# Addition, subtraction and equivalent fractions

Purpose

The purpose of this series of lessons is to develop understanding of equivalent fractions and the operations of addition and subtraction with fractions.

Specific Learning Outcomes
• Add and subtract fractions with like denominators.
• Explore and record equivalent fractions for simple fractions in everyday use.
• Recognise that equivalent fractions occupy the same place on the number line.
• Recognise and apply the multiplicative relationship between simple equivalent fractions.
• Understand that fractions can have an infinite number of names.
• Understand the reciprocal nature of multiplication and division and how this knowledge can be used to simplify fractions.
• Apply knowledge of equivalent fractions to solving problems which involve comparing, adding and subtracting fractions with different denominators.
Description of Mathematics

This series of lessons builds upon students’ understanding the relationship between proper and improper fractions and where fractions fit amongst whole numbers on a number line. Students have been using regional, set and linear representations of fractions and working between these.

This emphasis on physical representations continues as the understanding of how to add and subtract everyday fractions with like denominators develops. These operations with whole numbers and like parts establish understanding of strategies for joining and partitioning proper fractions and mixed numerals before the added complexity of working with unlike parts is introduced. In working with like parts, students learn to add the numerators whilst keeping the denominators the same.

In comparing, combining or separating fractions with unlike denominators, the demand to understand and work with equivalent fractions increases with the need to ‘make the denominators the same’ before you can operate with the fractions. Whilst students have been using repeated addition and subtraction, and simple multiplication and division when working between improper fractions and mixed numerals, understanding and creating equivalent fractions requires a sound operating knowledge of multiplication and division facts. It is essential that the multiplicative relationships inherent in equivalent fractions are understood if students are to successfully recognise, create and work with equivalent fractions when carrying out number operations with any fractions within contextual problems.

The progression of the teaching model is inherent in these lessons. Students make, show and explore fractions in a range of ways. These physical representations with materials then become points of reference as the relational aspects of fractions are explored and number properties are increasingly relied upon. The importance of early practical modeling to build fundamental conceptual understanding of fractions becomes evident as working with fractions becomes more abstract and ‘number based’.

Key understandings which underpin these lessons include knowing the place of fractions amongst whole numbers (understanding improper fractions and mixed numerals), understanding that equivalent means having the same value, recognising that equivalent fractions have multiplicative relationships with each other, understanding how and why division undoes multiplication, recognizing that the value of a number (fraction) does not change when it is multiplied or divided by one, and understanding that a fraction can have an infinite number of names.

These ideas are presented in five sessions however, as they include complex concepts that are fundamental to a student’s success with fractions, these sessions can be extended over a longer period of time.

Games are included and can be made by the students. Whilst the games are introduced and used within sessions to consolidate ideas, they can also be added to the class or group independent activities, or be sent home for family challenges and enjoyment.

#### Links to the Number Framework

Stages 6 - 7

This unit supports teaching and learning activities in the Student Fractions e-ako 7 and 7+, 8 and 8+, 9 and 9+, and complements the learning activities in Book 7 Teaching Fractions, Decimals and Percentages.

Required Resource Materials
Unifix cubes

Fraction circle sets

Fraction strip sets

Packs of playing cards

Strips of paper

Hundreds flip board

Activity

#### Session 1

SLOs:

• Consolidate understanding of where fractions fit on a number line.
• Add and subtract fractions with like denominators.

Activity 1

1. Begin this session by skip counting forwards and backwards in common fractions and mixed numerals.
For example: “one quarter, two quarters, three quarters, one, one and one quarter, one and two quarters, one and three quarters, two, two and one quarter…”

2. Ask “How do we write 1 using quarters?” (4/4). How do we write 2 (8/4), 3 (12/4) and 4 (16/4)?
Write on the class/group chart in words and symbols:
Four quarters is the same as one: 4/4 = 1,
Eight quarters is the same as two: 8/4 = 2, etc.
Highlight the equals sign.
Ask students to discuss the meaning of = (is the same as, the amount on one side is equivalent in value to the amount on the other side).
Record their ideas.

Activity 2

1. Pose and write this problem on the class/group chart. “There was a pizza party at Elijah’s house. These bits of pizzas were left over: 1 1/4 vegetarian, 3/4 meat lovers, 1 3/4 pepperoni, 1/4 Hawaiian, and 2 1/4 seafood. Altogether, how much pizza was left over?”

2. Have students work in pairs to reach a solution and share these. Summarise their strategies on the class chart.

Activity 3

1. Make available to each student pair: sticky tape and two strips of paper of A3 length.
Have each pair make their own fraction number line to 8. Each strip should be folded 4 times, opened and taped at one end to the other strip. This is now numbered from 0 at the left hand edge and 8 at the right with whole number and quarter fractions written in place.

2. Refer to the fractions in the pizza problem posed in Activity 2.
Have students begin at 1 1/4 ( the vegetarian pizza left over) on the number strip and record a jump of 3/4 to 2, another of 1 3/4 to 3 3/4, another of 1/4 to 4 and the final jump of 2 1/4 to 6 1/4.
Have students confirm if 6 1/4 was the total number of left over pizzas they calculated.

3. Pose another problem orally, slowly: eg. 2 1/2 + 3/4 + 3/4 + 1 3/4 + 3/4 = ☐
Have the students solve it by following with their eyes along the number line then sharing their result with their partner.
Repeat with other examples.

4. Suggest the students close their eyes and pose one more problem orally and slowly, giving the students the opportunity to try to image this: eg. 1/4 + 1 1/4 + 3/4 + 1 1/4 = ☐.
Have partners share and discuss their results.
This may prove challenging for some.

Activity 4

1. Have each pair of students collect at least 30 unifix cubes (interlocking cubes), if possible, of the same colour. Make one ‘rod’ of 4 connected cubes, stating that the ‘rod’ represents 1.
Ask “So what part of the rod is each cube?”

2. Have students use the cubes to make each of the fractions from the pizza problem and talk about ways they could add the cubes together. Before they do so record their ideas on the class/group chart and number their suggestions. For example:
2. Join them all into one long line (make an improper fraction) then break them into groups of 4 (1 rods) making mixed numerals.
3. Add them (mixed numerals) consecutively as they did on the number line.

3. Have students solve the problem using the cubes. Check their solution was 6 1/4 as was found previously.

4. Pose another simple problem for each pair to investigate using one of the strategies suggested. Have them share their results.

5. Have pairs make the original amounts once again and try each of the other two strategies and decide which is their favourite and why. Emphasise that all strategies are valid.

6. Pose a subtraction problem. For example: 5 1/4 - 2 3/4 . Have pairs explore and reach a solution that they then pair share. On the class/group chart, record the strategies used and find which was the most popular and why.
1. Subtract the whole numbers first then subtract the fractions using either the number line or the cubes
2. Join the minuend into one long line (make 5 1/4 into the improper fraction of 21/4) then subtract the subtrahend 11/4 (2 3/4) to find the difference, or in this case the remainder.

Pose several other problems using simple fractions with like denominators and have students apply their preferred strategy.

7. Tell the students they are now going to solve some problems without materials but that they should decide on their best strategy.
Suggest the students close their eyes and picture the cubes and/or the number line, as you pose several subtraction problems (which are within the limits of the numbers on their number strip). Compare solutions.

Activity 5

1. Discuss and chart key learning from this session. Guide questioning to have students recognise that in the examples explored, the denominators are the same. They can work with the whole numbers first then the numerators of the fractions.

2. Have the students write for their partners to solve, several of their own addition and subtraction problems involving like denominators, in contexts that are familiar to them. The videos below complement this lesson. Right click and choose "Save As" to save them to your computer.

#### Session 2

SLOs:

• Recognise that denominators must be the same to add or subtract fractions.
• Understand the meaning of equivalent.
• Explore and record equivalent fractions for simple fractions in everyday use.
• Add and subtract simple fractions that have unlike denominators.

Activity 1

Begin the session by posing a subtraction problem in which the denominators are the same.
For example: The builder has a 3 3/4 metres length of wood. He cuts off 1 2/4 metres. What length of wood is left?
Review strategies from Session 1.

Activity 2

Pose a problem in which the denominators are not the same.
For example: The builder has a 3 3/4 metres length of wood. He cuts off 2 1/3 metres. What length of wood is left?
Have students work in pairs to (attempt to) solve this problem, making available number lines and cubes. Have students share their methods and discuss these.
Elicit from the students their recognition of the need to work with a common denominator.

Activity 3

1. Write on the class/group chart: ‘What is an equivalent fraction?’
Record student responses, including examples they give: eg. 1/2 = 2/4
Refer to the class/group chart recording from Session 1:
4/4 = 1, 8/4 = 2, 12/4 = 3, 16/4 = 4, and the highlighted = sign which they agreed means ‘is the same as’ or ‘that the amount on one side is equivalent in value to the amount on the other side.’

2. Make available to the students pairs Fraction Circle pieces or plastic Fraction Strips.
Have them explore common fractions, finding and recording at least ten expressions of equivalence.
For example: 1/2 = 2/4, 1/3 = 2/6, 1/5 = 2/10, 1/6 = 2/12, 4/8 = 6/12

3. Provide student pairs with Fraction Strips (MM 7-7), scissors, glue and chart paper.  Have students work in pairs to make a poster showing and explaining equivalent fractions, by cutting and pasting equivalent regions onto their poster, writing expressions of equivalence and stating what is the same.
For example:
• =  • 1/2 = 2/4 = 4/8
• These fractions are equivalent
• They are the same area
• The fractions have the same value

Activity 4

1. Return to the problem posed in Activity 2: The builder has a 3 3/4 metre of length of wood. He cuts off 2 1/3 metres. What length of wood is left?
Challenge students to identify which fractional part they used to show equivalence with both quarters and thirds.
If it is not obvious, model 3/4 = 9/12 and 1/3 = 4/12.

2. Have student pairs solve the problem and pair share their solutions.

3. Conclude the session by highlighting that by using twelfths as the common denominator it was easy to find a solution to this problem.

#### Session 3

SLOs:

• Recognise that equivalent fractions occupy the same place on the number line.
• Recognise and apply the multiplicative relationship between simple equivalent fractions.

Activity 1

1. Have students share their equivalent fractions posters. Together record on the class/group chart what the students notice about equivalent fractions highlighting that equivalent fractions:
• have the same value
• have the same area
• have different numerators and denominators.

2. Provide students with a blank piece of paper each and have them fold this to demonstrate (‘prove’) these points.  and 3. Write on the class/group chart:
“Equivalent fractions are found in the same place on a number line. True or false?”
Make long strips of paper available but do not insist they use these.
Have student pairs answer the question then have them pair share and demonstrate their thinking by showing how they know.

4. Use the number line from Session 1 and have students mark equivalent fractions on the number line: eg. 1/2, 4/8, 3/6, in the same place.

5. Add to the points from the start of the activity:
Equivalent fractions:
• have the same value
• have the same area
• have different numerators and denominators
• occupy the same place on a number line.

Activity 2

Write on the class/group chart: 1/2 = 2/4 = 4/8
Ask the students what they notice happening to the numerator (it is doubling each time) and what is happening to the denominator (it is doubling each time).
Ask students how this could be shown. Accept their suggestions, reaching an understanding that it could be shown in this way.  Together recognise that the value of 2/2  is 1 and that when any whole number or fraction is multiplied by 1 its value remains unchanged. We can multiply a fraction by 2/2, 3/3, 4/4 etc. to find an equivalent fraction without changing the value.

Activity 3

Have pairs of students record these non-unit fractions 2/3, 3/4, 2/5, 4/5.
Have them multiply each by 2/2 and 3/3 and show this using fraction pieces cut from MM7-7.
For example: 2/3 x 2/2 = 4/6 2/3 x 3/3 = 6/9 They are demonstrating by their area models that the equivalent fractions are the same (have the same area).

Activity 4

Have the students play Fractions Fish. (Attachment 1)
(Purpose: to recognise the multiplicative relationship between equivalent fractions)
This game is for two or three players.
Each player has one complete set (6) of blue cards (5 showing fractions equivalent to 1, and one showing ‘other equivalent fractions’). They place these cards face up in front of them. Each player is dealt five cards. The remainder of the cards is placed face down in a pile in the centre of the players.
Players first check their hands to see if they have been dealt any equivalent fractions. If so, these pairs are placed face up below the appropriate blue card (the multiplier).
The decision about the placement is: The first fraction multiplied by the fraction on the blue card makes the second equivalent fraction in the pair.
For example: 4/4 and 8/16 are placed below the 2/2 card because 4/8 x 2/2 = 8/16
1/2 and 6/12 are placed below the 6/6 card because 1/2 x 6/6 = 6/12
There are equivalent fractions for which the ‘given multipliers’ do not apply. For example: 4/8 and 6/12. These should be placed face up below the card reading ‘other equivalent fractions’.
Players then take turns to ask another player for a specific fraction card equivalent to one held in their hand. If the player who is asked does not have the requested card, the ‘asker’ is told, “Go Fish”. The ‘asker’ then takes a card from the centre pile.
If the asker does receive a card from another player the asker can make another request until they are told to “Go fish”.
Once a pair of cards is placed on the table, another player may not request them. A player can however shift their own fraction cards from beneath one blue card to another as the game proceeds. (This will most often be using cards from the ‘other equivalent fractions’ pairs to make other blue card pairs.)
The winner is the first player to have (at least) 2 different pairs of cards for each blue card in front of them. (12 different pairs in total).

Activity 5

Conclude the session by recording on the class/group chart:
When we multiply a number by 1 (2/2, 3/3, 4/4…) we do not change the value of that number.

#### Session 4

SLOs:

• Understand that fractions can have an infinite number of names.
• Understand the reciprocal nature of multiplication and division and how this knowledge can be used to simplify fractions.
• Use multiplication and division to calculate equivalent fractions.

Activity 1

1. Begin this session by reading together the final statement made in Session 3.
Refer to this expression and have the students suggest how the pattern of equivalent fractions continues. Ask, “How long can this pattern of making fractions that are equivalent to one half continue?” Have students discuss this in pairs and agree on their answer. “Forever. We can just keep on going”.
Discuss and conclude that there are an infinite number of names for any fraction.

2. Have one student choose another fraction to write and explore in the same way on the class chart, with the class/group contributing. For example: 2/3 x 3/3 = 6/9= 18/27 = 54/81 = …

Activity 2

Write 32/64 and ask if there is a simpler way to write this fraction. This should be obvious but have pairs of students investigate the question, record what they did and report back. Some suggestions may include:

• We went ‘backwards’ (divided the numerator and denominator by 2 several times).
• 32/64 ÷ 8/8 = 4/8 because we could see that you could divide both numbers by 8.
• 32/64 ÷ 32/32 = 1/2 because we could see that you could divide both numbers by 32/32 which is the same as 1.
• 32/64 ÷ 32/32 = 1/2 if you multiply both the numerator and the denominator by 2 x 2 x 2 x 2 x 2, which is 32, then you can divide by the same thing.

Explore all suggestions thoroughly and draw a conclusion.
For example:

• To simplify a fraction you can divide both the numerator and the denominator by the same number, which is the same as dividing by 1.
• Multiplication and division by whole numbers are opposite (or reciprocal) operations.

Activity 3

Have pairs of students play Simplify Secrets.
(Purpose: to simplify simple fractions and to identify which fractions can be simplified)
Student pairs share a deck of shuffled playing cards, Ace (1) to 10, and have a pencil and paper each.
Players take turns to turn over two cards. The first card is the numerator and the second is the denominator. The player records their fraction. Beside the fraction they write a simpler equivalent fraction if they can, showing what they divided both numbers by. If their partner agrees they tick their recording.
Fractions for which they can find no simpler fraction are recorded and nothing further is added.
For example:
If a player draws a 6 and then an 8, they write 6/8 ÷ 2/2 = 3/4 and once their partner agrees they can add a tick ✔.
If a player draws a 7 and then a 9, they write 7/9 but no tick is given.
If the fraction is an improper fraction that is changed to a mixed numeral without simplification, no tick is given as this is not simplifying the fraction. For example: 7/2 = 3 1/2.
Students play and record 10 rounds (at least). They should then look closely at the fractions they were able to simplify (✔) and the ones they were not.
They must discuss, agree and write the ‘Secret to simplifying fractions’:

• Both numbers must be divisible by the same number and leave no remainder.

Activity 4

Have students complete MM8-9 part 1, Equivalent fractions: Equation and Explanation.

Activity 5

On the class/group chart summarize what has been learned in this session.
For example:

• A fraction can have an infinite number of names if we keep on multiplying the numerator and denominator by the same number.
• Some fractions can be simplified by evenly dividing the numerator and denominator by the same number.

#### Session 5

SLOs:

• Find the common multiple and use this to create equivalent fractions.
• Apply knowledge of equivalent fractions to solving problems which involve comparing, adding and subtracting fractions with different denominators.

Activity 1

1. Begin by posing the question: “When would it be useful to know about equivalent fractions?’’ Record the student’s suggestions and ask them to provide examples: They may suggest problems like these:
• “If we were comparing different kinds of fractions to see who had more. Like you might have 3/4 of metre of fabric and you need 2/3 metre. Will you have enough?
• “If we were adding different kinds of fractions to see how much altogether. Say you had 3/4 of a pizza and another 1/3 of a pizza. Does that make 1 whole pizza?”
• “If you had an amount and gave away a fraction and you wanted to work out what you had left. Like you had 5/8 of something and you gave away 1/4.”

2. Revisit ideas explored in Session 2 and 4; when adding, subtracting or comparing fractions we need to work with a common denominator.

Activity 2

1. Pose the question: “How do we find a common denominator. In other words how do we find a common name for two fractions, like 3/4 and 1/3?”
Record the students’ ideas then write a summary statement such as: ‘We look for the lowest common multiple (LCM) which can become the new name for both of our fractions.’

2. Have the class play Beep first then Buzz-Beep, using the numbers 3 and 4. Ensure that the students understand that in this game multiples are identified by Beep and common multiples by Buzz-Beep.
Have one student work with a hundreds flip board leaving multiples ‘unflipped’. The students stand in a circle. Decide on a multiple of two or five that will be the “beep”numbers. Select a student to start counting from one. It is important that all the students count aloud. For example, for counting in fives: “1, 2, 3, 4, beep, 6, 7, 8, 9, beep, 11 ...” When a student says “beep”, they sit down. The game continues until only one student is left standing. This activity can be used to reinforce the forwards and backwards counting sequences. Use a hundreds board to assist the students to visualise the patterns. Flip over the spoken numbers but leave the “beep” numbers unflipped.

Extension Activity Have two multiples going at the same time. For example, threes (say “beep”) and fives (say “buzz”). If the number is a multiple of both three and five, then the person says “buzz-beep”. So the sequence goes “1, 2, beep, 4, buzz, beep, 7, 8, ... 11, beep, 13, 14, buzz-beep ... “Begin the counting sequences at different starting numbers. For example, “3, 7, 11 ...” or “100, 97,94, 91 ...” These patterns will help the students to recognise algebraic relationships.

3. Record the multiples of 4: 4, 8, 12, 16, 20…. And the multiples of 3: 3, 6, 9, 12, 15, 18, 21………. Have a student highlight the first ‘Buzz-Beep number.’ Point out that this is the lowest common multiple.

Activity 3

Refer to problem 1 in Activity 1 above. Record on the class/group chart: 3/4 x ☐/☐ = 9/12 and 2/3 x ☐/☐ = 8/12 and discuss. Here is a suggested format for discussion. Complete together the student problems posed in Activity 1 above.

Activity 4

Have students work with Common Multiples (Attachment 2)

1. Provide each student with either a blank multiplication table to complete, or the completed table. Have students use the table and pose: “What are the common multiples of 8 and 12.”

2. Have students follow along the 8 row and down the 12 column highlighting numbers common to both rows. (see page 2) Have them highlight or circle to first number that is common to both rows (24) and write in words and using the abbreviation. For example: The Lowest Common Multiple, LCM , for 8 and 12 is 24.

3. Pose several more examples such as 3 and 4, 5 and 4, 8 and 2 and each time have the students record the statements. Have students complete the LCM chart, using the multiplication table if needed.

Activity 5

1. Have students play in pairs, I can make it. (Attachment 3).
(Purpose: to apply knowledge of equivalent fractions to solving problems)
This is a game for two players. Story cards and fraction cards are shuffled and placed face down in two piles. Each player takes two story cards and 6 fraction cards and places them face up in front of them.
If Player One has fraction cards that allow them to answer either of their problems, the player says, “I can make it”, places the fraction cards beside the problem, operates on the fractions and gives the answer to the problem. Player Two checks. If they agree, this becomes a ‘completed’ story card for Player One who then takes another story card. If Player One is not correct he must give Player Two the story card and return the fraction cards to the bottom of the pile.
Players take turns to either take a card from the pile or ask the other player for a fraction card they need. If declined, they pick one card up from the fraction pile and, if necessary, also discard one fraction card by placing it at the bottom of the pile. A player cannot have more than 6 fraction cards in their hand front of them. If a card is given by the other player, the asker can ask again or pick up. The winner is the player with the most completed story cards.
NB. There will be 7 fraction cards left over because some are ‘trick’ cards.

2. Have each student write at least three contextual addition or subtraction fraction problems (as in 1.i. above) for their partner to solve. Swap these and compare results.

Activty 6

Conclude the lesson by summarizing key learning on the class/group chart: Have students complete e-ako modules 7 and 7+, 8 and 8+, 9 and 9+ to consolidate learning of concepts developed in these lessons.

Extension: Have students independently complete MM8-9 - part 2, explanation of which fraction is larger. Have them pair share and discuss.