CHAPTER 15 Developing Fraction Concepts 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 For students to really understand fractions, they must experience fractions across many constructs, including part of a whole, ratio, and division. Three categories of models exist for working with fractions—area, length, and set or quantity. Partitioning and iterating are ways for students to understand the meaning of fractions, especially numerators and denominators. Equal sharing is a way to build whole-number knowledge to introduce fractional amounts. Equivalent fractions are ways of describing the same amount by using different-sized fractional parts. Fractions can be compared by reasoning about the relative size of fractions. Estimation and reasoning are important in teaching understanding of fractions.

Fraction Understanding Common Core State Standards ( CSSO, 2010) Fractional experiences should begin early- 1st and 2nd grade partition shapes and refer to fractional amounts 3rd grade fractions are major emphasis, attention to using fraction symbols, exploring unit fractions 4th grade focus on fraction equivalence and begin with fraction operation

Fractional Meaning Division Part-whole Idea of sharing with an amount of something Often not thought of with fractions Operator Used to indicate an operation Ratio the probability of an event is one in four Part-whole Shading a region Part of a group of objects Measure Identifying a length and using that length as a measurement piece Focus on how much rather than how many parts

Why Fractions are Difficult; Misconceptions Thinking of numerator and denominator as separate and not as a single value Not recognizing equal parts--thinking ¾ green instead of ½ green Thinking that fraction 1/5 is smaller than 1/10 because it has a smaller denominator Using the operation rules from whole numbers to compute with fractions

Models for Fractions

Fraction Area Models Area models are good to begin with because they work well for equal sharing and partitioning.

Fraction Length Models Length models are physical materials that are compared on the basis of length, number lines are subdivided.

Fraction Set Models Set models, the whole is understood to be a set of objects and the subsets of the whole make up fractional parts.

Try this one Activity 15. 3 Class Fractions Materials- students Directions –

Construct the Idea of Fractional Parts Fraction size is Relative A fraction by itself does not describe the size of the whole or the parts A fractions tells us only about the relationship between the part and the whole. Comparing two fractions with any representation can be made only if both fractions are parts of the same size whole.

Construct the idea of Partitioning Partitioning Sectioning a shape into equal-sized parts Introduce words- halves, thirds, fourths, fifths… eighths before symbols

Partitioning with Area Model Fractional parts must be the same size, though not necessarily the same shape The number of equal-sized parts that can be partitioned within the unit determines the fractional amount

Partitioning with Area Model cont. Same shape, same size: a and f (equivalent) Different shape, same size: e and g (equivalent) Different shape, different size: b and c (not equivalent) Same shape, different size: d (not equivalent)

Partitioning with Length Models Students need to be able to partition a number line into parts and realize what part of the whole the section represents Activity 15.7 Materials- Partitioned number lines

Partitioning with linear models Partitioning is a strategy commonly used in Singapore for solving story problems TRY THIS ONE A nurse has 54 bandages. If 2/9 are white and the rest are brown, how many are brown? A bar diagram is used by a student to partition into nine parts and then figure out the equal shares of the bandages for each partition.

Partitioning with Set Models Set models can be confusing when students are determining the number of shares. When equal parts are not already figured out students may not see how to partition. Understanding that parts of the whole must be partitioned into equal-sized parts across different models in important.

Sharing Tasks Students need experiences to build an informal understanding of sharing and proportionality to develop fractional concepts (Siegler et al., 2010) Student thinking about sharing brownies and pizza.

Iterating Counting fractional parts, or iterating, helps students understand the relationship between the parts and the whole. The top number (numerator) counts The bottom number (denominator) tells what is being counted ¾ is a count of three parts called fourths TRY THIS ONE Count to 4/5- What is being counted? How many of them do you have? Student should be able to tell how many fifths in one whole. Iterating applies to all models but is particularly connected to the length model A ribbon that is ¼ of a yard long

Iterating Fractions Greater Than One Whole Term improper fraction is used to describe fractions greater than one. Substitute the word improper for fraction or fraction greater than one 12/5 is a fraction greater than 1 Models will support this understanding

Conceptual focus on equivalence Area models for equivalent fractions help students create understanding.

Try this one Activity 15.17 Apples and Bananas Materials- 24 counters in all- two colors, 16 red and 8 yellow Directions- Group the counters into different fractional parts of the whole and use parts to create fraction names for the fractions that are apples and fractions that are bananas. Ask- If we make groups of four, what part of the set is read?

Developing Equivalent-Fraction Algorithm Area model is a good visual to connect concept of equivalence to the standard algorithm. The approach would be to look for a pattern in the way that the fractional parts in both the part and whole are counted.

Try this one Activity 15.19 Garden Plots

Student responses from Slicing Squares

Writing Fractions in Simplest Terms A fraction in simplest terms means that the numerator and denominator have no common whole-number factors. Reducing fractions implies that the fraction is getting smaller and this is not true. Avoid that term. Accept all equivalent fractions and do not say that an answer is “incorrect” if not in simplest terms.

Comparing Fractions Ways that the fractions could have been compared; Same- size whole Same number of parts but different wholes More than/less than one-half of one

Estimating with Fractions Benchmarks of zero, one-half, and one Using number sense to compare About how much Name a fraction for each drawing and explain why you chose that fraction. Focus on the infinite number of fractions that can be used to explain between 0, 1, and 1/2

Teaching Considerations for Fractions Concepts Emphasize number sense and meaning of fractions Provide a variety of models and contexts Emphasize that fractions are numbers Dedicate time for understanding of equivalence (concretely, symbolically) Link fractions to key benchmarks and encourage estimation