, NKY RegionKentucky Department of Education 1

Children’s Ideas about Fractions: Show me where ½ could be on the

number line below:

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0 1 2 3

Why do students sometimes choose this part of the number line?

Children’s Ideas about Whole Numbers:

3 > 2 ALWAYS. 1 = 1 ALWAYS.

So…how can it be that 1/3 > ½ ?

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When students can’t ‘remember’ a procedure, they resort to performing any operation they know they can do…• Estimate the answer: 12/13 + 7/8

• A) 1• B) 2• C) 19• D) 21• E) I don’t know.

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National Assessment of Educational Progress (NAEP) results show an apparent lack of understanding of fractions by 9, 13, and 17 yr olds.

Estimate the answer: 12/13 + 7/8

Only 24% of the 13-yr-olds responding chose the correct answer, “2”.

55% selected 19 or 21These students seem to be operation on the

fractions without any mental referents to aid their reasoning.

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Results from the 2nd Mathematical Assessment of the National Assessment of Educational Progress

Perhaps you’ve seen this reasoning…

1/2 + 1/3 = 2/5

If students have an understanding of the value of the fractions on a number line, or as parts of a whole, then they can argue the unreasonableness of this answer.

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How can students learn to think quantitatively about fractions?• Based on research…• “…students should know something about the relative

size of fractions. • They should be able to order fractions with the same

denominators or same numerators as well as• to judge if a fraction is greater than or less than 1/2. • They should know the equivalents of 1/2 and other

familiar fractions. • The acquisition of a quantitative understanding of

fractions is based on students' experiences with physical models and on instruction that emphasizes meaning rather than procedures.” (Bezuk & Cramer, 1989)

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Hands on experiences help students develop a conceptual understanding of fractions’ numerical values.

FRACTION MANIPULATIVES

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Learning Activity: Fraction Circles

The white circle is 1. What is the value of each of these pieces?

1 yellow

3 reds

1 purple

3 greens    

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Now…change the unit: The yellow piece is 1. What is the value of those pieces?

Learning Activity: Using Counters

Eight counters equal 1, or 1 whole.

What is the value of each set of counters? 1 counter 2 counters 4 counters 6 counters

12 counters   

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Now, change the unit: Four counters equal 1. What is the value of each set of counters?

Learning Activity: Cuisinaire Rods

The green Cuisenaire rod equals 1.

What is the value of each of these rods? red black white dark green    

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Change the unit: The dark green rod is 1.Now what is the value of those rods?

Learning Activity: Number Lines

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A “new” way of thinking/teaching…

“Many pairs of fractions can be compared without using a formal algorithm, such as finding a common denominator or changing each fraction to a decimal.”

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Comparing without an algorithm• Pairs of fractions with like denominators:   1/4 and 3/4 3/5 and 4/5 • Pairs of fractions with like numerators:   1/3 and 1/2 2/5 and 2/3 • Pairs of fractions that are on opposite

sides of 1/2 or 1:   3/7 and 5/9 3/11 and 11/3 • Pairs of fractions that have the same

number of pieces less than one whole:   2/3 and 3/4 3/5 and 6/8

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Comparing 3/7 and 5/9…a student’s response:• The fractions in the third category are on

"opposite sides" of a comparison point.• One fourth-grade student compared 3/7

and 5/9 in the following manner (Roberts 1985):

• "Three-sevenths is less. It doesn't cover half the unit. Five-ninths covers over half."

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Comparing 6/8 and 3/5: A student’s response…• A fourth-grade student compared 6/8 and 3/5 in this

way (Roberts 1985): • "Six-eighths is greater. When you look at it, then you

have six of them, and there'd be only two pieces left. • And then if they're smaller pieces like, it wouldn't have

very much space left in it, and it would cover up a lot more.

• Now here [3/5] the pieces are bigger, and if you have three of them you would still have two big ones left. So it would be less."

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Conceptual Understanding Notice that each child's reasoning from the

previous two examples is based on an internal image constructed for fractions.

Hands-on experiences with fractional parts, both smaller than and greater than one, helps to create this conceptual knowledge, so that procedures that they develop make sense.

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Exploring fractions with the same denominators

• Use circular pieces. The whole circle is the unit. – A. Show 1/4 – B. Show 3/4

Are the pieces the same size?

How many pieces did you use to show 1/4?

How many pieces did you use to show 3/4?

Which fraction is larger? How do you know?

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Comparing fractions to ½ or 1

• Use circular pieces. The whole circle is the unit.

• A. Show 2/3 B. Show 1/4 • Which fraction covers more than one-half of the

circle? • Which fraction covers less than one-half of the

circle? • Which fraction is larger? How do you know?

• Compare these fraction pairs in the same way. – 2/8 and 3/5– 1/3 and 5/6– 3/4 and 2/3

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Resources for Activities

Illuminations (NCTM) Rational Number Project nzmaths Mars/Shell Centre Teaching Channel www.jennyray.net

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