Developing a Meaningful Understanding of Fractions and Operations with Fractions Scott Adamson, Ph.D.

Unpacking Fraction Ideas Suppose that $6070 is 5/7 the price of a car. What is the price of the car? After you solve the problem, fully unpack your thinking and understanding of what it takes to solve and understand this problem. In other words, if a student was to have a profound understanding of the mathematics needed to solve this problem, what would this entail? Open this in IE http://tpc2.net/Courses/Func1F05/ConceptConv.html#

Fraction Frustrations Share with your team the challenges/frustrations you have experienced (or what you anticipate) in teaching fractions to children. What makes fractions so difficult for children?

Fraction Frustrations - 1 Children have difficulty internalizing that the symbol for a fraction represents a single entity. When asked if 2/3 was one or two numbers, many children would say that the symbol represented two numbers. When students consider 2/3 as two numbers then it makes sense to treat them like whole numbers. For example, when students add two fractions by adding the numerators and then denominators, they are interpreting the symbols as four numbers, not two. Many errors with fractions can be traced to students’ lack of mental images for the quantity the symbol represents. From the Rational Number Project

Fraction Frustrations - 2 Ordering fractions is more complex than ordering whole numbers. Comparing 1/4 and 1/6 conflicts with children’s whole number ideas. Six is greater than four, but 1/4 is greater than 1/6. With fractions, more can mean less. The more equal parts you partition a unit into, the smaller each part becomes. In contrast, 3/5 is greater than 2/5 because 3 of the same-size parts are greater than 2 of the same-size parts. In this case, more implies more. Being able to order plays an important part in estimating fraction addition and subtraction. Ideally when a student adds, for example,1/4 +1/3, she should be able to reason from her mental images of the symbols that (a) the answer is greater than 1/2, but less than one and (b) 2/7 is an unreasonable answer because it is less than 1/2. From the Rational Number Project

Fraction Frustrations - 3 Understanding fraction equivalence is not as simple as it may seem. Some children have difficulty noting equivalence from pictures. Imagine a circle partitioned into fourths with one of those fourths partitioned into three equal parts. Some children we worked with were unable to agree that 3/12 equals 1/4 even though they agreed that physically the two sections were the same size. Children said that once the lines were drawn in, you could not remove them. [Therefore 3/12 ≠ 1/4 ]. In reality, that is just what must be done to understand fraction equivalence from a picture. From the Rational Number Project

Fraction Frustrations - 4 Difficulties children have with fraction addition and subtraction come from asking them to operate on fractions before they have a strong conceptual understanding for these new numbers. They have difficulty understanding why common denominators are needed so they revert to whole number thinking and add numerators and denominators. From the Rational Number Project

Fraction Frustrations Santa Barbara City College Mathematics Diagnostic Testing Project Item Description % of students who answered correctly Add simple fraction and a decimal 19 Order four numbers (2 fractions, 2 decimals) 21 Add two improper fractions 24 Find the largest of 4 fractions 33 Multiply 2 simple fractions 37 5830 students

Beliefs Children learn best through active involvement with multiple concrete models, Physical aids are just one component in the acquisition of concepts: verbal, pictorial, symbolic and real world representations also are important, Children should have opportunities to talk together and with their teacher about mathematical ideas Curriculum must focus on the development of conceptual knowledge prior to formal work with symbols and algorithms. From the Rational Number Project

Big Initial Fraction Ideas What is a fraction? Equivalent fractions Comparing fractions What represents “1 unit”? Investigating student thinking

Your Mind’s Eye

Your Mind’s Eye Part-Whole as in “three out of seven” How would one interpret 7/3? How would one interpret 8/(3/7)? Ratio as in “there are 3 boys for every 7 girls” Implies a multiplicative (proportional) relationship that may not be explicit for students Division as in “3 divided by 7” Somewhat limited if this is the sole conception

What is a Fraction? Suppose that this bar represents 3/8. Create a bar that is equivalent to 4/3. Do Activity 1

What is a Fraction? Suppose that this bar represents 3/8. Create a bar that is equivalent to 4/3.

What is a Fraction? Suppose that this bar represents 3/8. Create a bar that is equivalent to 4/3.

What is a Fraction? Describing a fraction requires: Partitioning “1/8 is the amount we get by taking a whole, cutting it up into 8 equal parts and taking 1 of those parts.” Iterating “1/8 is the amount such that 8 copies of that amount, put together, make a whole.”

What is a Fraction? Describe 5/8 as… Partitioning Iterating “5/8 is 5 one-eighths, where 1/8 is the amount we get by taking a whole, cutting it up into 8 equal parts and taking 1 of those parts.” Iterating “5/8 is 5 one-eighths, where 1/8 is the amount such that 8 copies of that amount, put together, make a whole.”

Initial Fraction Ideas Exploring with Fraction Circles Do Activity 2 What is a fraction? Equivalent fractions

Fraction Names - Multiplicatively 6-8 band ended here on Tuesday Name each. Express each in equivalent ways. What is a fraction?

Beliefs Children learn best through active involvement with multiple concrete models, Physical aids are just one component in the acquisition of concepts: verbal, pictorial, symbolic and real world representations also are important, Children should have opportunities to talk together and with their teacher about mathematical ideas Curriculum must focus on the development of conceptual knowledge prior to formal work with symbols and algorithms. From the Rational Number Project

Which Is Larger? Do Activities 3, 4 and 5. Thinking about mathematics as a search for patterns, as something to be done rather than trivia to know, how do these activities work? How do you see activities like these being used in your own teaching? Comparing fractions

Fraction Strips Practice folding the paper strips into 2, 3, 4, 6, 8, and 12 parts. Use the paper strips to show these fractions. Which is largest? Comparing fractions

Fraction Challenge Find three fractions between 7/11 and 7/12. Explain your thinking. From Susan Lamon – Marquette University Comparing fractions

Fraction Challenge “You can get fraction in between those two by making the bottom numbers in between 11 and 12. You can make any number of fractions you want between them.” Comparing fractions

Fraction Challenge “7/12=0.583 and 7/11=0.636, so let’s pick 7/(11 ½). If you double the top and the bottom, you get 14/23, and that’s 0.608, so it’s in there.” Comparing fractions

Fraction Challenge Find three fractions between 1/8 and 1/9. Explain your thinking. From Susan Lamon – Marquette University Comparing fractions

Fraction Challenge Martin first rewrote the fraction 1/8 so that it had 9 in the denominator. He did this by noticing that 1/9=1÷9 and 1/8=9/8÷9.

Fraction Challenge Now he was ready to form new fractions between 1/9 and (1 1/8)/9. All the denominators would be 9.

Fraction Challenge To form numerators, he stayed between 1 and (1 1/8) by adding to 1 fractions smaller than 1/8 (1+1/9, 1+1/10, 1+1/11, etc.)

Focus on the Unit If this is one unit: What is: What represents “1 unit?”

Focus on the Unit

Focus on the Unit

Orange Slices Suppose you have 2 ½ oranges. If a serving consists of ¾ of an orange, how many servings (including parts of a serving) can you make? What represents “1 unit?”

Can You See? Can you see 3/5 of something in this picture? Be specific. Can you see 5/3 of something in this picture? Be specific. Can you see 2/3 of something? Be specific. What represents “1 unit?” Taken from Pat Thompson – Arizona State University

Investigating Student Thinking

Investigating Student Thinking Following are two statements Ally made when comparing fractions. For each statement, identify the incorrect rule she is applying, hypothesize how she may have developed the rule, and describe how you might help her develop a deeper understanding of the concepts.

Investigating Student Thinking In comparing 1 with 4/3, Ally said, “One is bigger because it's a whole number. It's one group of-it's one number.” In comparing 1/7 with 2/7 , Ally said, “I thought it was just the smallest number. Because usually you go down to the smallest number to get to the biggest number.”

Sharing Cookies Solve the following problem in two ways: Five children want to share two cookies fairly. How much cookie will each person get? Pay attention to your thinking about this problem as you solve it. What important ideas emerged? How might 4th grade students struggle with this problem? What mistakes might they make?

Sharing Cookies

Sharing Cookies Consider the question Felisha was asked. How could 1/5 be correct for the amount each person gets when 5 people fairly share two cookies? How could 1/5 be an incorrect answer? How could 2/10 be a correct answer? How could 2/10 be an incorrect answer? What are some things you might do to help students understand that the whole can be comprised of more than one unit?

Unpacking Fraction Ideas Suppose that $6070 is 5/7 the price of a car. What is the price of the car? After you solve the problem, fully unpack your thinking and understanding of what it takes to solve and understand this problem. In other words, if a student was to have a profound understanding of the mathematics needed to solve this problem, what would this entail? Open this in IE http://tpc2.net/Courses/Func1F05/ConceptConv.html#

Unpacking Fraction Ideas http://tpc2.net/Courses/Func1F05/ConceptConv.html#

Unpacking Fraction Ideas http://tpc2.net/Courses/Func1F05/ConceptConv.html#

Thinking About Fraction Operations When we add or subtract fractions, we have to find a common denominator, but not when we multiply or divide…

Thinking About Fraction Operations And once we get a common denominator, we add or subtract the numerators, but not the denominators…

Thinking About Fraction Operations But when we multiply, we multiply both the numerators and denominators…

Thinking About Fraction Operations And when we divide, we divide neither the numerators nor the denominators!

Operations With Fractions Work to develop a story problem that would result in each of the following computations. Then for each, estimate an answer to your problem. Explain your thinking.

What Do We Teach First? Traditionally, multiplication and division of fractions is taught before addition and subtraction. Why do you think this is the case? Conceptually, which operation(s) are easier to understand? I think that, if initial fraction experiences are conceptually understood, addition/subtraction should occur before multiplication/division.

Addition We shall take addition to mean “combining quantities” William ate 1/4 of a pizza for dinner. The next morning he ate a piece that equaled 1/8 of the pizza. How much of a pizza did he eat?

Addition William ate 1/4 of a pizza for dinner. The next morning he ate a piece that equaled 1/8 of the pizza. How much of a pizza did he eat? Did William eat more or less than ½ of the pizza? Explain how you know. Model the problem with fraction circles. Some people would say that 1/8+1/4 is 2/12 . Does that make sense?

Addition Use your fraction circles to answer the problems in Activity 7.

Addition 3 + −5 = −2 − − + − + − + −

Addition

Addition

Subtraction We shall take subtraction to mean “taking away a quantity from another quantity” Alice noticed that there was 3/4 of a pizza left after the party. She ate a slice of pizza that was the size of 1/8 of a whole pizza. How much pizza was left after Alice ate a slice?

Subtraction Alice noticed that there was 3/4 of a pizza left after the party. She ate a slice of pizza that was the size of 1/8 of a whole pizza. How much pizza was left after Alice ate a slice? Will there be more or less than ½ of the pizza left? Explain how you know. Try picturing the ¾ pizza in your mind. Does this help in your estimate? Explain your thinking. Some people would say that 3/4 – 1/8 is 2/4. Does that make sense?

Subtraction Use your fraction circles to answer the problems in Activity 8.

Subtraction −5 − (− 3) = −2 − − − − −

Subtraction 5 − (−3) = 8 − + + + + − + + − + + − +

Subtraction

Multiplication We shall take multiplication to mean “if mxn, then take m copies of n or we shall think of something that is m times as large as n” Terry noticed that there was 3/4 of a cake left after the party. He notices that 1/8 of the remaining cake had all of the frosting taken from it. What part of the original cake remaining still has frosting?

Multiplication Terry noticed that there was 3/4 of a cake left after the party. He notices that 1/8 of the remaining cake had all of the frosting taken from it. What part of the original cake remaining still has frosting? Will there be more or less than ½ of the cake with frosting? Explain how you know. Try picturing the ¾ pizza in your mind. Does this help in your estimate? Explain your thinking.

Multiplication with Patty Paper

Multiplication with Patty Paper

Multiplication with Patty Paper

Multiplication with Number Lines

Multiplication with Number Lines

Multiplication Work on Activities 12 and 13

Multiplication Algorithm The algorithm for multiplying fractions is In the context of patty paper and/or number lines, explain what information ac provides. Explain what information bd provides.

Multiplication

Multiplication = 15 We need to remove an amount that is equivalent to a number that is 5 times as big as −3. − − + − + + − + − − + − + + − + − + − − + − + + − − + + − − − + + + − + − +

Multiplication

Division We will take a÷b to mean to find the number of copies of quantity b needed to make quantity a. 24÷6 asks “how many copies of 6 are needed to make 24?”

Division First, interpret the following then use your fraction circles to find the result.

Division You have 3 ½ pound of peanuts. You will make smaller bags each containing ¾ pound of peanuts. How many full bags can you make? How can you describe the remaining peanuts? Express this Pictorially Using a number sentence

Division Compute Now, think about Is it more or less than ? How do you know?

Division

Division Write a story problem that will lead to the computation 2/3 divided by 1/4.

Division Algorithm The traditional algorithm for dividing fractions is How can we make sense of this algorithm?

Division Do Activity 14

Division Algorithm Make sense of the algorithm based on your previous work.

Not On The Test http://www.notonthetest.com/statement.html

Math is Boring? www.youtube.com/watch?v=bRJiCEgfZ3Q

Irrational Numbers

Complex Numbers