Fraction Models: More Than Just Pizzas SARIC RSS Mini-Conference 2014 Laura Ruth Langham Hunter AMSTI-USA Math Specialist

Everyday Fractions In our every day lives, we see and hear about fractions. Brainstorm with your small group or partner to list of all the ways you hear about fractions in the real world. Place each idea on a separate sticky note. Keep at desk

Learning Outcomes Identify features of different fractional models Define partitioning and what it looks like for different fractional models Define iterating and what it looks like for different fractional models

CCRS Math Practice Standards Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.

CCRS Content Standards Grade 3: Number and Operations – Fractions Develop understanding of fractions as numbers. Grade 4: Number and Operations – Fractions Extend understanding of fraction equivalence and ordering. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. Understand decimal notation for fractions, and compare decimal fractions.

CCRS Content Standards Grade 5: Number and Operations – Fractions Use equivalent fractions as a strategy to add and subtract fractions. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

Types of Models Area Set Linear

Area (Region) Model for Fractions A set area or space divided into smaller fractional pieces Manipulatives: Fraction circles, pattern blocks, paper folding, geoboards, fraction bars, fraction strips/kits Have participants place all ideas from engage on the area poster.

Linear (Length) Model for Fractions The length of the whole is divided into equal lengths. A fraction is identified as being a particular distance from the 'start' of the whole. Manipulatives: Number lines, rulers, (fraction bars, fraction strips/kits) Have participants place linear model ideas on poster

Set (Discrete) Model for Fractions A group of countable items More than one item to be shared Manipulatives: Chips, counters, painted beans, candy Have participants place all set models on poster. If any questions about a specific example, facilitate a group discussion to decide where to put it.

Sort your models! Have each group sort a page from the list. They’ll tape it to the poster it best fits. After all are sorted, take a gallery walk and discuss why each one is where it is placed.

Activity Reflection What are some difficulties students might have transitioning from one model to the next? In what ways do the set models differ from one another? In what ways do the area models differ from one another? In what ways do the linear models differ from one another?

Big Ideas for Fraction Models The use of models should permeate instruction, and not just be an incidental experience, but a way of thinking, solving problems, and developing fraction concepts. (Post, 1981; Clements, 1999) Students should interact with a variety of models that differ in perceptual features, which causes them to rethink and ultimately generalize the mathematical concepts being investigated in the models. (Dienes, cited in Post & Reys, 1979) Modeling is a means to the mathematics, not the end. (Post, 1981; Clements, 1999) Over time, if students are allowed to interact with models whose perceptual attributes vary as well as construct their own models to solve problems, their mental images of, and understandings derived from, the models will be sufficient to solve problems. (Petit, Laird, Marsden)

Which models are best suited when students first solve problems like these… Explain your choices. Helping students understand a fraction as a number/quantity Solving problems like—There are 24 students in a class. Three-fourths of the students are girls. How many students are girls? Which one is larger? ¾ or 1/3 Determining the which fraction is closer to 1. 2/3 4/5 5/4 6/8

“Students who experience a variety of ways to represent fractions, and are asked to move back and forth between them develop more flexible notions of fraction.” (Lesh, Landau, & Hamilton, 1983)

Virtual Manipulatives Illuminations “Fraction Models” http://illuminations.nctm.org/Activity.aspx?id=3519 Illuminations “Patch Tool” (Pattern Blocks) http://illuminations.nctm.org/Activity.aspx?id=3577 Illuminations “Equivalent Fractions” http://illuminations.nctm.org/Activity.aspx?id=3510

Partitioning and Iterating Partitioning consists of creating smaller, equal-sized amounts from a larger amount. Iterating consists of making copies of a smaller amount and combining them to create a larger amount. Read Article at https://mathed.byu.edu/~peterson/Fractions%20Unit%20Sec%201.pdf

Partitioning Different Models Area Model Fraction Cookies Pictorial Representation #37-40 Linear Model Fraction Strips Number Line “Ants Marching” Set Model Red Hots “Apple Farming” Pictorial Representation

Partitioning What fraction of this square is shaded?

Partitioning the Number Line Use cuisinaire rods to partion the number line.

Partitioning the Number Line

Reasoning about the Number Line Why are sixths smaller than fourths? What numbers are the same distance from zero as two-thirds? What number is halfway between zero and one- half? What would you call a number halfway between zero and one-twelfth?

Apple Picking Oscar picked 12 apples. He gave 1/3 of the apples to Gil and 1/3 of the apples to Becky. How many apples did each of them get? Pilar picked 12 apples. She gave ¼ of the apples to Dewayne, ¼ of the apples to Murphy, and ¼ of the apples to Kelley. How many apples did each of them get? Chiang picked 12 apples. She gave 1/6 of the apples to each of her 5 friends. How many apples did each of them get?

Unitizing Unitizing is the ability to identify subgroups within groups—for fractions– unitizing helps to see fractional parts and equivalent fractions. Let’s look at a few examples…

Transition between Area and Set Halves, fourths, sixths, twelfths

Transition between Area and Set Halves, fourths, sixths, twelfths

Iterating Area Model Linear Model Set Model Fraction Circles Pictorial Representation Linear Model Fraction Strips Number Line Representation Set Model Red Hots “Apple Farming” Pictorial Representation #33

Iterating on a Number Line Draw a line on a piece of paper. If your line is ½, what would 1 whole look like? If your line is 1/10, what would 1 whole look like? If your line is 2/5, what would 1 whole look like? If you line is ¾, what would 2 wholes look like? If your line is 5/4, what would ½ look like?

Challenge Suppose this bar represents 3/8. Create a bar that is equivalent to 4/3. Partition Iterate Also, a challenge is available on their worksheet if time.

Learning Outcomes Identify features of different fractional models Define partitioning and what it looks like for different fractional models Define iterating and what it looks like for different fractional models

Thank You for Attending! Contact Information Laura Ruth Langham Hunter LLangham@southalabama.edu Review outcomes from today’s session.