# fraction progressions

of 22 /22
Presented by: Jenny Ray, Mathematics Specialist Kentucky Department of Education Northern KY Cooperative for Educational Services Jenny C. Ray Math Specialist, NKY Region Kentucky Department of Education 1

Author: odessa-love

Post on 31-Dec-2015

47 views

Category:

## Documents

Embed Size (px)

DESCRIPTION

Presented by: Jenny Ray, Mathematics Specialist Kentucky Department of Education Northern KY Cooperative for Educational Services. Fraction Progressions. Children’s Ideas about Fractions :. Show me where ½ could be on the number line below:. - PowerPoint PPT Presentation

TRANSCRIPT

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:

Kentucky Department of Education 2

0 1 2 3

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

3 > 2 ALWAYS. 1 = 1 ALWAYS.

So…how can it be that 1/3 > ½ ?

Kentucky Department of Education 3

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.

Kentucky Department of Education 4

Kentucky Department of Education 5

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.

Kentucky Department of Education 6

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.

Kentucky Department of Education 7

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)

Kentucky Department of Education 8

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

FRACTION MANIPULATIVES

Kentucky Department of Education 9

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

Kentucky Department of Education 10

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

Kentucky Department of Education 11

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

Kentucky Department of Education 12

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

Learning Activity: Number Lines

Kentucky Department of Education 13

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.”

Kentucky Department of Education 14

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

Kentucky Department of Education 15

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."

Kentucky Department of Education 16

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."

Kentucky Department of Education 17

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.

Kentucky Department of Education 18

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?

Kentucky Department of Education 19

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

Kentucky Department of Education 20

Kentucky Department of Education 22