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### Operations with Fractions

Learning goals:

Know the key teaching strategies for elementary mathematics

Understand the depth of the content regarding fractions in 4th and 5th grades

Consider additional teaching strategies for engagement and scaffolding higher order thinking

In which of the following are the three fractions arranged from least to greatest?

NAEP 8th Grade, 49% correct

Why so few?

Key Points

- Use manipulatives and drawings.
- Build knowledge of fraction operations on the underlying structures of word problems.
- Help students reason with fractions.
- Focus on the reasons behind operations, to develop the procedures.

Problems that represent key content

Procedures

- Adding or subtracting by finding equivalent fractions

- How to find equivalent fractions

- Why add numerators when the denominators are the same

Procedures

- Multiplying fractions by multiplying the numerators and multiplying the denominators.

- Where does this come from?

Marty made two types of cookies. He used 1/5 cup of flour for one recipe and 2/3 cup of flour for the other recipe. How much flour did he use in all? Is it greater than 1/2 cup or less than 1/2 cup?Is the amount greater than 1 cup or less than 1 cup?

- Explain your reasoning in writing.

When first learning…

- Allow students to use manipulatives or drawings to figure this out.
- Use fraction circles or fraction bars to determine whether 1/5 + 2/3 is less than or greater than ½ or 1.
- Eventually the image of the manipulatives or drawings will become second nature so students can “see” in their heads the fraction relationships.

Basic concepts

- A fraction is a part of a whole…
- The numerator means… the denominator means…
- Unit fractions get smaller as their denominators get larger.
- Fractions are numbers on the number line (1/2 is half the way from 0 to 1…)
- Fractions that are the same size are called equivalent fractions.

Reasoning questions

- Which is larger, 2/8 or 5/8? Why?
- Which is larger, 2/4 or 2/6? Why? How can you prove this?
- Which is larger, 2/3 or 3/4? 2/5 or 5/10?

3.NF.3 d. Compare two fractions with the same numerator or the same denominator by reasoning about their sizes.

4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2.

Reasoning questions

- Where would you place 5/6 on the number line?
- Can you use other fraction pieces with different denominators to show 1/2? 1/4? 3/4?

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Illuminations fraction game (Fraction Tracks)

http://illuminations.nctm.org/ActivityDetail.aspx?ID=18

- How can playing a game like Fraction Tracks help a student build understanding about the relative sizes of fractions?
- How can playing a game like Fraction Tracks help a student build understanding about the equivalence of fractions?
- What characteristics of the classroom environment would support students as they use a game like Fraction Tracks to help them deepen their understanding of fractions?

Smarter Balanced Assessment items

- Other virtual manipulatives on our Elementary Math Resources web pages

Which approach?

Is it A… or B…

A: “You can find equivalent fractions by multiplying the numerator and denominator by the same number.” (Teacher explains procedure, shows worked out examples, students practice with new problems)

4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models

Fraction Addition and Subtraction

Understanding the reasons behind a procedure is just as important as being able to do the procedure.

Adding or subtracting fractions by finding equivalent fractions… What’s the procedure and why does it work?

Learning Progression

4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.

c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100 and add 3/10 + 4/100 = 34/100. (Addition and subtraction with unlike denominators in general is not a requirement at this grade.)

Estimation and visualization are important. These abilities will help students monitor their work when finding exact answers.

Using reasoning about size

- For each of the following problems, explain if you think the answer is a reasonable estimate or not.

Estimation and visualization are important. These abilities will help students monitor their work when finding exact answers.

- Students need to experience acting out addition and subtraction concretely with an appropriate model before operating with symbols.

Learning Progression

- Step 1: Learn what it means to add fractions with the same denominator.
- Pictures, analogies, methods, etc.
- How does this generalize into adding fractions with different denominators?
- Step 2: one is a multiple of the other
- Step 3: both scale up to a common multiple

5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7 by observing that 3/7 < 1/2.

You try it

… by using visual fraction models or equations to represent the problem

Estimation and visualization are important. These abilities will help students monitor their work when finding exact answers.

- Students need to experience acting out addition and subtraction concretely with an appropriate model before operating with symbols.
- Making connections between concrete actions and symbols is an important part of understanding. Students should be encouraged to find their own way of recording with symbols.

Using circle fractions

- Page 494, last paragraph 1stcolumn, through end.
- Mark the text.
- For the problems in Figure 6, see the packet with rulers (like number lines).

For students who struggle …

- Manipulatives and drawings
- Partner work
- Explicit teaching:
- Teacher verbalizes thought processes
- Works together with student
- Allows for practice with guided feedback

Pair up and try this with

Start with “Can you show me how to make 2/5 from the fraction circle pieces?”

Collaborative cards

- What do you think of this game as a teaching tool?

What about decimals?

- What does the common core say?
- How would you sequence this in a learning progression?
- What manipulatives and visual representations are helpful?
- How are decimals related to fractions?

NLVM Place Value Number Line (3-5 Number and Operations)

4.NF.5 Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100 and add 3/10 + 4/100 = 34/100. (Addition and subtraction with unlike denominators in general is not a requirement at this grade.)

4.NF.6 Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

4.NF.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

5.NBT.3 Read, write, and compare decimals to thousandths.

5.NBT.4 Use place value understanding to round decimals to any place.

5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Decimal place value

- Why do we line up the decimal point when adding or subtracting?
- How is this like adding or subtracting three-digit whole numbers?
- See the NLVM simulation Base Blocks Decimals

Multiplying with fractions

- Multiplication of a fraction by a whole number.4.NF.4
- 10 people at a party eat a half sandwich each. How many whole sandwiches is that? 10 x ½ (Count on the number line. This is repeated addition.)
- Multiplication of a fraction or whole number by a fraction. 5.NF.4
- A bag has 20 apples in it. You want to give ¼ of the bag to a friend. How much is ¼ of 20? ¼ x 20(How else can you solve this? How much is ¾ of 20?)

It is important to emphasize the underlying structure of these word problems.

- 10 people at a party eat a half sandwich each. How many whole sandwiches is that? 10 x ½ (How could you solve this with a number line? What kind of problem is it?)
- A bag has 20 apples in it. You want to give ¼ of the bag to a friend. How much is ¼ of 20? ¼ x20

Multiplication of a fraction or whole number by a fraction.

- You have 2/3 of a pumpkin pie left over from Thanksgiving. You want to give 1/2 of it to your sister. How much of the whole pumpkin pie will this be? (Use fraction pieces or drawings to figure this out, or mental reasoning.)

3/4 of a pan of brownies needs to be divided equally among three classes. How much does each class get?

- You have 3/4 of a pan of brownies left over from a party. You want to give 1/3 of that to your neighbor. How much does your neighbor get?

Fair shares division

You have 3/4 of a pan of brownies left over from a party. You want to give 2/3 of that to your neighbor. How much does your neighbor get?

or

There is 1/2 of a pan of brownies left. You want to give 2/3 of it to your sister. What fraction of the whole pan of brownies will she get?

- Try this with fraction circles or fraction bars. What did you have to do to get the answer?
- What drawing might you make that could help?
- We see 2/6 as the answer from the drawings? Can you also see 1/3 in the drawings? Is it necessary to force the answer to be 1/3?

Look at the previous problem. Do we need to force the reduced fraction as the answer?

What operation can you do with 2/3 and 3/4 to get your answer? View the C-R-A to develop the answer.

- Then create a C-R-A for 2/3 x 3/4.
- Read and do pp. 23-25 in Operations with Fractions.

Try these(drawings and symbols)

- You have 7/8 of a cup of sugar. You need 2/3 of this for a recipe. How much is that?
- A piece of wood is 2 1/8 feet long. How much is 3/4 of that?

Solve real world problems

- 5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
- Write a problem for this multiplication.

Here’s one for you

- It takes ¾ of a yard of fabric to make one pillow case. How many can be made from 12 ½ yards?
- How much is left over?

16 and 1/2 of a yard

16 and 2/3 of a pillowcase

Division involving fractions

- Thirteen big cookies need to be divided equally among 4 people. How much does person get?

5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3 and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4.If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get?

Divide a fraction by a whole number to solve problems.

- Divide a whole number by a fraction to solve problems.
- Think up problems to go with each of these.

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

Partitive division – equal shares. Is this the same as 1/4 of 1/2?

- 4 runners on a relay team will run equal portions of a race that is 1/2 mile long. How far does each race?
- How many 1/4 cup servings are in 10 ounces of cereal?

Measurement division – how many times does the fraction go into the whole number?

Homework

- Is there a Number Talks in this?

- Bring back stories to share about work with whole numbers or fractions. Samples of student work are nice, too!

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