The purpose of this unit is to introduce fractions as answers to sharing division (partitive division) problems. Students are expected to generalise the relationship between the dividend and divisor in division and the fraction quotient, for example 3 ÷ 4 = 3/4.
- Apply the understanding that fractions can be quotients (or the result of division), for example, 3 ÷ 5 = 3/5.
- Represent sharing division using diagrams, words, and symbols.
- Anticipate the results of a sharing situation using knowledge of the dividend and divisor, that is, a ÷ b = a/b (the idea not the algebra).
- Recognise when two, or more, different sharing situations produce equivalent shares, for example, 3 ÷ 4 = 6 ÷ 4.
- Choose appropriate dividends and divisors to get a fraction share, for example, How many pirates share 10 gold bars if the equal shares are two thirds of a bar?
Using appropriate mathematical language for the numbers and symbols in division equations is an important learning intention. Mathematical language for division allows students to clearly describe the meaning of the symbols, and succinctly express generalisations. In division, the dividend is partitioned by the divisor and this results in the quotient.
dividend ÷ divisor = quotient
for example, 15 (dividend) ÷ 3 (divisor) = 5 (quotient).
Division models the operation of equal partitioning in two different, but connected, situations. The first situation is sharing or partitive division, which often involves answering the question, “If a objects are equally shared among b parties, how many objects does each party get?” The second form of division is the measurement interpretation (sometimes referred to as quotative division). Here the number in the group, or size of each measure is known. That group or quantity becomes the unit of measure. For example, “How many fours can be made from 15?” is a measurement division problem.
This unit is entirely devoted to division as equal sharing. In general, it is always true that the dividend and divisor become the numerator and the denominator of the quotient, for example, 3 ÷ 4 = 3/4. In context, this might answer a problem like, “Four cows share three bales of hay equally. How much of one bale does each cow get?” The answer equals three quarters of one hay bale.
Students can be supported through the learning opportunities in this unit by differentiating the nature and complexity of the tasks, and by adapting the contexts. Ways to support students include:
- Providing a physical model, particularly continuous objects (paper strips and shapes, plasticine or play dough), so students can represent relationships and think with those relationships.
- Modelling how to represent division problems with fractions diagrammatically and with equations. Equal sharing diagrams are important in this unit.
- Use mathematical vocabulary to identify common features in situations and the equations that represent those situations. Use words like fraction, numerator, denominator, dividend, divisor, and quotient.
- Seek generalisation of properties of operations from whole numbers, and apply the generalisations to fractions. One interpretation of 12 ÷ 3 = 4 is sharing 12 objects equally among three students. Connect whole number contexts to the more challenging but similar situations in this unit.
- Encouraging students to work collaboratively, and share their ideas. Work hard on establishing norms that encourage risk taking and valuing the ideas of others.
Tasks can be varied in many ways including:
- Alter the complexity of the problems by simplifying the difficulty of the fractions and whole numbers that are used. Begin with fractions, such as halves, quarters, thirds, and fifths that students may be most familiar with. Limit the dividends and divisors at first, moving to more difficult numbers as strategies become efficient.
- Provide access to physical representations so students can make sense of the actions in a situation, alongside using symbols.
The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. This unit uses two situations, children sharing trampoline time and pirates sharing gold. If these contexts are not meaningful and appropriate to your students then find similar contexts that will work, for example, sharing laptop time. Fractions can be applied to a wide variety of problem contexts, including making and sharing food, constructing items, travelling distances, working with money, and sharing earnings. The concept of equal shares and measures is common to collaborative settings. Be conscious that fair sharing in families is not necessarily equal, for example, bigger people get more food, and discuss the differences between the real world and the world of mathematics.
Note that most of the fractions in these teachers’ notes are displayed as typed text, meaning that the vinculum (the line between the numerator and the denominator) is diagonal rather than horizontal When writing fractions for students it is recommended that you use a horizontal vinculum.
In this session students explore sharing division among two, four, and eighth parties. The divisors are constrained to make equally partitioning easier and allow students to attend to equal shares and naming the shares as fractions.
- Motivate students about the first context, trampoline time!
Has anyone in the class been on a trampoline?
What is it like?
Is it good to have a trampoline for your whānau? Why?
- Use PowerPoint One to introduce the scenario of equally sharing trampoline time. It is worthwhile discussing equal shares since the size of shares is seldom equal in real family situations.
- After Slide Five, ask students to discuss “Three hours shared between two people.” Encourage them to act out the process of sharing, or draw diagrams, to justify their solutions. Two strategies are likely:
- Give each person one whole hour and spilt the last hour into halves.
- Split each hour into two half hours and give each person one half from each hour.
- Slide Six shows an animation of both strategies. There is also the possibility that students might convert 3 hours to 180 minutes and use whole number division.
Are the shares for each person the same, no matter which strategy you use?
Students should agree that 1 1/2 hours equals three lots of 1/2 an hour (3/2 hours). Be aware that some students may believe that fractions are always less than one, and not be aware that a fraction is a collection of counts of unit fractions, for example, 3/2 equals 1/2 + 1/2 + 1/2.
Record the problem using the equation 3 ÷ 2 = 3/2 or 1 1/2. Students might suggest that three times of half an hour will be better practically than one and one half hours straight.
- Use Slide Seven to present the problem of three hours shared equally among four people.
What is the equation for this situation? (3 ÷ 4 = [ ])
- Ask students to solve the problem in pairs or threes. Provide copies of Copymaster One and scissors so students can act out the equal sharing. Roam to see how students approach the problem. Expect strategies like:
- Cutting two hours (circles) into halves, giving each person half an hour, cutting the remaining hour into quarters, and allocating each person one quarter of an hour. Each share is one half and one quarter.
- Cutting each hour into quarters and giving each person one quarter from each hour. Each share is three quarters.
- Halving three hours as per the two person scenario, then halving the result, one and one half, to get equal shares for four people. Each share is one half and one quarter.
- Discuss whether or not one half and one quarter is the same amount as three quarters.
- Provide your students with Copymaster Two to work from in pairs or three. As students work roam the room to look for the following:
- Are students able to equally partition the circles?
- Can they determine the equal share for each child?
- Can they name the amount of time using fractions?
- Do the anticipate the equal share before physically partitioning the circles?
- After a suitable time gather the class to share answers. Record the answers given in the form of equations such as:
3 ÷ 3 = 1 or 3/3 3 ÷ 5 = 3/5 or 1/2 + 1/10 3 ÷ 8 = 3/8 or 1/4 + 1/8
4 ÷ 3 = 1 1/3 or 4/3 2 ÷ 5 = 2/5 5 ÷ 4 = 5/4 or 1 + 1/4
- Discuss the meaning of the symbols, particularly the division sign as “shared equally among”.
Is there any pattern in these answers? It would be good to know the equal share before we cut up the hours.
Students may notice that for some equations the dividend becomes the numerator in the fraction answer and the divisor becomes the denominator. For example, in 3 ÷ 8 = 3/8 three becomes the numerator and eight becomes the denominator.
- Act out sharing five hours equally among two children. Cut out five circles from Copymaster One. Stack the circles like pancakes, cut them into halves, then give the stacks of halves to two students.
How much time has each person got? Give you answer in fractions.
Students might explain that since there were five hours to start, each person will get five halves.
How do I write that as an equation? (5 ÷ 2 = 5/2)
Is there another name for five halves? (2 ½)
- Work through two other examples using the ‘pancake technique’ so students have opportunity to generalise.
For example, Seven hours shared equally among four children, and Five hours shared equally among eight children.
In this session students explore variations to numbers within the sharing situations. For example, “What is the effect on the quotient when there are twice as many people?” Students should find out that the shares of time are half as big as previously. Varying the dividend (number of hours) and divisor (number of people) gives students an opportunity to build on their anticipation of a ÷ b = a/b.
- Introduce the following problem:
Let’s recap. Previously you worked out the equal shares if two children shared five hours of trampoline time.
What do you expect the share to be?
Can someone show us the shares using these paper circles? (Made from Copymaster One).
- Act the problem out with class members and paper circles. Record 5 ÷ 2 = 5/2 = 2 1/2 and ask students about the meaning of the symbols.
- Imagine that two more children join the whānau. There are four children now.
How much time would each person get then?
- Ask students to work out the problem in pairs or threes. Have paper circles and scissors available.
- Gather the class to share and justify their solutions. Record 5 ÷ 4 = 5/4 = 1 1/4.
Have students used the theorem a ÷ b = a/b?
Have students recognised that the equal shares are half the previous shares?
- Draw students’ attention to the equations, written one under the other.
5 ÷ 2 = 5/2 = 2 1/2
5 ÷ 4 = 5/4 = 1 1/4
What do you notice? (Twice as many people results in half the share)
- Extend the problem to five hours shared among eight children. (5 ÷ 8 = 5/8)
What has happened to the shares this time? Why has this happened?
Predict the shares for 16 children sharing five hours equally. (5 ÷ 16 = 5/16)
- Provide students with Copymaster Three to work on in small collaborative teams.
- Roam the room to look for the following:
- Are students anticipating the equal shares based on a ÷ b = a/b or are they enacting the sharing with materials or diagrams?
- Do they notice equivalence? For example, 8 ÷ 12 = 8/12 = 2/3 and 8 ÷ 24 = 8/24 = 1/3
- Do they generalise the effect of doubling the dividend or the divisor?
- Gather the class to process the answers and bring out the points above. Looking for patterns in the equations, connected to the meaning of the symbols in context, is important. For example:
6 ÷ 4 = 6/4
3 ÷ 4 = 3/4
1 ½ ÷ 4 = 1 ½/4 = 3/8
In this example, halving the available time halves the equal shares.
- Encourage the students to generalise:
What happens if the hours are the same, but the number of people doubles?
What happens if the hours are the doubles, but the number of people stays the same?
What happens if you double both the number of hours and the number of people? (This results in equivalent amounts)
- Provide the students with an assessment task to attempt individually. Read the questions aloud before students begin and support students who find reading the text difficult. Their work samples will establish who has made generalisations about equal sharing situations. Use PowerPoint Two as the stimulus for the task.
In this session the context of dividing up time is changed to pirates sharing gold bars. The purpose of this change is to see if students can recognise similar structure in the problems, irrespective of the representation.
- Use the first three of slides of PowerPoint Three to introduce the topic of pirates. Many students will have seen the Pirates of the Caribbean films. You could show a trailer of one film as motivation, locate the Caribbean on a map or globe, and discuss the business of pirates. Theft!
- Present the scenario on Slide Four. Read the story.
How heavy are the bars? (1000g or 1kg)
Is Rewa right about the shares?
How could the pirates get equal shares?
You might suggest that pirates can cut a gold bar with a swipe of their cutlass. Gold is a soft metal, but very heavy.
- Before there is a massive fight among the pirates, please work out with a partner, how much an equal share is for each person. Please draw a picture to show how you shared the loot.
- Watch as students work. Look for recognition that the pirate situation is alike the trampoline scenarios from the previous session.
Can students name the equal shares they create using fractions?
- Gather the class to share their solutions. Ask questions about ensuring the parts are equal, naming the parts, and working out how much of a gold bar each pirate gets.
- Use Slides Five, Six, and Seven, if necessary, to discuss possible strategies. Ideally student have used similar strategies, and you can say “[Student], you used a strategy like this. Please explain it.”
- Slide Five: Whole bars are shared first, then halves of bars, then thirds of halves. The total for each pirate equals 2 + 1/2 + 1/6 (This amount is the same as 2 2/3).
- Slide Six: Whole bars are shared first, then the remaining two bars are partitioned into thirds. The amount for each pirate equals 2 2/3.
- Slide Seven: Each bar is cut into thirds and a pirate receives one third from each bar. The total amount for each pirate equals 8/3 (This amount is the same as 2 2/3).
How can we record this operation as an equation? (8 ÷ 3 = 8/3 = 2 2/3)
- Use Slides Eight and Nine to present other pirate sharing scenarios. Ask students to work in pairs or threes, and to record their solution strategies. Encourage high achievers to find more than one way to solve the problems and compare the shares they get from each method. That will encourage them to rename the amounts.
Watch out for students who transfer the generalisation, a shared among b equals a/b to the pirate situation. Some students may get confused by naming the equal shares. For example, to share three bars among five pirates they might partition each bar into fifths and give each pirate three pieces. However, they might name the amount as 3/15 rather than 3/5. Remind those students that the shares need to be expressed as “How much of one whole bar does each pirate get?”
- Play the game Pirate Shares with students. Here are the rules:
To be played in fours.
You need: Three standard 1-6 dice with 1 changed to 8 (Use a sticker or tape over that face), counters, one board per player (Copymaster Four).
Players take turns to:
- Roll all three dice.
- Select one number to be the dividend (number of gold bars) and another number to be the divisor (number of pirates). Both numbers can be the same if they show on different dice.
- Place the dice numbers in the appropriate square on the top right of the board.
- Calculate the equal share and that quotient on their board.
Note that players need to be strategic in their choice of dice numbers and can use equivalence if they see it. For example, 6 ÷ 8 can be used to cover 6/8 and 3/4.
The first player to cover all nine numbers on their board wins the game.
Some students will need physical materials to support them to play the game, but the aim is for students to anticipate equal shares without needing materials.
- Gather the class after an appropriate time to discuss the following:
How do you work out the equal shares easily?
Which fractions are hardest to get? Why?
Which fractions are easiest to get? Why?
Is there a good strategy for winning this game?
In this session students continue with the pirates and gold scenario. They explore the idea of fairness and equivalence in sharing scenarios.
- Introduce the first problem using Slides 1-3 of PowerPoint Four. The problem is about determining the size of the divisor to get equal shares of one half. Be aware that one half is a very ‘intuitive’ fraction so using it as the quotient is designed to keep the problem simple. Students should find the following:
Santa Maria: 5 ÷ □ = ½ so □ = 10 (Ten pirates are needed)
Santa Katarina: 7 ÷ □ = ½ so □ = 14 (14 pirates are needed)
Santa Barbara: 3 ÷ □ = ½ so □ = 6 (Six pirates are needed)
Santa Henrietta: 12 ÷ □ = ½ so □ = 24 (24 pirates are needed)
- Let students solve the problem in pairs or threes and share their solutions as a class.
Students are likely to find the answers using multiplication, for example, How many lots of one half make three? For Santa Barbara.
Recording the calculations as division is important since many students will believe that division can only involve dividing a larger number by a smaller number.
- Use Slides 4-5 to introduce a problem where the equal shares are three quarters of a bar per pirate. Encourage students to use diagrams to find the number of pirates needed rather than reason from the numbers.
- Slide 6 shows a diagram of how 6 ÷ □ = 3/4 so □ = 8 can be worked out (Santa Aroha). Use that slide only if students do not create viable strategies.
Roam the room to look for:
Do students recognise that the number of pirates must be greater than the number of bars? Why?
Do they recognise that from three bars, four pirates get an equal share? (A ratio of 3 bars:4 pirates)
Do they see that sharing five bars will be problematic?
- Gather the class to share their solutions. Record division equations to represent the answers:
Santa Arabella: 9 ÷ □ = 3/4 so □ = 12 (12 pirates are needed)
Santa Rosa: 12 ÷ □ = 3/4 so □ = 16 (16 pirates are needed)
Santa Melissa: 3 ÷ □ = 3/4 so □ = 4 (Four pirates are needed)
Santa Aroha: 6 ÷ □ = 3/4 so □ = 8 (Eight pirates are needed)
Santa Mika: 5 ÷ □ = 3/4 so □ = ? (? pirates are needed) Leave this unsolved.
Why do we need twice as many pirates for the Santa Rosa as for Santa Aroha?
Why do we need half as many pirates for the Santa Melissa as for Santa Aroha?
How many pirates do we need to attack Santa Mika?
Students might note that using 7 pirates requires 7 x ¾ = 5 ¼ bars but using 6 pirates requires 6 x ¾ = 4 ½ bars.
- Use Slides 7-9 to present the next problem in the series. Two thirds are more challenging to work with than three quarters. Use Slide 9 only if students need support to represent the problem. Roam the room to look for:
Do students recognise that from each two bars, three pirates get an equal share? (A ratio of 2 bars:3 pirates)
Do they see that sharing 11 bars will be problematic?
Do they suggest a solution?
- Gather the class to process their solutions and represent each problem as an equation.
Is there an easy way to work out Captain Pegleg’s problems? He may need to change the share again.
Some students may notice that the ratio for three quarters of bar per pirate was 3 bars: 4 pirates. Similarly, for two thirds of a bar per pirate the ratio was 2 bars:3 pirates.
- If Captain Pegleg changed the share per pirate to three fifths of a bar, what would happen? (4 bars would provide for 5 pirates)
- Change the share per pirate a few times and see if students can find a generalisation. Diagrams like those on Slides 6 and 9 will support students if necessary.
This session provides students with a chance to demonstrate their understanding of fractions as quotients. It is an opportunity for you to assess their progress.
- Provide each student with Copymaster Five. Ask them to solve each problem independently and to record as much as they can. Recording will help you to see what they have learned.
- As students finish pair up students to share their answers. When all students have sufficient time gather the class to process the answers. It is important that the strategies to solve each problem are discussed so the experience contributes to learning rather than just assessment.
- Answers are as follows:
- Q1: 2 ÷ 5 = 2/5 (Ideally students use a ÷ b = a/b rather than solve the problem by drawing)
- Q2: As above
- Q3: 7 ÷ 3 = 7/3 = 2 1/3 (Look for students to express the equal shares as improper fraction and mixed number)
- Q4: As above
- Q5: 2 ÷ 3 = 2/3 and 4 ÷ 5 = 4/5. Since 4/5 is greater than 2/3 the five pirates each get a greater share. Note that students may solve this problem in other ways, probably using ratio. For example, 2 ÷ 3 is the same equal share as 4 ÷ 6. This must be less than 4 ÷ 5 since there are more pirates sharing the same number of gold bars.
- Q6: 15 ÷ □ = 3/4. Each three bars give four pirates an equal share of three quarters. Since there are five threes in 15, 5 x 4 = 20 pirates are needed.
- Q7: As above
- Q8: 5 ÷ 8 = 5/8 (Ideally students use a ÷ b = a/b rather than solve the problem by drawing). Do students see the same division structure in this problem when the context is changed?
- Q9: □ ÷ 10 = 3/5. Since the shares are three fifths of a bar for each pirate three bars will provide for five pirates. Since there are ten pirates, and two times five equals ten, there must be 2 x 3 = 6 bars. Do students draw their solutions or reason from the numbers?
- Choose a few examples of problems composed by students (Question 10). Solve those problems together as a class.
- Play a game of Pirate Shares with the students to finish the unit. Discuss how the game might be changed to make it easier or harder to play.
Can students create a harder version of the same game?