Day 3 Equivalence and Addition of Fractions...Day 3 Equivalence and Addition of Fractions I can name a fraction number in different ways. ... 5.1.3.2 Model addition and subtraction

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Day 3 Equivalence and Addition of Fractions

I can name a fraction number in different ways. I can draw pictures of equivalent fractions. I can write true-false sentences and open number sentences to represent equivalent fractions. I can explain how to create a set of equivalent fractions. I can estimate sums of fraction numbers. I can find the sum of two fraction numbers using fraction circles. I can find the sum of two fraction numbers using the algorithm. I can find the sum of two fraction numbers using a number line.

Minnesota Mathematics Standards (2007) Understand meanings and uses of fractions in real-world and mathematical situations. 3.1.3.3 Order and compare unit fractions and fractions with like denominators by using models and an understanding of the concept of numerator and denominator. Represent and compare fractions and decimals in real-world and mathematical situations; use place value to understand how decimals represent quantities. 4.1.2.1 Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions. 4.1.2.2 Locate fractions on a number line. Use models to order and compare whole numbers and fractions, including mixed numbers and improper fractions. 4.1.2.3 Use fraction models to add and subtract fractions with like denominators in real-world and mathematical situations. Develop a rule for addition and subtraction of fractions with like denominators. Add and subtract fractions, mixed numbers and decimals to solve real-world and mathematical problems. 5.1.3.1 Add and subtract decimals and fractions, using efficient and generalizable procedures, including standard algorithms. 5.1.3.2 Model addition and subtraction of fractions and decimals using a variety of representations. 5.1.3.3 Estimate sums and differences of decimals and fractions to assess the reasonableness of results. 5.1.3.4 Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed numbers, including those involving measurement, geometry and data.

Equal Sharing Problem

Eight students share six cookies evenly. How much of a cookie does each student receive?

Day 3 – Equivalence and Addition page 2

Making Connections Among Student Strategies

A

B

C

D

Questions that could be asked

Is

(student A) the same amount as

(student D)? Why or why not?

Is

(student D) the same amount as

(student C)? Why or why not?

How is the amount

(student C) the same as

(student B)?


Why are we here?

Student Problem Student knows … Student does not know …

Madison (5th

grade)

#356

25 seconds

Sean (5th

grade)

#370

1 min 35 seconds

Jacky (5th

grade)

#344

1 min 33 seconds

Johanna (3rd

grade)

#352

1 min 3 seconds

Felicia (2nd

grade)

#329

1 min 39 seconds

Leah (5th

grade)

1 min 40 seconds


Why is

equivalent to

?

4.1.2.1 Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions.

I. Drawing Pictures

Johanna (3rd

grade) explains in her video that

is the same as

. Use a variety of pictures

and symbols to show that

is equivalent to

.

II. Equal Sharing Problems

Write an equal-sharing problem to encourage a discussion comparing the amounts

and

.

III. True-False Sentences / Open Number Sentences

Determine if these number sentences are true or false. Explain your response.

a)

b)

c)

Find the value of the variable that makes the equation true.

d)


Equivalence Using Pictures of Fraction Circles I. Use the fraction circles to show and name two additional fractions that are equivalent to one-half.

one-half

II. Draw in lines to show how to find the value of the variable that makes the equation true.


Why is

equivalent to

?

4.1.2.1 Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions.

I. Drawing Pictures

In the last video, Leah (5th grade) explains using a number line that

is the same as

. Use

a number line to show that this is true.

II. Recording Leah’s Thinking

Write some true-false sentences that describe Leah’s thinking.

III. Equal Sharing Problems

Write an equal-sharing problem to encourage a discussion comparing the amounts

and

.

IV. True-False Sentences / Open Number Sentences

Write some true-false sentences or open number sentences that might illustrate student

thinking in this discussion.


Equivalence Using Paper Strips

Fold and shade three paper strips that each represent

of a strip shaded in. Make

additional folds to create a different equivalent fraction for each strip. Record your actions on the strips below. Write appropriate true-false sentences and/or open number sentences for each strip you make.


Drawing Number Lines to Find the Missing Number

A.

B.

C.

D.

E.


Capture Half or More Game

Capture Half or More For two players/teams Materials: Capture Half or More Fraction Cards Paper for Fraction Strips

1. Each player/team makes a separate fraction strip for halves, fourths, eighths, thirds and sixths. All strips

from both players/teams are placed in the middle. (They may trace over their fold lines to make the strips

easier to determine but should not write fraction names on the strips.)

2. Player/Team 1 draws a fraction card and searches for a strip where they can shade in that fraction. If they

have shaded half or more of a strip they can keep the shaded strip. If the strip is less than half shaded, they

return the shaded strip to the middle pile.

3. Player/Team 2 draws a fraction card and can either shade that fraction on a new strip or add onto a

previously shaded strip. If they have now shaded half or more of a strip they can keep the shaded strip. If

the strip is less than half shaded, they return the shaded strip to the middle pile.

4. Play continues, alternating turns.

5. They must “prove” any equivalences they use. (e.g. They draw the fraction four eighths but all the eighths

strips are gone. They might say: “Four eighths is the same as one half because I can hold a strip divided into

eighths next to a half and 4 of the eights cover the same amount as one of the halves so I can color one half

of this strip and that is the same amount as four eighths.”)

6. Player/Team who ends up with capturing the most shaded strips wins the game.

VARIATION-for beginning fraction learners: Initially play the game with only strips folded in halves, fourths and eighths. Perhaps have both players/teams make 2 of each kind of strip to make a longer game and reinforce equivalences using connections between visual and numerical representations. VARIATION-Capture Half or Less: Use all the Capture cards. Play to capture half or less. Any fraction card drawn that would result in shading more than half of a strip is thrown out of the game and play passes to other player/team.


Generating Equivalent Fractions using Paper Strips and Fraction Circles I can make connections among equivalent fractions represented by paper strips, fraction circles, and symbols.

fraction circle paper strip symbol

Why does this work?


Why does this work? Assume that c divides both a and b.


Review of Informal Ordering Strategies (from Day 2) Determine the larger number by picturing them in your mind. Circle the larger number in each pair.

I. Same Numerator II. Transitive

III. Residual IV. Same Denominator

Formal Ordering Strategies

V. Using Equivalent Fractions

VI. Using Cross-Multiplication


Estimating Sums I can estimate sums of fraction numbers using mental images. Use Landmark Number Lines here.

11

2

1

2210

11

2

1

2210

11

2

1

2210

11

2

1

2210

11

2

1

2210

11

2

1

2210

0


Finding Exact Sums I can find exact sums of fractions using fractions circles, the algorithm, and the number line. I can make connections between the different strategies for finding sums of fractions. 4.1.2.3 Use fraction models to add and subtract fractions with like denominators in real-world and mathematical situations. Develop a rule for addition and subtraction of fractions with like denominators. 5.1.3.1 Add and subtract decimals and fractions, using efficient and generalizable procedures, including standard algorithms. 5.1.3.2 Model addition and subtraction of fractions and decimals using a variety of representations. 5.1.3.3 Estimate sums and differences of decimals and fractions to assess the reasonableness of results. 5.1.3.4 Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed numbers, including those involving measurement, geometry and data.

Problem

Jayna has

of a cup of flour. Sarah gives her

of a cup of flour. How much flour does

Jayna have now?

Estimate

Exact Using Fraction Circles Exact Using Algorithm

Exact Using a Number Line

11

2

1

2210


Finding Exact Sums using Fraction Circles http://www.cehd.umn.edu/ci/rationalnumberproject/RNP1-09/RNP1-09_20.pdf

Name Lesson 20 / Student Page A

Fraction Addition and Estimation: Finding the Exact Answer

1. Marty ate some candy. He ate 1-half of a whole Hershey bar before lunch. He ate 1-fourth of a whole Hershey bar after lunch. About how much of one candy bar did he eat? With your fraction circles, find out the exact amount of a Hershey bar that Marty ate. Draw pictures to show what you did with the circles. Estimate first!!! Estimate: _____________________ 2. Terri ate 1-half of a small pizza and 5-twelfths of another small pizza. About how much of a whole pizza did she eat? With your fraction circles, find out the exact amount. Draw pictures to show what you did with the circles. Estimate first!!! Estimate: _____________________ 3. Allie rode her bicycle 7-eights of a mile to school. Then she rode 1-fourth of a mile to her friend’s house. About how far did she ride altogether? With your fraction circles, find out the exact amount. Draw pictures to show what you did with the circles. Estimate first!!! (Use back of the page for your drawing). Estimate: _____________________


Name Lesson 20 / Student Page B

Fraction Addition Continued

4. Because of a rainstorm, the water level in a swimming pool rose

2

3 of

an inch. The following day it rained again. The pool rose another

11

12 of

an inch. About how high did the water level increase? With your

fraction circles, find out the exact amount.

Estimate _______________

5. Alex used

1

4 cup of flour in one recipe and

3

8 cup of flour in another

recipe. Together about how much flour did he use? ? With your

fraction circles, find out the exact amount.

Estimate _______________

6. With your fraction circles, find the exact answers.

1

3 +

1

6

1

8 +

3

4

4

10 +

1

5

1

6 +

3

12

1

2 +

3

4


Connecting Fraction Circle Addition with Algorithm 5.1.3.1 Add and subtract decimals and fractions, using efficient and generalizable procedures, including standard algorithms.

I. Randi drinks

of a cup of water in the morning. She drinks

of a cup of water for

lunch. How much water did Randi drink during the day? Estimate: _______________________

Draw pictures of the fraction circle pieces. Write symbols and connect with pictures.

Annotate the connections.

How does a common denominator help name the sum?


II. Johnny runs

of a mile before school. He runs

miles after school. How many

miles did he run for the day? Draw pictures and connect with symbols. Estimate: ___________________

Draw pictures of the fraction circle pieces. Write symbols and connect with pictures.

Annotate the connections.


http://www.cehd.umn.edu/ci/rationalnumberproject/RNP2/Lesson04.pdf

Name Lesson 4 / Student Page B

Fraction Addition

Use your fraction circles to model each problem. Record what you do with the circles with symbols.

1. Mario ran of a mile and then rested a

few minutes. He then ran of a mile

more. How far did he run altogether?

2.

+

3. of the whole class finished the

spelling test and went to the

playground. of the whole class

finished soon after and joined the others in the playground. What fraction of the class is now on the playground? What fraction of the class is left inside?

4.

+


http://www.cehd.umn.edu/ci/rationalnumberproject/RNP2/Lesson05.pdf

Name Lesson 5 / Student Page A

Adding Fractions with Fraction Circles Do you think the answer will be greater than one or less than one?

Use fractions circles to solve and then describe what you did with the circles to find the answer.

Now record the steps you used with the fraction circles to solve this problem with symbols.

Solve this problem using equivalent fractions with 8ths as the common denominator.

Verify using the fraction circles.

Solve in two ways. Use 9 ths as a common denominator and use 18 ths as the common denominator. Why are the answers equivalent?


Fraction Addition on a Number Line http://www.cehd.umn.edu/ci/rationalnumberproject/RNP2/Lesson16.pdf


5.1.3.2 Model addition and subtraction of fractions and decimals using a variety of representations.


Fraction Fill Directions: The teacher randomly selects a numeral card and shows it to the students. Each student chooses to shade that amount on one of the circles. They can only shade one representation for that

amount. For example, if the

card is selected the student can shade in one the section of the circle divided

into fourths or three sections of the circle divided into twelfths . The student may not draw an extra line in the circle divided into half to shade in the one-fourth. The teacher continues to show numeral cards. The student who is first to completely shade two circles wins and says “Fraction Fill.”

A

B

C

D


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Day 3 Equivalence and Addition of Fractions...Day 3 Equivalence and Addition of Fractions I can name a fraction number in different ways. ... 5.1.3.2 Model addition and subtraction