Teaching unit · Grade 4 (ages 9 to 10)

Comparing and ordering fractions

Comparing with the benchmark 1/2, comparing with common denominators, and putting fractions in order

About three to four lessons of 45 to 60 minutes

Start here · hook

Who ate more pizza?

Two friends buy the same size pizza. Mia cuts hers into 3 equal slices and eats 2 of them, so she ate 2/3. Ben cuts his into 4 equal slices and eats 3 of them, so he ate 3/4. They both have one slice left, so it feels like a tie. But who actually ate more pizza, 2/3 or 3/4? You cannot just say 3 is more than 2, because the slices are not the same size.

In Grade 3 you compared fractions when the tops matched or the bottoms matched. Today you will settle any argument, even when all four numbers are different. You will get three tools: a quick check against one half, a way to give both fractions the same size parts, and a number line that lines up a whole list of fractions from smallest to largest.

Mia ate 2/3, Ben ate 3/4 of the same pizzasame size pizza, so we can compare, and 3/4 turns out to be more
One cup 2/5 full, another 5/8 full2/5 is under half, 5/8 is over half, so 5/8 has more
3/8 of a bar or 3/4 of the same barsame 3 pieces taken, but quarters are bigger than eighths, so 3/4 is more
Ribbons 1/2, 2/3 and 5/6 of a meter longput them on one line to see the order from shortest to longest

Learning objective

What students will be able to do

Students will compare two fractions with unlike numerators and denominators by reasoning against the benchmark 1/2 and by rewriting them with a common denominator, record the comparison with the symbols for greater than, less than and equal to, and order three or more fractions on a single number line, always checking that the fractions refer to the same whole.

Success criteria

I can decide whether a fraction is less than, equal to, or greater than one half.
I can rewrite two fractions with a common denominator so their parts are the same size.
I can compare two unlike fractions and record it with the greater than, less than, or equal to symbol.
I can put three or more fractions in order on one number line.
I can explain why two fractions can only be compared when they describe the same whole.

Curriculum anchor

Standards this unit teaches

4.NF.A.2Common Core (US)
Compare two fractions
Compare two fractions with different numerators and different denominators, e.g. by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognise that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols greater than, equal to, or less than, and justify the conclusions, e.g. by using a visual fraction model.
AC9M4N03Australian Curriculum v9 (ACARA)
Locate fractions on a number line (Year 4)
Count by fractions, including mixed numerals, and locate and represent them as points on a number line. Placing each fraction on one shared line is exactly how this unit orders them.
AC9M5N03Australian Curriculum v9 (ACARA)
Use equivalence to compare and order fractions (Year 5 bridge)
Use equivalence to compare, order and represent common fractions on the same number line and justify the order. ACARA places explicit comparing and ordering of unlike fractions at Year 5, so this US Grade 4 unit reaches toward that Year 5 descriptor.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Numerator: the top number, how many equal parts you have
Denominator: the bottom number, how many equal parts the whole is cut into
Benchmark fraction: a friendly fraction such as 1/2 that you compare others against
Common denominator: the same bottom number given to two fractions so their parts are the same size
Equivalent fractions: different fractions that name the same amount, such as 2/3 and 8/12
Order: arrange a list of fractions from least to greatest, or greatest to least

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Fair comparing means the same whole

Concrete

Start with the pizza question from the hook using two identical paper circles. Fold one into 3 equal parts and shade 2 of them for 2/3. Fold the other into 4 equal parts and shade 3 of them for 3/4. Lay them side by side. Because the two circles started the same size, it is fair to compare the shaded amounts, and the class can see the 3/4 circle has slightly more colored in.

The one rule that must come first: a comparison is only fair when both fractions are parts of the same whole. Half of a small cookie is not more than a quarter of a large cake. Keep the two wholes the same size and the comparison means something.

The eye can settle 2/3 against 3/4 when the pieces are close, but it cannot be trusted for trickier pairs. So we build tools that never need a perfect drawing.

Mia ate 2/3 of her pizza: 2 of 3 equal slices.

Ben ate 3/4 of the same size pizza: 3 of 4 equal slices, a little more than 2/3.

Check for understanding, ask

Why can we compare these two pizzas but not a slice of pizza against a slice of cake?
Which looks like a bit more here, 2/3 or 3/4?

2. Compare against one half

Pictorial

The fastest tool is a check against the benchmark 1/2. A fraction is exactly one half when the numerator is exactly half of the denominator, such as 4/8. It is more than half when the top is more than half of the bottom, and less than half when the top is less than half of the bottom. If one fraction is under a half and the other is over a half, you are done without any other work.

Take 2/5 and 5/8. Half of 5 is 2.5, and 2 is less than 2.5, so 2/5 is less than 1/2. Half of 8 is 4, and 5 is more than 4, so 5/8 is more than 1/2. One is under half, the other is over half, so 2/5 is less than 5/8.

The number line makes this visible: put a marker at 1/2 and drop each fraction onto the line. Anything left of the 1/2 mark is smaller than anything right of it.

2/5 lands left of the 1/2 mark and 5/8 lands right of it, so 2/5 is less than 5/8.

Worked example

Use the benchmark 1/2 to compare 3/8 and 4/6.

Half of 8 is 4. The numerator 3 is less than 4, so 3/8 is less than 1/2.
Half of 6 is 3. The numerator 4 is more than 3, so 4/6 is more than 1/2.
One fraction is under a half and the other is over a half.

Answer: 3/8 is less than 4/6, written 3/8 < 4/6.

Check for understanding, ask

Is 5/12 more or less than a half? How do you know?
Why does comparing to 1/2 not help when both fractions are more than a half?

3. Give them the same size parts

Pictorial

When the benchmark is not enough, for example both fractions are over a half, give both fractions the same denominator so the parts are the same size. Then the fraction with more parts is greater. Return to the pizza: compare 2/3 and 3/4. A denominator that both 3 and 4 divide into is 12.

Rewrite each fraction in twelfths using equivalent fractions: 2/3 = 8/12 because you multiply top and bottom by 4, and 3/4 = 9/12 because you multiply top and bottom by 3. Now the parts are identical twelfth-size pieces, so just compare the tops: 9 is more than 8.

So 3/4 is greater than 2/3, and Ben ate more pizza. The bars below shade the same length whole into twelfths so the extra twelfth in 9/12 is easy to see.

2/3 rewritten as 8/12: eight of twelve equal parts.

3/4 rewritten as 9/12: nine of twelve equal parts, one more part than 8/12, so 3/4 is greater.

Worked example

Compare 5/6 and 3/4 with a common denominator.

A denominator both 6 and 4 divide into is 12.
5/6 = 10/12 (multiply top and bottom by 2). 3/4 = 9/12 (multiply top and bottom by 3).
Same size twelfth parts, so compare the numerators: 10 is more than 9.

Answer: 5/6 is greater than 3/4, written 5/6 > 3/4.

Check for understanding, ask

What denominator would you use to compare 2/3 and 1/6? Does 6 work?
After you rewrite both fractions with the same bottom number, what do you compare?

4. When the tops already match

Pictorial

Sometimes there is a shortcut. If two fractions have the same numerator, you do not need a common denominator. The same number of parts is taken from each whole, so the fraction with the smaller denominator has the bigger parts and is therefore the greater fraction.

Compare 3/4 and 3/8. Both take 3 parts. Quarters are bigger pieces than eighths, because cutting a whole into 8 makes smaller pieces than cutting it into 4. So 3 big quarter pieces beat 3 small eighth pieces: 3/4 is greater than 3/8.

This is the case that feels backward, so lean on the circles: the same 3 shaded pieces cover more of the quarters circle than of the eighths circle.

3/4: three of four equal parts, big pieces.

3/8: three of eight equal parts, small pieces, so 3/4 is greater than 3/8.

Check for understanding, ask

Which is greater, 2/5 or 2/9, and why?
Why does a bigger denominator make each part smaller?

5. Putting a list in order

Abstract

Ordering is just comparing done to a whole list. The cleanest way is to rewrite every fraction with one common denominator, then read the numerators in order. Take 1/2, 2/3 and 5/6. A denominator all three divide into is 6.

Rewrite each in sixths: 1/2 = 3/6, 2/3 = 4/6, 5/6 stays as 5/6. Now the parts are all sixths, so order the tops: 3, then 4, then 5. That gives 1/2, then 2/3, then 5/6 from least to greatest.

Place them on one number line to check. The marks step to the right in the same order, which is the meaning of least to greatest: further right on the line means the greater number.

1/2, 2/3 and 5/6 placed on one line. Reading left to right gives the order from least to greatest.

Worked example

Order 3/4, 2/3 and 5/6 from least to greatest.

A common denominator for 4, 3 and 6 is 12.
3/4 = 9/12, 2/3 = 8/12, 5/6 = 10/12.
Order the numerators: 8, then 9, then 10.

Answer: From least to greatest: 2/3, 3/4, 5/6.

Check for understanding, ask

When every fraction is written over the same denominator, how do you read off the order?
Which end of the number line holds the greatest fraction?

Watch for

Common misconceptions and how to address them

MisconceptionThe fraction with the bigger numbers is bigger, so 3/8 is greater than 3/4.

Why it happens: Students carry over whole-number thinking, where 8 beats 4, and read the denominator as if a bigger bottom means a bigger fraction.

How to address it: Hold up 3/4 and 3/8 circles. The same 3 pieces cover more of the quarters circle. Say it in sharing language: cut a cake for 4 people or for 8, and the smaller crowd gets the bigger slice.

3/8 shades less than 3/4: more parts in the whole means smaller parts, so 3/8 is the smaller fraction.

MisconceptionJust compare the numerators, so 4/9 is greater than 1/2 because 4 is more than 1.

Why it happens: Comparing only the tops works when the denominators match, and students over-apply it to unlike fractions.

How to address it: Check against a half first. Half of 9 is 4.5, and 4 is less than 4.5, so 4/9 is less than 1/2. The bare top number is not enough when the parts are different sizes.

MisconceptionTo make a common denominator, add the two bottom numbers, so 2/3 and 3/4 become fractions over 7.

Why it happens: Adding the denominators feels like a natural move and produces a single new bottom number.

How to address it: A common denominator must be a number both denominators divide into, found by equivalent fractions, not by adding. For 3 and 4 use 12, then 2/3 = 8/12 and 3/4 = 9/12. Adding to 7 would break the equivalence.

MisconceptionWhen you rewrite a fraction with a new denominator, change only the bottom number, so 2/3 becomes 2/12.

Why it happens: Students remember that the denominator changes but forget the numerator must scale by the same factor to keep the value.

How to address it: Multiply top and bottom by the same number. Going from thirds to twelfths multiplies by 4, so 2/3 = 8/12, not 2/12. Check on a bar that 8/12 shades the same length as 2/3.

2/3 = 8/12: the top scaled by 4 as well, so the shaded length is unchanged.

MisconceptionYou can compare any two fractions straight away, even from different wholes.

Why it happens: Students treat a fraction as a bare number and forget it always refers to a whole.

How to address it: Ask which is more, 1/2 of a grape or 1/4 of a watermelon. The laughter makes the point: a comparison is only fair when both fractions are parts of the same size whole.

MisconceptionOn a number line, the fraction on the left is the greater one.

Why it happens: Some students read a line right to left, or confuse being first in the list with being largest.

How to address it: Anchor the direction: numbers grow as you move right, so the rightmost mark is the greatest. Trace from 0 rightward and read the order as least to greatest.

Do it together

Guided practice (with answers)

1. Is 3/5 more or less than one half?
Answer: More. Half of 5 is 2.5, and 3 is more than 2.5, so 3/5 is greater than 1/2.
2. Compare 2/3 and 3/4 using twelfths.
3/4 = 9/12, one twelfth-part more than 8/12.
Answer: 2/3 = 8/12 and 3/4 = 9/12. Since 9 is more than 8, 3/4 is greater, written 3/4 > 2/3.
3. Which is greater, 5/8 or 1/2?
Answer: 5/8. Half of 8 is 4, and 5 is more than 4, so 5/8 is over a half.
4. Which is greater, 4/5 or 4/9?
Answer: 4/5. Same numerator, and fifths are bigger parts than ninths.
5. Put 1/2, 2/3 and 5/6 in order from least to greatest.
Answer: In sixths they are 3/6, 4/6, 5/6, so the order is 1/2, 2/3, 5/6.
6. True or false: to compare 2/3 and 3/4 you can add the denominators to get sixths.
Answer: False. A common denominator is a number both 3 and 4 divide into, which is 12, not the sum 7.

On their own

Independent practice worksheets

Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with fraction comparing, then mix in visual fractions for students who still need the picture.

Grade 4 fractions worksheet (set A)compare and order unlike fractions Grade 4 fractions worksheet (set B)more comparing and ordering practice Grade 4 visual fractions worksheetname and shade fractions to support comparing All Grade 4 fraction worksheetsthe full set, six sets each All visual fraction worksheetsnaming and shading across grades

Reach every student

Differentiation

Support

Keep the paper circles and strips on desks so a stuck student can fold and compare the real amounts.
Limit early pairs to comparisons the benchmark 1/2 settles in one step (one under half, one over half).
Give a part-built common denominator (the shared bottom already written) so the student only scales the numerators.
Colour code the two fractions being compared so the numbers do not blur together.

Extension

Compare using a common numerator as well as a common denominator, and decide which is faster for a given pair.
Order a list of four or five fractions, including one that equals 1/2 exactly.
Insert a fraction between two given fractions, such as one between 2/3 and 3/4.
Write a same-whole trap question, like the grape and the watermelon, for a partner to catch.

Check it stuck

Assessment: exit ticket

A three-question exit ticket for the last five minutes. It samples the benchmark, the common denominator, and ordering, the three tools of the unit.

1. Is 3/8 more or less than one half? Explain.
Answer: Less. Half of 8 is 4, and 3 is less than 4, so 3/8 is under a half.
2. Compare 2/3 and 3/4. Write < or >.
Answer: 2/3 < 3/4, because 2/3 = 8/12 and 3/4 = 9/12.
3. Order 1/2, 5/6, 2/3 from least to greatest.
Answer: 1/2, 2/3, 5/6 (in sixths: 3/6, 4/6, 5/6).

For the teacher

Teacher notes and timings

Rough timing across three to four lessons: Lesson 1 the same-whole rule and the benchmark 1/2 (sections 1 to 2), Lesson 2 common denominators (section 3), Lesson 3 same-numerator reasoning and ordering (sections 4 to 5 plus the exit ticket).
Teach the benchmark before common denominators. It settles many pairs in one step and builds number sense, so students reach for the heavier common-denominator method only when they need it.
Language to keep saying: same whole, same size parts, further right is greater. These three phrases pre-empt most of the misconceptions.
The number-line diagrams mark only 0, 1/2 and 1 so young readers are not distracted by decimals. Tell the class to read the fraction labels above the line and ignore any decimal thinking for now.
US and AU alignment: the US asks for comparing unlike fractions at Grade 4 (4.NF.A.2). ACARA introduces locating fractions on a number line at Year 4 (AC9M4N03) but places explicit comparing and ordering of unlike fractions at Year 5 (AC9M5N03), so for Australian classes this unit runs slightly ahead of the Year 4 descriptor and into Year 5 work. That divergence is deliberate and noted for AU teachers.
Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.

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