Making Sense of Fractions: Laying the Foundation for Success in Algebra Nadine Bezuk and Steve Klass NCTM Annual Conference--Salt Lake City April 10, 2008

2 What makes fractions so difficult for students? What do students need to know and be able to do so they can reason with fractions? How can developing fraction reasoning help students to reason algebraically? The Big Questions

3 Connecting Arithmetic and Algebra “If students genuinely understand arithmetic at a level at which they can explain and justify the properties they are using as they carry out calculations, they have learned some critical foundations of algebra.” Carpenter, Franke, and Levi, 2003, p. 2

4 Laying the Foundation for Algebra Encourage young students to make algebraic generalizations without necessarily using algebraic notation. NCTM Algebra Research Brief

5 Fraction concepts and number sense about fractions Equivalence Order and comparison Meaning of whole number operations Students need to understand these topics well before they can be successful in operating with fractions. Students need to be successful with fraction reasoning and operations if we want them to have success in transitioning to algebraic thinking. Foundation for Fraction Reasoning

6 Grade 3 - Foundational fraction concepts, comparing, ordering, and equivalence... They understand and use models, including the number line, to identify equivalent fractions. Grade 4 - Decimals and fraction equivalents Grade 5 - Addition and subtraction of fractions Grade 6 - Multiplication and division of fractions Grade 7 - Negative integers Grade 8 - Linear functions and equations From the NCTM Focal Points: Relating Fractions and Algebra

7 Area/region Fraction circles, pattern blocks, paper folding, geoboards, fraction bars, fraction strips/kits Set/discrete Chips, counters, painted beans Length/linear Linear Number lines, rulers, fraction bars, fraction strips/kits Types of Models for Fractions

8 What Should Students Understand about Fraction Concepts Meaning of the denominator (number of equal-sized pieces into which the whole has been cut) Meaning of the numerator (how many pieces are being considered) The more pieces a whole is divided into, the smaller the size of the pieces

9 What is Equivalence, Anyway? “Equivalence” means “equal value” A fraction can have many different names Understanding that 1/2 is equivalent to many other fractions helps learners to use that benchmark Simplify: when and why: (does “simplify” mean “reduce”?)

10 Ordering Fractions Fractions with the same denominator can be compared by their numerators.

11 Ordering Fractions Fractions with the same numerator can be compared by their denominators.

12 Ordering Fractions Fractions close to a benchmark can be compared by finding their distance from the benchmark.

13 Ordering Fractions Fractions close to one can be compared by finding their distance from one.

14 Strategies for Ordering Fractions Same denominator Same numerator Benchmarks: close to 0, 1, 1/2 Same number of missing parts from the whole (”Residual strategy”)

15 “Clothesline” Fractions Activity,,

16 “Clothesline” Fractions Activity,

17 “Clothesline” Fractions Activity,,

18 “Clothesline” Fractions Activity,,

19 “Clothesline” Fractions Activity,,

20 “Clothesline” Fractions Activity X,,

21 “Clothesline” Fractions Activity, (where x ≠0),

22 “Clothesline” Fractions Activity,,

23 “Clothesline” Fractions Activity,, (where x ≠0)

24 “Clothesline” Fractions Activity (where x ≠0)

25 “Clothesline” Fractions Activity (where x ≠-1)

26 The Number Line Helps Develop Fraction sense Benchmarks Relative magnitude of fractions Algebraic connections

27 Fractions aren’t just between zero and one; they live between all the numbers on the number line; A fraction can have many different names; There are more strategies than just “finding a common denominator” for comparing and ordering fractions; Fractions can be ordered on a number line just like whole numbers. The thinking involved when placing fractions on a number line can be symbolized algebraically. What Should “Kids” Know?

28 Contact Us : Slides and Fraction Tents Master are available at: (click on “PDC Presentations”)