Introduction to
Fractions
A One Week Unit of Study for Third Graders
Laura Sharpless
December 2011
Math Methods-Elementary
2!
Introduction to Fractions
A One Week Unit of Study for 3rd Graders
Table of Contents
Overview……………………………………………………………………………………. 4
Lessons:
Sharing Sub Sandwiches…….……………………….……………………………..
6
Fraction Strips…………………………………………………………………………
9
Family Snacks…………………………………………………………………………. 12
Quilt Blocks……………………………………………………………………………. 15
Menu of Independent Activities…………………………………………………….. 20
Scope and Sequence…………………………………………………………………….. 25
Resources………………………………………………………………………………….. 28
3
Introduction to Fractions
A One Week Unit of Study for 3rd Graders
Overview
Grade Level: Third
Time Frame: One week (about five hours of math classes)
Topic: An introduction to fractions
I chose this topic because during My Work With a Child assignment I worked with a
fourth grader who had a good understanding of fractions. I was curious to explore
how he might have constructed that understanding. In the Common Core
Standards, fractions are first given dedicated attention in third grade. While third
graders may think more abstractly than second graders, the abstractions necessary
for constructing fractional knowledge are still going to be challenging for eight and
nine year olds. Therefore I have grounded the problems students will be solving in
real situations that they can model with manipulatives.
Standards Addressed:
Common Core Content Strand - Number
Domain - Fractions
Clusters:
• Fractions are representations of numbers
• Fractional Quantities
*See the Scope and Sequence included at the end of this unit for a more detailed
description of the learning landmarks within the fraction domain.
NCTM Process Standards:
• Connections: Students are able to make connections between mathematics
and the rest of the world and across mathematics disciplines.
• Problem Solving: Students work on good problems that require mathematical
thinking.
• Communication: Students talk to each other about their thinking and are able
to effectively express themselves.
• Representation: Students express their thoughts visually (diagrams, tables,
number lines, pictures, etc.)
• Reasoning and Proof: Students find ways to prove their assertions.
Learning Expectations:
This unit is designed to be the first formal introduction to fractions for third graders. The
central concepts that this unit helps students construct are the following:
• Fractions represent a part of a whole.
• The size of the whole matters.
• The whole can be a measured length, a set of objects, or an area.
• There are multiple ways of representing the same fraction.
4!
Overview
Lesson Descriptions:
The lessons that follow are designed to give students real world problems that will guide
them to construct knowledge about fractions and the tools they need to solve and
communicate about those problems. The lessons use three models of fraction
problems: length, area, and set. Each of these different models is used in order to help
the students make connections between multiple contexts in which fractions can be
used. The unit concludes with an opportunity for the students to work independently
and choose among a number of activities in order to give the students time and space
to self-differentiate and work on problems that interest them.
Lesson 1: Family Sub Sandwich Lunch
In this lesson students are asked to divide whole sandwiches into equal shares for
different sized families. The concept of equal shares will be used as the foundation
for student knowledge about fractions.
Lesson 2: Fraction Strips
In this lesson students get a formal introduction to fraction language and notation by
creating their own fraction strip sets. Students also continue to develop the concept
of fractions as equal parts of a whole and may notice patterns of fraction
equivalency.
Lesson 3: Family Snacks
In this lesson students solve more equal share problems using a set model of
fractions. This is a very similar problem to the one presented in Lesson 1 but the
shift in model provides a new learning context for the students. Students will be
encouraged to think of groups of self-contained objects as a single whole.
Lesson 4: Quilt Blocks
In this lesson students gain flexibility with how fractions are represented, order
fractional quantities and begin to construct the idea of fractional equivalencies by
designing quilt blocks and comparing their blocks to those of other students.
Lesson 5: Menu of Independent Activities
In this lesson students continue to develop the big ideas about fractions that have
already been introduced: fractions represent part of a whole, the size of the whole
matters, and there are multiple ways of representing the same fraction by using
area, set, and length models of their choosing.
Overview!
5
Lesson Plan 1: Sharing Sub Sandwiches
Activity adapted from lesson taught by Carol Mosesson as described by Fosnot and Dolk in Young
Mathematicians At Work: Constructing Fractions, Decimals, and Percents, p.2
Standards Addressed:
Content: Understand that fractions apply to situations where a whole is
decomposed into equal parts.
Process: Problem Solving, Representation, Connections
Objectives:
Students will think of the fractions as equal shares of a whole using a length model
of fractions. The fair sharing context will encourage them to see the whole as a
constant; they are unlikely to consider throwing away pieces of sandwich.
Materials:
• Sharing Sub Sandwiches worksheet, one for each pair of students
• Pens (Have students record their work with pens so that none of their thinking is
lost through erasing.)
Lesson Description: (1 hour total)
Introduction: (10 minutes)
Bring the class together to introduce the lesson. Explain that this past weekend
you were at a sandwich joint for lunch. While you were there you saw three
different families order themselves lunch. Each family got sub sandwiches to
share amongst themselves. Describe the three situations outlined on the
worksheet. Emphasize that each family wants to share their sandwiches fairly.
Ask the students to help the families to figure out how to cut their sandwiches.
Worksheet Work: (35 minutes)
Divide the students into pairs, give each pair a Sharing Sub Sandwiches
worksheet and have them find a workspace.
While students work on the problems on the worksheet, circulate through the
room, asking them about their thinking. “Is this way of dividing the sandwiches
fair? How do you know? Show me.” Encourage them to use manipulatives if
they think that would be helpful.
Don’t worry about using proper fraction language at this time. Focus your
questions and the students’ attention on equal shares.
If some students are finished earlier than others, ask them to order the families
by how much sub each member got to eat.
Reflection: (15 minutes)
Bring the group back together to talk about their work. Have select students
present their work, being sure that they are explaining their thinking. Ask
students to explain what other students did, checking for comprehension. Note
misconceptions.
6!
Lesson 1
Possible discussion questions:
• Highlight different strategies students used: “What do you notice about how
(students A) and (students B) divided the sandwiches?”
• Emphasize the importance of equal parts: “To be sure that the way you cut
the sandwiches was fair, how did you cut the sandwiches?” Model a cut that
is not exactly in half and ask, “Would this be fair? Why not?”
• “Did the people in the different families get the same amount of sandwich?”
• “Who ate the most? Who ate the least? How can you tell?”
Assessment Criteria:
• Student can divide drawing of sandwich into equal shares.
• Student labels and speaks about the pieces of sandwich accurately.
Lesson 1!
7
Sharing Sub Sandwiches
Work with your partner to think about the following problem. Be sure to show your work!
Three different families are getting together for lunch. They will be enjoying sub
sandwiches today. Each family wants to share their subs fairly. How should each family
cut their subs so that each member of the family gets to eat the same amount?
The Harris Family:
4 people, 2 sandwiches
The Blake Family:
8 people, 2 sandwiches
The Swartz Family:
6 people, 2 sandwiches
8!
Lesson 1
Lesson Plan 2: Fraction Strips
Activities adapted from “The Fraction Kit” and “Cover-Up” in Marilyn Burns’, About Teaching Mathematics:
A K-8 Resource, p. 271
Standards Addressed:
Content: Understand that fractions apply to situations where a whole is
decomposed into equal parts. Understand that a unit fraction corresponds to a point
on the number line. Understand that fractions are built from unit fractions.
Process: Representation, Communication
Objectives:
Students will learn how to accurately express the fractions ½, ¼, and ⅛. Students
will explore fractional relationships: unit fractions as equal parts of a whole, ordering
unit fractions, and relationships of unit fractions to each other.
Materials:
• Strips of paper in four different colors, enough for each student to have one of
each color. Note that larger strips may be easier for students to handle.
• Envelopes large enough to fit the strips of paper inside without folding, one for
each student.
• Scissors, one for each student
• Black crayons, one for each student
• Fraction dice, enough for each pair of students to have one
• Blank recording paper
Lesson Description: (1 hour total)
Introduction: (10 minutes)
Bring the class together to introduce the lesson. Ask the students, “What do you
know about fractions?” Record their answers on a class chart. Children may
refer to the previous day’s lesson on sharing sub sandwiches. If they don’t make
the connection, don’t point it out to them; wait for them to make the connection
themselves. Explain that today they will be exploring fractions and that everyone
will make their own fraction kits.
Making the Fraction Kits: (15 minutes)
1. Hand out supplies to students. Each student will get four strips of paper, each
of a different color, one envelope, a pair of scissors, and a crayon. Have the
students label their envelopes with their names.
2. Ask the students, “What do you notice about the strips?” Make sure that they
see that all the strips of paper are the same size.
3. Identify one of the colored strips as the whole. Have the students all label the
same colored strip, “1.”
4. Chose a color to be the halves. Model folding the strip in half. “What do you
notice about how I’m folding this strip?” Make sure they notice that you are
carefully lining up the ends of your strip. Have the students fold their strips
the same way.
Lesson 2!
9
1. Before opening the folded strips, ask the students how many folds you have
made. Ask them to predict how many pieces their strips are now divided into.
Record their answers on a table like this one:
Folds
Pieces
2. Have the students unfold their strips, cut along the fold line, and label each
piece “½.” Write “½” on the board for all the students to see so that they know
the proper notation and tell them this number is called “one half.”
3. Repeat steps 4-6 for quarters and eighths.
4. Have the students lay their strips out in front of them like this:
1
1/2
1/2
¼
⅛
¼
⅛
⅛
¼
⅛
⅛
¼
⅛
⅛
⅛
Ask the students, “What do you notice about your sets of strips? What
patterns do you see? Are there patterns in the size of the strips? How about
in the way the fractions are written? Do you see a pattern in the number of
folds we made in our strips and the number of pieces we ended up with?”
Students may notice:
• The number on the bottom of the fraction (denominator) is equal to the
number of parts the whole was divided into.
• The more parts there are, the smaller each part is.
• Larger denominators make smaller fractions.
• ½, 2/4, and 4/8 all make up the same part of the whole.
• Folding the halves in half gives us fourths and folding the fourths in half
gives us eighths.
Cover Up Game: (20 minutes)
Explain that the students will now be using their fraction strips to play a game,
Cover Up. Explain the rules of the game while modeling the way the game is
played with your set of fraction strips.
• Students take turns rolling the dice and placing the fraction rolled on top of
their whole strip.
★ Be sure that as you place your strips on the whole that the edges are
right next to each other without overlapping.
• The first person to cover their whole strip exactly wins. If you roll a
fraction that would cover more than the whole, you must pass.
• Record how you get to one on a recording sheet.
10!
Lesson 2
After explaining the rules and modeling the game, split the class into pairs
(groups of three can work also), give each group a fraction die and tell them to
find a place to play.
While the students play the game, circulate through the room making sure that
they are playing the game properly, asking them about their thinking, and
reminding them to record their work on recording sheets.
Reflection: (10 minutes)
Bring the group back together to talk about their work. Ask select students to
share how they reached one or a whole. Have them use their strips to explain
themselves. Ask students to explain what other students did, checking for
comprehension. Note misconceptions.
Ask, “Do you think you’ve ever used fractions before today? Where do we see
fractions in the world?” Record their answers on a class chart. If they have not
yet connected fractions to the equal sharing of sandwiches lesson, point it out to
them.
Homework:
Have the students write in their math journals to help them begin to generalize
different fractional contexts*. “What relationship do you see between the way you
solved the sandwich problem and the way you made your fraction kits? Is there any
relationship between the family of eight sharing two sandwiches and the strip that we
cut into fourths or eighths? We’ll start math workshop tomorrow with a discussion of
your entries.”
*Assignment given by Jake Robinson as described by Fosnot and Dolk (2002), p. 80.
Assessment Criteria:
• Students will be able to decompose wholes into two, four, and eight equal parts.
• Students will be able to use fraction strips to exactly cover a whole with halves,
quarters, and eighths.
• Students will be able to communicate about fractions verbally, using the
expressions “one half,” “one fourth” and “one eighth.”
• Students will be able to write fractions using correct mathematical symbols: ½, ¼,
and ⅛.
• Students will be able to communicate their thinking verbally and through written
representations.
Possible Extension:
Continue constructing student understanding of fractions through more paper
folding. Try doing some origami with your students, emphasizing the importance of
exact folds to get precise halves etc. and to make sculptures looks correct. Try
folding and unfolding paper to see how the paper is divided up by different folds.
The Math Expressions grade 4 Teacher’s Guide recommends Math in Motion:
Origami in the Classroom by Barbars Pearl. You could also use origami as a
gateway into learning about Japanese culture.
Lesson 2!
11
Lesson Plan 3: Family Snacks
Original activity created by Laura Sharpless
Standards Addressed:
Content: Understand that fractions apply to situations where a whole is
decomposed into equal parts. Understand that fractions give meaning to the
quotient of division problems (4th grade).
Process: Problem Solving, Connections, Communication
Objectives:
Students will think of fractions as fair shares of a whole using a set model of
fractions. Students will unitize a group of objects and think of parts of that group as
a fraction of the whole.
Materials:
• Sharing Snacks at Lunch worksheet, one for each pair of students
• Pens (Have students record their work with pens so that none of their thinking is
lost through erasing.)
Lesson Description: (70 minutes total)
Homework Discussion: (10 minutes)
Bring the class together to discuss the entries they made in their math journals
last night. Have select students share their ideas. Highlight cross-context
relationships.
Introduction to Today’s Activity: (15 minutes)
Start by introducing a set model of fractions. Show them a pack of something
that holds two self-contained units. I suggest a set of two to keep the model
simple and to have a conversation about a fraction that the students are already
familiar with, one half. Examples: an Almond Joy candy bar, a salt and pepper
shaker set, a two-pack of Reece's Peanut Butter Cups, etc.
Ask the students how many packs or sets you have. Emphasize that you are
holding a single set. Then ask the students how many items are in the set. Two.
“So how many parts are in the whole?” Two. “We learned a lot about fractions
yesterday. What do you think we would call one piece of this whole that has two
pieces in it?” One half.
“Do you think things can come in groups with more than two pieces in them?”
Brainstorm other items that come in groups and how many parts are in the
whole. Record the classes ideas on a chart.
Explain that the same families you saw at the sandwich joint the other day also
got snacks to go with their sandwiches. The snacks need to be divided fairly, just
like the sandwiches. How can the amount that each person ate be expressed as
a fraction?
Worksheet Work: (20 minutes)
Split the class into pairs, give each pair a Sharing Snacks At Lunch worksheet
and have them find a place to work.
12!
Lesson 3
While students work on the problems on the worksheet, circulate through the
room, asking them about their thinking. “Is this way of dividing the snacks fair?
How do you know? Show me. What is the whole for this family? How many
parts are in that whole?” Be sure that the students are attaching a fractional
quantity (not just a count) to their groups of snack items. Encourage them to use
manipulatives if they think that would be helpful.
Reflection: (15 minutes)
Bring the group back together to talk about their work. Have select students
present their work, being sure that they are explaining their thinking. Ask
students to explain what other students did, checking for comprehension. Note
misconceptions.
Class Fractions: (10 minutes)
Go back to your chart of things that come in sets that you created during your
introduction. Ask if they have any more ideas to add. If they don’t bring it up, ask,
“What about our class? Are we a set? How many parts are in our set?” Count off
around the group to find out how many people are in your whole class. “So what
fraction of the whole class is one person?” Experiment with what fraction of the
class has certain properties: girls and boys, eye colors, likes broccoli, etc. (This
section of the lesson can be moved to another time if you are short on time or want
to extend it.)
Assessment Criteria:
• Students will be able to evenly divide the pieces of a whole into equal parts and
name the fraction of the whole represented by one group of pieces.
• Students will be able to explain their answers
Possible Extensions:
1. Continue to use the construction of class fractions to take attendance each day.*
Count off at your morning meeting then ask, “There are 28 people in our class.
26 of us are here today. What fraction of us is here today? What fraction is
absent?”
*Activity designed and taught to me by Paula Denton, professor at Antioch University New
England.
2. Fractional quantities other than ½, ¼, and ⅛ have likely come up in your class
discussions since you made your fraction kits. Have the students add to their kits
by making strips for ⅓, ⅙, 1/12, and 1/16. Let them figure out how to do this on
their own rather than guiding them through it as you did with the original kit.
Know that folding the strips into exact thirds is likely to be challenging. Be
prepared to help with this or give the students the time to figure it out on their
own. Students will often trim off an end of extra paper to make thirds without
regard for the size of the whole. So that students who do this reach
disequilibrium, have a master set of fraction strips available for the students to
check their strips against. If their strips are not the same size, they need to
rethink their strategy.
Lesson 3!
13
Sharing Snacks at Lunch
Work with your partner to think about the following problem. Be sure to show your work!
The same three families that I saw having sub sandwiches together, also each got
snacks to go with their sandwiches. Just like the sandwiches, the families want to share
their snacks equally. How should they split their cookies, carrot sticks and grapes?
What fraction of the whole bag does each person get?
The Harris Family:
4 people, one bag of 4 cookies
The Blake Family:
8 people, one bag of 32 carrot sticks
The Swartz Family:
6 people, one bunch of 30 grapes
14!
Lesson 3
Lesson Plan 4: Quilt Blocks
Original activity created by Laura Sharpless
NCTM Common Core Standards Addressed:
Content: Understand that fractions apply to situations where a whole is
decomposed into equal parts. Compare and order fractional quantities. Understand
that two fractions are equivalent when both fractions correspond to the same point
on the number line.
Process: Reasoning and Proof, Problem Solving, Connections, Communication,
Representation
Objectives:
Students will use an area model of fractions to continue developing their
understanding of fractions. Students will gain flexibility with how fractions are
represented. Students will construct understanding of equivalent fractions.
Materials:
• Quilt Block Design and Quilt Block Fractions worksheets, one for each student.
• Colored tiles, at least ten (ideally 20) each of red, blue, green and yellow for each
student.
• Colored pencils and crayons
Lesson Description: (1 hour total)
Introduction: (5 minutes)
“My mother is a quilter. She wants to make a beautiful quilt out of some scraps
of fabric that she has. All of her scraps are cut into squares and she doesn’t
want to do any more cutting. The squares are four different colors: red, blue,
green and yellow. The final blocks need to all be the same size: six by six. She
would love your help in coming up with ideas for how to arrange her fabric to
make her quilt. Please make a beautiful design with the colored tiles and record
your design on a piece of grid paper.”
It would be great if you could bring in examples of real scrap quilts and/or
squares of fabric that might be used for a project like the one described above.
This will make the problem more real for the students and give them an idea of
what a finished quilt can look like. If you have a quilt, such the one on the cover
of this unit, to bring in take time to look closely at it and ask the students what
they notice about it. They may see that some squares are broken up into smaller
pieces and that sometimes those pieces are half triangles and sometimes they
are quarter triangles. They may notice the grid pattern that the squares make on
the quilt and want to count the number of squares in the whole quilt. These are
all examples of great fractional thinking.
You can change the size of the quilt block to meet the needs of your students. I
chose 6x6 because 36 is a square number that can be divided into many familiar
fractions: halves, thirds, fourths, sixths, ninths, and twelfths. However, smaller
3x3 blocks could be given to some students and used in the boarders of the final
Lesson 4!
15
class design. Larger 9x9 blocks could be given to students in need of a greater
challenge. In fact, having students work on differently sized blocks may give you
the opportunity to highlight the importance of the size of the whole. Say a
student working on a 3x3 block makes their block ⅓ red and a student working
on a 9x9 block also makes their block ⅓ red. Ask the students, “Whose design is
going to need more red? How do you know? If they need different amounts of
red fabric, why are the fractions the same? Show me.”
Design Blocks: (15 minutes)
Give each student a Quilt Block Design worksheet. Make colored tiles, colored
pencils and crayons available to everyone. As the students work, circulate
through the room observing the patterns that students create, particularly blocks
that use the same fraction of a color but arrange the squares differently. Let this
be a quiet work time. Avoid interrupting the students to ask mathematical
questions. If some students are finished before others, you can have them
design a second block.
Mid-Activity Check-in: (5 minutes)
Bring students together once they have all completed at least one design. Show
them a simple design you made with just a 3x3 grid. Have the design made with
tiles and colored on paper. Ask the students what fraction of your block is
different colors. Model rearranging the tiles of your block to put all the like colors
together. This can help with seeing all the squares of a single color as one
fraction of the whole block. Explain that you want them to do the same thing with
their blocks. Encourage them to use the tiles too. Hand out the Quilt Block
Fractions worksheet and have them return to their work spaces.
Find Quilt Block Fractions: (20 minutes)
While the students work, circulate through the room observing and offering
guiding questions. “What is the whole? How many parts is it divided into? You
decided that your block is 18/36 red. Is there another fraction we could use to
describe how much red is in your block?” Take note of students who are
constructing equivalent fractions; you may want one or two of them to share
during the reflection session. Notice again blocks that use the same fraction of a
color but with the squares arranged differently. You may want to have some of
these students share during the reflection also to highlight that the fractions can
be represented in different ways.
Reflection: (15 minutes)
Bring the students back together to share their work. Have select students
present their work, being sure that they are explaining their thinking. As noted
above, you may choose to have certain students share in order to highlight
different representations of the same fraction and the idea of equivalent fractions.
Ask students to explain what other students did, checking for comprehension.
Note misconceptions.
16!
Lesson 4
Assessment Criteria:
• Students will be able to identify the fractional quantity of each color in their quilt
block designs.
• Students will order fractional quantities from greatest to smallest accurately.
• If students are beginning to construct the idea of fractional equivalency, they will
identify multiple fractions that refer to the same quantity of squares.
Possible Extensions:
• Piece the students’ paper quilt blocks together and hang your class quilt
somewhere public so that others can enjoy it.
• Use the students’ designs to make an actual quilt. This project would be a major
undertaking but offers many opportunities for developing student’s knowledge
about fractions, geometry and measurement. Making a quilt can also be used as
a gateway into learning about the many different cultures around the world that
make quilts. In fact, an entire theme study could be designed around quilts as
was done by Lesley Chapman, a fellow student of mine at Antioch University
New England. Making a quilt together and deciding who to gift it to would be an
excellent culminating activity for the year.
Lesson 4!
17
Quilt Block Design
Use colored tiles (red, blue, green and yellow) to make a beautiful quilt block design.
Your final design should have six squares on all its sides. Record your design in the
grid bellow.
18!
Lesson 4
Quilt Block Fractions
When you are finished designing your quilt block answer the following questions:
1. How many squares of fabric will your design use? How do you know?
2. What fraction of those squares are red? Green? Yellow? Blue?
3. Place the colors of squares in order from the color that my mother will need the
most of to make your design to the color that she will need the least of. How do
you know that this is the correct order?
4. Compare your design to someone else’s design. Who used more red? Green?
Yellow? Blue? How do you know?
Lesson 4!
19
Lesson Plan 5: Menu of Independent Activities
Activities adapted from Marilyn Burns’ About Teaching Mathematics: A K-8 Resource, p. 280-282
NCTM Common Core Standards Addressed:
Content: Understand that fractions apply to situations where a whole is
decomposed into equal parts. Understand that a unit fraction corresponds to a point
on the number line. Understand that fractions are built from unit fractions.
Understand that two fractions are equivalent when they both correspond to the same
point on the number line.
Process: Problem Solving, Representation, Connections, Communication,
Reasoning and Proof
Objectives:
Students will work independently on activities of their choice to deepen their
understanding of fractions. Students will have time to think deeply about the
problems and use manipulatives to help them construct their ideas.
Materials:
See the materials requirements for each activity on the following pages. Set up
stations around the room with materials and instructions for each activity.
Lesson Description: (1-3 hours depending on the model you choose)
Introduction: (5 minutes)
Gather the students to introduce the menu of activities to them. Let them know
that they will have an opportunity to choose from a number of activities that all
use fractions. Set clear expectations for how you would like them to work during
this time (i.e. “choice time” does not equal “free for all”). Briefly describe each of
the activities that the students have to chose from. Avoid going into too much
detail at this time. All of the instructions are at each station already. You just
want to give the students an idea of what their choices are.
Activities: (20 - 150 minutes)
Possible Menu Activities:
• Cover up,
• Rod Relationships,
• Building Rectangles,
• Build the Yellow Hexagon,
• Class survey
Each of these activities could take children 20-45 minutes to complete.
Depending on student interest and your time constraints, you could have
students do all the activities, choose just one activity or something in between.
You may choose to offer the same menu of activities over several days or start
with just a couple and add in others on later days.
You do not need to feel like you need to offer all or only these activities. Others
may be added or some could be taken away depending on your judgement of
your students’ needs. I chose these activities because they are based on ideas
that the students have already been introduced to in the unit. Menu items should
20!
Lesson 5
be offered as an opportunity for students to build up confidence in areas that they
already have a foundation in rather than a time to introduce new ideas.
While the students are working, circulate through the room observing and
offering guiding questions. If you saw students quickly gravitate towards an
activity, ask them why. “What about this activity attracted you?” “What are you
noticing as you work on this activity?” “What do you think about __? Show me.”
Reflection: (10 minutes for each activity)
How you structured the activity time will determine how you have the students
reflect on their work. If not all the students will be doing an activity, decide on a
way for just the students who have done an activity to share their work with each
other. Small group or partner sharing as students finish an activity can work well.
If you have decided that all the students will do all the activities, wait until all of
the students have had an opportunity to finish an activity before conducting a
reflection period. So that you are not sitting in a refection meeting for an
extremely long period of time, space your reflection out over a number of days.
In any case, have select students share their work. Have other students explain
their thinking. Listen for comprehension. Note misconceptions.
Assessment Criteria:
• Students will improve their understanding of fractions.
• Students will communicate about fractions accurately in writing and verbally.
• Students will record and explain their thinking.
Lesson 5!
21
Independent Activity Menu
Cover Up
Materials:
• 2-3 players
• A fraction kit for each player
• A fraction die
Instructions:
• Students take turns rolling the dice and placing the fraction rolled on top of their
whole strip.
★ Be sure that as you place your strips on the whole that the edges are right
next to each other without overlapping.
• The first person to cover their whole strip exactly wins. If you roll a fraction that
would cover more than the whole, you must pass.
• Record how you get to one on a recording sheet.
Rod Relationships
Materials:
• Cuisenaire rods, 1 set
• Blank paper for recording
• Colored pencils
Instructions:
The yellow rod is half as long as the orange rod. Prove this to yourself with the
rods. This can be recorded by drawing the rods like this:
yellow
yellow
orange
Find all the other pairs of halves you can with the rods and build them. Record
each.
Then do the same with thirds. For example, it takes three light green rods to make
a train as long as the blue rod, so light green is one-third of blue. Prove it with the
rods. Record like this:
light green
light green
light green
blue
Find all the fractional relationships you can for halves, thirds, fourths, fifths, and so
on up to tenths. Explain why you think you’ve found them all.
22!
Lesson 5
Building Rectangles
Materials:
• color tiles, about ten of each color
• half inch grid paper, several sheets
• markers, crayons or colored pencils, one each of red, yellow, green and blue
Instructions:
Use the tiles to build a rectangle that is ½ red, ¼ yellow, and ¼ green. Record and
label it on grid paper. Find at least one other rectangle that also works. Build and
record.
Now use the tiles to build each of the rectangles below. Build and record each in at
least two ways.
• ⅓ green, ⅔ blue
• ⅙ red, ⅙ green, ⅓ blue, ⅓ yellow
• ½ red, ¼ green, ⅛ yellow, ⅛ blue
• ⅕ red, ⅘ yellow
• ⅛ red, ⅜ yellow, ½ blue
Building the Yellow Hexagon
Materials:
• pattern blocks, 1 set
• Blank paper for recording
• Colored pencils
Instructions:
Find all the ways you can build the yellow hexagon from different assortments of
blocks. Count only different combinations of blocks. For example, if you use two
blues and two greens, that combination counts as only one way even if the
arrangement looks different. These count as one way:
Use fractions to record the different ways you found. For example, the green
triangle is ⅙ of the hexagon and the blue rhombus is ⅓ of the hexagon. Therefore,
if you build the hexagon as pictured above, you can record the following:
⅓+⅓+⅙+⅙=1
Or shortened, it would be: ⅔ + 2/6 = 1
Record each of the ways you build the yellow hexagon, recording each in different
ways.
Lesson 5!
23
Class Survey
Materials:
• Class list
• Half inch grid paper
• Colored pencils
Instructions:
1. Come up with a question that you would like to conduct a class survey with. You
can pick one of the questions bellow or come up with your own. If you come up
with your own, decide on a question that only has a certain number of answers
so that your data can be easily categorized. Check in with the teacher about
your question before you begin surveying.
2. Ask everyone in the class your question. Record your answers on the class list
to make sure that you don’t miss anyone.
3. Once you have gathered all your data, decide on a way to represent your it so
that others will be able to see what fraction of the class answered the question in
different ways. For example, if your question was, “Which do you like better,
dogs or cats?,” you could make a bar graph that showed the number of people
who answered “dogs” and the number of people who answered “cats,” and label
each bar with the fraction of the whole class represented by each bar.
Survey Question Ideas:
1. Which do you like better, dogs or cats?
2. How many people are in your family?
3. What color are your eyes?
4. What is your favorite subject in school?
5. If you have free time to yourself, what would you rather do: read, play outside or
watch TV?
6. If you could know what was going to happen in the future, would you want to
know? Yes or no.
7. What is your favorite season of the year?
8. If you had to pick between chocolate, vanilla or strawberry ice cream, which
would you pick?
24!
Lesson 5
COMMON CORE STANDARDS FOR FRACTIONS
KINDERGARTEN
•10 ones = a “ten”
•Decade words = 1, 2, 3, 4, 5, 6, 7, 8, or 9 tens.
•Shapes put together = other shapes.
GRADE 1
•Can create shapes by putting together other shapes.
•Can take apart circles and rectangles into 2 and 4 equal parts. Describe the parts using the words
halves, fourths, and quarters, and using the phrases half of, fourth of, and, quarter of. Describe the
whole as two of, or four of the parts. Understand that decomposing into more equal shares creates
smaller shares.
•Shapes can be broken into 2 and 4 equal parts (shares).
GRADE 2
•10 tens = a “hundred.”
•Units of measurement can be decomposed into smaller units, e.g., feet to inches, meter to
centimeters. A small number of long units might compose a greater length than a large number of
small units.
•Can take apart circular and rectangular objects into 2, 3, or 4 equal parts. Describe the parts using
the words halves, thirds, half of, a third of, etc.; describe the wholes as 2 halves, 3 thirds, 4 fourths.
Recognize that a half, a third, or a fourth of a circular or rectangular object - a graham cracker, for
example - is the same size regardless of its shape.
GRADE 3
FRACTIONS REPRESENT NUMBERS:
•Unit fraction = point on number line (Ex. ! = point between 0 and 1 when that space is broken
into 3 equal parts)
•Fractions are built from unit fraction (Ex. 5/4 = point that is 5 lengths of ¼ on umber line)
•Fractions are equivalent when they mark the same place in the number line
•Recognize & generate equivalent fractions w/denominators 2, 3, 4 and 6
•Whole numbers can be expressed as fractions (Ex. 1 = 4/4, 6 = 6/1, 7 = (4*7)/4)
FRACTIONAL QUANTITIES:
•A fraction = a whole number split into equal parts
•Use fractions to describe parts of whole
•Compare & order fractions w/equal numerators or denominators
Scope and Sequence!
25
GRADE 4
OPERATIONS ON FRACTIONS:
•Understand addition of fractions:
-Adding or subtracting fractions with the same denominator = adding or subtracting copies of
unit fractions (Ex. # + 4/3 means 2 copies of ! plus 4 copies of !)
-Related fractions can be added by making them equivalent fractions (Ex. # + ⅙ = 4/6 + ⅙)
•Add & subtract fractions w/like denominators
•Multiplying a fraction by a whole number = repeated addition (Ex. 3 x ⅖ = ⅖ + ⅖ + ⅖)
•Solve word problems that involve multiplying fractions by whole numbers.
•Fractions = quotient of a whole number divided by a non-zero whole number
•Solve word problems that involve non-whole number quotients of whole numbers
DECIMAL CONCEPTS:
•2-digit decimal = sum of fractions w/denominators 10 and 100 (Ex. 0.34 = 3/10 + 4/100)
•Use decimals to 100ths to describe parts of wholes
•Compare & order decimals to 100ths based on meanings of digits
•write fractions of the form a/10 or a/100 in decimal notation
GRADE 5
FRACTION EQUIVALENCE:
•Understand fraction equivalence:
- Multiplying the numerator and denominator by the same non-zero whole number produces an
equivalent fraction
- Equivalent fractions correspond to the same point on the number line
- Quotients of equivalent fractions are equal
•ID pairs of equivalent fractions; given 2 fractions w/unlike denominators, find 2 equivalent
fractions w/common denominators
•Compare & order fractions w/like and unlike denominators
OPERATIONS ON FRACTIONS:
•Understand that sums & differences of fractions w/unlike denominators can be found by
replacing each w/equivalent fractions that have common denominators
•Compute sums & differences of fractions w/like & unlike denominators. Be able to solve word
problems and estimate answers.
26!
Scope and Sequence
•Understand that multiplying a fraction by a/b means taking a parts of a decomposition of the
fraction into b equal parts
•Area of rectangle w/side lengths a/b and c/d = a/b x c/d
•Explain & justify the properties of operations w/fractions
•Understand the division of unit fractions by whole number and the division of whole numbers by
unit fractions:
- 1/b divided by a (a whole number) = 1/a x b (a smaller unit fraction)
- a (a whole number) divided by 1/b = a x b (a greater whole number)
•Calculate products of fractions and quotients of unit fractions and nonzero whole numbers (with
either as divisor).
•Understand that a mixed number such as 3 ! represents the sum of a whole number a fraction less
than one. Write fractions as equivalent mixed numbers and vice versa
GRADE 6
OPERATIONS ON FRACTIONS:
•Understand that properties of operations apply to, and can be used with, addition &
multiplication of fractions
•Understand that division of fractions is defined by viewing a quotient as the solution for an
unknown-factor multiplication problem. (Ex. (#) / (5/7) = 14/15 because (5/7) x (14/15) = #)
•Solve word problems requiring arithmetic w/fractions, using properties of operations and
converting between forms as appropriate; estimate to check reasonableness of answer.
Scope and Sequence!
27
Mathematics Education Resources
Annenberg Learner, teacher resources for mathematics: http://www.learner.org/
resources/series32.html?pop=yes&pid=871#
Video library of mathematics lessons being taught w/analysis questions at end.
Burns, Marilyn. (2007). About Teaching Mathematics: A K-8 resource. Sausalito, CA:
Math Solutions Publications.
A guide to mathematics education methods including rationale for using a constructivist approach,
sample lessons, and activities.
Fosnot, Catherine Twomney and Maarten Dolk. (2001 & 2002). Young Mathematicians
at Work. Portsmouth, NH: Heineman.
This three volume series describe how children construct mathematical ideas in three areas: number
sense, addition and subtraction, multiplication and addition, and fractions, decimals and percents.
Parallels are drawn between how humans originally constructed these ideas and the evolution of
thinking in today’s children through analysis of classroom activity, discussions, and student work.
Fuson, Dr. Karen C. (2006). Math Expressions: Teacher’s Guide, Grade 4. Boston, MA.
Houghton Mifflin.
Math Expressions is a curriculum guide developed by The Children’s Math Worlds Research Project.
The curriculum is based on NCTM standards with a focus on inquiry and fluency.
Math Learning Center: http://www.mathlearningcenter.org/
A non-profit organization “offering innovative and standards-based curriculum, professional
development, and supplemental materials to support learning and teaching.” Includes links to the
Bridges curriculum resources.
Math Solutions: http://www.mathsolutions.com/
Professional development and resources to improve mathematics instruction. Sponsored by Marilyn
Burns. Includes listings of resources for sale and quite a few free resources such as Math Talk
videos and discussion, Q&A’s on a variety of topics, and articles.
National Council of Teachers of Mathematics: http://www.nctm.org/
Non-members have access to current news and information put out by NCTM. Members have online
access to NCTM publications including the journalTeaching Children Mathematics and discounts on
books.
Pearl, Barbara. (2005). Math in Motion: Origami in the Classroom. Muze, Inc.
Pearl strives to make math fun by explaining ways that origami and language arts can be used in
mathematics lessons. Website: http://www.mathinmotion.com/
Stenmark, Jean Kerr, Virginia Thompson and Ruth Cossey. (1986). Family Math.
Berkley, CA, University of California.
An activity guide for parents and children to learn mathematics together organized by topic and grade
level. Also contains a guide for organizing a Family Math class and a extensive resource list.
The Teaching Channel: http://www.teachingchannel.org/
Videos, lessons and other resources for teachers conveyed through a video library of high quality
teachers teaching their classes. You can search by subject area and grade level. Includes a “New
Teachers Survival Guide” video series.
Van de Walle, John A. (1994). Elementary School Mathematics: Teaching
Developmentally. Second Edition. White Plains, NY, Longman.
Van de Walle provides a framework for thinking mathematics and children learning mathematics,
descriptions of how children develop knowledge in mathematics organized by content strand, activity
ideas, and resources for implementing these ideas.
28!
Resources