by liza dallavalle and michele ziegler, 2015 math resource teachers for carroll county public...

*What is a fraction?

*A fraction is ___.

By Liza Dallavalle and Michele Ziegler, 2015

Math Resource Teachers for Carroll County Public Schools

WELCOME to

Developing the Meaning of Fractions

in Grades 1-3

*Who is who?

*What is a fraction?

*A fraction is ___.

*Today we will learn how the…

- partitioning standards in first, second and third grade build the foundation for understanding fractions as numbers. 

*Prove It!(True or False)

Standard for Mathematical PracticeConstruct viable arguments and critique the reasoning of others.

I can’t share 13 equally among 4

people.

* Prove It!(True or False)

I can share 2 equally among 4

people.

* Prove It!(True or False)

When an object is cut into equal shares, the shares have to be identical in shape.

* Prove It!(True or False)

One half is always greater

than one fourth.

* Prove It!(True or False)

*This is not a fraction.

* Prove It!(True or False)

A fraction is a number.

* Prove It!(True or False)

https://www.plickers.com/

* Think about your grade level’s Geometry partitioning standard.

* In your own words describe what this standard entails.

*Cohesion in Grades 1 -3 Geometry Standards

CCSS.Math.Content.1.G.A.3Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

CCSS.Math.Content.2.G.A.3Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

CCSS.Math.Content.3.G.A.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

The Progression of Equal Shares

Cohesion in Grades 1 - 3*Read each problem.

*Solve and discuss.

*Place in order of progression

for teaching that builds on students’

understanding of whole number quantities

and informal partitioning strategies.

The Notions of PartitioningFirst Grade

CCSS.Math.Content.1.G.A.3Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

1st Grade Big Idea

1. Items can be partitioned into equal shares.

2. More shares of the same whole leads to a smaller fraction for each.

Using a Familiar Context

• Read The Doorbell Rang

• Record how the cookies are shared

on a class chart.

• Talk to a Shoulder Partner:

What do you notice?

What happens to the shares as more kids come into the kitchen?

Standard for Mathematical PracticeLook for and express regularity in repeated reasoning.

* First Grade

Partitioning a Set

Students are given manipulatives to model (2, 3, 4, 6, and 12)

fair shares as they create a class book.

Standard for Mathematical PracticeLook for and express regularity in repeated reasoning.

When creating the book page, the students used wooden blocks to represent the set of items that were equally partitioned.

Standard for Mathematical PracticeModel with mathematics.

* How do first and second grade students naturally deal with equally sharing one

whole cookie?

2 children and 13 cookies

4 children and 13 cookies

2 bears share 1 block of cheese

* First Grade

Partitioning of One Whole

* First Grade

Partitioning of a Whole

Standards for Mathematical PracticeConstruct viable arguments and critique the reasoning of others.

Look for and express regularity in repeated reasoning.

The Progression of Equal Shares

Cohesion in Grades 1 - 3Think about the Big Ideas addressed by the learning opportunities we have shared.

Talk About It

Which three situations best highlight the progression of equal shares in first grade.

Be prepared to support your decision.

*The Progression of Equal Shares

First Grade

-All objects MUST be shared and shared equally: 15 grapes shared with 3 children

-Relate their understanding of whole numbers to fractions and find them easier to solve than solutions that are less than one: Two children share 5 apples (deal and continue to deal)

- 2, 4, 8 still capitalize on halving strategies , just a little harder because it is less than one: Four children and one pie

* First Grade

Partitioning of a Whole

• Develop precision of vocabulary related to partitioning by beginning with what they know.

• Use a variety of manipulatives in familiar contexts to represent equal shares.

• Connect the representations to clarify language.

Standards for Mathematical PracticeAttend to precision.

Look for and make use of structure.Look for and express regularity in repeated reasoning.

Connect language to prior knowledge

Use models to clarifythe language

Connect models with prior knowledge

Fold shapes to modelhalves/fourths

Create models usingdifferent manipulatives

Standard for Mathematical PracticeReason abstractly and quantitatively.

Standard for Mathematical PracticeModel with mathematics.

Each group was given a

predeterminedset of “dishes”

(aka cubes).

Standard for Mathematical PracticeModel with mathematics.

* Coherence in First Grade

Connections to Other Standards

1.G.2. Compose two-dimensional shapes…to create a composite shape…

*What if this piece represented half of a cookie? What would the whole cookie look like?

*What if this piece represented a quarter of the cookie? What would the whole cookie look like?

- Give students a set of different sized pieces and ask them to compose the cookie.

- Give students a half/quarter of a circle to trace and the students sketch the other half/quarters.

1.MD.3 Tell and write time in hours and half-hours using analog and digital clocks.

*When is the hour hand half way around the clock?- Clock face fold it in half

* If an hour is when the hour hand makes a full rotation around the clock, when does the hand on a clock show half of an hour? - Are there other ways to show half an hour?

The Notions of PartitioningSecond Grade

CCSS.Math.Content.2.G.A.3Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

2nd Grade Big Ideas

1. Recognize equal shares.

2. Check for identical measures rather than identical shapes.

3. Notice that the size of the pieces relates to the size of the whole.

*2nd GradeRecognizing Equal Shares

Equal Shares

vs

Not Equal SharesSort It Out

Standards for Mathematical PracticeAttend to precision.

Look for and make use of structure.

*2nd GradeRecognizing Equal Shares

Dividing Cakes“Each group will divide rectangular cakes so that groups of 2 – 10 people will have equal portions.”

“Draw lines on each cake to partition it into equal shares for different numbers of people. You may want to fold the paper first. Also be prepared to explain for each of your cakes why you think the shares are equal.”

Standards for Mathematical PracticeModel with mathematics.

Look for and make use of structure.

*Talk About It

*Sort your cakes into “Easy to Partition” vs “Harder to Partition”

- Discuss why.

*For each of your cakes, how did you know the shares were equal?

* 2nd GradePartitioning of a whole into Equal Shares

*“I’ve bought some apples for you. I am going to put you into groups of three . Then you’ll talk to your group about how you might share the apple equally, so each get the same amount.”

*“ After you share your ideas, agree on one plan that you think would work. Then write down the plan. You may want to include drawings and words.”

*“When you show me your plan, I’ll follow it and cut your apple.”

*“Also I would like you to write how much each person gets.”

Each person gets ___.

Standard for Mathematical PracticeConstruct viable arguments and critique the reasoning of others.

*Talk About It

*What insight in student thinking do these “plans” provide?

- What do the students know?

- What misconceptions might they have?

The Progression of Equal Shares

Cohesion in Grades 1 - 3Think about the Big Ideas addressed by the learning opportunities we have shared (2nd grade).

Talk About It

Which two situations best highlight the progression of equal shares in second grade.

Be prepared to support your decision.

*The Progression of Equal Shares

Second Grade

-Children have to anticipate how to slice or cut the objects. Many children will attempt to use a repeated halving strategy or trial and error. Moving to 3 will force children to confront their tendencies to use halving strategies.: 3 children share 10 sticks of clay (deal and continue to deal)

-Same as above just less than one: 3 children share 1 candy bar

The Progression of Equal Shares

Cohesion in Grades 1 - 3Take a look at the remaining two situations.

Talk About It

Which situation do you think would come first?

What makes these situations more challenging than the others presented to first and second grade students?

*The Progression of Equal Shares

Third Grade

-Less than one, but not a unit fraction: Four children share 3 cookies

-Less than one, not a unit fraction, with thirds: Three children share 2 pizzas

*2nd Grade

Area Matters

Which brownie do you think is the largest?

Would you have more brownie to eat if you selected that one?

Which brownie is the smallest? Would you have less brownie to eat if you selected that one?

Are any brownies equal in size?Standards for Mathematical PracticeMake sense of problems and persevere in solving them.

Construct viable arguments and critique the reasoning of others.

Brownies

Cutting the Cake

The baker wants to cut his square cake into fourths a different way

every day, how many ways can he cut his cake?

Standard for Mathematical PracticeConstruct viable arguments and critique the reasoning of others.

*2nd Grade

Area Matters

*2nd Grade

MINGLE*Mingle.

*Compare the shaded portion of your square.

*Record your findings.

*Mingle and repeat.

*Make a group of all the people with the same

share of the square shaded.

*Prove it!

Standard for Mathematical PracticeConstruct viable arguments and critique the reasoning of others.

*I wonder…(Pulling it together)

*2nd Grade

The Whole Matters

Ring Around the

Halves

Standard for Mathematical PracticeConstruct viable arguments and critique the reasoning of others.

Are all halves equal ?

*2nd Grade

The Whole Matters

Standard for Mathematical PracticeConstruct viable arguments and critique the reasoning of others.

* Coherence in Second Grade

Connections to Other Standards

2.OA.3 Determine whether a group of objects has an even or odd number of members…

2.MD.2 Measure the length of an object twice, using length units of different lengths for the two measurements. Describe how the two measurements relate to the measurement chosen.

2.MD.7 Tell and write time…to the nearest five minutes.

2.MD.8 Solve word problems involving money.

2.MD.9 Show the measurements by making a line plot

2.G.2 Partition a rectangle into rows and columns of same size squares…

* Coherence in Second Grade

Connections to Other Standards2.OA.3 Determine whether a group of objects has an even or odd number of members…

-partition a set into two equal groups or into groups of 2

2.MD.2 Measure the length of an object twice, using length units of different lengths for the two measurements. Describe how the two measurements relate to the measurement chosen.

- the smaller the unit the more of the units needed

- the more you partition the smaller the piece

2.MD.7 Tell and write time…to the nearest five minutes.

- partition the clock in half and quarter hours

- partition an hour into 60 minutes or 12 groups of 5 minutes

2.MD.8 Solve word problems involving money.

- understanding units of pennies, nickels, dimes, and quarters

- the more you partition the smaller the amount

2.MD.9 Show the measurements by making a line plot

-partitioning the number line into equal units of 1

2.G.2 Partition a rectangle into rows and columns of same size squares…

- partitioning

*3rd Grade

*CCSS.Math.Content.3.G.A.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.

3rd Grade Big Idea

1. Fractions are numbers.

* Fractions as Numbers

Standards for Mathematical PracticeLook for and make use of structure.

Look for and express regularity in repeated reasoning.

* Third Grade

How do the geometry standards support understanding of fractions in third grade?

1. Items can be partitioned into equal shares.

2. More shares of the same whole leads to a smaller fraction for each.

3. Checking for identical measures rather than identical shapes.

4. Noticing that the size of the pieces relates to the size of the whole.

5. Fractions as numbers.

* Third GradeShow how the BIG IDEAS of the geometry standards

in 1st and 2nd grade support understanding of “fractions as numbers” in third grade.

*What is the learning progression from here?

Coherence in Grades 3-53rd Grade: CCSS.Math.Content.3.NF.1 through 3

*Develop understanding of fractions as numbers.

4th Grade: CCSS.Math.Content.4.NF.1 through 7

*Extend understanding of fraction equivalence and ordering.

*Build fractions from unit fractions by applying and extending previous understandings of operations and whole numbers.

*Understand decimal notation for fractions, and compare decimal fractions.

5th Grade: CCSS.Math.Content.5 .NF.1 through 7

*Use equivalent fractions as a strategy to add and subtract fractions.

*Apply and extend previous understanding of multiplication and division to multiply and divide fractions.

*Let’s Go Back to the Beginning

I can’t share 13 equally among 4

people.

* Prove It!(True or False)

I can share 2 equally among 4

people.

* Prove It!(True or False)

*When an object is cut into equal shares, the shares have to be identical in shape.

* Prove It!(True or False)

One half is always greater

than one fourth.

* Prove It!(True or False)

*This is not a fraction.

* Prove It!(True or False)

A fraction is a number.

* Prove It!(True or False)

*Thank you for attending our

session.

Please feel free to contact us at the following:

Michele Ziegler- maziegl@carrollk12.orgLiza Dallavalle- eadalla@carrollk12.org

*We would like to give special thanks

to…*Meghan Engle and Shelley Frock (first grade teachers from Charles Carroll Elementary School), Sandy Johnson and Adelle King (second grade teachers from Linton Springs Elementary School), Shelly Boblits (second grade teacher from Spring Garden Elementary School ), Ingrid Hiltz (first grade teacher from Spring Garden Elementary School) for contributing to this presentation by implementing the lessons, collecting work samples, providing feedback, and submitting photos.