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Unless otherwise noted, SFUSD Math Core Curriculum is licensed under the Creative Commons Attribution 4.0 International License
Unit 3.8: Fractions
Big Idea: Fractions are numbers that describe the division of a whole (region, set, segment) into equal parts. Unit fractions are the building blocks of all fractions.
Teacher-facing pages are green
Student-facing pages are white
notes for teachers are in the speaker notes
Emphasized Standards in this unit:
Number and Operations—Fractions* Develop understanding of fractions as numbers. 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. 3.NF.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. 3.NF.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. 3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 3.NF.3a Understand two fractions are equivalent (equal) if they are the same size, or the same point on a number line. 3.NF.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4 and 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model. 3.NF.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. 3.NF.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or < and justify the conclusions, e.g., by using a visual fraction model. *Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. |
Additional Gr. 3 Standards:
Re-engagement with Standards from Gr. 2:
Fraction Notation
Geometry Reason with shapes and their attributes. 3.G.2 Partition 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. |
Geometry Reason with shapes and their attributes. 2.G.3 Partition 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. |
This Fraction Templates slide from By Eric Curts - ControlAltAchieve.com - @ericcurts provides a template for writing fractions with tables in a Google Slideshow or Google Drawing. |
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New Learning in this Unit:
A note about reading composite fractions
Note: It is helpful to use this language when students are first learning to read composite fractions:
one ⅛
two ⅛ s
three ⅛ s
four ⅛ s
five ⅛ s
etc.
A note about equivalent fractions
The 3rd grade standards ask students to see equivalent fractions using visual models. Students may also see patterns in the numbers, but they are NOT asked to algorithmically determine equivalent fractions.
Students will delve further into the patterns in equivalence in grades 4 and 5.
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The Progression of Fraction Ideas
Grades 1 & 2 | Grade 3 | Grade 4 and beyond |
Although there are no specific standards regarding fractions in Grades 1 & 2, students have had experiences partitioning shapes (circles and rectangles) into two, three, or four equal parts, and naming the parts using the words halves, thirds, fourths, half of, a third of, a fourth of, and so on. Students have also worked with fractions of an hour when studying time. Students were introduced informally to equivalent fractions, describing the whole as two halves, three thirds, and four fourths. Finally, they recognized that equal shares of identical wholes need not have the same shape. | Students develop a more formal understanding of fractions, beginning with unit fractions. They use visual fraction models to represent parts of a whole, seeing fractions as parts of regions and locations on a number line. Students understand that the size of a fractional part is relative to the size of the whole, and use fractions to represent numbers equal to, less than, and greater than 1. Students recognize equivalent fractions and solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators. *Grade 3 expectations are limited to fractions with denominators 2, 3, 4, 6, & 8. | In Grade 4, students will develop their understanding of equivalent fractions and operations with fractions. They will extend their understanding of fraction equivalence to develop methods for generating and recognizing equivalent fractions. Students will use their knowledge of how fractions are built to compose (e.g., ⅛ + ⅛ + ⅛ = ⅜) and decompose (e.g., ⅔ = ⅓ + ⅓) fractions. They will develop understanding of fraction multiplication as repeated addition, understanding 3 x ⅛ as 3 instances of ⅛, or ⅜. |
This progression video from Graham Fletcher explores meaning, equivalence, and comparison of fractions. Another helpful video about early fraction ideas can be found here.
Synchronous and Asynchronous Teaching Options:
Use a combination of Synchronous and Asynchronous approaches
| Launch | Explore | Summarize |
Synchronous (live) Whole class or small group |
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Asynchronous (time-delayed) Individual |
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Groupwork during distance learning
If you haven’t yet put students in breakout groups during math instruction, we recommend you do so in this unit, starting with pairs and working up to larger groups.
Either Jamboards or Google Slides can be used by pairs/groups.
You may use these short instructional videos (one for Chromebooks, one for Macs/PCs) to teach students how to split their screens between Zoom and another window. You can: (Note that splitting screens cannot be done on iPads.)
Continuing Daily Routines:
Extensions and Continuing Activities are listed in each lesson
Add fractions to your number of the day routine! (SEE THIS SLIDE)
Suggested Lesson Sequence
Week 1
Unit Warm-up Part 1: Cookie Sharing Revisited
Unit Warm-up Part 2: Faith Ringgold’s Quilts
Lesson 1: Entry Task
Continuing Activities: Fraction Apps and finding Fractions in the Environment
Week 2
LS 1 Day 1: Mario’s Sandwich
LS 1 Day 2: Make a Fraction Kit
LS 1 Day 3: Fractions on a Number Line
Week 3
LS 2 Day 1: Equivalent Fractions with Kits
LS 2 Day 2: Equivalent fractions on a number line
Expert Task: Equivalent Fractions on a Number Line
Week 4
LS 3 Day 1: Fractions greater than 1
LS 3 Day 2: Comparing fractions with the same numerator
LS 3 Day 3: Comparing fractions with the same denominator
The School Garden
Unit Warm-ups
To help students get INTO this unit, we offer two warm-ups:
1 - Sharing Cookies Again:
Students revisit the Sharing Cookies problems from Unit 3.7, this time sharing 1 cookie among a few students.
2 - Faith Ringgold’s Quilts:
Students learn about the great artist Faith Ringgold and her quilts and look at quilts as a way to visualize multiplication, division, and fractions
Sharing Cookies Again
Remember the story about sharing cookies? Today we will look at what happens when we only have 1 cookie!
Can 2 kids share this cookie fairly? How?
Can 3 kids share this cookie fairly? How?
Can 4 kids share this cookie fairly? How?
One cookie, 2 kids.
How much does each kid get?
One cookie, 3 kids.
How much does each kid get?
One cookie, 4 kids.
How much does each kid get?
Faith Ringgold’s Quilts
Who is Faith Ringgold?
Faith says: When somebody says you can’t do something, do more of it, accelerate it, that’s the way you get it done!
Faith Ringgold’s Quilts
Faith was inspired by the colorful patterns, rhythms, and repetitions of African Textiles
These cloth patterns are both from the Shoowa people of Congo
Faith Ringgold’s Quilts
The Shweshwe Pattern on the left is from a southern African fabric, and the Ashanti pattern on the right is a pattern called Nkyimkyim.
Faith Ringgold’s Quilts
Faith made quilts in the tradition of her ancestors and added words to create a new form:
The Story Quilt
← This quilt is called “The Women: Mask Quilt #1”. Faith made it in 1986
Faith Ringgold’s Quilts
This tanka → is a painting that Faith made in 1974. It’s called Windows of the Wedding #2: Breakfast in Bed, 1974
Faith Ringgold’s Quilts
In 2015, Faith made a game that is like Sudoku but with quilt patterns!
Faith Ringgold’s Quilts
Each panel is made of little squares.
How many rows are in the panel?
How many squares are in each row?
What fraction of the panel is each square?
Lesson 1 (Entry Task)
Core Math |
CCSS-M Standard(s) |
Geometry
Reason with shapes and their attributes.
(2nd Grade) 2.G.3 Partition 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.
(3rd Grade) 3.G.2 Partition 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 Entry Task answers the question: What do you already know?
Note: Students worked with fractions informally in grades 1 & 2 when they partitioning shapes (circles and rectangles) into two, three, or four equal parts, and named the parts using the words halves, thirds, fourths, half of, a third of, a fourth of, and so on. They also recognized that equal shares of identical wholes need not have the same shape. Formal fraction notation and language may come from students in the warm up or this lesson. If it does not, it will be introduced in the summary.
Lesson 1 (Entry Task)
Whole Class or Groups:
Independent or Group work:
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
Finding and sharing fractions in your environment
Ask students to photograph or draw fractions in their environment and share with the class. This can be a show-and-tell type of activity in which students discuss each other’s artifacts and what fractions they see in them.
Give students time to explore The Math Learning Center’s Fractions Environment, which will be used in a number of lessons later in the unit.
Continuing Activities
Math Norms
26
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
LAUNCH |
1 |
At Ramona’s elementary school they are finishing a renovation of the playground. They will divide the playground in half. One half will be for playground equipment and the other half will be grass for playing games.
What might each half of the playground look like?
LAUNCH |
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Some students drew these plans for the playground. Which ones show that half of the playground is grass?
Design A
Design B
Design C
Design D
How do you know?
EXPLORE |
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Divide into four equal parts | Divide into three equal parts | Divide into eight equal parts |
The school will have 3 gardens. The gardens are all the same size, but each garden will be divided into a different number of equal parts.
Show how to divide each garden into equal parts:
Explain how you know the parts are equal.
SUMMARIZE |
3 |
EXPLORE |
2 |
Options for student work:
Options for monitoring and sharing work:
Divide the rectangle into four equal parts
SUMMARIZE |
3 |
What do you notice?
What do you wonder?
Divide the rectangle into three equal parts
SUMMARIZE |
3 |
What do you notice?
What do you wonder?
Divide the rectangle into eight equal parts
SUMMARIZE |
3 |
What do you notice?
What do you wonder?
Reading and Writing Fractions: Notation and Language
It is possible that formal fraction notation and language have already come up in the warm up or this task. In this case, the next slide will serve as a review.
If formal fraction notation and language have not yet come up in your discussion, use this summary to introduce it.
Core Math:
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Writing and Naming Fractions
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Today we divided rectangles into equal parts called fractions
What does it look like?
How do we write it?
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What do you notice? • What do you wonder?
one whole
What is it called?
one half
one third
one fourth
one eighth
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
SUMMARIZE |
3 |
Our class of mathematicians knows that fractions name equal parts of shapes.
What is a way that a friend divided a shape up that you hadn’t thought of?
What do you already know about fractions?
Math Norms
38
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Options for Continuing Activities
Following this slide there are sample slides for “Show and Tell” with fractions in the environment.
Finding and sharing fractions in your environment
Ask students to photograph or draw fractions in their environment and share with the class. This can be a show-and-tell type of activity in which students discuss each other’s artifacts and what fractions they see in them.
Give students time to explore The Math Learning Center’s Fractions Environment, which will be used in a number of lessons later in the unit.
Just as with physical manipulatives, it’s important to give students time for free exploration and guided exploration before asking them to use the manipulatives for solving problems.
What fractions do you see?
The jar looks about ½ full.
It’s also about ½ empty.
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What fractions do you see?
The window shade covers ½ of the window.
½ of the window doesn’t have a shade.
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What fractions do you see?
The pizza is cut into 8 equal pieces.
⅛ of the pizza is being eaten by someone!
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What fractions do you see?
1 out of 3 bananas has a sticker. ⅓
2 out of 3 bananas don’t have a sticker. ⅔
What fractions do you see?
The fence has 8 posts.
Each post is ⅛ the length of the fence.
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What fractions do you see?
How many parts are these dragons broken into?
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What fractions do you see?
I broke the stick in two.
Then I broke each half in two.
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Lesson 2 (LS1 Day 1)
Core Math |
CCSS-M Standard(s) |
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. Students will work with halves, fourths, and eighths in this lesson series. Thirds and sixths will be introduced in lesson series two.
Geometry
Reason with shapes and their attributes.
3.G.2 Partition 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.
Lesson 2 (LS1 Day 1)
Whole Class or Groups:
Independent or Group work:
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
Give students time to explore the The Math Learning Center’s Fractions Environment. There are two slides to guide students to do free exploration and guided exploration of the Fractions Environment
Continuing Activities
Math Norms
49
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Mario’s Sandwich
LAUNCH |
1 |
What do you notice?
What do you wonder?
Mario’s Sandwich
LAUNCH |
1 |
Mario has a peanut butter and jelly sandwich for lunch every day.
He always cuts his sandwich into four equal pieces.
He eats three pieces during lunch and saves the rest for after school.
What is this story about?
Mario’s Sandwich
LAUNCH |
1 |
Mario has a peanut butter and jelly sandwich for lunch every day.
He always cuts his sandwich into four equal pieces.
He eats three pieces during lunch and saves the rest for after school.
What are the quantities in the situation?
Quantities: |
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Mario’s Sandwich
LAUNCH |
1 |
Mario has a peanut butter and jelly sandwich for lunch every day.
He always cuts his sandwich into four equal pieces.
He eats three pieces during lunch and saves the rest for after school.
What mathematical questions can we ask about this?
Mario’s Sandwich
LAUNCH |
1 |
Mario has a peanut butter and jelly sandwich for lunch every day.
He always cuts his sandwich into four equal pieces.
He eats three pieces during lunch and saves the rest for after school.
Today we will answer this question:
How much of Mario’s sandwich does he save for after school?
Mario’s Sandwich
EXPLORE |
2 |
How much of Mario’s sandwich does he save for after school?
Mario has a peanut butter and jelly sandwich for lunch every day.
He always cuts his sandwich into four equal pieces.
He eats three pieces during lunch and saves the rest for after school.
SUMMARIZE |
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EXPLORE |
2 |
Options for student work:
Options for monitoring and sharing work:
Mario’s Sandwich:
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
Mario has a peanut butter and jelly sandwich for lunch every day.
He always cuts his sandwich into four equal pieces.
He eats three pieces during lunch and saves the rest for after school.
How much of Mario’s sandwich does he save for after school?
I divided the sandwich into four parts. Since he ate 3, he will have 1 left over.
What are the parts called?
Mario’s Sandwich:
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
Mario has a peanut butter and jelly sandwich for lunch every day.
He always cuts his sandwich into four equal pieces.
He eats three pieces during lunch and saves the rest for after school.
How much of Mario’s sandwich does he save for after school?
I’m not sure the parts are equal. One looks smaller and one looks bigger.
Mario’s Sandwich:
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
Mario has a peanut butter and jelly sandwich for lunch every day.
He always cuts his sandwich into four equal pieces.
He eats three pieces during lunch and saves the rest for after school.
How much of Mario’s sandwich does he save for after school?
There are 4 parts. Each part is one fourth. If he ate 3, there is one left. ¼
Mario’s Sandwich:
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
Mario has a peanut butter and jelly sandwich for lunch every day.
He always cuts his sandwich into four equal pieces.
He eats three pieces during lunch and saves the rest for after school.
How much of Mario’s sandwich does he save for after school?
I drew the sandwich and divided it up a different way. There is one piece left.
Writing and Naming Fourths
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Today we divided squares into fourths
What does ¼ look like?
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What do you notice? • What do you wonder?
This is called the numerator
This is called the denominator
The numerator tells the number of parts being counted.
The denominator tells the number of parts that the shape is divided into.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Faith Ringgold’s Quilts
Now do you know what fraction of the panel each square is?
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SUMMARIZE |
3 |
Our class of mathematicians knows that we can use pictures to help us understand a new idea.
What is a picture a friend drew that helped you understand fractions?
How do you know when parts of a whole are equal?
Math Norms
65
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Options for Continuing Activities
Give students time to explore the The Math Learning Center’s Fractions Environment.
Following this slide there are two slides to guide students to do free exploration and guided exploration of the Fractions Environment
Just as with physical manipulatives, it’s important to give students time for free exploration and guided exploration before asking them to use the manipulatives for solving problems. The following slides give some suggestions for guided explorations.
Two Jamboards from Kentucky Center for Mathematics give students visual practice with fractions of rectangles
Explore the Fractions App
Today you will explore manipulatives for learning about fractions!
Go to tinyurl.com/grade-3-fractions-app
Try out the different ways to make fractions!
If you have questions or want more ideas, click this button →
When you are done, you can take a screenshot to share.
Make fractions in the Fractions App
Today you will try to make some fractions in the fraction app.
Go to tinyurl.com/grade-3-fractions-app
Can you make ½ ? Can you make it a different way?
Can you make ⅓ ? Can you make it a different way?
Can you make ¼ ? Can you make it a different way?
What other fractions can you make?
Lesson 3 (LS1 Day 2)
Core Math |
CCSS-M Standard(s) |
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. Students will work with halves, fourths, and eighths in this lesson series. Thirds and sixths will be introduced in lesson series two.
Geometry
Reason with shapes and their attributes.
3.G.2 Partition 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.
Note: In this lesson, students make a fraction kit. Fraction kits are a great tool for exploring many of the important fraction ideas of 3rd grade, like unit fractions, composite fractions, and equivalent fractions. Making a fraction kit is tricky, but it is as important as using a fraction kit for developing this understanding.
In the classroom, fraction kits are made by folding strips of colored paper. In the online environment, this might not be possible. We recommend either using strips of paper or online manipulatives or both. Each method has its benefits.
See next slide for instrucdtions.
Lesson 3 (LS1 Day 2)
Whole Class or Groups:
Independent or Group work:
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
This movie shows how, and can be shared with students: Making a Fraction Kit in the Math Learning Center.
Students will need to add thirds and sixths to their fraction kits before the following lesson.
Continuing Activities
Making fraction kits with paper
Making fraction kits online
Each student will need:
Directions
You can demonstrate how to make a fraction kit in the Math Learning Center.
This movie shows how, and can be shared with students: Making a Fraction Kit in the Math Learning Center.
Here are some helpful videos for making paper fraction kits:
https://www.youtube.com/watch?v=_gZGAZErHmQ
Math Norms
72
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Making Fraction Kits with Paper
LAUNCH |
1 |
Today we will make fraction kits together.
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Making Fraction Kits Online
LAUNCH |
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Fraction Kits
EXPLORE |
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Picture | How many parts? | What is each part called? |
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Which unit fraction is the smallest?
Which unit fraction is the largest?
What else do you notice?
Draw your fraction kit here, then answer the questions:
SUMMARIZE |
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EXPLORE |
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Options for student work:
Options for monitoring and sharing work:
Fraction Kits
Which unit fraction is the largest?
SUMMARIZE |
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Which unit fraction is the smallest?
What do you notice?
What do you wonder?
Fraction Kits
SUMMARIZE |
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We can count the unit fractions:
There are 2 of the unit fraction ½. Thats one ½, two ½s
There are 4 of the unit fraction ¼. Thats one ¼, two ¼s, three ¼s, four ¼s.
There are 8 of the unit fraction ⅛. Thats one ⅛, two ⅛s, three ⅛s, four ⅛s, five ⅛s, six ⅛s, seven ⅛s, eight ⅛s.
SUMMARIZE |
3 |
Our class of mathematicians knows that being precise is important when we are working with fractions.
What is a comment a friend made that helped you be more precise when talking about fractions.
How did building a fraction kit help you understand fractions?
Math Norms
81
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
We can count unit fractions!
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Let’s count halves!
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
We can count unit fractions!
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Let’s count fourths!
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
We can count unit fractions!
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Let’s count eighths!
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Options for Continuing Activities
Students will need to add thirds and sixths to their fraction kits.
These can be a little harder to fold. These will be used in Lesson 5.
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This game, Fraction Splat, gives students practice in identifying fractions as equal parts of shapes.
This Seesaw Activity - Making Fractions of Shapes - gives students practice dividing rectangles and circles into halves, thirds, and fourths, then prompts them to explore these fractions with other shapes.
Lesson 4 (LS1 Day 3)
Core Math |
CCSS-M Standard(s) |
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
3.NF.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8. Students will work with halves, fourths, and eighths in this lesson series. Thirds and sixths will be introduced in lesson series two.
Lesson 4 (LS1 Day 3)
Whole Class or Groups:
Independent or Group work:
Either demonstrate for students or assign students to watch the video Using Fraction Bars to Put Fractions on a Number Line
Options for student work:
Note that the Fractions on a Number Line Mini Lesson BLM .S. .C. is not in student workbooks.
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
Before the next lesson, give students time to explore the ToyTheatre Fraction Strips. The slide following this guides students to do free exploration and guided exploration of the Fraction Strips.
Continuing Activities
Math Norms
88
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
A cat on a fence
LAUNCH |
1 |
What do you notice?
What do you wonder?
How far has the cat gone?
A cat on a fence & fractions
LAUNCH |
1 |
How far has the cat gone?
1 |
8 |
2 |
8 |
3 |
8 |
4 |
8 |
5 |
8 |
6 |
8 |
7 |
8 |
8 |
8 |
Fractions on a number line
What numbers could be on a number line between 0 and one whole ?
0
1
LAUNCH |
1 |
Fractions on a number line
Today we will use fraction bars to label a number line in the Math Learning Center.
EXPLORE |
2 |
SUMMARIZE |
3 |
EXPLORE |
2 |
Either demonstrate for students or assign students to watch the video Using Fraction Bars to Put Fractions on a Number Line
Options for student work:
Note that the Fractions on a Number Line Mini Lesson BLM .S. .C. is not in student workbooks.
Options for monitoring and sharing work:
Fractions on a number line
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
Fractions on a number line
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
What do you notice about the size of each section on the number line?
Fractions on a Number line
96
1
1
2
1
2
1
4
1
4
1
4
1
4
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
0
1
1 |
2 |
2 |
2 |
0 |
2 |
1 |
4 |
2 |
4 |
3 |
4 |
4 |
4 |
0 |
4 |
1 |
8 |
2 |
8 |
3 |
8 |
4 |
8 |
5 |
8 |
6 |
8 |
7 |
8 |
8 |
8 |
0 |
8 |
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Cat on a Number line
97
1 |
4 |
2 |
4 |
3 |
4 |
4 |
4 |
0 |
4 |
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Cat on a Number line
98
1 |
8 |
2 |
8 |
3 |
8 |
4 |
8 |
5 |
8 |
6 |
8 |
7 |
8 |
8 |
8 |
0 |
8 |
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
SUMMARIZE |
3 |
Our class of mathematicians knows that we can use ideas we already know to understand new ideas.
What is a way that a friend explained their understanding of the number line that helped you understand it?
How are fractions on a number line like whole numbers (1, 2, 3) on a number line?
Math Norms
101
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Options for Continuing Activities
Extensions
Students can make other fractions in the app. They can also extend the number line to 2 or more.
Students can create their own animation of an animal walking on a number line with this slide. You can teach kids how to do this or reference a tutorial such as this.
Before the next lesson, give students time to explore the ToyTheatre Fraction Strips. The slide following this guides students to do free exploration and guided exploration of the Fraction Strips.
Just as with physical manipulatives, it’s important to give students time for free exploration and guided exploration before asking them to use the manipulatives for solving problems. The following slides give some suggestions for guided explorations.
Explore Fraction Strips
Today you will explore manipulatives for learning about fractions!
Go to toytheater.com/fraction-strips/
When you are done, you can take a screenshot to share.
Can you make ½ ? Can you make it a different way?
Can you make ⅓ ? Can you make it a different way?
Can you make ¼ ? Can you make it a different way?
What other fractions can you make?
Lesson 5 (LS2 Day 1)
Core Math |
CCSS-M Standard(s) |
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.3a Understand two fractions are equivalent (equal) if they are the same size, or the same point on a number line.
3.NF.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4 and 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
3.NF.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Note: The 3rd grade standards ask students to see equivalent fractions using visual models. Students may also see patterns in the numbers, but they are NOT asked to algorithmically determine equivalent fractions.
Lesson 5 (LS2 Day 1)
Whole Class or Groups:
Independent or Group work:
Either demonstrate for students or assign students to watch the video Exploring Equivalent Fractions with Fraction Strips
Options for student work:
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
Math Norms
106
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Who has more?
LAUNCH |
1 |
Who got more cookie?
How do you know?
I got two ¼ pieces
I got one ½ piece
Equivalent Fractions
and are equivalent. They are the same size.
LAUNCH |
1 |
Two fractions are equivalent if they are the same size.
Equivalent means the same thing as equal.
1
4
1
4
1
4
1
4
1
2
1
2
2
4
1
2
Equivalent fractions are fractions that are the same size
1 |
2 |
1 |
2 |
1 |
4 |
1 |
4 |
1 |
4 |
1 |
4 |
LAUNCH |
1 |
Do you see two fractions that are the same size?
Equivalent fractions are fractions that are the same size
LAUNCH |
1 |
Do you see two fractions that are the same size?
1
3
1
3
1
3
1
6
1
6
1
6
1
6
1
6
1
6
1
1
2
1
2
1
4
1
4
1
4
1
4
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
Exploring Equivalent Fractions
Today we will explore equivalent fractions in Toy Theatre.
Here are all the fractions in Toy Theatre →
EXPLORE |
2 |
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
EXPLORE |
2 |
Either demonstrate for students or assign students to watch the video Exploring Equivalent Fractions with Fraction Strips
Options for student work:
Options for monitoring and sharing work:
NOTE: notation for equivalent fraction equations is introduced in the summary. The emphasis during the exploration is on developing visual understanding of equivalent fractions. After students are introduced to the notation, you may want them to return to their work and add equations.
Equivalent Fractions
SUMMARIZE |
3 |
Are any of these fractions equivalent? How do you know?
Equivalent Fractions
SUMMARIZE |
3 |
How could we write an equation to show that these are equivalent?
1 |
2 |
=
equals two
1 |
4 |
2 |
4 |
1 |
2 |
Remember: The denominator tells the number of parts that the shape is divided into. The numerator tells the number of fractions being counted.
Equivalent Fractions
SUMMARIZE |
3 |
How could we write an equation to show that these are equivalent?
1 |
2 |
=
equals two
1 |
8 |
2 |
8 |
1 |
4 |
1
4
1
8
1
8
Equivalent Fractions
SUMMARIZE |
3 |
How could we write an equation to show that these are equivalent?
2 |
4 |
=
two equals four
1 |
8 |
4 |
8 |
1 |
4 |
1
4
1
4
1
8
1
8
1
8
1
8
Equivalent Fractions
SUMMARIZE |
3 |
How could we write an equation to show that these are equivalent?
2 |
2 |
=
two equal one whole
1 |
2 |
1
1
2
1
2
1
Equivalent Fractions
SUMMARIZE |
3 |
How could we write an equation to show that these are equivalent?
3 |
3 |
=
three equal one whole
1 |
3 |
1
1
2
1
2
1
1
3
1
3
1
3
SUMMARIZE |
3 |
Our class of mathematicians knows that tools like the fraction kit help us learn new ideas.
What is something a friend showed with their fraction kit that helped you understand a new idea?
What does it mean when two fractions are equivalent?
Math Norms
121
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Options for Continuing Activities
Extension
Since you just introduced notation for equivalent fraction equations, you will want to give students a chance to return to their work and add equations if they did not do so originally.
Fraction Circles from Toy Theatre are another place for students to explore equivalent fractions.
This 3 Act Task: Butter Believe It provides another opportunity to explore the idea of equivalent fractions.
Lesson 6 (LS2 Day 2)
Core Math |
CCSS-M Standard(s) |
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
3.NF.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.3a Understand two fractions are equivalent (equal) if they are the same size, or the same point on a number line.
3.NF.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4 and 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
3.NF.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Note: The 3rd grade standards ask students to see equivalent fractions using visual models. Students may also see patterns in the numbers, but they are NOT asked to algorithmically determine equivalent fractions.
Lesson 6 (LS2 Day 2)
Whole Class or Groups:
Independent or Group work:
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
Math Norms
125
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Glenn and Maggie’s Chocolate Bars
LAUNCH |
1 |
What do you notice?
What do you wonder?
Glenn and Maggie’s Chocolate Bars
LAUNCH |
1 |
Glenn and Maggie each bought the same sized chocolate bar.
Glenn broke his bar into two equal pieces and ate one piece.
Maggie broke her chocolate bar into six equal pieces and ate three pieces.
What is this story about?
Glenn and Maggie’s Chocolate Bars
LAUNCH |
1 |
Glenn and Maggie each bought the same sized chocolate bar.
Glenn broke his bar into two equal pieces and ate one piece.
Maggie broke her chocolate bar into six equal pieces and ate three pieces.
What are the quantities in the situation?
Quantities: |
|
|
|
Glenn and Maggie’s Chocolate Bars
LAUNCH |
1 |
Glenn and Maggie each bought the same sized chocolate bar.
Glenn broke his bar into two equal pieces and ate one piece.
Maggie broke her chocolate bar into six equal pieces and ate three pieces.
What mathematical questions can we ask about this?
Glenn and Maggie’s Chocolate Bars
LAUNCH |
1 |
Glenn and Maggie each bought the same sized chocolate bar.
Glenn broke his bar into two equal pieces and ate one piece.
Maggie broke her chocolate bar into six equal pieces and ate three pieces.
Glenn thinks Maggie ate more than him but Maggie argues that they ate the same amount.
Who do you agree with? Why?
Glenn and Maggie’s Chocolate Bars
Glenn and Maggie each bought the same sized chocolate bar.
Glenn broke his bar into two equal pieces and ate one piece.
Maggie broke her chocolate bar into six equal pieces and ate three pieces.
Glenn thinks Maggie ate more than him but Maggie argues that they ate the same amount.
Who do you agree with? Why?
EXPLORE |
2 |
SUMMARIZE |
3 |
EXPLORE |
2 |
Options for student work:
Options for monitoring and sharing work:
Note: The classwork student page does not include number lines, whereas the slides and seesaw page do.
SUMMARIZE |
3 |
What do you notice?
What do you wonder?
Glenn and Maggie each bought the same sized chocolate bar.
Glenn broke his bar into two equal pieces and ate one piece.
Maggie broke her chocolate bar into six equal pieces and ate three pieces.
Glenn thinks Maggie ate more than him but Maggie argues that they ate the same amount.
Who do you agree with? Why?
I think Maggie got more because she ate 3 pieces.
SUMMARIZE |
3 |
I think they ate the same amount. One half is the same length as 3 pieces of one sixth.
What do you notice?
What do you wonder?
1 |
2 |
=
3 |
6 |
Equivalent Fractions
135
Equivalent fractions are fractions that are the same size.
2 |
4 |
=
4 |
8 |
1
4
1
4
1
8
1
8
1
8
1
8
1 |
4 |
2 |
4 |
3 |
4 |
4 |
4 |
0 |
4 |
1 |
8 |
2 |
8 |
3 |
8 |
4 |
8 |
5 |
8 |
6 |
8 |
7 |
8 |
8 |
8 |
0 |
8 |
Equivalent fractions are located at the same point on a number line.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
SUMMARIZE |
3 |
Our class of mathematicians knows that fractions can be represented as parts of a shape and as points on a number line.
What is something a friend said that helped you understand equivalent fractions?
How can you show equivalent fractions on a number line?
Math Norms
138
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Options for Continuing Activities
Extension
Ask students to find other ways that Maggie and Glenn could break apart a chocolate bar into equal parts and eat half.
Lesson 7 (Expert Task)
Fractions are equivalent if they are the same point on a number line.
Core Math |
CCSS-M Standard(s) |
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.3a Understand two fractions are equivalent (equal) if they are the same size, or the same point on a number line.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Note: The 3rd grade standards ask students to see equivalent fractions using visual models. Students may also see patterns in the numbers, but they are NOT asked to algorithmically determine equivalent fractions.
Lesson 7 (Expert Task)
Whole Class or Groups:
Independent or Group work:
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
Math Norms
142
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
The Swim Race
LAUNCH |
1 |
What do you notice?
What do you wonder?
The Swim Race
LAUNCH |
1 |
Four friends are racing in a pool.
The pool is divided into four lanes.
They are swimming one pool length, from one side of the pool to the other.
What is this story about?
length
The Swim Race
LAUNCH |
1 |
The following fractions show how far each swimmer has gone after 10 seconds.
What is this story about?
Name | Alan | Fiona | Maria | Benjamin |
Fraction of 1 pool length | | | | |
3 |
4 |
1 |
2 |
6 |
8 |
1 |
3 |
The Swim Race
LAUNCH |
1 |
Four friends are racing in a pool. The pool is divided into four lanes. They are swimming one pool length, from one side of the pool to the other.
The following fractions show how far each swimmer has gone after 10 seconds.
Name | Alan | Fiona | Maria | Benjamin |
Fraction of 1 pool length | | | | |
3 |
4 |
1 |
2 |
6 |
8 |
1 |
3 |
What are the quantities in the situation?
Quantities: |
|
|
|
Swimmers in the pool
LAUNCH |
1 |
How far has the each swimmer gone?
Alan
Maria
The Swim Race
LAUNCH |
1 |
Four friends are racing in a pool. The pool is divided into four lanes. They are swimming one pool length, from one side of the pool to the other.
The following fractions show how far each swimmer has gone after 10 seconds.
Name | Alan | Fiona | Maria | Benjamin |
Fraction of 1 pool length | | | | |
3 |
4 |
1 |
2 |
6 |
8 |
1 |
3 |
What mathematical questions can we ask about this?
The Swim Race
Four friends are racing in a pool. The pool is divided into four lanes. They are swimming one pool length, from one side of the pool to the other.
The following fractions show how far each swimmer has gone after 10 seconds.
Name | Alan | Fiona | Maria | Benjamin |
Fraction of 1 pool length | | | | |
3 |
4 |
1 |
2 |
6 |
8 |
1 |
3 |
Show where each swimmer is after 10 seconds. Who is winning so far? How do you know?
EXPLORE |
2 |
The Swim Race Part 2
To prepare for a race swimmers practice by doing laps in the pool.
At the end of each practice the swimmers record the total distance they have swum.
The table below shows how far each friend swam after one practice.
Name | Alan | Fiona | Maria | Benjamin |
Swim Practice Distances | of a mile | of a mile | of a mile | of a mile |
2 |
4 |
4 |
8 |
1 |
2 |
3 |
4 |
Use the number lines to show how far each friend swam during practice. What do you notice about their practice distances?
EXPLORE |
2 |
SUMMARIZE |
3 |
EXPLORE |
2 |
Options for student work:
Options for monitoring and sharing work:
The Swim Race
SUMMARIZE |
3 |
Name | Alan | Fiona | Maria | Benjamin |
Fraction of 1 pool length | | | | |
3 |
4 |
1 |
2 |
6 |
8 |
1 |
3 |
What do you notice?
What do you wonder?
The Swim Race
SUMMARIZE |
3 |
How do you know that Alan and Maria are “tied”?
3 |
4 |
1 |
2 |
6 |
8 |
1 |
3 |
3 |
4 |
=
6 |
8 |
The Swim Race
SUMMARIZE |
3 |
What do you notice?
What do you wonder?
I think that Benjamin has swam the most. The other three are tied.
1 |
2 |
4 |
8 |
3 |
4 |
2 |
4 |
SUMMARIZE |
3 |
Our class of mathematicians knows that fractions can show parts of shapes and distance on the number line.
What is one thing a friend explained that helped you understand fractions.
How can you show that two fractions are equivalent?
Math Norms
157
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Lesson 8 (LS3 Day 1)
Whole numbers and numbers between whole numbers can be represented as fractions.
Core Math |
CCSS-M Standard(s) |
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Lesson 8 (LS3 Day 1)
Whole Class or Groups:
Independent or Group work:
Note that the Fractions Greater than One Number Line BLM .S. .C. is not in student’s workbook
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
A Note About Fractions Greater Than 1
In the Standards, the word “fraction” is used to refer to a type of number, which may be written in the form numerator over denominator (“in fraction notation” or “as a fraction” in conventional terminology), or in decimal notation (“as a decimal”), or—if it is greater than 1—in the form whole number followed by a number less than 1 written as a fraction (“as a mixed number”).
Re. fractions greater than 1, one form of notation is not privileged over another. That is, 7/4 is no more or less correct than 1 ¾. (As an illustration of this, the term “improper fraction” has been replaced with “fraction greater than 1). However, because of the emphasis in grade 3 on unit fractions, students are more likely to see and use 7/4 (seven fourths or, preferably, 7 one-fourths.) (See Bill McCallums Blog and the Fractions Progression for more information about this.)
Mixed numbers are not introduced formally until grade 4. However, in this lesson, students will have the opportunity to make sense of the idea that eg. 3/2 is the same place on the number line as 1 ½ and therefore they are equivalent. This understanding will support students in the unit 3.10, Measurement and Graphs, and will help them work with fractions and mixed numbers in grades 4 and 5. There is NO intention to develop an algorithmic understanding of this equivalence, though students may certainly notice patterns in the equivalence.
Math Norms
161
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Fractions Greater Than 1
LAUNCH |
1 |
0
1
2
3
What are some fractions you could put on this number line?
Do you think there are fractions greater than one?
Fractions Equal To 1
LAUNCH |
1 |
How did we write 1 as a fraction?
1
4
1
4
1
4
1
4
1 |
4 |
2 |
4 |
3 |
4 |
4 |
4 |
0 |
4 |
1
1
2
1
2
0
1
1 |
2 |
2 |
2 |
0 |
2 |
4 |
4 |
1
2 |
2 |
=
=
Fractions Greater Than 1
LAUNCH |
1 |
0
1
2
3
How could you use your fraction kit to make fractions equal to 2?
Investigating Fractions Greater than One
EXPLORE |
2 |
Use fraction strips or a number line to solve this problem!
How many fourths are in two wholes?
1
4
1
4
1
4
1
4
1 |
4 |
2 |
4 |
3 |
4 |
4 |
4 |
0 |
4 |
1
SUMMARIZE |
3 |
EXPLORE |
2 |
Options for student work:
Note that the Fractions Greater than One Number Line BLM .S. .C. is not in student’s workbook
Options for monitoring and sharing work:
Fractions Greater Than 1
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
What are some fractions you could put on this number line?
Fractions Greater Than 1
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
What are some fractions you could put on this number line?
1
2
1
2
1
2
1
2
1
2
1
2
Different Ways to Write Fractions Greater than 1
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
3 |
2 |
=
1 |
2 |
1
We can call this number 1½ or one and a half!
Fractions Greater Than 1
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
How many fourths are in two wholes?
Fractions Greater Than 1
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
How many fourths are in two wholes?
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
0
1
2
Fractions Greater Than 1
172
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1 |
8 |
2 |
8 |
3 |
8 |
4 |
8 |
5 |
8 |
1
4
1
4
1
4
1
4
6 |
8 |
7 |
8 |
8 |
8 |
1 |
4 |
2 |
4 |
3 |
4 |
4 |
4 |
0 |
4 |
0 |
8 |
1
1
2
1
2
0
1
1 |
2 |
2 |
2 |
0 |
2 |
1
2
1
2
1
1
4
1
4
1
4
1
4
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
9 |
8 |
10 |
8 |
11 |
8 |
12 |
8 |
13 |
8 |
14 |
8 |
15 |
8 |
16 |
8 |
5 |
4 |
6 |
4 |
7 |
4 |
8 |
4 |
3 |
2 |
4 |
2 |
2
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
SUMMARIZE |
3 |
Our class of mathematicians knows that we can use patterns to help us decide if our answer makes sense.
What is a pattern a friend saw that helped you understand fractions greater than 1?
What is an example of a fraction greater than 1?
Math Norms
175
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Options for Continuing Activities
Lesson 9 (LS3 Day 2)
Core Math |
CCSS-M Standard(s) |
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or < and justify the conclusions, e.g., by using a visual fraction model.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Note: Today’s lesson focuses on comparing two fractions with the same numerator. The next lesson will focus on comparing two fractions with the same denominator.
Lesson 9 (LS3 Day 2)
Whole Class or Groups:
Independent or Group work:
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
Math Norms
179
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
How do we know that something is larger or smaller than something else?
How do we know that there are more or fewer of something?
How do we know that one number is greater and one is less?
LAUNCH |
1 |
300
1
14
2,000
½
How do we know that one number is greater and one is less?
LAUNCH |
1 |
300
1
14
2,000
½
We could show them with base ten blocks!
We could show them on a number line!
300
14
1
14
What does it mean to compare two numbers?
LAUNCH |
1 |
Let’s review using the > and < symbols!
300
200
> or < ?
greater than
less than
17
59
We say: 300 is greater than 200.
We say: 17 is less than 59.
> or < ?
We can compare fractions!
LAUNCH |
1 |
What do you notice?
Which do you think is greater?
½
¼
greater than
less than
1
3
1
6
1
1
2
1
4
1
8
We can compare fractions!
LAUNCH |
1 |
Which do you think is greater?
3 |
2 |
3 |
3 |
1
3
1
6
1
1
2
1
4
1
8
1
3
1
6
1
1
2
1
4
1
8
1
3
1
6
1
1
2
1
4
1
8
What do you notice?
Numerators and Denominators
The numerator tells the number of parts being counted.
1 |
4 |
This is called the numerator
This is called the denominator
What fraction of this square is green?
The denominator tells the number of parts that the shape is divided into.
2 |
4 |
What fraction of a mile did Alan swim?
Sharing Brownies: Part 1
LAUNCH |
1 |
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 8 equal pieces. Shawn sliced his into 6 equal pieces.
What is this story about?
Sharing Brownies: Part 1
LAUNCH |
1 |
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 8 equal pieces. Shawn sliced his into 6 equal pieces.
What are the quantities in the situation?
Quantities: |
|
|
|
Sharing Brownies: Part 1
LAUNCH |
1 |
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 8 equal pieces. Shawn sliced his into 6 equal pieces.
What mathematical questions can we ask about this?
Sharing Brownies: Part 1
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 8 equal pieces. Shawn sliced his into 6 equal pieces.
Today we will answer the question:
Would you want a brownie from Robin or Shawn? Why?
EXPLORE |
2 |
SUMMARIZE |
3 |
EXPLORE |
2 |
Options for student work:
Note that the Comparing Fractions with the Same Numerator .S. .C. is incorporated into the summary of this lesson
Options for monitoring and sharing work:
SUMMARIZE |
3 |
Compare the unit fractions
I notice that ⅛ is the smallest and one whole is the largest.
1
3
1
6
1
1
2
1
4
1
8
I notice that ¼ is larger than ⅙.
What do you notice?
What do you wonder?
We write:
1 |
4 |
1 |
6 |
And ⅛ is less than ¼
1 |
8 |
1 |
4 |
SUMMARIZE |
3 |
1
6
1
1
4
I think that the fourths are bigger because there are fewer of them in a whole than sixths.
Why is
?
1 |
4 |
1 |
6 |
1
4
1
4
1
4
1
6
1
6
1
6
1
6
1
6
SUMMARIZE |
3 |
Compare the fractions
I notice that 3 halves is larger than 1 and also larger than 3 thirds.
What do you notice?
What do you wonder?
We write:
3 |
2 |
3 |
3 |
And 3 eighths is less than 3 fourths
3 |
8 |
3 |
4 |
1
3
1
6
1
1
2
1
4
1
8
1
3
1
6
1
1
2
1
4
1
8
1
3
1
6
1
1
2
1
4
1
8
SUMMARIZE |
3 |
Sharing Brownies Part 1:
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 8 equal pieces. Shawn sliced his into 6 equal pieces.
Would you want a brownie from Robin or Shawn? Why?
Shawn
Robin
I divided the rectangle up and the sixths look bigger than the eighths.
That makes sense because there are fewer pieces, so each piece is larger.
What do you notice?
What do you wonder?
Who Ate More?
Glenn ate ¼ of this chocolate bar
Maggie ate ½ of this chocolate bar
SUMMARIZE |
3 |
Comparing Fractions
196
We can compare fractions by thinking about their size.
1 |
4 |
1 |
6 |
1
6
1
1
4
1
4
1
4
1
4
1
6
1
6
1
6
1
6
1
6
If a whole is divided into more pieces, the pieces will be smaller. If a whole is divided into fewer pieces, the pieces will be larger. The larger the denominator, the more pieces there are and the smaller each piece is.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
SUMMARIZE |
3 |
Our class of mathematicians knows that pictures, diagrams, and models can help us see patterns in mathematics.
What is a diagram or picture a friend made that helped you understand a new idea?
How can you convince someone that ½ is greater than ¼ ?
Math Norms
199
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Options for Continuing Activities
These practice problems can be found here in student workbooks: Comparing Fractions with the Same Numerator .S. .C.
Write a number sentence using <, >, or = to compare:
Put these fractions in order from least to greatest:
3. 4.
1 |
4 |
1 |
8 |
1 |
6 |
1 |
3 |
1 |
6 |
1 |
2 |
1 |
8 |
1 |
3 |
1 |
1 |
1 |
4 |
3 |
4 |
3 |
2 |
3 |
6 |
3 |
3 |
Lesson 10 (LS3 Day 3)
Core Math |
CCSS-M Standard(s) |
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or < and justify the conclusions, e.g., by using a visual fraction model.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Note: Today’s lesson focuses on comparing two fractions with the same denominator.
Lesson 10 (LS3 Day 3)
Whole Class or Groups:
Independent or Group work:
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
Math Norms
203
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
More about comparing fractions!
LAUNCH |
1 |
What do you notice?
Today we will compare fractions with the same denominator.
1
1
2
1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1 |
4 |
2 |
1 |
2 |
2 |
2 |
3 |
2 |
Comparing Sixths
LAUNCH |
1 |
What do you notice?
1
1
1 |
4 |
6 |
1 |
6 |
2 |
6 |
3 |
6 |
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
5 |
6 |
6 |
6 |
7 |
6 |
Sharing Brownies: Part 2
LAUNCH |
1 |
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 6 equal pieces. Shawn also sliced his into 6 equal pieces.
How is this story different than the other brownie story?
Sharing Brownies: Part 2
LAUNCH |
1 |
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 6 equal pieces. Shawn also sliced his into 6 equal pieces.
What are the quantities in the situation?
Quantities: |
|
|
|
Sharing Brownies: Part 2
LAUNCH |
1 |
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 6 equal pieces. Shawn also sliced his into 6 equal pieces.
What mathematical questions can we ask about this?
Sharing Brownies: Part 2
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 6 equal pieces. Shawn also sliced his into 6 equal pieces.
After the party, Robin had more brownies left to take home than Shawn did.
What fraction of the whole pan might each person take home?
EXPLORE |
2 |
SUMMARIZE |
3 |
EXPLORE |
2 |
Options for student work:
Options for monitoring and sharing work:
SUMMARIZE |
3 |
Compare the fractions
What do you notice?
What do you wonder?
I notice that 3 halves is more than 2 halves, and 4 halves is more than 3 halves.
3 |
2 |
2 |
2 |
1
1
2
1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
I notice that ½ is the smallest.
I notice that 4 halves is the same as two wholes.
SUMMARIZE |
3 |
Compare the fractions
I notice that 6 sixths is the same as one whole.
What do you notice?
What do you wonder?
When they are in order, the number in the numerator goes up by 1.
1 |
6 |
2 |
6 |
I notice that on sixth is the smallest.
1
1
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
1
6
3 |
6 |
4 |
6 |
5 |
6 |
6 |
6 |
7 |
6 |
SUMMARIZE |
3 |
Robin
Shaw
Shawn has 1 sixth left and Robin has 4 sixths left. Robin has more left.
Sharing Brownies Part 1:
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 6 equal pieces. Shawn also sliced his into 6 equal pieces.
After the party, Robin had more brownies left to take home than Shawn did.
What fraction of the whole pan might each person take home?
X
X
X
X
X
X
X
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
Robin
Shaw
Shawn has 3 sixth left and Robin has 4 sixths left. Robin has more left.
Sharing Brownies Part 1:
Robin and Shawn each made a pan of brownies.
Their pans were exactly the same size.
Robin sliced her brownies into 6 equal pieces. Shawn also sliced his into 6 equal pieces.
After the party, Robin had more brownies left to take home than Shawn did.
What fraction of the whole pan might each person take home?
X
X
X
X
X
What do you notice?
What do you wonder?
Comparing Fractions
215
We can compare fractions by thinking about their size.
2 |
6 |
4 |
6 |
1
6
1
1
6
1
6
1
6
1
6
1
6
If a whole is divided into the same number of pieces, the more pieces there are, the larger the fraction or part will be.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
SUMMARIZE |
3 |
Our class of mathematicians knows that some problems have many solutions.
What is a solution a friend thought of that you did not?
If you have more than ¼ of a cookie, how much cookie could you have?
Math Norms
218
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Options for Continuing Activities
These practice problems can be found here in student workbooks: Comparing Fractions with the Same Denominator .S. .C.
Write a number sentence using <, >, or = to compare:
Put these fractions in order from least to greatest:
3. 4.
3 |
4 |
6 |
4 |
2 |
6 |
4 |
6 |
2 |
3 |
5 |
3 |
1 |
3 |
4 |
3 |
6 |
3 |
3 |
3 |
3 |
2 |
1 |
2 |
4 |
2 |
2 |
2 |
Students can solve problems on the Seesaw Greater Than, Less Than, or Equal to?
The Milestone includes core math from the entire unit.
Core Math |
CCSS-M Standard(s) |
Number and Operations—Fractions*
Develop understanding of fractions as numbers.
3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
3.NF.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.3a Understand two fractions are equivalent (equal) if they are the same size, or the same point on a number line.
3.NF.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4 and 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
3.NF.3c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
3.NF.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or < and justify the conclusions, e.g., by using a visual fraction model.
*Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
The Milestone answers the question: Did you learn what was expected of you from this unit?
Whole Class or Groups:
Independent or Group work:
Whole Class or Groups:
Core Math to Emphasize
Strengths to highlight
Note: Provide access to any tools or manipulatives students have used throughout this unit.
Note: The Milestone Task has 3 parts. You may want to assign them over the course of 3 sessions.
See this slide for information about scoring.
Math Norms
222
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
What did we learn about FRACTIONS in this unit?
223
Look at each picture and share what you notice!
Each square is one ninth of the quilt panel
1 |
4 |
2 |
4 |
3 |
4 |
4 |
4 |
0 |
4 |
1
4
1
4
1
8
1
8
1
8
1
8
Equivalent fractions are fractions that are the same size or the same place on a number line.
1 |
4 |
1 |
6 |
I notice...
1
1
1
2
1
2
1
2
3 |
2 |
2 |
1
6
1
4
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Gardens in San Francisco
Today you will show what you understand about fractions
There are about 42 community gardens and 32 school gardens in San Francisco!
Does your school have a garden?
This garden has raised beds.
LAUNCH |
1 |
You will solve some problems about a school garden.
Garden beds
The School Garden
LAUNCH |
1 |
Part 1: Preparing the Garden
There are four 3rd grade classes at Sunny Elementary School.
Each class is planting an equal part of a garden they share.
Use the rectangles to show four different ways they could divide the garden into equal parts. Label the parts with fractions.
What is this story about?
How could you divide this in 4 equal parts?
What are the quantities in the situation?
The School Garden
LAUNCH |
1 |
Part 2: Planting the Garden
The students plant carrots in a row.
They plant one seed at every interval.
Divide the line into 8 equal parts and show where the fractions in the box go.
What is this story about?
0
1
What are the quantities in the situation?
1 |
8 |
The School Garden
LAUNCH |
1 |
Part 2: Planting the Garden
Some students are planting seeds in rows that are twice as long.
Use the number line to show where the fractions in the box are located.
What is this story about?
What are the quantities in the situation?
0
2
1
The School Garden
LAUNCH |
1 |
Part 3: Harvesting the Garden.
The garden has four different vegetables: carrots, lettuce, peas, and cucumbers.
This chart shows the fraction of a garden row used by each vegetable.
What is this story about?
What are the quantities in the situation?
Carrots | Lettuce | Peas | Cucumbers |
| | | |
5 |
6 |
2 |
6 |
2 |
4 |
3 |
4 |
Write the symbol < or > or = to compare these fractions.
2 |
4 |
2 |
6 |
3 |
4 |
2 |
4 |
Is the same as ? Explain.
2 |
4 |
1 |
2 |
The Milestone Task has 3 parts!
EXPLORE |
2 |
Part 1. Use the rectangles to show four different ways they could divide the garden into equal parts. Label the parts with fractions.
Part 2. Divide the lines into equal parts and show where the fractions in the box go.
Part 3. Write the symbol < or > or = to compare these fractions.
Is the same as ? Explain.
0
1
0
2
1
2 |
4 |
1 |
2 |
SUMMARIZE |
3 |
EXPLORE |
2 |
Options for monitoring and sharing work:
Provide access to any tools or manipulatives students have used throughout this unit. Note: The Milestone Task has 3 parts. You may want to assign them over the course of 3 sessions.
Options for student work:
Use your observation of student work and struggles to pick one point to emphasize in the summary.
The google slides are given as an option but not recommended for this task.
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
How many parts are there? How do we know the parts are equal?
Part 1. Use the rectangles to show four different ways they could divide the garden into equal parts. Label the parts with fractions.
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
Part 2. The students plant carrots in a row.
They plant one seed at every interval.
Divide the line into 8 equal parts and show where the fractions in the box go.
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
Part 2. Some students are planting seeds in rows that are twice as long.
Use the number line to show where the fractions in the box are located.
1 |
0 |
2 |
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
Part 3: Harvesting the Garden.
The garden has four different vegetables: carrots, lettuce, peas, and cucumbers.
This chart → shows the fraction of a garden row used by each vegetable.
Carrots | Lettuce | Peas | Cucumbers |
| | | |
5 |
6 |
2 |
6 |
2 |
4 |
3 |
4 |
Use the symbol < or > or = to compare these fractions.
Explain your thinking!
I know that fourths are bigger than sixths. So 2 fourths is bigger than 2 sixths.
1
6
1
4
1
6
1
4
I used my fraction kit.
What do you notice?
What do you wonder?
SUMMARIZE |
3 |
A student said that the fraction of peas, , is the same as . Is this true?
2 |
4 |
1 |
2 |
Explain your thinking with a drawing and words.
1
4
1
4
1 |
4 |
2 |
4 |
3 |
4 |
4 |
4 |
0 |
4 |
1 |
2 |
2 |
2 |
0 |
2 |
1
2
I used my fraction kit.
I used my a number line.
SUMMARIZE |
3 |
Our class of mathematicians knows that when we are learning about a new idea, we can have lots of questions.
What is a question that a friend asked that you answered? What is a question that you asked that a friend answered?
What is a question that you still have about fractions?
Math Norms
238
Errors are gifts that promote discussion.
Answers are important, but they are not the math.
Talk about each other’s thinking.
Ask questions until ideas make sense.
Use multiple strategies and multiple representations.
SAN FRANCISCO UNIFIED SCHOOL DISTRICT
Milestone Scoring
See this list for continuing activities for Fractions.
Consider giving students the opportunity to revise their work after the summary.
Extensions
Cyberchase Ways of Looking at Half is a game that challenges students to find many different ways to show half of a shape.