Grade 4 – Unit 2: Comparing Fractions

This is a teacher's guide made by Illustrative Mathematics. It has ideas and practice pages for teaching fractions.

What This Unit Is About

In this unit, kids learn about equivalent fractions. Equivalent means "equal" — so these are fractions that show the same amount, even though they look different. For example, $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent fractions. They are the same size!

Kids also learn to compare fractions — to figure out which one is bigger or smaller. They will work with fractions that have these bottom numbers (called denominators): 2, 3, 4, 5, 6, 8, 10, 12, and 100.

What Kids Already Know

Last year, in grade 3, kids learned to:

• Cut shapes into equal parts
• Name each part with a unit fraction (like $\frac{1}{3}$, which means "1 out of 3 equal parts")
• Build bigger fractions using unit fractions
• Show fractions on strips of paper and on number lines
• See that fractions can be the same point on a number line, even if they look different

What's New This Year

Now, kids will use the same tools — fraction strips, diagrams, and number lines — but with more denominators: 5, 10, and 12.

Here's the big idea kids will learn: If you split each part of a fraction into smaller pieces, you get an equivalent fraction.

For example: Take $\frac{3}{5}$. If you split each fifth into 2 smaller parts, you now have 6 shaded parts instead of 3. So $\frac{3}{5}$ is the same as $\frac{6}{10}$. You doubled the number of parts, so each part became half as small — but the total amount shown is still the same!

Kids will also use benchmark fractions — friendly fractions like $\frac{1}{2}$ and 1 — to help them guess where a fraction sits on a number line, and to compare fractions with each other.

Building Math Muscles All Unit Long

All through the unit, kids practice doing multiplication in their heads. This helps them get faster and smarter with numbers. They will do short "Number Talks" (quick number games) using the numbers 2, 4, 5, 6, 8, 10, and 12.

Kids will practice tricks like doubling and halving numbers. This connects to folding fraction strips in half, or splitting diagrams into smaller and smaller parts.

Section A: How Big Is a Fraction, and Where Does It Go?

In this section, kids look again at ideas from grade 3, but now with new denominators: 5, 10, and 12.

They use:

• Fraction strips (strips of paper split into equal parts)
• Diagrams
• Number lines

...to understand how big a fraction is, and how fractions relate to each other.

Kids will look at fraction pairs where one denominator is a multiple of the other. For example, 4 is a multiple of 2, so $\frac{1}{2}$ and $\frac{2}{4}$ can be compared this way.

Kids will also learn to compare fractions to benchmarks like $\frac{1}{2}$ and 1. This gets them ready to compare and order fractions in later lessons.

Practice Check: What Kids Should Know

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

Example Answers:

• If a number line is split into 8 equal parts, and the point is on the 7th mark, the fraction is $\frac{7}{8}$.
• If a number line has 5 equal parts per whole, and the point is on the 8th mark, the fraction is $\frac{8}{5}$.

Watch for This: Some kids might count the tick marks wrong. They might think the marks between 0 and 1 tell them the top number (numerator), when really the marks show how many equal parts make up the whole.

Some kids might not yet understand that a fraction can be greater than 1 — meaning the top number is bigger than the bottom number. For example, they might mix up $\frac{8}{5}$ with $\frac{5}{8}$.

What Teachers Can Do: Ask kids to explain what the tick marks mean. Ask how a fraction number line is like — and different from — a whole number line. Remind kids what the top number (numerator) and bottom number (denominator) each tell us.

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Task: Is $\frac{7}{12}$ greater than or less than $\frac{8}{12}$? (Note: exact fractions from source are unclear, but reasoning follows below.)

Sample Answer: One fraction is greater. Here's how to think about it: Half of 12 is 6. If a fraction has more than 6 parts (like 7 out of 12), it's more than $\frac{1}{2}$.

Watch for This: Some kids may pick the right answer without really understanding why. They might think a fraction is bigger just because both numbers in it are bigger — that's not correct reasoning!

What Teachers Can Do: Have kids play a comparing game. Watch how they use $\frac{1}{2}$ and 1 as benchmarks, or how they find a common denominator (making both fractions have the same bottom number) to compare. Have them draw pictures to show their thinking.

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Task: Explain why $\frac{1}{3}$ is equivalent to $\frac{4}{12}$.

Sample Answer: If you split each third into 4 equal pieces, each new piece is one-twelfth. So $\frac{1}{3}$ lands on the 4th mark out of 12 — showing it's the same as $\frac{4}{12}$.

Watch for This: Some kids may not yet see how the top and bottom numbers relate when finding equivalent fractions.

What Teachers Can Do: Have kids play the comparing game and look for equivalent fractions with denominators that are multiples of each other (like 3 and 12). Connect drawing extra tick marks on a number line to finding a common denominator.

Practice Problems

Problem 1: What fraction of each shape is shaded?
Answer: Part of the circle and part of the square are shaded (based on how many equal parts each shape is split into).

Problem 2: Explain why the shaded part shows $\frac{1}{8}$ of the whole rectangle.
Answer: The rectangle has 8 equal parts, and only 1 part is shaded.

Problem 3: Label each tick mark with the correct number. Explain your thinking.
Answer: There are 4 equal parts, so each part equals $\frac{1}{4}$. (Watch out: some kids may mix up the label!)

Problem 4: Explain or show why $\frac{2}{4}$ and $\frac{1}{2}$ are equivalent fractions.
Answer: Picture a shape split into 4 equal parts, with 2 shaded. That's $\frac{2}{4}$. Now picture the same shape split into just 2 equal parts, with 1 shaded. That's $\frac{1}{2}$. Both show the same amount shaded!

Problem 5:

• a. Shade a diagram to show $\frac{3}{4}$.
• b. Would you shade more or less to show $\frac{3}{6}$? Explain.

Answer: Less! When a whole is split into 6 parts instead of 4, each part is smaller.

Problem 6:

• a. What fraction does the shaded part show? Answer: $\frac{7}{10}$ — the rectangle has 10 equal parts, and 7 are shaded.
• b. Shade a diagram to show a different fraction.

Problem 7: Circle the bigger fraction in each pair. Explain your thinking using pictures if it helps.