Comparing and Ordering Fractions: a week of ready-to-teach maths

Five days of lessons for Year 6 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet, plus twenty fraction cards, a line-up mat, a mini-check and every answer.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Cut out the fraction cards once; one set and one line-up mat per pair serves Days 2 to 5.
4. Open the free interactive unit on your board. Every plan tells you which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and answers are all on the one page.

No maths background needed

This pack is written for the busy generalist teacher. Each plan explains the idea in plain words, lists the misconceptions children bring, and gives the answers. The big idea of the week: one amount has many fraction names. Two quarters, three sixths and six twelfths all name one half, because cutting the same whole into more pieces makes each piece smaller by the same amount. Renaming fractions over a common denominator lets any two be compared, and a number line cut into twelfths holds halves, thirds and quarters together, so a set of them can be put in order and the order explained.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 50 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point if your timetable runs two shorter sessions.

Words to watch

The numerator is the top number, how many pieces; the denominator is the bottom number, how many equal pieces make one whole. A common denominator is a denominator both fractions can be renamed over. Every comparison in this pack is between parts of the same whole: half of a big pizza is more than half of a small one.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 finds the many names of one amount; Day 2 puts halves, thirds and quarters on one line; Day 3 renames two fractions over a common denominator to compare them; Day 4 adds quicker ways and the words to justify them; and Day 5 orders whole sets. It builds on the Year 5 work with related denominators, such as thirds and sixths, and on last unit’s common multiples. It prepares the next unit, adding and subtracting fractions with different denominators, and Year 7, where all kinds of rational numbers go on one number line.

Materials for the week (one trip)

• From the classroom: scissors, pencils, coloured pencils, rulers, this pack printed.
• For Day 1: a ruler or a straight strip of card to lay down the fraction wall.
• Cut out once, use all week: the twenty fraction cards and one line-up mat per pair. No maths equipment to buy.

Words for the word wall

numerator, denominator, equivalent fraction, simplest form, common denominator, number line, mixed numeral, improper fraction, greater than, less than, justify

Dear families

This week in maths, Year 6 compares and orders fractions. We find many names for one amount (one half is also two quarters and six twelfths), put halves, thirds and quarters on the same number line, and rename fractions so they can be compared. Each answer comes with a reason.

Try this at home

• Fold three paper strips of the same length into halves, thirds and quarters. Which is longer, two thirds or three quarters?
• Measuring cups: is a third of a cup more or less than a quarter of a cup? How do you know?
• Share one pizza or cake two ways in your head: would you rather have 3/4 or 5/6 of it? Why?
• Spot fractions on packets and signs, and say another name for each one.

My fractions this week

Fill one row a day. Write a fraction you found, another name for it, and a fraction it is bigger than.

Printed from the free seegongsik Comparing and Ordering Fractions teaching pack · seegongsik.com/au/y6/number/AC9M6N03/pack

Many names, one amount

Two fractions are equivalent when they name the same amount of the same whole. Children read equivalent fractions off a fraction wall, then make them without the wall by multiplying or dividing the top and the bottom by the same number.

We are learning to

• find equivalent fractions on a fraction wall,
• make an equivalent fraction by multiplying or dividing the numerator and the denominator by the same number,
• write a fraction in its simplest form.

Success criteria

• I can write three other names for one half and two for two thirds.
• I can explain why multiplying the top and the bottom by the same number keeps the amount the same.

You need

A ruler or a straight strip of card per child, coloured pencils, and the worksheet, one per child.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the missing numbers in Part B.

Watch for these ideas

• Adding instead of multiplying: adding 1 to the top and the bottom of 1/2 gives 2/3, a different amount.
• Thinking more pieces means more: 6/12 has more pieces than 1/2, but each piece is smaller.
• Changing only one number: 3/4 and 3/12 are not equal. The top and the bottom change together.

Answers

• Part A: 1/2 = 2/4 = 3/6 = 6/12; 1/3 = 2/6 = 4/12; 2/3 = 4/6 = 8/12; 1/4 = 3/12; 3/4 = 9/12; 5/6 = 10/12.
• Part B: 4, 6, 15, 10, 2, 4.
• Part C: 6/12 = 1/2; 4/12 = 1/3; 9/12 = 3/4; 10/12 = 5/6; 8/12 = 2/3; 15/12 = 1 1/4.
• Part D: no. On the wall 2/3 reaches further than 1/2 (8/12 against 6/12). Adding the same number to the top and the bottom changes the amount; multiplying both by the same number does not.

Many names for one amount

A. Lay a ruler down the wall

B. Find the missing number

C. Write each fraction in its simplest form

D. Is Ella right?

Ella says 1/2 = 2/3: add 1 to the top and 1 to the bottom. Is Ella right? Use the wall to explain.

______________________________________________________________________

Halves, thirds and quarters on one line

A fraction is a place on the number line as well as a piece of a shape. Children mark quarters on one line and thirds on another, then find the line that holds both: one cut into twelfths, because 12 is a common multiple of 2, 3 and 4.

We are learning to

• mark halves, thirds and quarters, and mixed numerals, on a number line from 0 to 2,
• rename a fraction in twelfths to find its place on the line,
• explain why one line in twelfths holds halves, thirds and quarters together.

Success criteria

• I can mark 2/3 and 3/4 on the same line and say which is further right.
• I can explain why twelfths work for halves, thirds and quarters.

You need

Rulers, the worksheet, one per child, and the cut-out fraction cards with a line-up mat, one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part B; start Session B with the twelfths line in Part C.

Watch for these ideas

• Counting marks instead of jumps: 1/4 is the end of the first jump, not the first mark at 0.
• Putting 1/3 at the third mark of a quarters line. Thirds need their own cuts, or a line in twelfths.
• Losing the whole after 1: 1 1/4 is one whole and one quarter more, so it sits past 1.

Answers

• Part A: 1/4 at the 1st mark, 3/4 at the 3rd mark, 1 1/2 at the 6th mark, 1 3/4 at the 7th mark after 0.
• Part B: 1/3 at the 1st mark, 2/3 at the 2nd mark, 1 1/3 at the 4th mark, 1 2/3 at the 5th mark after 0.
• Part C: 1/4 = 3/12, 1/3 = 4/12, 1/2 = 6/12, 2/3 = 8/12, 3/4 = 9/12, 1 1/4 = 15/12, 1 1/3 = 16/12, 1 1/2 = 18/12.
• Part D: A 1 1/2 (18/12); B 1/4 (3/12); C 2/3 (8/12); D 1 5/6 (22/12).
• Part E: 12 is the lowest common multiple of 2, 3 and 4, so each one is a whole number of twelfths; 24ths work too.

One line for all of them

A. The quarters line

Mark and label 1/4, 3/4, 1 1/2, 1 3/4.

B. The thirds line

Mark and label 1/3, 2/3, 1 1/3, 1 2/3.

C. One line for all of them: twelfths

Mark all eight: 1/4, 1/3, 1/2, 2/3, 3/4, 1 1/4, 1 1/3, 1 1/2. Write the halves and quarters above the line and the thirds below it, each with its twelfths name.

D. Read the line

Write each letter as a fraction in its simplest form.

A ________B ________C ________D ________

E. Why twelfths?

Why do halves, thirds and quarters all fit on the twelfths line? Name another line that would hold them all.

______________________________________________________________________

Rename to compare

Fractions with different denominators are hard to compare as they stand. Renamed over a common denominator, they become pieces of the same size, and the larger numerator is the larger fraction.

We are learning to

• find a common denominator for two fractions,
• rename both fractions over it and compare the numerators,
• record a comparison with <, > or =.

Success criteria

• I can show that 3/4 is larger than 2/3 by renaming both in twelfths.
• I can choose the lowest common denominator, or say why another one also works.

You need

The worksheet, one per child, and the cut-out fraction cards, one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the problems in Part B.

Watch for these ideas

• Comparing the denominators: 3/5 is not smaller than 1/2 just because 5 is bigger than 2.
• Renaming only one fraction. Both must be over the same denominator before the numerators can be compared.
• Multiplying the denominators every time: 24 works for 5/6 and 3/4, but 12 is smaller and easier.

Answers

• Part A, renamed in order: 8/12 < 9/12; 10/12 > 9/12; 6/10 > 5/10; 15/24 < 16/24; 4/6 = 4/6; 14/20 < 15/20; 5/12 > 4/12.
• Part B: (1) Jo, by 1/12 of a lap (8/12 and 9/12). (2) No: 6/8 and 5/8, so 1/8 of a cup more is needed. (3) The tank that is 5/6 full, by 1/18 of a tank (15/18 and 14/18).
• Part C: both work (18/24 and 20/24; 9/12 and 10/12); 12, the lowest, is easier to picture and to check. Face-off: 5/12 beats 1/3 = 4/12.

Rename to compare

A. Rename, then compare

B. Same whole, which is more?

C. Which common denominator?

To compare 3/4 and 5/6, Sam used 24 and Tia used 12. Do both work? Which is easier, and why?

________________________________________________________

Quick ways and good reasons

Renaming always works, but some pairs can be compared at a glance: the same denominator, the same numerator, one fraction on each side of a half, or two fractions each one piece short of a whole. Children learn the quick ways and give every comparison a reason, not just a sign.

We are learning to

• compare fractions with the same numerator by the size of the pieces,
• use one half and one whole as landmarks,
• justify a comparison in a sentence.

Success criteria

• I can say why 3/4 > 3/5 without renaming.
• I can choose a quick way for a pair, or rename when no quick way works.

You need

The worksheet, one per child, and the cut-out fraction cards, one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part B; start Session B with the missing pieces in Part C.

Watch for these ideas

• Bigger denominator, bigger fraction: with the same numerator it is the other way round.
• Using the half-way mark when both fractions are past a half, like 2/3 and 3/4.
• A sign with no reason. Every comparison this week comes with a because.

Answers

• Part A: 3/4 > 3/5; 5/8 < 5/6; 2/9 < 2/7. The same number of pieces, so the bigger pieces win.
• Part B: 3/8 < 4/7; 4/9 < 5/8; 2/3 > 5/12. One is past a half and the other is not.
• Part C: 5/6 < 7/8; 3/4 > 2/3; 11/12 > 9/10. Each is one piece short of 1, so the smaller missing piece wins.
• Part D: 4/7 > 4/9 (same numerator); 6/11 > 3/7 (half-way mark); 5/12 < 7/12 (same denominator); 2/5 > 3/8 (16/40 and 15/40); 7/8 < 11/12 (missing piece); 5/12 < 5/9 (same numerator).
• Part E: 2/3 = 10/15 and 3/5 = 9/15, so 2/3 > 3/5. The denominators alone say nothing when the numerators differ.

Quick ways and good reasons

A. The same top number

B. Past a half or not?

C. The missing piece

D. Choose a quick way, or rename

E. Help Sam

Sam says 2/3 < 3/5 because 3 < 5. Explain the mistake and write the right sign.

______________________________________________________________________

Order and justify

Ordering is comparing, done for a whole set. Children place each fraction on one line from 0 to 2 in twelfths, read the order off from left to right, write it with < signs, and justify the hardest step.

We are learning to

• order a set of fractions, mixed numerals and improper fractions,
• place them on one number line to check the order,
• justify an order with equivalence, piece size or a landmark.

Success criteria

• I can order 2/3, 1/2, 3/4 and 5/12, and show the order on a line.
• I can explain why 7/6 < 5/4 in two different ways.

You need

The worksheet, one per child, and the cut-out fraction cards with a line-up mat, one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the line in Part B.

Watch for these ideas

• Ordering by the denominators alone: in set (c), 3/8 has the biggest denominator but is not the smallest.
• Forgetting the wholes: 5/4 is more than 1, so it comes after every fraction less than 1.
• Mixing up smallest first and largest first. Read the question, then check the first and the last on the line.

Answers

• Part A: (a) 5/12 < 1/2 < 2/3 < 3/4; (b) 7/6 < 5/4 < 1 1/3 < 1 1/2; (c) 1/4 < 1/3 < 3/8 < 1/2; (d) 2/3 < 5/6 < 1 1/4 < 3/2.
• Part B: 5/6 at 10/12, 1 1/4 at 15/12, 3/2 at 18/12, 2/3 at 8/12, 7/4 at 21/12.
• Part C: Cleo 5/6 (10/12), Ari 3/4 (9/12), Ben 2/3 (8/12), Dev 7/12 (7/12).
• Part D: renamed in twelfths, 7/6 = 14/12 and 5/4 = 15/12. Or: 7/6 = 1 1/6 and 5/4 = 1 1/4, and one sixth is a smaller piece than one quarter.
• The whole set: 1/6, 1/4, 1/3, 4/12, 5/12, 3/6, 2/4, 1/2, 7/12, 2/3, 9/12, 3/4, 5/6, 1 1/4, 5/4, 1 1/3, 3/2, 1 1/2, 1 2/3, 1 3/4 (equal cards share a place).

Order and justify

A. Order each set, smallest first

(a) 2/3, 1/2, 3/4, 5/12
________  <  ________  <  ________  <  ________

(b) 1 1/3, 5/4, 1 1/2, 7/6
________  <  ________  <  ________  <  ________

(c) 3/8, 1/3, 1/4, 1/2
________  <  ________  <  ________  <  ________

(d) 3/2, 5/6, 1 1/4, 2/3
________  <  ________  <  ________  <  ________

B. Mark and label each one on the line

5/6, 1 1/4, 3/2, 2/3, 7/4

C. Four pizzas

Four friends each had part of a pizza. The pizzas were all the same size. Ari ate 3/4, Ben ate 2/3, Cleo ate 5/6, Dev ate 7/12. Order the friends from the most pizza to the least.

________________________________________________________

D. Two ways

Explain in two different ways why 7/6 < 5/4.

1. ______________________________________________________

2. ______________________________________________________

Fraction cards

Cut out the twenty cards once. Pairs use them on Days 2 to 5.

Line-up (Days 2 and 5)

1. Shuffle the cards face down. Take turns to turn one over.
2. Lay it at its mark on the line-up mat and say its twelfths name: “3/4 is 9/12.”
3. If a card is already on that mark, the two are equivalent. Say both names and keep the pair.

Face-off (Days 3 and 4)

1. Deal the cards out. Each player turns one card over at the same time.
2. The larger fraction wins both cards, once its owner says how they know.
3. Equal fractions: say both names, then put both cards under your piles.

Teacher note: the equivalent cards are 1/3 = 4/12; 1/2 = 3/6 = 2/4; 3/4 = 9/12; 1 1/4 = 5/4; 1 1/2 = 3/2. Every other card has a mark of its own.

Line-up mat

Print one mat for each pair. Each line is cut into twelfths. Lay each card above or below its mark.

Teacher note: halves, thirds and quarters all land on a mark, because 12 is a common multiple of 2, 3 and 4.

What we know: Comparing and Ordering Fractions

Work on your own. Show your thinking if you can.

1. Write two fractions equivalent to 3/4. ______________ ______________
2. 2/3 = ______/12
3. Mark and label 1/4, 2/3, 1 1/2 on the line.
4. Which is larger, 3/4 or 5/6? Show how you know. ______________________________
5. Write <, > or =:   3/5 ____ 1/2  4/6 ____ 2/3  7/8 ____ 5/6
6. Order from smallest to largest: 1 3/4, 5/3, 1 1/2, 11/6. ______________________________
7. Leo says 1/3 > 1/2 because 3 > 2. Explain the mistake. ______________________________
8. Two jugs hold the same amount when full. One is 2/3 full and the other is 5/8 full. Which holds more? How do you know? ______________________________

Answers and marking guide

Answers

1. Any two of 6/8, 9/12, 12/16, ... (the top and the bottom multiplied by the same number).
2. 8.
3. 1/4 at 3/12, 2/3 at 8/12, 1 1/2 at 18/12.
4. 5/6: 9/12 and 10/12.
5. 3/5 > 1/2; 4/6 = 2/3; 7/8 > 5/6.
6. 1 1/2 < 5/3 < 1 3/4 < 11/6 (in twelfths: 18, 20, 21, 22).
7. Thirds are smaller pieces than halves, so one third is less than one half: 1/3 = 2/6 and 1/2 = 3/6.
8. The jug that is 2/3 full: 16/24 and 15/24.

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say why, with a renaming, a place on the line or a quick way.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: equivalent names, one line for halves, thirds and quarters, renaming to compare, quick ways, and ordering with reasons.