Adding and Subtracting 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, fraction strips, 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 and the fraction strips once; one set per pair serves Days 1 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: fractions can only be added or taken away when their pieces are the same size. The denominator names the size of the pieces, so when the denominators differ, both fractions are renamed over a common denominator first, using equivalent fractions. Then the numerators are added or subtracted and the denominator stays. Mixed numerals work the same way, with the wholes and the parts kept apart until the end.

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

Only the numerators are added: 1/2 + 1/3 is not 2/5. A common denominator is a common multiple of the denominators, and the lowest one keeps the numbers small. A mixed numeral such as 1 3/4 means 1 + 3/4; an improper fraction such as 7/4 names the same amount.

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 adds and subtracts pieces of the same size; Day 2 makes the pieces match with a common denominator; Day 3 takes away and compares; Day 4 carries it all to mixed numerals; and Day 5 solves problems. It builds on the Year 5 work with related denominators and on this year’s equivalent fractions and common multiples. It leads to finding fractions, decimals and percentages of a quantity, and to Year 7, where all four operations work with fractions.

Materials for the week (one trip)

• From the classroom: scissors, pencils, coloured pencils, this pack printed.
• Cut out once, use all week: the fraction strips (one set per pair) and the twenty fraction cards. No maths equipment to buy.
• Optional: real measuring cups and a jug for the recipe problems on Day 5.

Words for the word wall

numerator, denominator, equivalent fraction, common denominator, lowest common denominator, simplest form, mixed numeral, improper fraction, regroup, sum, difference

Dear families

This week in maths, Year 6 adds and subtracts fractions. We learn why the pieces must be the same size before fractions can be added, how to rename them so they match, and how to solve problems with mixed numerals such as 1 1/2.

Try this at home

• Cooking: a recipe needs 1/2 a cup of milk and 1/4 of a cup of cream. How much liquid is that altogether?
• Pizza or cake: if one person eats 1/3 and another eats 1/4, how much is left?
• Time: half an hour and a quarter of an hour make what fraction of an hour?
• Ask your child to explain why 1/2 + 1/3 is not 2/5.

My fractions this week

Fill one row a day. Find two fractions at home (recipes, time, sharing), add them and take one from the other.

Printed from the free seegongsik Adding and Subtracting Fractions teaching pack · seegongsik.com/au/y6/number/AC9M6N05/pack

Same-size pieces

When two fractions have the same denominator, the pieces are the same size, so the numerators are added or taken away and the denominator stays. When one denominator is a multiple of the other, renaming one fraction makes the pieces match.

We are learning to

• add and subtract fractions with the same denominator,
• rename one fraction when one denominator is a multiple of the other,
• write a sum greater than 1 as a mixed numeral.

Success criteria

• I can work out 3/4 + 3/4 and write the answer as a mixed numeral.
• I can explain why 1/2 + 1/4 is not 2/6.

You need

Coloured pencils, the worksheet, one per child, and the cut-out fraction strips, one set per pair.

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with the fraction strips for Part B.

Watch for these ideas

• Adding the denominators: 2/5 + 1/5 is not 3/10. The pieces are still fifths.
• Forgetting to rename: 1/2 + 1/4 needs 2/4 + 1/4 first.
• Leaving 6/4 when a mixed numeral is asked for: 6/4 = 1 2/4 = 1 1/2.

Answers

• Part A: 2/5 + 1/5 = 3/5; 3/8 + 4/8 = 7/8; 7/10 − 3/10 = 2/5; 5/6 − 1/6 = 2/3.
• Part B: 1/2 + 1/4 = 2/4 + 1/4 = 3/4; 1/3 + 1/6 = 2/6 + 1/6 = 1/2; 3/4 − 3/8 = 6/8 − 3/8 = 3/8; 5/6 − 1/3 = 5/6 − 2/6 = 1/2.
• Part C: 3/4 + 3/4 = 3/4 + 3/4 = 1 1/2; 5/6 + 2/3 = 5/6 + 4/6 = 1 1/2; 7/8 + 1/2 = 7/8 + 4/8 = 1 3/8.
• Part D: Max added the tops and the bottoms. 2/6 is less than 1/2, so it cannot be the sum. 1/2 + 1/4 = 2/4 + 1/4 = 3/4.

Same-size pieces

A. Same denominator

The first fraction is shaded. Shade the second in another colour to add, or cross it out to take it away.

B. Rename one, then add or take away

C. Over one whole

Write each answer as a mixed numeral.

3/4 + 3/4 = ______ + ______ = ______ = ________

5/6 + 2/3 = ______ + ______ = ______ = ________

7/8 + 1/2 = ______ + ______ = ______ = ________

D. Is Max right?

Max says 1/2 + 1/4 = 2/6. Is that right? Explain, then work it out.

______________________________________________________________________

Make the pieces match

When neither denominator is a multiple of the other, as with thirds and quarters, both fractions are renamed. The new denominator is a common multiple of both, found by listing multiples as in the factors unit, and the lowest one keeps the numbers small.

We are learning to

• find the lowest common denominator by listing multiples,
• rename both fractions over it and add the numerators,
• write the sum in its simplest form.

Success criteria

• I can work out 2/3 + 1/4 and show it on a bar in twelfths.
• I can find the lowest common denominator of two fractions.

You need

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

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Multiplying the denominators every time: 24 works for 1/4 + 1/6, but 12 is smaller.
• Renaming the denominator but not the numerator: 2/3 is 8/12, not 2/12.
• Stopping before the simplest form: 6/12 is better written as 1/2.

Answers

• Part A: 2 and 3: 6; 3 and 4: 12; 4 and 6: 12; 6 and 8: 24; 2 and 5: 10.
• Part B: 1/2 + 1/3 = 3/6 + 2/6 = 5/6; 2/3 + 1/4 = 8/12 + 3/12 = 11/12; 1/4 + 1/6 = 3/12 + 2/12 = 5/12; 3/8 + 1/6 = 9/24 + 4/24 = 13/24; 2/5 + 1/2 = 4/10 + 5/10 = 9/10.
• Part C: 8 twelfths, then 3 more: 11/12.
• Make 1, pairs: 1/12 + 11/12; 1/8 + 7/8; 1/6 + 5/6; 1/4 + 3/4; 1/3 + 2/3; 3/8 + 5/8; 5/12 + 7/12; 1/2 + 1/2. Three cards: 1/4 + 1/3 + 5/12; 1/6 + 1/3 + 1/2; 1/6 + 1/4 + 7/12; 1/12 + 1/4 + 2/3; 1/12 + 1/6 + 3/4.

Make the pieces match

A. Find the lowest common denominator

B. Rename both, then add

1/2 + 1/3 = ______ + ______ = ______

2/3 + 1/4 = ______ + ______ = ______

1/4 + 1/6 = ______ + ______ = ______

3/8 + 1/6 = ______ + ______ = ______

2/5 + 1/2 = ______ + ______ = ______

C. Show it on a bar

Shade 2/3 in one colour and 1/4 in another on this bar in twelfths. What is 2/3 + 1/4? ______

Take away and compare

Subtraction follows the same path: rename both fractions over a common denominator, then take one numerator from the other. A difference also says how much more one amount is than another, and a missing part in an addition is found by subtracting.

We are learning to

• subtract fractions with different denominators,
• find how much more one fraction is than another,
• find a missing part in an addition or a subtraction.

Success criteria

• I can work out 3/4 − 1/3.
• I can find the missing fraction in 1/3 + ? = 5/6 and check it by adding.

You need

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

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Subtracting the denominators as well: 3/4 − 1/3 is not 2/1.
• Renaming only one fraction when both need it: 3/4 and 1/3 both become twelfths.
• Adding to fill a gap: 1/3 + ? = 5/6 needs 5/6 − 1/3.

Answers

• Part A: 3/4 − 1/3 = 9/12 − 4/12 = 5/12; 5/6 − 1/4 = 10/12 − 3/12 = 7/12; 2/3 − 1/2 = 4/6 − 3/6 = 1/6; 7/8 − 1/6 = 21/24 − 4/24 = 17/24; 9/10 − 2/5 = 9/10 − 4/10 = 1/2.
• Part B: 3/4 is 1/12 more than 2/3; 5/8 is 1/8 more than 1/2; 4/5 is 1/20 more than 3/4.
• Part C: 1/2, 5/12, 7/12 (check by adding).
• Part D: 7/8 − 1/3 = 21/24 − 8/24 = 13/24 of the bottle is left.
• Take-away Make 1: 5/12 needs 7/12 more, because 5/12 + 7/12 = 1.

Take away and compare

A. Rename both, then take away

3/4 − 1/3 = ______ − ______ = ______

5/6 − 1/4 = ______ − ______ = ______

2/3 − 1/2 = ______ − ______ = ______

7/8 − 1/6 = ______ − ______ = ______

9/10 − 2/5 = ______ − ______ = ______

B. How much more?

How much more is 3/4 than 2/3? ______________________

How much more is 5/8 than 1/2? ______________________

How much more is 4/5 than 3/4? ______________________

C. Find the missing part

Check each one by adding or taking away.

D. The water bottle

Lena’s bottle was 7/8 full. Then Lena drank 1/3 of a bottle. How much of the bottle is left?

______________________________________________________________________

Mixed numerals

A mixed numeral is a whole number and a fraction together. To add, add the wholes, then the fractions, and turn a fraction part over 1 into another whole. To subtract, if the top fraction is too small, regroup one whole into the fraction part first.

We are learning to

• change between mixed numerals and improper fractions,
• add mixed numerals, carrying a whole when the fractions pass 1,
• subtract mixed numerals, regrouping a whole when it is needed.

Success criteria

• I can work out 2 3/4 + 1 1/2 and 2 1/3 − 3/4.
• I can explain where a regrouped whole goes.

You need

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

Lesson flow (about 50 minutes)

Two half-sessions instead? Split after Part A; start Session B with Sam’s working in Part D.

Watch for these ideas

• Leaving a fraction part over 1: 2 3/4 + 1 1/2 is 4 1/4, not 3 5/4.
• Taking the smaller fraction from the larger whichever is on top: 3 1/4 − 1 5/6 is not 2 7/12.
• Turning 17/6 into 1 11/6. Count every six in 17: there are two.

Answers

• Part A: 1 1/2 + 2 1/3 = 3 5/6; 2 3/4 + 1 1/2 = 4 1/4; 3 2/3 + 1 5/6 = 5 1/2; 1 3/8 + 2 1/4 = 3 5/8.
• Part B: 3 1/2 − 1 1/4 = 2 1/4; 4 5/6 − 2 1/3 = 2 1/2; 2 1/3 − 3/4 = 1 7/12; 5 − 1 2/5 = 3 3/5; 3 1/4 − 1 5/6 = 1 5/12.
• Part C: 7/4 = 1 3/4; 11/3 = 3 2/3; 17/6 = 2 5/6; 2 3/5 = 13/5; 3 1/6 = 19/6; 1 7/8 = 15/8.
• Part D: Sam took one whole from the 2 and wrote it as 12 twelfths: 2 1/3 = 2 4/12 = 1 16/12. Then 1 16/12 − 9/12 = 1 7/12.

Mixed numerals

A. Add the wholes, then the parts

B. Take away (regroup a whole when you need to)

C. Change the form

D. Sam’s working

To work out 2 1/3 − 3/4, Sam wrote 1 16/12 − 9/12. Where did 16/12 come from? Finish Sam’s working.

______________________________________________________________________

______________________________________________________________________

Fraction problems

Problems do not say “add” or “subtract”. Children decide which operation the story needs, work it with a common denominator, and check the answer against a benchmark: is it more or less than a half, or than one whole?

We are learning to

• choose addition or subtraction for a fraction problem,
• solve problems with fractions and mixed numerals, giving the unit,
• check an answer with a benchmark such as 1/2 or 1.

Success criteria

• I can say which operation a problem needs and why.
• I can decide whether an answer is possible before I work it out exactly.

You need

The worksheet, one per child; measuring cups and a jug are a bonus for the recipe problem.

Lesson flow (about 50 minutes)

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

Watch for these ideas

• Adding every pair of numbers in a story. “How much further” and “how much was used” are differences.
• Giving an answer with no unit: 7/12 of what? Kilometres.
• Not checking: a sum of two amounts less than 1 can never be more than 2.

Answers

• Part A: (1) 1 1/2 + 3/4 = 2 1/4 cups; (2) 2 1/3 − 1 3/4 = 7/12 km; (3) 5/6 − 1/4 = 7/12 of the tank.
• Part B: 1/3 + 1/4 = 7/12 of an hour, which is 20 + 15 = 35 minutes.
• Part C: no: 3/4 + 1 1/3 = 2 1/12 litres, more than 2. Rounding to 1 + 1 says about 2, too close to decide, so this one needs the exact sum.

Fraction problems

A. Add or take away? Write the operation, then solve.

B. Parts of an hour

Nina spent 1/3 of an hour reading and 1/4 of an hour on maths. What fraction of an hour is that altogether? How many minutes?

Fraction of an hour: ________    Minutes: ________

C. Possible or not?

A 2 litre bottle of juice was full. Could 3/4 of a litre be poured at lunch and 1 1/3 litres at dinner? Show how you know.

______________________________________________________________________

D. Your own problem

Write a problem for a partner that needs 2/3 − 1/4. Solve it on the back.

______________________________________________________________________

______________________________________________________________________

Fraction cards

Cut out the twenty cards once. Pairs use them on Days 2 to 4.

Make 1 (Days 2 to 4)

1. Shuffle the cards. Lay eight face up; the rest are the pile.
2. Take turns to find two or three face-up cards that add to exactly 1. Say the sum with a common denominator: “1/4 + 1/3 + 5/12 = 3/12 + 4/12 + 5/12 = 1.”
3. Keep the cards and fill the gaps from the pile. No set? Turn over one more card.
4. When the pile runs out, the player with more cards wins.

Take-away round (Day 3): turn over one card and say how much more makes 1. Teacher note: the pairs that make 1 are 1/12 + 11/12; 1/8 + 7/8; 1/6 + 5/6; 1/4 + 3/4; 1/3 + 2/3; 3/8 + 5/8; 5/12 + 7/12; 1/2 + 1/2. Three cards with three different denominators: 1/4 + 1/3 + 5/12; 1/6 + 1/3 + 1/2; 1/6 + 1/4 + 7/12; 1/12 + 1/4 + 2/3; 1/12 + 1/6 + 3/4.

Fraction strips

Cut along the dashed lines: each strip into its pieces. Lay pieces under the whole to see a sum or a difference.

Teacher note: for 1/2 + 1/3, lay a half and a third under the whole, then look for the strip whose pieces fit both exactly. The sixths do: that strip is the common denominator.

What we know: Adding and Subtracting Fractions

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

1. 2/7 + 3/7 = ________
2. 1/2 + 1/3 = ________
3. 3/4 − 2/3 = ________
4. 5/6 + 3/4 = ________ (write it as a mixed numeral)
5. 1 2/3 + 2 3/4 = ________
6. 5 1/3 − 2 4/5 = ________
7. A recipe uses 2/3 of a cup of oil and 3/4 of a cup of water. How much liquid is that altogether? ________
8. Kim says 3/4 − 1/2 = 2/2. Explain the mistake and give the right answer. ______________________________

Answers and marking guide

Answers

1. 5/7.
2. 5/6.
3. 1/12.
4. 1 7/12.
5. 4 5/12.
6. 2 8/15 (regroup a whole).
7. 1 5/12 cups.
8. Kim took the tops and the bottoms away. 2/2 = 1, more than 3/4, so it cannot be a difference. 3/4 − 1/2 = 1/4.

A quick three-level guide

Eight questions, four ideas. A child at standard answers most questions and can say why, with a common denominator, a regrouped whole or a benchmark.

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: same-size pieces, common denominators, taking away, mixed numerals, and problems.