Fractions! Research-Based Teaching Strategies for the Key to Algebra Success

The Common Core Standards of Mathematics devote a strand of standards for grades 3-5 on fractions. Why Fractions?

”A child’s knowledge of fractions in fifth grade predicts performance in high school math classes, even after controlling for IQ, reading achievement, working memory, family income and education, and knowledge of whole numbers.” Siegler, 2012 Why Fractions?

Fraction Progression from Illustrative Mathematics Grade 3Grade 4Grade 5 The meaning of fractions Fractions on the number line Equivalent fractions Comparing fractions Adding and subtracting fractions Multiplying a fraction by a whole number Multiplying and dividing fractions Decimal fractionsMultiplication as scaling

: 44 teachers : 24 teachers From Akron Public Schools and surrounding districts. Showed statistically significant gains in conceptual understanding of fraction knowledge. Fractions Workshop

Five Recommendations from Panel

1.Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts. (2 nd grade) Recommendations from “Doing What Works” (IES)

Three children want to share 12 cookies. 13 cookies (Sharing a set of objects) Two children want to share 5 apples. Three children want to share. What happens to the pieces? (Partitioning multiple & single objects.) Examples:

Interpreting the remainder: review/29139 Jillian has 26 brownies which she will put onto plates that can each hold exactly 4 brownies. How many plates can be completely filled? Interpreting the Remainder

Ignore the remainder. The problem does not make use of the remainder, so it can be ignored. Jillian has 26 brownies which she will put onto plates that can each hold exactly 4 brownies. How many plates can be completely filled? Remainder Choices

Add 1 to the whole number quotient. In order to take care of all parts of the problem, one must be added to the whole number quotient. Julia has 26 brownies which she will put onto plates that can each hold exactly 4 brownies. How many plates will she use to hold all the brownies? Remainder Choices

The remainder is the answer. The question asks what is left after something has been divided. Reed has 26 brownies which he will put onto plates that can hold exactly 4 brownies. After he fills as many containers as he can, how many brownies will be left over? Remainder Choices

Share the remainder. Share the remainder so it becomes part of the quotient. This will result in a fraction or decimal. Victoria has 26 brownies. She will put 4 brownies on a plate and any remaining brownies on plate. How many plates will be filled and what fractional part will another plate hold? Remainder Choices

2.Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward. Recommendations from “Doing What Works” (IES)

Why? - Okimoto video - er-lines-a-key-representational-tool er-lines-a-key-representational-tool How? Fraction Track Game Fractional Clothesline Number Lines as Central Tools

How? Fractional Clothesline Number Lines as Central Tools

How? Fraction Track Game Number Lines as Central Tools

3.Help students understand why procedures for computations with fractions make sense. Recommendations from “Doing What Works” (IES)

Use a number line instead of circles (What does a common denom mean?) Comparing/Adding: VisualFractions.com,VisualFractions.com Oh No Fractions App Changing mixed to improper and back again Comparing/Adding/Subtracting

Use an array: Multiplying Fractions

-Why? -We haven’t done this in Western math curriculum. E.g. Liping Ma (1999): Asked 72 Chinese and 24 American elementary teachers, “Divide 1 ¾ by ½ and make up a story for the problem.” Only 9 Amer teacher gave a correct answer in the appr form. All Chinese tchrs gave correct answer. Only one Amer teacher gave a technically correct problem.Liping Ma -How? -Dividing: “How many _ go into _?” -Common denominator Dividing Fractions

4.Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross- multiplication as a procedure to use to solve such problems. Recommendations from “Doing What Works” (IES)

39 __=__ X36 How do you solve?

What’s the independent variable? Dependent? Build Up Strategies

-Why? -Instead, emphasize scale factor: -How? -e.g. to make purple paint, one company mixes 2 pints of red with 3 pints of blue. How many pints are needed to make 10 pints of purple? Postponing Cross-Multiplication RedBlueTotal 23 10

Unit on “Equations in Two Variables” McGraw Hill ●Constant rate of change ○ proportional linear relationship ●slope of a line ○ slope = rise/run Moving into 8th Grade