CHAPTER 16 Developing Fraction Operations Elementary and Middle School Mathematics Teaching Developmentally Ninth Edition Van de Walle, Karp and Bay-Williams Developed by E. Todd Brown /Professor Emeritus University of Louisville

Big Ideas The meanings of each operation for fractions are the same as the meanings of operations with whole numbers. For addition and subtraction, the numerator tells the number of parts and the denominators the unit. Repeated addition and area models support development of concepts and algorithms for multiplication of fractions. Partition and measurement models lead to two different thought processes for division of fractions Estimation should be an integral part of computation development to keep students’ attention on the meanings of the operations and the expected sizes of the results.

Understanding Fraction Operations Conceptual development takes time CCSS Developmental process Grade 4 Adding and Subtracting of fractions with like denominators and multiplication of fractions by whole numbers Grade 5 Fluency with addition and subtraction of fractions and developing an understanding of multiplication of fractions and division of fractions in limited cases (unit fractions ÷ by whole numbers and whole numbers ÷by unit fractions ) Grade 6 Completing understanding of division of fractions Grade 7 Solve real-world and mathematical problems involving the four operations with rational numbers (including complex fractions)

Problem-Based Number Sense Approach Goal- Help students understand why procedures for computations make sense ( Siegler et al., 2010) Use contextual tasks Explore each operation with a variety of models Let estimation and invented methods play a big role in the development of strategies Address common misconceptions regarding computational procedures

Addition and Subtraction of Fractions Contextual Examples - Contexts need to be interesting- examples

Addition and Subtraction of Fractions Area models Circles models are often used and not always carefully drawn. Circles models offer an effective visual and allow students to develop mental images.

Addition and Subtraction of Fractions Linear Model Try this one Activity 16. 2 Jumps on the Ruler Materials: Ruler Directions: Use the ruler as a visual and find the results of these three problems without applying the common denominator.

Addition and Subtraction of Fractions Linear Model Cuisenaire rods are linear models. First decision is what to use as the whole. Determine the smallest rod that will represent the sixths. Use the models to find the sum and difference.

Estimation Methods Ways to estimate fraction sums and differences: Benchmarks- Decide whether fractions are closest to 0, ½, or 1 Relative size of unit fractions-Decide how big the fraction is based on its unit (denominator)

Try This One Activity 16.3 Over or Under 1 Materials- List of sums or differences Directions- project a sum or difference for 10 seconds. Ask students to write “over or under” one. Discuss each problem and discuss how they decided on their estimates.

Developing the Algorithms Develop side by side with visual and situations Like denominators- focus on the key ideas units are the same so they can be combined ¾ + 2/4 How many fourths altogether? 3 7/8 – 1 3/8 is counting back eighths or finding the number of eighths between the two quantities Unlike denominators- invite students to use models to determine equal parts

Are Common Denominators “Required”? The use of invented strategies will show students that there are methods to find correct solutions without finding a common denominators. Looks for ways different fractional parts are related i.e. halves , fourths, and eighths Number lines are a tool that can be used to mentally add and subtract fractions.

Addressing Misconceptions Adding both numerators and denominators Or is this correct? Can you support this answer with a rectangle partitioned into eighths? What would you want the students to discover about the model and equation?

Addressing Misconceptions Cont. Failing to Find Common Denominators Students tend to ignore the denominator and just add numerators e.g. 4/5 + 4/10 = 8/10. Number line or fraction strip can help students pay attention to the relative size of the fraction. Difficulty finding Common Multiples Skill requires a command of multiplication facts, does not require less common multiple--any multiple will work. Activity 16. 6 Common Multiple Flash Cards Materials- flash cards with number pairs directions- students work in partners and turn over a card and state a common multiple.

Multiplication of Fractions Foundational ideas of iterating fractional parts and partitioning are the core of understanding multiplication of fractions. Start with finding fraction of a whole. There are 15 cars in Michael’s matchbox car collection. Two-thirds of the cars are red. How many red cars does Michael have? Partitioning (finding part of the whole) 15 into three groups and then see how many are in two parts. 2/3 of 15 or 15 ÷ 3 x 2

Fractions of Fractions No Subdivisions Choose problems that do not require additional partitioning.

Fractions of Fractions No Subdivisions- Models Examples of the Models

Fractions of Fractions Subdividing Unit Parts

Fractions of Fractions Area Model Start with a model Model how to subdivide the parts Articulate the model as the algorithm

Fractions of Fractions Addressing Misconceptions Treating denominator the same as in addition/subtraction- Inability to estimate the approximate size of the answer. Matching multiplication situations with multiplication (and not division)

Division Contextual Examples and Models Whole number divided by whole number- partitive and sharing context. 5 ÷ 4 = ¼ x 5 = 5/4 Partitive (sharing problems) you asking, “How much is the share for one friend?”

Division Contextual Examples and Models Try this one Activity 16.13 Sandwich Servings

Developing the Algorithm

Invert-and-Multiply Algorithm Provide a series of tasks and have students look for patterns. Pose problems beginning with the divisor as a unit fraction. Move to a problem with second fraction not a unit fraction. Articulate the model as the algorithm

Addressing Misconceptions Thinking the answer should be smaller true if the fraction divisor is greater than one not true if the fraction divisor is less than one Connecting illustration with the answer 1½ ÷ ¼ How many fourths in 1½? By counting they find 6, but record 6/4. Writing remainders 3⅜ ÷ ¼ How many fourths in 3⅜? By counting they find 13 but do not know what to do with the ⅛.