In this unit students develop important reference points or benchmarks for zero, one half and one. They use these benchmarks to help compare the relative sizes of fractions, through estimating, ordering and placing fractions on a number line.
This unit builds on the following key conceptual understandings about fractional parts.
An understanding of fractional parts supports students to develop sense for the size of fractions. This unit helps students develop an intuitive feel for zero, onehalf and one, as useful benchmarks for ordering fractions. Understanding why a fraction is close to zero, one half or one helps students develop a number sense for fractions.
Students can be supported through the learning opportunities in this unit by differentiating the complexity of the tasks, and by adapting the contexts. Ways to support students include:
The contexts for this unit are purely mathematical but can be adapted to suit the interests and cultural backgrounds of your students. 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 in collaborative settings. Students might also appreciate challenges introduced through competitive games or through stories.
Session 1
In this session students begin to develop benchmarks for zero, half and one.
Session 2
In this session the students continue to develop their sense of the size of fractions in relation to the benchmarks of zero, half and one by coming up with fractions rather than sorting them.
Session 3
In this session students estimate the size of fractions.
Session 4
In this activity students identify which fraction of a pair is larger. The comparisons rely on an understanding of the top and bottom number (numerator and denominator) in fractions and on the relative sizes of the fractional parts. Equivalent fractions are not directly introduced in this unit but if they are mentioned by students they should be discussed.
Session 5
In this session the students draw on their conceptual understanding of fraction benchmarks (0, 1/2, 1) and their understanding of the relative sizes of fractional parts to line fractions up on a number line.
Family and whānau,
This week we are learning about where fractions live on number lines. We have been especially interested in fractions that are close to zero, close to one half and close to one. Ask your child to put the following fractions on a number line and explain to you how they made up their mind about where to place them. Try to find some more fractions in newspapers or magazines and place them on the number line too.
1/35 2/99 44/85 99/100 25/60 3/25 77/80
This unit explores the relationships between fractions and decimals. A decimat model is used to explore how fractions arise from equal sharing and how the decimal place value system is a restricted form of equal sharing. The main objective is to link students’ knowledge of fractions with the decimal system.
This unit builds on the idea that the need for fractions and decimals arises from division situations in which ones (wholes) do not give an adequate degree of precision. Lack of closure of whole number under the operation of division creates the need for rational numbers. The division situations can be either partitive (sharing) or quotative (measuring). In this unit sharing of a decimat model is used to connect fractions and decimals.
A key idea is that decimals are a restricted form of equivalent fractions. For example, three quarters has decimal representation of 0.75 because 3/4 = 75/100. As with whole numbers the place values in decimals are connected although separate columns are used to write numbers. For example, 0.75 has can be expressed in different decimal forms, such as 7 tenths and 5 hundredths, 75 hundredths, 750 thousandths, 7.5 tenths, etc. Flexibility in the way students think about decimal place value supports their fluency with calculation. Central to fluency is students’ understanding of how decimal place value units can by partitioned and combined. For example, 2.3 – 0.7 requires a one in 2.3 to be partitioned into 10 tenths if subtraction is used, or 10 tenths to be combined to form a one if adding on from 0.7 is used.
The learning opportunities in this unit can be differentiated by providing or removing support to students, by varying the task requirements. Ways to support students include:
The context for this unit is chocolate bars. Students usually find the context engaging. Sharing food is a common practice across cultures worldwide. Investigating the origins of chocolate will lead to finding out about Aztec culture from Central America. However, using foods as a context is sometimes not appropriate for students. In that case change the story to a different situation about area, such as land (kumara patches), precious sheets of gold, or tablets of clay (as in an Indiana Jones film). Vary the context while retaining the important decimat model.
Students would benefit from previous experience with fractions, particularly using partitioning of areas to form equal parts and the naming of collections of those as nonunit fractions. This unit uses an area model, the decimat, that is based on tens frames used for whole number place value. It is expected that students will understand whole number place value in a flexible way. They are expected to see place value as a nested system, in that, place value units are nested in others. For example, 230 can be renamed as 23 tens. It is also expected that students will have a range of strategies for solving addition and subtraction problems with whole numbers, that include using standard place value (hundreds, tens and ones, etc.) and tidy numbers (rounding and compensating).
The aim of this session is to connect addition and subtraction of decimals with the place value understanding students have built up over the previous three lessons.
Dear parents and caregivers
This week we are beginning work on decimals. We will be using a paper model of a Breakway Chocolate Bar. Don’t worry we won’t be eating lots of real chocolate. The simplest chocolate bar is divided into ten equal parts like this:
By snapping the bar along the dark lines, the bar can be shared equally among two, five or ten people. The shares can be written as decimals, e.g. one fifth is two tenths or 0.2. With this model we will explore the relationships between simple fractions and decimals. By the end of the week we may even be adding and subtracting decimals.
In this unit students work with growing patterns made from square tiles. Students represent the relationships between pattern number and number of tiles using tables, graphs and rules, in order to predict further terms of the pattern.
A linear number pattern is a sequence of numbers for which the difference between consecutive terms is always the same. If plotted on a number plane the graph of a linear pattern is a straight line.
A progression in the way students process linear patterns is well established in the research. That progression is as follows:
Click to download a PDF with further information.
This unit is aimed at achievement of Level 3 in The New Zealand Curriculum, which requires students to develop recursive rules. Level 3 involves progression from phase 2 to phase 3 of the above progression.
For students that need to consolidate understanding of the first two phases, access to materials and restrictions on the terms for prediction makes the level of challenge appropriate. Asking for the number of items in the tenth member of a pattern is a good guideline.
Students progressing to the third phase need support in representing the relationship between term numbers and the number of items, using tables, graphs and diagrams.
To extend students to progress to the fourth phase (direct rules), tasks need to push beyond terms that are easily found using recursive rules. Asking for the number of items for the twentieth, fiftieth, or hundredth term is a good guideline.
The unit is based around patterns with square tiles, which is relatively context neutral. It may be that situations from real life motivate your learners. The patterns could be contextualised as buildings made of sections, stone paintings (kohutu peita), planting of trees, or fruit ripening in a tray. You might like to discuss situations in which everyday patterns grow in consistent way, such as saving the same amount of money each week, stack of items in the supermarket, shoes related to the number of people, or chairs on a bus or aircraft.
The unit begins by looking at the growth patterns of even and odd numbers. It is important that students ‘see’ even numbers as multiples of two, and odd numbers as multiples of two plus or minus one. The lesson also looks at generalisations about what happens when even and odd numbers are added.
Term  1  2  3  4  10 
Number of tiles 
2  4  6  8  ? 
For students who develop the rule quickly provide these challenges:
In this session students use a spatial pattern made with square tiles to investigate how relationships that exhibit constant difference are represented with graphs.
● Not yet ? Maybe P Yes

Continue a linear growth pattern from a few terms. 
Make a table of values. 
Draw a graph. 
Use a table or graph to find a term in the pattern. 
Create a rule for finding any term in the pattern. 
Annie 
P 
P 
P 
P 
P 
Tariq 
P 
P 
P 
P 
P 
Tipene 
? 
? 
? 
P 
P 
Veyun 
● 
● 
● 
P 
P 
Sione 
● 
● 
? 
P 
? 
In this session you differentiate the class into two groups, those that feel they need more help with patterns, and those who think they can attempt a challenging pattern investigation independently.
Copymaster Five provides a task that can be used to assess your students. They will need access to a calculator. You might also provide the students with square grid paper to make sketching the yacht pattern easier.
Let your students work independently and use the data to check their achievement against the criteria in Copymaster Three. Students might exchange worksheets so you can mark the task collectively.
Dear parents and whānau,
This week in algebra we have been looking at patterns made with squares tiles and how these patterns can be continued. We have looked at different ways to predict how the patterns continue. A recursive rule tells you how to go from the number of squares in one figure to the number of squares in the next. A general rule is a rule that gives you the number of squares for any term in the pattern.
In this unit we will explore the idea of having benchmarks of 1 litre and ½ litre or 500 millilitres, to aid in estimating the volume of given objects.
Volume is the measure of space taken up by a threedimensional object. The space within a container is known as its capacity but as the thickness of many containers is negligible, it has become acceptable to refer to the space inside a container as volume too.
Students need to develop personal measurement benchmarks. A benchmark is an understanding or a “feel” for the size of a measurement unit, which is useful when working with measures in daily life. Often these benchmarks are linked to familiar items such as a one litre milk bottle or a Pyrex jug.
This unit supports students to develop personal benchmarks for a litre and ½ litre, and also strengthens students’ understandings of the relationship between litres and millilitres.
The learning opportunities in this unit can be differentiated by altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:
This unit is focussed on measuring the volume of containers. Use a range of containers that are familiar to your students to encourage engagement.
For this session you will need plenty of bottles and containers of a range of sizes, including several that hold 1 litre. Fruit juice bottles, shampoo bottles, and yoghurt containers are particularly good containers for this task. You could either ask students to bring bottles and containers to school with them or collect them yourself.
The following activities are to provide students with experiences to compare volumes of different objects and to create a benchmark for a container that holds one litre.
In this session students compare their benchmarks for one litre and try to estimate one litre.
In this session students work with volume as the amount of space that an object takes up.
Family and whānau,
At the start of this week we would like each child to bring a plastic container or empty bottle to school. We are collecting as many different sizes and shapes of containers as we can. During the week draw your child's attention to the labels of containers around your home and ask them to tell you how many litres or millilitres there are in each.
In this unit students build on previous experiences with litres and millilitres. Work is carried out in the context of planning a party with students measuring volumes accurately as part of the planning process.
When students can measure areas effectively using nonstandard units, they are ready to move to the use of standard units. The motivation for moving to this stage, often follows from experiences where the students have used different nonstandard units for the same volume. This allows them to appreciate that consistency in the units used allows for easier and more accurate communication.
The usual sequence used in primary school is to introduce the litre as a measurement of volume before using cubic centimetres and cubic metres.
Students’ measurement experiences must enable them to:
Students also need to be able to read a scale to be able to measure volume accurately.
The standard units can be made meaningful by looking at the volumes of everyday objects. For example, the litre milk carton, the 2litre icecream container and the 100millilitre yoghurt pottle.
The learning opportunities in this unit can be differentiated by varying the scaffolding provided to make the learning opportunities accessible to a range of learners. Ways to support students include:
The context for this unit can be adapted to suit the interests and experiences of your students. The unit begins with a discussion of students’ experiences of birthday parties. Following this discussion you could work with the students to adapt the activities at the stations to reflect their experiences. Alternatively, you may like to choose a social gathering other than birthday parties as the context for the measuring tasks. For example, the school disco, or a morning tea for whānau.
This unit uses the context of birthday parties. Begin by discussing students' experiences at birthday parties and compare differences between families. Explain that this week they will be working at different stations to help prepare for a party. In each of these stations they will need to measure volume accurately. Points that may need to be discussed as work progresses include:
In this station students accurately measure the volume of a variety of different drinking glasses.
Student instructions (Copymaster 1)
In this station you need to estimate and measure the volume of different glasses for drinks at the party.
Which glass do you think will hold the most?
Which will hold the least?
Which glasses will hold a similar amount?
Compare your results with your estimates. How close were your estimates?
Which glass held the most?In this station students accurately measure the volume of a variety of bowls that could be used to make jelly.
Student instructions (Copymaster 2)
In this station you need to estimate and measure the volume of different bowls used to make jelly for the party.
Which bowl do you think will hold the most?
Which will hold the least?
Which bowls will hold a similar amount?
Which bowl held the most?
Which held the least?
Which bowls held a similar amount?
If each packet of jelly made 1 litre how many packets would be needed for each bowl?
How many packets would be needed for all the bowls?
In this station students make baskets to hold lollies and measure the volume of the baskets they have made.
Student Instructions (Copymaster 3)
In this station you will make baskets to hold lollies for the party and measure the volume of the baskets you have made. Can you make three baskets that hold different amounts?
Which basket held the most?
Which held the least?
In this station students measure the volume of a variety of cake tins and predict which recipe would be best to use for each tin.
Student Instructions (Copymaster 4)
In this station you need to measure the volume of the different cake tins, then decide which recipe mix would be best for each tin. Remember that the cakes will rise when they are cooked!
Which tin held the most?
Which held the least?
Recipes
Absurdly Easy Chocolate Cake
Ingredients
3 cups flour (750 mL)
2 cups sugar (500 mL)
6 tablespoons cocoa (90 mL)
2 teaspoons baking soda (10 mL)
1 teaspoon salt (5 mL)
3/4 cup vegetable oil (190 mL)
2 tablespoon vinegar (30 mL)
2 teaspoon vanilla (10 mL)
2 cup cold water (500 mL)
Directions
Mix the dry ingredients. Add the wet ingredients. Stir until smooth. Bake at 180ºC for at least 30 minutes.
One Mix Chocolate Cake
Ingredients
1 cup self raising flour (250 mL)
1 cup sugar (250 mL)
50 grams melted butter (50 mL)
1/2 cup milk (125 mL)
2 eggs
2 Tbsp. cocoa (30 mL)
1 tsp. vanilla (5 mL)
Method
Mix all ingredients together in a large bowl with a wooden spoon. Bake at 180ºC for about 30 minutes.
Daisy’s Easy Chocolate Cake
Ingredients
1 1/2 cups sugar (375 mL)
1 cup cold water (250 mL)
125g butter (125 mL)
2 Tablespoons cocoa (30 mL)
1/2 teaspoon baking soda (2.5 mL)
2 eggs, well beaten
1 1/2 cups selfraising flour (375mL)
Method
Put sugar, water, butter, cocoa and soda into a large pot.
Stir over low heat until butter has melted, then bring to the boil.
Simmer for 5 minutes and remove from heat.
When mixture has cooled, stir in beaten eggs. Sift in the flour and beat well.
Bake at 180°c for 5060 minutes.
In this station students calculate and measure the volume of sauce needed for cheerios at the party.
Student Instructions (Copymaster 5)
Family and whānau,
This week at school we are estimating and measuring volumes using litres and millilitres. Please help find a container in your home with an estimated volume of 600 ml. Please send it to school so we can measure its volume accurately.
The purpose of this series of lessons is to develop understanding of the connection between division and fractions. In the unit both types of division, sharing and measurement are explored to establish a need for fractions and generalisations about division and fractions.
Students need to understand and use the appropriate mathematical language for the numbers and symbols in division equations. Students understand and apply addend plus addend equals the sum for addition, and factor (multiplier) times factor (multiplicand) equals product for multiplication. Mathematical language for division allow for clear definition of the symbols, i.e. the meaning of those symbols, and allows for succinct expression of generalisations. In division, the dividend is partitioned by the divisor and this results in the quotient.
a (dividend) ÷ b (divisor) = c (quotient)
Division models the operation of equal partitioning in two different, but connected, types of 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. The unknown in measurement division, is the number of those units that can be made from a given amount. This interpretation is often associated with repeated subtraction, as one way to solve this kind of problem is to keep removing the given equal groups (measures) from the whole amount, until nothing is left. Counting each repeated subtraction gives the solution to the question.
Measurement division situations are the easiest contexts for division by fractions. Problems with division by a fraction involve finding how units of a given (fractional size) ‘fit within’ another fraction quantity. For example, 1 1/2 ÷ 1/4 is interpreted as how many units of one quarter measure 1 1/2?
When students carry out the operation of division with whole numbers, their expectation is that the quotient will be smaller than the dividend, for example, 20 ÷ 2 = 10, and sometimes smaller than both the dividend and the divisor, for example, 20 ÷ 5 = 4. It is a conceptual shift for students to come to understand that when they are dividing a fraction by a fraction the quotient may be larger than both the dividend and the divisor, for example 1/2 ÷ 1/4 = 2.
There are many reallife occasions when we divide fractions by fractions, however, the fractions are frequently presented as decimals in measurement situations. For example, if you pay $15.00 for three quarters of a kilo of prime lamb and you want to know the price per kilo, that will involve your dividing a fraction by a fraction, i.e. 15 ÷ 3/4 =$20. Providing simple fractional problems in context, and asking students to consider and create their own contexts, is helpful to their connection of mathematics to the real world.
Links to the Number Framework
Stages 7  8
This unit supports and complements the learning activities in Book 7 Teaching Fractions, Decimals and Percentages.
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:
Task can be varied in many ways including:
The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. 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 in collaborative settings. Be conscious that fair sharing in families is not necessarily equal, e.g. bigger people get more food.
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 on the board or in students’ books it is recommended that you use a horizontal vinculum.
Activity 1
Activity 2
In this session students explore division as sharing into equal parts. They learn that sharing situations can also be answered using measurement division.
Activity 1
Activity 2
In this session division as equal sharing is applied to problems with continuous whole, rather than the sets situations used in the previous two sessions. Students come to understand that quotients can sometimes be fractions, the result of division in which ones (wholes) are not sufficiently accurate.
Problem 
dividend 
divisor 
quotient 
I have this much pizza (4½). I want to share the pizza equally among the three people at my party. How much pizza should each person get? 



Problem 
dividend 
divisor 
quotient 
I have 4½ metres of cloth. Each bag is made from ¾ of one metre of cloth. How many bags can I make? 
4 ½ 
3/4 
6 
In this session students explore division where the dividend is a whole number and the divisor is a fraction, e.g. 1 ÷ 2/5 = 5/2 = 2 ½. Fraction strips (See Copymaster Three) are used as the physical representation since length is the simplest attribute for students to use. The problems involve measurement division since a fractional number of shares makes no sense. For example, 1 ÷ 2/5 means “How many quantities of two fifths equal one?”
Dear parents and whānau,
We have been exploring the connection between division and fractions in class.
You might enjoy solving these two problems together to see what they have been learning.
Problem One: Four people equally share three doughnuts. How much of one doughnut does each person get?
Problem Two: If each person needs three eighths of a big pizza, how many people can be fed with one and one half pizzas?
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