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FRACTIONS & RATIOS UNIT Kristin Herderich

MARCH 1, 2014 OAKLAND UNIVERSITY

MATEE 11

Herderich, Kristin Fractions & Ratios Unit

1

FRACTIONS & RATIOS UNIT

Unit Title: Fractions and Ratios Content Area/Grade Level(s): 5th Implementation Time Frame: 16 days @ 60 minutes a day Common Core State Standards: CCSS.MATH.CONTENT.5.NF.A.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. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) CCSS.MATH.CONTENT.5.NF.A.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. CCSS.MATH.CONTENT.5.NF.B.4.A

Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) CCSS.MATH.CONTENT.5.NF.B.4.B

Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. CCSS.MATH.CONTENT.5.NF.B.5.A

Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. CCSS.MATH.CONTENT.5.NF.B.5.B

Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1. CCSS.MATH.CONTENT.5.NF.B.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.

http://www.corestandards.org/Math/Content/5/NF/A/1/



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CCSS.MATH.CONTENT.5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by

whole numbers and whole numbers by unit fractions.1 CCSS.MATH.CONTENT.5.NF.B.7.A

Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. CCSS.MATH.CONTENT.5.NF.B.7.B

Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. CCSS.MATH.CONTENT.5.NF.B.7.C

Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? CCSS.MATH.CONTENT.5.OA.A.1

Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. CCSS.MATH.CONTENT.6.RP.A.3.C

Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Stage 1 – Desired Results

Established Goals: Students will be able to use multiple different methods to add, subtract, multiply, and divide fractions and mixed numbers.

Students will Understand…

How to rename fractions as equivalent fractions

How to use equivalent names for fractions and mixed numbers to perform operations

How to introduce algorithms for the multiplication of fractions and mixed numbers

How to practice estimating and calculating a percent of a number

Essential Questions

What is an equivalent fraction?

How do you add and subtract mixed numbers?

How can you use the area model to represent fraction multiplication?

How do you multiply fractions using partial products? Improper fractions?

How do you divide fractions?

What strategies do you implement when solving a Fraction division problem?


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How to divide fractions using visual models

Students will know….

That you have to find a common denominator in order to add or subtract mixed numbers

That equivalent fractions are equal to each other

The Partial Products method for multiplying fractions

Alternate strategies for multiplying fractions (Traditional & Area Model)

That you need to “Keep, Change, Flip” in order to divide fractions

Students will be able to…

Demonstrate fraction multiplication using the area model

Compare fractions

Rename fractions as equivalent fractions

Use equivalent names for fractions and mixed numbers to perform different operations (Addition, Subtraction, Multiplication, & Division)

Estimate and calculate a percent of a number

Explain visual models to multiply fractions

Solve addition and subtraction problems using fractions

Solve multiplication and division problems using fractions

Stage 2 – Assessment Evidence

Performance Task(s):

End of unit exam

Quarter 3 District Test

Other Evidence

Study Links

Math Journal work

Fractions packet

Fractions worksheets

St. Patrick’s Day worksheet

Stage 3 – Learning Plan

Date 2/25 2/26 2/27 2/28 3/3 3/5 3/6

Learning Activities: 8.1 Review: Comparing Fractions 8.2 Adding Mixed Numbers 8.3 Subtracting Mixed Numbers Math Activity and Fraction Packet/Review with Low Students 8.4 Calculator Practice: Computation with Fractions 8.5 Fractions of Fractions 8.6 An Area Model for Fraction Multiplication


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3/7 3/10 3/11 3/13 3/14 3/17 3/18 3/19 3/20

8.7 Multiplication of Fractions and Whole Numbers 8.8a Multiplication of Mixed Numbers 8.8b Multiplication of Mixed Numbers 8.9 Finding a Percent of a Number 8.10 Relating Fractional Units to the Whole 8.12a Fraction Division & St. Patrick’s Day Worksheet 8.12b Fraction Division Review for Test Unit 8 Test


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8.1 Comparing Fractions Lesson Plan

Name: Kristin Herderich Date: 02/25/14 Grade Level: 5th

Unit Topic: Fractions & Ratios Lesson Focus: Comparing Fractions

Common Core Standards:

CCSS.MATH.CONTENT.5.NF.A.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. For example, 2/3 + 5/4 = 8/12 + 15/12 =

23/12. (In general, a/b + c/d = (ad + bc)/bd.)


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.

CCSS.MATH.CONTENT.5.NF.B.5.B

Explaining why multiplying a given number by a fraction greater than 1 results in a

product greater than the given number (recognizing multiplication by whole numbers

greater than 1 as a familiar case); explaining why multiplying a given number by a fraction

less than 1 results in a product smaller than the given number; and relating the principle of

fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

Objective(s):

To review the use of equivalent fractions in comparisons

Anticipatory Set:

State each fraction and ask students to indicate whether it is closest to the benchmarks of 0,

½, 1, 1½, or 2. Have students use thumbs to indicate choices.



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Instruction & Guided Practice:

On the board, show the fractions 2

3 and

3

5. Ask students to compare the two fractions.

Which one is greater? How do they know? Accept all answers and explanations, but

do not comment on them (you’ll come back to this later).

Have students open up to page 248 in their Math Journal. Using the Smart Board and

the pie charts that the system offers, show students how to compare the two

fractions. Each problem has a different aspect to it. Model each problem using the

pie charts.

Ask volunteers how they would determine whether 3

5 or

5

8 is greater. The methods

outlined above don’t work in this case. A method that does work is to use the

Fraction-stick and Decimal Number-Line Chart on Student Reference Book page 399.

Explain the following: Locate 3

5 and

5

8 on the chart. Since the line for

5

8 is to the right of

the line for3

5, we know that

5

8 >

3

5.

Do the following problems using the chart on page 399:


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Lastly, explain that the Quick Common Denominator (QCD) method that we talked

about yesterday is another way to compare fractions. Going back to the fractions 2

3

and 3

5, show the problem on the board and have the students walk through it with

you. Then, go back to the first bullet point and have students explain the answer.

Checking Understanding:

Ask students to give a thumbs up (got it!) or thumbs down (don’t get it) throughout

the lesson.

Walk around to check if students are getting the correct answers or working towards

the correct answers.

Check the previous night’s study link to ensure understanding of concepts.

Closing:

Before students work on their independent work, say the following: Today, we talked a lot

about different ways we can compare fractions. Can someone name one that we talked about?

What about another? Remember these methods for tomorrow because we are going to use

them for the next math lesson!

Independent Practice:

Students will work on pages 248 – 250 of the Math Journal either independently or

with a partner.

Students will also have Study Link 8.1 for homework

Materials Used:

Everyday Mathematics Math Journal 2 (pgs. 248 – 250)

Everyday Mathematics Student Reference Book (pgs. 399 – 401)

Smart Board lesson

Reflection:

This lesson went really well. I felt that I taught it really fast, but I knew the students

understood the concept and that I didn’t need to spend as much time on the concept.


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8.2 Adding Mixed Numbers Lesson Plan


Unit Topic: Fractions & Ratios Lesson Focus: Adding Mixed Numbers








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.

Objective(s):

To develop addition concepts related to mixed numbers

Anticipatory Set:

Have students rename each fraction as a whole number or mixed number and each mixed

number as an improper fraction.


On the board, do problems 1-9 on page 251 of their Math Journal as a class.


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Pull sticks and have random students come up and solve a specific problem – help

where needed.

Ask students to share their strategies with the class as they are solving the problem

Next, do problems #10 & #11 on the board, showing them how to add Mixed

Numbers together with the same denominator. Have students tell you how to do

problem #12.

Talk about what to do with Mixed Numbers that have unlike denominators. Write

these two problems on the board and do the first one as a class (see if students can

figure out what to do). Then have students do the second problem in their Math

Spirals.

Show students how to rename Mixed Numbers with top-heavy fractions

Then have students practice in their Math Spiral with the following fractions:

85

3 , 9

9

4 , 5

10

6 , and 5

12

8



the lesson.





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Closing:

Ask students how adding Mixed Numbers is the same as adding fractions. How is it

different? Tell them that tomorrow we will be learning how to subtract mixed numbers.


Students will complete page 251 and work on pages 252-253 of the Math Journal

either independently or with a partner.


Materials Used:

Everyday Mathematics® Math Journal 2, pp. 264 – 266

Math Spiral

Smart Board Lesson

Reflection: This lesson went incredibly smoothly. Students understood all concepts and ideas presented. I liked that I left myself extra time to work independently with low students.


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8.3 Subtracting Mixed Numbers Lesson Plan


Unit Topic: Fractions & Ratios Lesson Focus: Subtracting Mixed Numbers













Objective(s):

To develop subtraction concepts related to mixed numbers

Anticipatory Set:

Have students name a common denominator for the following:


On the smart board, demonstrate how to subtract mixed numbers using problems #1-

3 on page 254 in the students’ Math Journal.


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Do questions #1-2 with the students. Have students tell you how to do question #3 to

ensure understanding.

Write the equation 31

3− 1

2

3 on the board. Ask students how this problem differs from

the problems we’ve already done. Ask students for solution strategies – accept all,

but move forward when somone names RENAMING. Use pie charts to help with this

strategy.

After walking through students on this one problem, give them two more sample

problems to work through on their own. Walk around and assist where needed.

o 8 − 32

3

o 43

5− 1

4

5

After students have completed the two problems above, write this equation on the

board 31

3− 1

1

2. Ask students how to solve the problem. They will most likely notice

that 1

3 is less than

1

2. Suggest renaming the fractions with common denominators.

Lastly, give students some real-life examples for them to solve in their Math Spiral


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o Daniel has a DVD set that provides 41

2 hours of viewing. So far, he has

watched 5/6 of an hour. How many more hours of the DVD set does he still

have to watch?

o Serena measured out 52

3 of rope to make a jump rope. Her friends told her

that it needed to be 61

2 feet long. How much more rope did she need to

measure out?



the lesson.




Closing:

Ask students how subtracting mixed numbers is different from subtracting fractions. Ask

students when they might need to subtract fractions in real life.


Students will complete page 254 and work on page 256 of the Math Journal either

independently or with a partner.


Materials Used:

Everyday Mathematics® Math Journal 2, pp. 254 & 256

Math Spiral

Smart Board Lesson

Reflection: I loved the use of the pie charts on the Smart Board. I was able to manipulate them and move them around to represent the subtracting of fractions for my visual learners.


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Supplemental Math Activity Lesson Plan


Unit Topic: Fractions & Ratios Lesson Focus: Addition & Subtraction of

Mixed Numbers













Objective(s):

To provide practice adding & subtracting fractions.

Anticipatory Set:

Ask students what they’ve learned about adding mixed numbers.

Ask students what they’ve learned about subtracting mixed numbers.

Have them discuss the similarities and differences.


Tell students that today is review over what we have learned the past two days.

Pass out the Fraction packet for students to work on.

Students will work as pairs on the fraction packet while you meet with struggling

students.

Meet with students who have been struggling on the study links first. Correct

misunderstandings and secure their understanding on the topic.

If time, circulate and assist students on the fraction packet.


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the lesson.



Check the previous night’s study link to ensure understanding of unit


Adding and Subtracting Fractions Packet

Materials Used:

Fractions packet

Reflection:

I enjoyed this lesson because it allowed me to work with students who were clearly

struggling with adding and subtracting mixed numbers.


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8.4 Computation with Fractions Lesson Plan


Unit Topic: Fractions & Ratios Lesson Focus: Fractions on a Calculator













Objective(s):

To provide practice adding fractions with unlike denominators and using a calculator to

solve fraction problems.

Anticipatory Set:


Tell students to get out their calculators (they should be a TI-15 or something similar)

and their Student Reference Book, pp. 260.


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Tell students that you are going to teach them how to check their addition and

subtraction of Mixed Numbers by using a calculator.

Have them follow along with you as you tell them what keys to press on their

calculator. Circulate and assist as necessary.

Once all students seem to feel comfortable with using the calculator, display the

following three problems for students to try on their calculator. Pull sticks to

determine who answers which question.

Assign students to work on pages 257 and 258 of their Math Journal.

While students are working, meet with students who have been struggling on the

last two study links. Fix misunderstandings and ensure further understanding.



the lesson.




Closing:

As you assign students class work, tell them that anyone who doesn’t feel comfortable with

adding and subtracting problems should come see you and you will work with them, one-on-

one or in small group.


Students will work on pages 257-258 of their Math Journal either independently or

with a partner.

Students will also have Study Link 8.4 for homework.


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Materials Used:

Everyday Mathematics® Math Journal 2, pp. 257-258

Student Reference Book, pp. 260 – 263

TI-15 Calculator

Smart Board Lesson

Reflection:

I wasn’t a fan of this lesson as I was creating it, but after teaching it, I really like the

fluffiness of it. It gave my kids extra work in adding and subtracting mixed numbers and it

allowed me to work independently with struggling students and they didn’t miss any

important instruction time.


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8.5 Fractions of Fractions


Unit Topic: Fractions & Ratios Lesson Focus: Fractions of Fractions


CCSS.MATH.CONTENT.5.NF.B.4.A

Interpret the product (a/b) × q as a parts of a partition of q into b equal parts;

equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual

fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the

same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) =

ac/bd.)CCSS.MATH.CONTENT.5.NF.A.2


Comparing the size of a product to the size of one factor on the basis of the size of

the other factor, without performing the indicated multiplication.

CCSS.MATH.CONTENT.5.NF.B.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.

Objective(s):

To introduce finding a fraction of a fraction.

Anticipatory Set:



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Have students open their Math Journal to page 259. Do questions #1-4 as a class.

Show students how to use the number-line to find a fraction of a number

Have students come up and do questions #5-10 for the class. Have the student tell the

class what they are doing and why they are doing it.

Pass out two sheets of white printer paper per student. Explain to them that you are

going to show them how to take a fraction of a fraction by folding and coloring blank

paper.

Using the Smart Board lesson, display this problem for the class:

o Larry has ½ of a fruit bar and wants to give half of it to his brother. How much

of the whole fruit bar will Larry give to his problem?

Ask students what the question is asking ( ½ of ½ )

Ask students to fold a sheet of paper into halves, vertically. Have students unfold

their sheets and show them how to orient the sheets so the halves are to the left and

right of each other. Shade the left half with a light color.

Ask the students to fold their papers into halves in the opposite direction

(horizontally). Have students unfold their sheets, orient them so the new halves are

above and below each other, and shade the bottom half with a darker color.

Ask a volunteer to give the answer and explain how this folded-paper model shows

how much of the fruit bar Larry gave to his brother. Remind students that the sheet

of paper represents the whole fruit bar, or ONE.

Explain to students the answer is where the colors OVERLAP.

Have students write 1

2𝑜𝑓

1

2𝑖𝑠

1

4.

On the Smart Board, display the second example, which shows how to take a fraction

of a fraction with different denominators.

o Ava has 1

2of a pizza. She eats

2

3of the pizza she has. How much of the whole

pizza did she eat?

Ask students how they would fold and shade a sheet of paper to find 2

3𝑜𝑓

1

2. Have

students fold their second sheet of paper with you:


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o Fold the sheet of paper vertically. Unfold it and shade the left half with a light

color.

o Fold the sheet approximately into thirds, horizontally. Unfold the sheet and

use a dark color to shard the bottom 2

3 of the entire sheet and also

2

3 of each

half.

o Write an X in each of the two parts of the sheet that was shaded twice.

Tell students that these parts represent 2

3𝑜𝑓

1

2.



the lesson.




Closing:

Tell students that we just started the process of multiplying fractions. Tell them to

remember what we did today for tomorrow’s lesson.


Students will finish page 259 and then work on pages 260-261 & 263 of their Math

Journal either independently or with a partner.


Materials Used:

Everyday Mathematics® Math Journal 2, pp. 259-261 & 263

Smart Board Lesson

White printer paper (2 sheets per student, but have extras on hand)

Markers/Crayons/Colored Pencils

Reflection:

I loved this lesson! The kids had fun with it and using the paper really helped students

understand the process of taking a fraction of a fraction. One thing that I would do

differently would be to have them unknowingly fold and color the paper as their Engage.

They would then have to put the pieces together that the paper they folded represented

Larry’s granola bar.


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8.6 Area Model for Fraction Multiplication Lesson Plan


Unit Topic: Fractions & Ratios Lesson Focus: Multiplying Fractions






same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)


Comparing the size of a product to the size of one factor on the basis of the size of

the other factor, without performing the indicated multiplication.






fraction equivalence




Objective(s):

To develop a fraction multiplication method

Anticipatory Set:

Using the Smart Board lesson, display these three questions have students write the

questions and answers in their Math Spiral. After everyone is done, ask for volunteers

to share their answers. Ask students to explain how the solved the problems.

o 3 sets of 4 coins. How many coins in all? (12)

o 8 rows of 8 stamps. How many stamps in all? (64)


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o 3 bags of 12 oranges for each of 3 classrooms. How many oranges in all? (108)


Tell students to work on question #1 of page 265 of their Math Journal.

Tell them to solve the problem by using the paper model we did yesterday.

After students are finished, have a student come up to the board and write their

answer. Have the student explain what they are doing to the class.

Show the equation 2

3∗

3

4 on the board. Ask students if they know how to solve the

problem. Have any students who do know how to solve the problem come up to the

board and show the class.

Correct any misconceptions and then re-show how to multiply fractions. (Numerator

* numerator and denominator * denominator then simplify).

Have students use the shading technique from the day before the represent the

multiplication problem.

Tell students that 2

3∗

3

4 is the same things as saying

2

3𝑜𝑓

3

4

Ask students: How does the diagram show the answer to the problem? (the over-

lapping areas represent the answer).

Ask students: How is this model similar to the paper folding from yesterday? Students

should notice that it is the exact same as the paper model from the day before.



the lesson.




Closing:

Tell students that they just learned how to multiply fractions!



with a partner.


Materials Used:

Everyday Mathematics® Math Journal 2, pp. 264-267

Smart Board Lesson

Math Spiral


24

Reflection:

This lesson went really well. I liked having done the paper folding model the day

before. This helped to introduce the concept of multiplying fractions with ease.


25

8.7 Multiplication of Fractions & Whole Numbers








same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)






fraction equivalence




CCSS.MATH.CONTENT.5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate

expressions with these symbols. Objective(s):

To provide experience finding a product of a whole number and a fraction

Anticipatory Set:


26


Have students open up to page 268 in their Math Journal.

o Ask students to scan the page.

o Ask students if they remember doing these types of problems in earlier

grades.

o Have student’s complete page 268 with a partner (it should take them about

5-10 minutes).

After students have completed the page, point out that 5th graders still solve the

problems, but they also write the number models for calculations that solve the

problems.

Review the problems by having volunteers rewrite each problem in the form 𝑎

𝑏∗

𝑐

𝑑 on

the board and then write the product.

Display this number story on the board. Have students solve the problem using both

the area model and multiplication.

o Sally made a cherry pie and a raspberry pie. Her family ate 2

3 of the pies. How

much of the pie did they eat? (11

3)

Refer students to the top of journal page 270 and ask how this algorithm could be

used to multiply a fraction and a whole number such as 2

3 * 2. Remind students that

any number can be thought of as a fraction with a denominator of 1.

Assign the rest of the page for the remainder of the lesson.



the lesson.


27




Closing:

Bring the class to a close by asking students to show you page 270 of their journals before

moving on to the next page.


Students will work on pages 268 - 271 of their Math Journal either independently or

with a partner.


Materials Used:

Everyday Mathematics® Math Journal 2, pp. 268 - 271

Smart Board Lesson

Reflection:

This was an easy lesson for students to understand and I liked how it connected to

the previous two days. The short lesson also allowed me time to work with my low students.


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8.8 Multiplication of Mixed Numbers

Name: Kristin Herderich Date: 03/10/2014 – 3/11/2014 Grade Level: 5th



CCSS.MATH.CONTENT.5.NF.B.4.A Interpret the product (a/b) × q as a parts of a partition of q into b equal parts;

equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) CCSS.MATH.CONTENT.5.NF.B.4.B

Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. CCSS.MATH.CONTENT.5.NF.B.5.A

Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. CCSS.MATH.CONTENT.5.NF.B.5.B

Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1. CCSS.MATH.CONTENT.5.NF.B.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. Objective(s):

To introduce multiplication with mixed numbers

Anticipatory Set:

DAY 1: Have students write each mixed number as a fraction.


29

DAY 2: Display the following problems on the board and have students solve the

problem using their favorite method from the day before.


DAY 1

Have students open up to page 272 and complete the problems on the page as a

class. Pull sticks for students to come up and answer the questions. Have students

explain their reasoning.

Present this math problem on the board and ask students how they think they would

solve the problem.

o 6 × 43

5=

o Accept all answers without comment, noticing the students that have the

correct answers.

The most common answer will be turning 43

5 into an improper fraction. Start with

this method first.

Write another multiplication problem on the board and have students practice in

their Math Spiral. Circulate and assist where necessary.

o 3 × 22

3=

Write the first multiplication problem on the board. Show students how to solve the

problem using the Partial Products method.


30

Have students practice the following problems in their Math Spiral. Circulate and

assist where necessary.

o 4 × 31

2=

o 8 × 31

4=

Assign pages 273 – 274A

DAY 2

After students solve the word problems, solve the following problems on the Smart

Board as a class using both methods from the day before.

Have students open up to page 275 of their Math Journal. Show students how to

solve multiplication problems using fraction sticks.

Assign pages 274B – 260. Assist struggling students while other students are

working.



the lesson.




31

Check the previous night’s study link to ensure understanding of concepts

Closing:

Exit Slip: Tell me one method you can use to multiply mixed numbers.



with a partner.


Materials Used:


Math Spiral

Smart Board Lesson

Reflection:

I felt that this was one of my stronger lessons of this unit. That being said, if I had to

do it again, I would have split up the days differently. I would introduce the improper

fractions method on day 1 and then introduce the partial products method on day 2. I feel

that this would help split up the concepts better and allow students adequate time to

practice both.


32

8.9 Finding a Percent of a Number


Unit Topic: Fractions & Ratios Lesson Focus: Percentages


CCSS.MATH.CONTENT.6.RP.A.3.C Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100

times the quantity); solve problems involving finding the whole, given a part and the percent. Objective(s):

To broaden students’ understanding of calculating percent’s to include discounts.

Anticipatory Set:

Have students write each percent as a decimal and a fraction


Show the following problem to the class. Have students talk with their islands about

the problem: It would cost $150,000 to rent a large amusement park for a private party.

Would you rather have the price reduced by $35,000 or discounted by 25%?

Ask each island what they chose. Have students explain their strategies to the class.

Pose the following question: The flu hit Rochester Elementary really hard. 40% of the

480 students were absent at least one day last week. How many students were absent

at least one day?

o Write 40% of 480 = students absent at least one day on the board.

o Ask students to share their solution strategies to this problem.

Students will give a variety of answers. Focus on the unit-percent approach first.

o Remind students that to find 1%, they could divide the whole by 100 or think:

what times 100 equals the whole?


33

Another possible strategy is the 10% approach:

o Remind students that to fine 10% they could divide the whole by 10 or think:

what times 10 equals 480?

Other possible strategies that students might mention:

Have students turn to page 277 of their Math Journal. Tell them that percent’s can be

thought of as fractions

o 43% = 43

100

o Have students figure out two on their own:

50% = (50

100𝑜𝑟

5

10𝑜𝑟

1

2)

25% = (25

100𝑜𝑟

1

4)

o Assign the first problem and circulate and assist where necessary.

Display this number story on the board: The list price for a toaster is $45. The toaster

sold at a 12% discount (12% off the list price). What are the savings?

o Ask students what the problem is asking (12% of $45).

o Have students do the problem.

o Remind students that a discount is the amount to be subtracted from a given

whole.

Assign the rest of page 277 through 278. Circulate and assist where necessary.



the lesson.





34

Closing:

Ask students where they might use percent discount in real life. Shopping should be a

popular answer.



with a partner.


Materials Used:


Math Spiral

Smart Board Lesson

Reflection:

This is a short(er) lesson and I inadvertently left myself a lot of time at the end of the lesson. This ended up working out really well because I was able to help my low students and work with them one-on-one or in small group to help them understand the concept.


35

8.10 Relating Fractional Units to the Whole Lesson

Plan


Unit Topic: Fractions & Ratios Lesson Focus: Relating Fractional Units to

the Whole






same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) =

ac/bd.)CCSS.MATH.CONTENT.5.NF.A.2




CCSS.MATH.CONTENT.6.RP.A.3.C

Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100

times the quantity); solve problems involving finding the whole, given a part and the

percent.

Objective(s):

To provide practice finding the whole, given a fraction or a percent of the whole.

Engage:

Pass out a small Dixie-cup of 20 skittles to each student (any other small type of

candy will work).

Ask each student to use their skittles to represent the following fractions:

o 2

5,

1

4,

3

5,

1

2,

1

5,

3

4&

4

5

Walk around and check students work after each fraction to ensure understanding

After they have successfully found each fraction using skittles, tell students that they

just found fractional units of a whole! And then they may eat their skittles.


36


Have students open up to page 280 in their Math Journals.

o Walk them through questions #1-4.

o Question #4: Emphasize the number the model needed to figure out the

answer. Encourage students to use number models from now on.

o Have students try question #5 on their own.

o Pull sticks and have students come up to the Smart Board to solve the

remaining questions. Have them tell the class what they are doing and why

they are doing it.

Ask students what the definition of a Unit Fraction is. What about a Unit Percent?

Once students understand the definitions, ask students which questions on page 280

are a unit fraction and unit percent.

Have students open up their Student Reference Book to page 75.

o Using the prepared Smart Board slide, walk students through the first

question using a Fraction Strip.

o Emphasize how the meaning of the fractions in the examples is sued to find

the value of a unit fraction.

o Show students the second example in their book and see if THEY can walk

YOU through it.

o Have students complete the following problem in their Math Spiral to ensure

understanding. Walk around to check students’ answers.

o After students have successfully completed the above problem, have them

work together as an island to complete Journal page 281 for about 10 – 15

minutes.

Open Student Reference Book to page 52. Ask students how the first example relates

to the other problems we worked on. Ask them how they are different.

o Work with the class on the first problem on the Smart Board.

o Ask students how to solve the problem using percentages instead of fractions

o If time allows, have students work together to complete the second example

in the book.

Assign students to complete Journal pages 282 and 283 for the rest of the class

period.

Meet with any students who are still unsure about the concept after the lesson is

over.



the lesson.


37




Closing:

Ask students to give thumbs up if they feel comfortable working on the rest of the lesson by

themselves. Ask students to give a thumbs down if they do not feel comfortable and work

small group with those who do not feel comfortable.


Students work on page 282 and 283 of the Math Journal either independently or with

a partner.


Materials Used:

Everyday Mathematics® Math Journal 2, pp. 254 & 256

Student Reference Book, pp. 52 & 75

Math Spiral

Smart Board Lesson

20 skittles per student

Dixie cups (to hold skittles)

Plates (to count skittles on)

Reflection:

This lesson was a semi-disaster. Instead of building up to the idea of algebra, I

instantly started with algebra and immediately confused most of my students. Instead, I

would slowly build up to it. Also, the skittles were fun, but it was hard to reign my students

back in after the activity.


38

8.12 Fraction Division

Name: Kristin Herderich Date: 03/17/2014 – 3/18/2014 Grade Level: 5th

Unit Topic: Fractions & Ratios Lesson Focus: Dividing Fractions


CCSS.MATH.CONTENT.5.NF.B.7.A Interpret division of a unit fraction by a non-zero whole number, and compute such

quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. CCSS.MATH.CONTENT.5.NF.B.7.B

Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. CCSS.MATH.CONTENT.5.NF.B.7.C

Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? Objective(s):

To introduce division of fractions and relate the operation of division to multiplication.

Anticipatory Set:

DAY 1: Chalk Talk

o You will have 4 large posters, each with a question:

What do you know about adding fractions?

What do you know about subtracting fractions?

What do you know about multiplying fractions?

What do you know about dividing fractions?

o Tell students that they are to take a marker and answer each question without

talking.

o If a student agrees with another student or their ideas are similar, they may

draw arrows to each other’s ideas.


39

DAY 2: Display the following questions on the Smart Board. Have students answer

each question in their head and then go through the answers. Have students explain

the last problem and how it is similar to the other problems.

Show the following YouTube video on dividing fractions:

https://www.youtube.com/watch?v=uMz4Hause-o


DAY 1

Have students open up to page 288 in their Math Journal and look at problems 1

and 2.

Discuss students’ solutions to the problems. On the Smart Board, show students

how to solve the problems using a ruler.

https://www.youtube.com/watch?v=uMz4Hause-o


40

Point out that each problem on the journal page asks: How many x’s are in y?

Ask students to translate each problem into a question of this form and record

their answers on the board.

Next, ask students what open number sentence fits each of the problems above.

Have students pull out their Student Reference Book and open up to page 79

Walk students through dividing a whole number by a fraction.

Draw three rectangles on the Smart Board and present the following problem:

o Jane has 3 loaves of bread to share with her friends. If she cuts each loaf

into 1

4’s, how many quarter loaves will she have to share with her friends?

o Ask students: How many

1

4’s are in 3?

12

o Ask students: What number model is used to represent this problem?


41

3 ÷1

4= 12

Have students open up to page 80A in their Student Reference Book. Talk about

what to do when dividing a fraction by a whole number.

Pose the following problem on the board and draw a square underneath the

problem.

o Three family members equally share 1

5 of a loaf of corn bread. How much

of the loaf of corn bread will each person get?

Split the square into fifths and shade one-fifth of the square. Tell students that

the shaded part represents the loaf of bread that the family members will share.

Have a student come up to the board and split the square into thirds. Tell

students that the thirds represent each family member.

Explain that because the family only has 1

5 of a loaf to begin with, when it is

divided into three equal parts, each part of the corn bread that is cut up is 1/15 of

the entire loaf. So each person will get 1

15 of the loaf of corn bread.

Ask: What number model represents this problem? 1

5÷ 3 =

1

15

Assign pages 288 – 289A as well as the St. Patrick’s Day worksheet for homework.

DAY 2

After students complete the mental math, have students work on three sample

problems. Emphasize the “Keep, Change, Flip!” method. Also encourage other

strategies from the day before.

o 1

10÷ 5 =

o 6 ÷1

5=

o 1

10÷

3

10 =

Explain the relationship between multiplication and division of fractions

Ask students what they would do if they had a division problem with mixed

numbers?! Pose a number sentence on the board and have students talk with a

partner about how to solve the problem.


42

o 31

3÷

5

6=

o Accept the answer that you must turn the mixed number into an improper

fraction and then “keep, change, flip!”

Have students open their Math Spiral and try a few on their own. Circulate and assist

where necessary.

Assign pages 289B – 290 and SL 8.12.

Work small group with low students who are struggling with the concept for the rest

of the time.



the lesson.




Closing:

Pull students attention back to the Chalk Talk they did on the 17th. Ask students what new

information they would add to the poster board on fraction division.



with a partner.

St. Patrick’s Day worksheet for homework (Day 1)

Students will also have Study Link 8.12 for homework (Day 2)

Materials Used:


Student Reference Book, pp. 79 & 80A

Math Spiral

Smart Board Lesson

4 Poster Boards with one question on each:

o What do you know about adding fractions?

o What do you know about subtracting fractions?


43

o What do you know about multiplying fractions?

o What do you know about dividing fractions?

Markers (different color for each student, if possible)

St. Patrick’s Day Worksheet

Flocabulary YouTube Video: https://www.youtube.com/watch?v=uMz4Hause-o

Reflection:

This was the first time I ever attempted a Chalk Talk and I absolutely LOVED it and

the students did too. However, I think I would start the second day with the Chalk Talk

instead of the first day. This way, division of fractions has already been introduced and

students will have more to write on the concept.

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