This unit applies students’ strengths in equal sharing to develop their fractions concepts.
- Equally partition a length into two, four, or eight parts using symmetry.
- Equally partition a length into three or five parts using estimation and iteration (copying of a unit).
- Name the result of equally sharing a number of objects among two, or four parties, extended to three, five and eight parties if possible.
- Use the language of fractions, particularly halves, thirds, quarters, fifths and eighths.
Usually fractions are introduced by partitioning a single whole into equal parts. While this approach has some merit it is now accepted in research circles that it restricts opportunities for students to integrate their whole number and fraction knowledge. The need for extension of the whole numbers system to include parts of wholes (fractions) arises in two types of situation, measurement and sharing, which involve division.
Measurement involves finding how many times a given units fits into a space. So 12 ÷ 4 = ? can be thought of as a question, “ How many times does four fit into 12?” This measurement with a unit is sometimes referred to as quotitive division. When the unit does not fit a whole number of times then fractions are needed. For example 15 ÷ 4 = ? can thought of as a question, “How many times does four go into 15?” The answer is 3¾ times.
This unit focuses on the sharing view of division, called partitive division. In this view a number of objects are equally shared among a number of parties. So 12 ÷ 4 = ? can be thought of as a question, “If 12 objects are shared equally among four people, how many objects does each person get?” When the sharing is not possible to complete with whole objects, equal parts of objects need to be created by partitioning. For example 15 ÷ 4 = ? can thought of as a question, “What is the share if 15 objects are equally shared among 4 people?” The answer is 3¾ objects. The advantages of using a partitive division approach are that whole numbers and fractions must be dealt with simultaneously, students create the parts that form the fractions, and the one (whole) is never lost. Equi-partitioning and iterating (copying) of a unit are critical understandings in the measurement process. This unit uses length as the attribute of choice, as opposed to regions like squares or circles, or other more complex attributes like mass, capacity or time. As length is the simplest attribute it allows students to attend to the equi-partitioning required and links to conceptual tools like the number line.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
- providing additional examples with whole number solutions before moving to fractions
- doing additional activities relating to reading and recording fractions, if required
- allowing students to continue to use the playdough worms if required, but encouraging those that are ready to solve problems without them.
The activities in this unit can be adapted to make them more interesting by adding contexts that are familiar to them. While the kiwi and worms context works well since the worms can be easily divided into fractions, you could easily change it to children and chocolate bars or rabbits and carrots, or encourage your students to come up with their own ideas.
You may like to make the play dough for this unit with your class.
This is the Plunket recipe for play dough. It lasts a long time if stored in an airtight container. Make sure you watch very young ones playing with this, as you don't want them to eat it due to the high salt content.
Prep time: 10 minutes
2 1/2 cups white flour
1/2 cup salt
2 tablespoons cream tartar
2 cups boiling water
2 tablespoons oil
couple drops of food colouring
Mix flour, with salt and cream tartar in a bowl.
Add boiling water and oil plus a couple drops of food colouring.
Mix quickly then knead on a floured board until firm.
Add extra flour if sticky.
Store in an airtight container.
The recipe involves the simple fraction one half and can be modified for larger or smaller amounts:
“How much of each ingredient would we need if we doubled the recipe?”
“How much of each ingredient would we need if we halved the recipe?”
The point of this session is to establish the single worm as a unit of measurement for length. Prepare a few play dough worms that are 12 centimetres long before the session.
- Discuss with your class what they know about kiwi, particularly what they like to eat. Worms are a delicacy for kiwi. New Zealand has 192 species of worms – believe it or not! Tell them this week they will feed kiwi with make-believe worms.
- Tell the students that you have some ready-made worms. Ask the students, “How long is a juicy worm?” Look for comparative descriptions like “As long as…” Check out the comparisons by putting a worm next to the object. Some students might want to measure the worms using a ruler. See if they recognise that the worm is 12 centimetres long, a fact that is potentially useful for later work.
- Ask the students to think about how long three worms would be if you put them end on end. Get them to show you with the gap between their two hands. Have one student lie on the ground with their hands apart while another measures the gaps with worms. “So if one kiwi ate three worms their meal would be this long.”
- Ask the students to make a single worm that is the same length as the ones you have. They may want to use rulers to do that or come up to check that their estimated length is correct then modify. Give them a list of things to measure using worms, e.g. “How long is your desk in worms? How tall are you in worms? Can you find something that is four worms long? How many worms can fit along the room?” Tell the students to record what they find out in their own way so they will remember their strategies and answers. Look for students to show you an understanding of the worm as a measurement unit by showing these principles:
- Units need to start at a baseline
- Units need to fit end-on-end with no gaps or overlaps
- Units must be the same size (length)
- In your class discussion ask the students to report various findings, highlighting the principles of measurement units above. Tell them that you want to make a worm ruler to check some of their answers. Ask, “How should we start?” Marking the baseline as zero is fundamental so if a student does not venture the idea start that way.
- Ask a child to place the first worm and ask “Where should you put the one mark?” Note that students count by pointing to the centre of the object so marking the endpoint is a departure from that practice. The endpoint idea is critical if fractions are be marked on the same ruler. Continue the worm ruler to 10 units. Ask the students to look for something that is four/six/ten worms long and get them to check with the rule.
- Put the ruler on display for students to use during the day, e.g. “How long is your shadow in worms?”, “How long is your step in worms?” Finally you might get the students to create a lot of worms to feed the kiwi tomorrow. Put the worms in airtight containers (e.g. old yoghurt pots) for safe keeping.
The focus of this session is on equal sharing. The problems have whole number answers so fractions are not needed unless students create more complex situations. Observe for the strategies students use. These strategies might be:
- One by one sharing of objects physically
- Imaged sharing of objects one by one
- Sharing in groups of two or more
- Use of number facts like 3 + 3 =6
- Begin with the story scenarios on Copymaster 1. Invite students to predict what the equal shares will be. The problems are designed to build on one another so look for students to make connections.
- Problems 1 and 2 are accessible by doubles facts though students may not recognise that. Encourage students to find non-counting strategies if they can, such as “With six worms they got three each. So with twelve worms they should get six each.” However, one by one sharing, or sharing in composites like twos, solves the problem and you should act the sharing out with play dough worms as required. You might interrupt a one-by-one sharing process to invite prediction, e.g. “You have given each kiwi three worms. How many will they have when you have shared the 12 worms?
- Problem Three invites the students to recognise that doubling the number of kiwi halves the share for each bird. If that idea is not forthcoming let the students solve the problem as they will and record the two results (Problems Two and Three). Ask the students if they see a pattern.
12 worms shared between 2 kiwi. They get 6 worms each.
12 worms shared among 4 kiwi. They get 3 worms each.
- Shared into three equal parts (Problem Four) is a considerable leap in difficulty as it does not allow use of doubles or symmetry without significant modification. So students may opt for equal sharing by ones. Ask, “Will they each get more or less than the four kiwi? Why? How many worms might they get each?” Some students may remember that the worms were 12 centimetres long so use their ruler to subdivide each worm equally.
- Send the students away to come up with their own kiwi sharing worms problems. Ask them to create an A3 sized problem complete with illustration and answer on the back. Tell them that you want to make a Kiwi Kai book of problems. Let the students use the play dough worms to solve their problem if they want.
- To complete the lesson you may solve a few chosen sharing problems together as a whole class. Make the student problems into a book or scan them to make a PowerPoint slide show.
The focus of this lesson is to create fraction shares where whole numbers of worms are inadequate to complete the sharing equally. If there are problems created by students from the previous day that have remainders you may like to use those problems rather than those on Copymaster 2.
- Start with Problem One that requires nine worms shared between two kiwi. When four worms are given to each kiwi there is one left over. Some students may suggest discarding that worm but you should point out that hunting is hard work. No kiwi just leaves a worm uneaten. Some students may suggest giving the whole worm to one kiwi but that creates unequal shares. Ask, “What could the Keri and Kenny do with the extra worm?” (cut it in half). “What would the share for each kiwi be then?” (four and one half worms). You might ask where four and one half worms comes to on the Session One ruler (equally spaced between 4 and 5). Record 4½ at that point. For each question above act out sharing the worms between two students named Keri and Kenny but encourage prediction at every step before acting.
- Let the students work in small groups with play dough worms to solve the other three problems in the Copymaster 2 collection of “tricky problems.” They will need plastic knives and a surface to cut them on, like pieces of card or paper plates. Make sure students record their solutions using pictures, words and symbols (if they can). Encourage them to look for patterns among the problems as they did in Session Two. For example, Problems 2-4 the dividend (starting amount) is eleven and the number of kiwi is varied. So look for students to realise that given 11 worms the more shares, the smaller the shares and the opposite with fewer shares.
- As a whole class discuss their solutions. Points to bring out are:
- Why do we need fractions for this problem? (ones are not adequate for the equal sharing)
- How do we write fractions as words and symbols?
- What do words like half, quarter, third, fifth, etc. mean? (indicate the number of equal splits of one whole)
- Why do the parts need to be equal?
- What do the numbers in fractions, like ½ and ¼ mean?
- How do we write “three quarters” and “two thirds”?
- What do the numbers mean in these symbols?
- See if the students can locate the equal shares for each problem on the number line from Session One. You may need to align the physical share of play dough worms from one group’s solution to find the correct endpoint. This physical enactment will help students see the measurement units that underpin the number line.
You might ask the students if they know other fractions and where those fractions live on the number line.
The purpose of this session is for students to use inverse thinking. So the lesson explores variation to what conditions can be given or be missing from a sharing scenario. Students are given the share for each kiwi and asked to recreate the total number of kiwi and the number of worms that were shared.
- Begin with a discussion about the first “really tricky problem” in Copymaster 3.
- There are many sharing scenarios that result in half shares. For example, two kiwi sharing one worm gives one half as the equal share. If this is the only possibility students suggest, ask “I wonder if there are other answers. What if there were ten kiwi? How many worms would there be?” You may need to act out the 5 worms shared among 10 kiwi scenario for students to believe it works. Record the possible answers as word stories:
1 worm shared between 2 kiwi. They get ½ of a worm each.
2 worms shared among 4 kiwi. They get ½ of a worm each.
3 worms shared among 6 kiwi. They get ½ of a worm each.
- Three examples are sufficient. Ask students to look for patterns and differences such as:
S: “The numbers keep changing except the one half.”
T: “Yes, but how do they change?”
S: “The next story would be 4 worms shared among 8 kiwi.”
T: “How do you know that? Does that work? Are the shares one half?”
- Depending on how well your students notice the pattern you might ask predictive questions such as,
If there were 12 kiwi, how many worms would there be to get one half each?
If there were 50 worms, how many kiwi would there be?
If students find the situations difficult you can always act out the sharing with students being hungry kiwi sharing play dough worms.
- Next let the students find solutions to the two other problems. They may need access to the play dough worms to reconstruct the ‘whole’ collection of worms. For example, knowing the share is three-quarters they may build up collections of three quarters until whole numbers of worms are found. So four lots of three quarters gives three whole worms so the scenario is 3 worms shared among 4 kiwi.
Look for signs of simple equivalence such as “Six quarters is the same as one and one half.” Also look for students who connect sharing with the symbols, e.g. 3 shared among 4 gives ¾. Feel free to suggest recording that finding as 3 ÷ 4 = ¾ if you feel students can understand the meaning of those symbols.
- For very capable students who complete the problems early ask for generalisation.
What has to be true about the number of kiwi and number of worms for the share to be …one half? …three quarters?