Grade 4 Teacher Guide

Unit 2

This book is for Grade 4.
It is a teacher guide.
It helps teachers teach fractions.

This book was made by Illustrative Mathematics.
It is a sample.
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Unit 2: Fraction Equivalence and Comparison

What Students Learn

Kids learn about fractions.
A fraction has a top number and a bottom number.
The bottom number is called a denominator.

Kids learn fractions with these denominators: 2, 3, 4, 5, 6, 8, 10, 12, and 100.
Kids learn to find fractions that are equal.
Kids learn to compare fractions.
Compare means to see which one is bigger or smaller.

The Story of This Unit

Last year, in grade 3, kids learned about fractions too.
They split shapes into equal parts.
Each equal part is called a unit fraction.

A unit fraction comes from splitting 1 whole into equal parts.
Kids used unit fractions to make bigger fractions.
Some fractions were even bigger than 1 whole.

Kids used fraction strips to show fractions.
A fraction strip is a paper strip cut into equal parts.
Kids also used tape diagrams.
A tape diagram is a drawing that shows parts of a whole.

Last year, kids only used denominators 2, 3, 4, 6, and 8.

Kids also put fractions on a number line.
A number line is a straight line with numbers on it.
Kids learned that fractions are numbers too.
Equal fractions sit on the same spot on the number line.

Now, in grade 4, kids do this again.
But now they use more denominators: 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Kids use fraction strips again.
Kids use tape diagrams again.
Kids use number lines again.

This helps them see how big a fraction is.
This helps them find equal fractions.
This helps them compare and put fractions in order.

Kids learn why two fractions can be equal.
This happens when each part is split into more equal pieces.
The pieces get smaller.
But there are more of them.

Kids also learn about benchmark fractions.
A benchmark is a fraction we use to compare, like one-half or 1 whole.
Kids use benchmarks to guess where a fraction goes on a number line.
Kids use benchmarks to compare fractions too.

Math Talk Time

All through this unit, kids practice multiplying in their heads.
They build on what they learned in grade 3.
They use math rules to help them multiply fast.

Kids do short math talks called Number Talks.
These talks use the numbers 2, 4, 5, 6, 8, 10, and 12.

Kids learn tricks like doubling and halving.
Doubling means making a number two times bigger.
Halving means making a number two times smaller.

These tricks connect to folding fraction strips.
These tricks connect to splitting tape diagrams into smaller parts.

The Number Talks happen in Lesson 5, Lesson 9, and Lesson 16.

These numbers are picked on purpose.
They help kids get better with fractions.
Kids start to see patterns.
This helps them compare fractions faster.

Getting Ready: What Teachers Need

Teachers gather tools for each lesson.
Some lessons need straightedges, like rulers.
Some lessons need fraction strip papers to copy.
Some lessons need card sets for sorting games.
Some lessons need sticky notes, tape, or colored pencils.
Some lessons need paper and paper clips.

Each lesson has its own list of tools.

Section A: Size and Location of Fractions

What Students Learn

In this part of the unit, kids look at fractions again.
They remember what they learned in grade 3.
Now they also use denominators 5, 10, and 12.

Kids use real fraction strips.
Kids use pictures of fraction strips.
Kids use tape diagrams.
Kids use number lines.

These tools help kids see how big fractions are.

Kids compare fractions where one denominator is a multiple of the other.
A multiple means one number fits inside the other evenly.

Kids also compare fractions to benchmarks, like one-half and 1 whole.

This work gets kids ready for the next lessons.
In those lessons, kids will learn more about equal fractions and comparing fractions.

Practice Problems

Kids solve problems like these:

Problem 1: What fraction of each shape is shaded?
A student looks at a circle and a square with shaded parts and names the fraction for each.

Problem 2: Why does a shaded part show one-eighth of a rectangle?
The rectangle has 8 equal parts. Only 1 part is shaded. That is why it shows one-eighth.

Problem 3: Label the marks on a number line.
There are 4 equal parts. Each part is one-fourth.

Problem 4: Show why two-fourths and one-half are equal fractions.
A picture can show this. The same shaded part can be named two-fourths or one-half, depending on how you group the parts.

Problem 5: Shade a picture to show three-fourths.
Then think about whether three-sixths would need more or less shading.
Three-sixths needs less shading. When a whole is split into more equal parts, each part is smaller.

Problem 6: Look at a shaded picture. What fraction does it show?
The rectangle has 10 equal parts. Seven parts are shaded. That shows seven-tenths.
Then shade a new picture to show four-fifths.

Problem 7: Circle the bigger fraction in each pair.
Show your thinking with pictures or words.

Section A Checkpoint

Task: Label the point on each number line with a fraction.

For one number line, there are 8 equal parts in 1 whole.
The point is on the 7th mark. So the fraction is seven-eighths.

For another number line, there are 5 equal parts in each whole.
The point is on the 8th mark. So the fraction is eight-fifths.

Note for Teachers:
Some kids may count the marks the wrong way.
Some kids may not yet understand that a fraction can be bigger than 1 whole.
Teachers should ask kids to explain what the top and bottom numbers mean.
This helps kids understand the marks and points on a number line.

Task: Is seven-twelfths bigger or smaller than one-half? Explain your thinking.

Seven-twelfths is bigger than one-half.
One way to know: seven-twelfths is past the halfway mark toward 1 on the number line.
Half of 12 is 6. Seven is bigger than 6. So seven-twelfths is bigger than one-half.

Note for Teachers:
Some kids may know the right answer but not explain it well.
Teachers can have kids play a comparing game to practice their thinking.
Kids should learn to compare using one-half and 1 whole as helpers.

Task: Explain why one-third is equal to four-twelfths.

One way to know: if you split each third into 4 equal pieces, you get twelfths.
One-third becomes four of those twelfth pieces.
So one-third equals four-twelfths.

Note for Teachers:
Some kids may not yet see how the size of parts connects to equal fractions.
Teachers can have kids play a comparing game to practice this idea.
Teachers can connect drawing more parts on a number line to finding equal fractions.