All Fractions are Not Created Equal SARIC RSS Mini-Conference 2014 Laura Ruth Langham Hunter AMSTI-USA Math Specialist

Flashback… Discuss your experiences with fractions in the classroom when you were going through school. What are some of the biggest challenges students face with fractions today?

Common Misconceptions Students… plot points based on understanding fractions as whole numbers instead of fractional parts. see the numbers in fractions as two unrelated whole numbers separated by a line. do not understand that when partitioning a whole shape, number line, or a set into unit fractions, the intervals must be equal. do not understand the importance of the whole of a fraction and identifying it. do not count correctly on the number line. do not understand there are many fractions less than 1. do not understand fractions can be greater than 1. think all shapes can be divided the same way.

Why are Fractions so Hard? Fraction notation – numbers must be considered in new ways Practices that simplify and/or mask the meaning of fractions Overreliance on whole number knowledge Many meanings of fractions

Learning Outcomes Identify the importance of the whole Determine the unit fraction and it’s purpose in working with fractions Identify different strategies for comparing and ordering fractions Evaluate the four types of fractions

CCRS Math Practice Standards 1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7. Look for and make use of structure. 8.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.

Mini Olympics Stations: – Mini Javelin: Stand behind the line and throw a cotton swab like a javelin. – Long Thump: Press the edge of a penny with your thumb onto the edge of the another coin to make the lower coin jump forward. – Discus Blast: Use a breath of air to blow a plastic bottle cap across a long table while remaining entirely behind the line. Use your teacher-provided measuring tool to figure out the distance your item traveled.

Shift from Whole Number Reasoning What allowed for more precision in your measurements? Why did you subdivide your “ruler”? What was the smallest subdivision that a unit was divided into? What is the smallest possible subdivision the unit could have been divided into? In conclusion, rational numbers are just an extension of whole numbers.

Attending to Precision Two important aspects of fractions provide opportunities for the mathematical practice of attending to precision: Specifying the whole. Explaining what is meant by “equal parts.”

Pizza Problem José ate 1⁄2 of a pizza. Ella ate 1⁄2 of another pizza. José said that he ate more pizza than Ella, but Ella said they both ate the same amount. Use words and pictures to show that José could be right.

Example from Illustrative Mathematics

What’s the Whole? Emily Raj Alejandra

What’s the Whole? What misconceptions do you think your students might have about the size of the whole? What questions could you ask students to get them to think deeper about the size of the whole?

What is a Unit Fraction? A fraction with the numerator of 1 ¼ is a unit fraction ¼ + ¼ + ¼ + ¼ = 4/4 = 1 whole 4/5 is a fraction made up of 4 pieces of size 1/5 or 4 copies of the unit fraction 1/5 1/5 + 1/5 + 1/5 + 1/5 = 4/5

Pattern Blocks: If… Then… If the yellow hexagon is one whole, then… If the red trapezoid is one whole, then… If the blue rhombus is one whole, then… If the green triangle is one whole, then…

Part-Whole Relationship “Looking at fractions as though they represent two separate whole numbers leads to misinterpretation of their meaning and an inability to assess the reasonableness of calculation results.” (Investigations in Number, Data, and Space “Finding Fair Shares”)

Equal Parts? The parts of the whole must be equal to one another, and all the parts combined must equal the whole.

Part-Whole Relationship One key idea throughout the study of fractions is that a fraction represents a quantity in relation to a unit whole. Examples of unit wholes could include a single object, an area, a linear measure, or a group of objects. Although ½ can represent many different quantities, depending on the size of the whole, ½ has the same relationship to any whole. It is one of two equal parts that compose the whole.

Let’s Estimate! Let’s reason about two numbers for a second… Estimate the sum of these two numbers A.1B. 2C. 19D. 21 Most students answered C or D! They were not able to see the relation of each number to 1 and thus missed the correct estimation of 2.

Compare unit fractions in context Kim, Les, Mario, and Nina each had a string 10 feet long. Kim cut hers into fifths. Les cut his into fourths. Mario cut his into sixths. Nina cut hers into thirds. After the cuts were made, who had the longest pieces of string?

Reasoning Strategies for Comparing Fractions Same Denominators – More same-sized parts Same Numerators – Same number but different-sized parts Comparison to Benchmarks – More or less than one-half of one whole Distance from Benchmarks – Closeness to one-half or one whole

Snow Day Alec and Felix are brothers who go to different schools. The school day is just as long at Felix' school as at Alec's school. At Felix' school, there are 6 class periods of the same length each day. Alec's day is broken into 3 class periods of equal length. One day, it snowed a lot so both of their schools started late. Felix only had four classes and Alec only had two. Alec claims his school day was shorter than Felix' was because he had only two classes on that day. Is he right?

Learning Outcomes Identify the importance of the whole Determine the unit fraction and it’s purpose in working with fractions Identify different strategies for comparing and ordering fractions Evaluate the four types of fractions

Thank You for Attending! Contact Information Laura Ruth Langham Hunter