The following six stations provide a range of problems for students to apply and interpret measurement of mass. Consider what would be the most effective method for introducing these to your class. You could work on them as a whole class and provide support to groups of students. Alternatively, you could use another relevant igniting activity to introduce the context for learning, before directing students to work on one or more of these stations independently, or in small groups. These stations could serve as the basis for learning in different sessions, or could be used as one session. At the conclusion of these stations, students draw on the problems presented and create their own stations to be used in other lessons.

**Station 1: A kilo of coins**

You have won a prize which can be just one of the following:

- 1 kilogram of $1 coins
- A 1.5 metre long trail of $2 coins (lying flat and touching)
- A 0.5 metre high stack of 50c coins

What is your choice?

__Answers:__

1 kg of $1 coins (1000 ÷ 8 = 125 coins, so $125)

1.5 metre of $2 coins (1500 ÷ 26.5 = 57 coins, so $114)

0.5 metre stack of 50c coins (500 ÷ 1.7 = 294 coins, so $147)

**Station 2: Largest Lasagne**

This problem could be adapted to reflect food that is meaningful to your students (e.g. the largest tray of pani popo).

The world’s largest lasagne was made in 2012 at a restaurant in Wieliczka, Poland.

It weighed 4865 kg and measured 25 m x 2.5 m.

The ingredients were:

2500kg of pasta, 800kg of mince, 400kg of mozzarella cheese, 100kg of peas, 100kg of carrots, and equal amounts of white sauce and tomato sauce.

- How much did the white sauce and tomato sauce weigh?
- What would be the size of a 500g piece from the lasagne?
- How many people could be fed with the whole lasagne?

Show how you arrived at your estimate.

__Answers:__

- The other ingredients total 3900kg so the sauces must weigh 4865 – 3900 = 965 kg.

500L of each sauce was used. Does that sound right? - A 500g piece would be about 1/10 000 of the whole lasagne. One way is to cut both the length and width into 100 parts, since 100 x 100 = 10 000. A single piece would measure 25cm x 2.5cm. That’s a bit skinny so 12.5cm x 5cm might work better.
- The lasagne was actually cut into 10 000 pieces so that’s how many people were fed. Each piece had a mass of 0.486 kg or 486 grams. That is a good serving of lasagne.

**Station 3: Weighing Tonnes**

Konsihiki was the largest active sumo wrestler in the world with a mass of 287 kg. Now he is retired.

How many Konishikis weigh as much as 1 tonne?

Make a table of tonne weights using objects in the classroom. Remember that 1000 kg is a tonne.

Object | Mass | Number in a tonne |

Konsihiki | 287 kg | |

School bag | 5 kg | 200 |

| | |

| | |

__Answers:__

The number of Konshihikis in 1 tonne equals 3.48, about 3 and ½ of him.

To find how many of any object make 1 tonne, divided 1 000 by the weight of the object in kilograms. For example, if a schoolbag weighs 5kg then 1 000 ÷ 5 = 200 make 1 tonne.

**Station 4: Jumbo facts**

Find out facts about the mass of very large animals and make a report about these animals for the class. To get you started here are some facts about the Blue Whale, which can be seen in New Zealand waters.

The blue whale is the largest animal living on Earth. It can reach up to nearly 30 metres in length and weigh up to 180 tonnes (t). Their tongues alone can weigh as much as some elephants and their hearts are huge, weighing a whopping 180kg. They have the largest babies on Earth. When they are first born they can be 8 metres (m) in length and weigh 4000kg. Imagine a jet engine that registers at 140 decibels. A blue whale, when it calls, registers at 188 decibels. Compare the facts about the Blue Whale with the large African elephant

The African elephant is the biggest animal on land. Fully grown the male can be 7 metres long, 3.2 metres tall at the shoulder and have a mass of 6500kg. Its tusks can weigh as much as 100kg each. The largest pair of tusks on record are in the British Museum and weigh 133kg each.

What combination of animals could be equal to the elephant's weight?

For example, it takes 6500 ÷ 5 = 1300 big domestic cats to weigh 1 elephant or 130 big dogs.

How many rhinoceroses, lions, giraffes, or hippopotamuses weigh the same as an elephant?

__Answers:__

Answers will vary depending on what other animals your students research.

**Station 5: Mass of water**

Measure out one litre (l) of water.

- What is the mass of one litre of water?

If 1L = 1000ml, what is the mass of 1mL of water? For each container, estimate the capacity of the container, measure it to check, estimate the mass of water when the container is full, and find the mass of the water using scales.

Record your results like this:

Container | Estimate capacity | Measure actual capacity | Estimate mass of water | Measure actual mass |

A | | | | |

B | | | | |

C | | | | |

D | | | | |

E | | | | |

- How many drops of water are needed to fill each container?
- What is the mass of a single drop?

__Answers:__

- 1 litre (l) of water has a mass of 1 kilogram (1000 grams). 1 millilitre (mL) of water has a mass of 1 gram.
Answers depend on the size of the containers. Here is an example:

Container | Estimate capacity | Measure actual capacity | Estimate mass of water | Measure actual mass |

A | 400 mL | 450 mL | 390 g | 450g |

- About 20 drops make 1 ml of water. Find the capacity of the container in mL then multiply by 20 to get the number of drops.
- A single drop has a mass of 1/20 of 1g, that’s 0.05g.

**Station 6: Frank’s arms**

*Counting on Frank *by Rod Clement (1990; Harper Collins Publishers: Sydney) has some great ideas for measurement investigations. You can view readings of the book on YouTube if you cannot source a copy of the book. One of the ideas introduced in the story is about Frank carrying a trolley load of cans to the supermarket.

- Trolleys measure 60 litres or 80 litres.

What do those measures mean? - How heavy do you think Frank’s load of cans is?
- How many cans are you able to carry in a reusable supermarket bag?

How did you work that out?

Here is another task based on Counting on Frank that you may choose to use.

__Answers:__

- 1 litre is a unit of capacity, but it is also used as a unit of volume. One litre measures 10cm x 10cm x 10cm. 60 litres is 60 times that size.
- Frank has 47 cans. They could be 420g cans so they would weigh 47 x 420g = 19 740g or 19.740 kg (about 20 kg). If the cans are bigger, say 820g, then the mass equals 47 times the mass of one can.
- You could get 24 x 420g cans in a supermarket bag, or 36 cans if you add another layer. The mass of the bag would be 24 x 420g = 10.080 kg or 36 x 420g = 15.12 kg.

## Representing 3D objects in 2D drawings

This unit develops students’ ability to represent three dimensional objects using two dimensional representations.

In this unit students learn to use two different two-dimensional drawings to represent three-dimensional shapes.

The first type of drawing is the use of plan views. These views are usually from the top, front, and side, as you would see in house plans. Such views are called orthogonal, meaning that the directions of sight are at right angles to each other. The images below show an example of plan views.

The second type of drawing is perspective, using isometric paper. True perspective shows objects getting smaller as they are further from the point of sight. Iso means “same” and “metric” means measure, so isometric paper shows every cube as the same size. Therefore, an isometric drawing shows a three-dimensional model from a single viewpoint, and distorts the perspective the eye sees. This can be seen in the image below.

In this unit students also work to develop flat patterns (nets) for simple solids, such as pyramids and prisms. A pyramid consists of a base, that names the solid, and triangular faces that converge to a single vertex, the apex. A hexagonal-based pyramid has a hexagonal base and six triangular faces. A prism has two parallel faces, that also name the solid, and parallelogram shaped faces. In right-angled prisms those faces are rectangles. Therefore, prisms have a constant cross-section when ‘sliced’ parallel to the naming faces.

Hexagonal pyramid Hexagonal prism

The flat surfaces of a three-dimensional solid are called faces. All faces must be connected in a net, the flat pattern from which the solid can be built. However, not all arrangements of faces will fold to the target solid. In the net for a hexagonal pyramid there needs to be one hexagon and six triangles, arranged in a way that means when folded there are no overlapping and no missing faces.

The learning opportunities in this unit can be differentiated by providing or removing support to students, and by varying the task requirements. Examples include:

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Select three-dimensional structures that are meaningful to your students. Mathematical solids are often used in construction, such as the shape of traditional Māori food stores (Pātaka), climbing equipment in playgrounds, and iconic structures around the world, e.g. pyramids of Egypt, high rise buildings.

Consider how you can make links between the learning in this unit, and other recent learning. For example if you have recently visited a local marae, your students might be engaged by the concept of drawing the floor plan of the marae.

Te reo Māori vocabulary terms such as āhua ahu-toru (three-dimensional shape), āhua ahu-rua (two-dimensional shape), āhua (shape), inerite (isometric), tukutuku inerite (isometric grid), raumata (net of a solid figure), tirohanga (perspective), tirohanga pūtahi (one point perspective), and the names of different shapes could be introduced in this unit and used throughout other mathematical learning.

Session 1SLO:

In this context, you might draw on the knowledge of community members (e.g. builders, architexts) and have them show architectural plans to your students.

What are these pictures used for?Discuss the idea of flat (two-dimensional) drawings used to show the structure of three-dimensional structures.

Sometimes more than one drawing is needed. Why?Tell the students that today they are becoming architects.

Show the class the building laying flat on your hand. After a few seconds of viewing time, place the building flat on a desktop. Move to the front of the desk, crouch down and take a digital photograph with the house at lens level.

Draw what you think the photograph looks like.How do the cubes show in the photograph?(as squares)Why do they appear that way?(Only one face of each cube is visible)What strategies did you use to get the viewpoint correct?(layers or columns, relative position, etc.)Be aware that many students are likely to have attempted to show depth in their pictures. Point out that the camera can only capture what it sees.

Are students understanding that information is lost when a 3D object is represented in 2D diagrams?

orthogonalbecause they are at right angles to faces of the model.How many cubes make up this building? How do you know?(9 cubes)Put your building on a desktop.Draw your building from the front, right side and top.Provide grid paper (Copymaster 1) to support students with drawing squares.

Can you find the model that goes with the plan?Which plan was the most useful?(A key point is that one viewpoint, often the top, is a good screening tool for possible models. Other views can be used to confirm the correct model).Discuss the use of the top view to organise the information from the other views.

Session TwoSLOs:

Before class gather at least five different shaped objects from around the classroom. The objects might be mathematical models (e.g. cube, pyramid, sphere, etc.) or common objects (book, cone, bottle, box, etc.) or a combination of things. It is better if the objects are different heights.

A spy took these photographs of an enemy city. She took four pictures, one from each of the compass points. After returning to her base she emailed the images.Show the students all four views of the ‘city’ on an electronic whiteboard or using a data projector.

Your job back at Kiwi Intelligence is to construct a plan map of the city. You know there are these buildings (objects). Look carefully at the photographs to work out where to put each building.How do these tricks work?(Objects that are further away look smaller, even to a camera. That is called perspective)How do artists adjust what they draw to allow for perspective?(Show Slide Two)Session ThreeSLO:

Draw what this model looks like on isometric dot paper. Before you start, you will need to decide whether to draw the front of the model angled to the left or right.Model looking at the model angled to the left and to the right.Let students attempt to draw the model. There are two possible perspectives depending on which direction the front is angled. Both drawings are shown on slide two.

Session FourSLO:

Where are you likely to see a shape like this?What is the shape in this picture called?What are the shapes of its faces?How many vertices (corners) and edges does it have?What do all three solids have in common?The common properties that define a prism are,

a solid that has two identical parallel faces and all other faces are parallelograms.Slide Four shows a loaf of bread being sliced.

How are a loaf of bread and a prism the same?Here is a rectangular prism shaped box that holds soap powder.Imagine that I open out the packet to form the flat pattern that makes it.Sketch what you think the net will look like.What is the same about all three nets?(Rectangular faces in a line)What is different about the three nets? (Parallel faces that create the cross section)How can you tell how many rectangular faces the prism needs? (The number of rectangular faces equals the number of sides on one of the parallel faces)Visualise the net for the pentagonal prism. What does that net look like?(Five rectangles in a line, with two pentagonal faces).What are these three-dimensional shapes called?In what way are the solids related?Look for students to discuss the properties of a pyramid; a base of a given shape, triangular faces that meet at an apex.

Let students investigate the problem in pairs and record their ideas.

Look at the table for prisms together (see below):

Number of faces

Number of edges

Number of vertices

Triangular Prism

5

9

6

Rectangular Prism

6

12

8

Pentagonal Prism

7

15

10

Hexagonal Prism

8

18

12

If you know the number of sides of the cross section, can you predict the number of faces, the number of edges, and the number of vertices?Students might notice that:

Why?Why?Why?Look at the table for pyramids together (see below):

Number of faces

Number of edges

Number of vertices

Triangular Pyramid

4

6

4

Square based Pyramid

6

8

5

Hexagonal Pyramid

8

12

7

Ask: If you know the number of sides of the base shape, can you predict the number of faces, the number of edges, and the number of vertices?Students might notice that:

Why?Why?Why?Session FiveSLOs:

Will this net fold to form a solid?Which solid will it create?How do you know?(Consider the number and shapes of faces, the result of folding)Imagine this net is folded. What other corners of the net will meet?Which other side meets this one when the net is folded?How do you know?## Balancing Acts

The unit involves students in solving problems that can be modeled with algebraic equations or expressions. Students are required to describe patterns and relationships using letters to represent variables.

Algebraic thinking is about generalisation, describing something that consistently occurs in a particular situation. In algebra, letters are used to express generalisations. Einstein’s famous equation

e=mcexpressed a relationship about the amount of energy, the mass of matter, and the speed of light.^{2}Letters may be used to express an important number. In Einstein’s equation

crepresents the speed of light, a number. Letters can also refer to a specific unknown. For example, “Mere gathers some pipi. Rawiri gives her seven more pipi. Now she has 18 pipi in her bucket.”x + 7 = 18expresses the unknown number of pipi she first collected.Letters are used more powerfully to express generalised relationships as Einstein did. The equation

v=d/texpressed velocity as a function of distance divided by time. All of the letters refer to variables, quantities that can change relative to each other.Algebra is loaded with conventions about how unknown variables are expressed. For example,

5xrefers toxmultiplied by five,t^{2}^{ }refers totmultiplied by itself, andn/mrefers to the division ofnbym. Decoding algebra is similar to learning to read. Students need lots of experience linking symbols with the meaning of those symbols.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:

5xmeans five times an unknown x, with a ‘missing’ multiplication signTasks can be varied in many ways including:

This unit is focussed on learning how to use algebraic expressions and equations, and the teaching sessions are not set in a real world context. You may wish to explore real world applications of algebra in the teaching sessions following the unit, for example bags of ingredients going into a manufacturing machine, servings of food at a hāngī, or ticket prices for an event.

## Session 1

Make up a set of three containers by trimming down 1.5 litre plastic soft drink bottles. It is preferable that these containers are of the same type. Straight-sided bottles are better than curved ones as they make it easier for the students to predict the relationships. The three cut down containers should hold different amounts of water when full. Make their capacity no more than 500 ml and ensure that these capacities are not multiples of each other, e.g. 120 ml, 230 ml, and 400 ml would be better than 100 ml, 200 ml, and 400 ml. Label the containers p, q, and r respectively.

With each problem students write an expression for the water level. Here are some examples. You can extend these as appropriate for your class.

How can you write an expression for what is in the bottle?(p + p + p + p or 4p, noting that 4p has an ‘assumed’ multiplication sign, 4 x p)Next pour from the bottle into container r and container q until the mark is reached.

What is left in the bottle? How might you write an expression for that amount?(This gives 4p – r – q which can also be written as 4p – (r + q))Pour out two lots of p (5r – 2p).

Pour out two lots of p.

(This gives 3r + 3q – 2p which could be written as 3(r + q) – 2p).

Repeat pouring 1, that is add two lots of r and one lot of p into the large bottle

(2r + p + 2r + p, or 2(2r + p) or 4r + 2)

Are these expressions equal or unequal? Why?Can students simplify the equality using inverse operations?

p + 2r = r + p + r

2r = r + r (subtracting p from both sides)

Which of these are correct and which are false? Explain your answers.Students could make up their own set of containers, label them with algebraic letters, and develop challenges for each other. Alternatively, the class containers could be left as a station for students to use independently.

## Session 2

In this session students look for patterns within each equation set and use these patterns to predict further equations in the set. They may do this using recursion, that is finding a relation between consecutive equations, rather than by looking for relationships within the equations across the equals sign. Highlight relationships that might be found between the numbers in each set of equations and encourage the students to look for ways to describe these relationships. It is important that students find the unknowns using mental calculation rather than with calculators, as their attention needs to be on the relative size of numbers.

Some learners will need support from physical materials to notice and describe the patterns. Connecting cubes in stacks make a useful representation.

Below are some suitable equation patterns:

2 – [ ] = 1

3 – [ ] = 1

4 – [ ] = 1

...

456 – [ ] = 1

Why is the right-hand side of the equation always one?What is the rule for all equations in this pattern?Use this pattern to solve:2000 – [ ] = 1

1001 – [ ] = 2

1 + 2 + 3 =

2 + 3 + 4 =

3 + 4 + 5 =

Is there anything in common with the numbers in the boxes?Why do you think this happens?(The number in thea + b + c = 300 , a, b and c are different numbers.What numbers could they be?What values for a, b and c would fit the pattern?A physical representation of the pattern, using connecting cubes, would look like this:

52 – 8 =

62 – 8 = [ ] – 10

...

92 – 8 =

712 – 8 =

What rule can you find for equations in this pattern?2 x 9 + 2 =

3 x 9 + 3 = [ ]

4 x 9 + 4 = [ ]

...

In[ ] x 9 + [ ] = 100, what is[ ]if both[ ]are the same number?What rule can you find for all the equations in this pattern?A physical model of this pattern, made with cube towers, looks like this

(example of 4 x 9 + 4 = 40 (4 x 10)):

2 =

3 = [ ] ÷ 11

4 = [ ] ÷ 11

...

In [ ] ÷ 11 = 7, what is [ ]?

In 9 = [ ] ÷ 11, what is [ ]?

What rule can you find for equations in this pattern?Would these equations be further down in the pattern?Give an explanation.

[ ] = 594 ÷ 11

[ ] = 682 ÷ 11

## Session 3

Students solve “What’s my Number?” problems and record how they found the final answer. At this stage trial and improvement are legitimate strategies though the problems encourage students to attend to structure, and apply their understanding of inverse operations.

Flowcharts can support students to apply inverse operations. Consider the problem; “Take my number, multiply it by three then subtract 4. The answer is 20. What is my number?” The problem is equivalent to 3

x- 4 = 20.A flowchart looks like this:

Applying inverse operations gives:

What is my number?

(15 since 15 x 3 + 7 = 52, or 3 x [ ] + 7 = 52, so 3 x [ ] = 45, and [ ] = 15)

What is my number?

(42 since 16 x 2 = 32 and 32 + 10 = 42)

What is my number?

([ ] + 24 = 3 x [ ], so 24 = 2 x [ ], and [ ] = 12)

What is my number?

([ ] ÷ 3 = [ ] - 40, so 40 = 2 x [ ], and [ ] = 60)

What is my number?

(2 since 2 x 2 = 4 and 2 + 2 = 4 and 0 for the same reason;

or [ ] x [ ] = [ ] + [ ], so [ ] x [ ] = 2 x [ ] , and [ ] = 0 or [ ] = 2.)

## Session 4

In this session students work out the functional rule for given input and output numbers. The functions increase in complexity as the week progresses. Each example offers the students three input/output pairs. to the right of these three pairs are three other pairs (typed in bold) that could be used if needed. Copymaster 1 can be used to make stimulus cards by cutting and folding the output of the three additional pairs behind so they can not be seen unless needed.

The rule is two times the input number less one gives the output number.

The rule is times three plus one.

The rule is times three take away nine.

The rule is divide by two plus one.

The rule is the input number multiplied by itself (squared).

Discuss how a rule might be found.

If the output numbers are always more than the input numbers what types of rules do you try first?Students are likely to suggest addition or multiplication. That strategy is useful but needs to be qualified. Multiplying by a fraction less than one or an integer will result in smaller output numbers.

How can you check ifonlyaddition is involved?The difference between input and output numbers must be constant. That generalisation is true of subtraction as well.

How can we check ifonlymultiplication is involved?The differences between input and output numbers change. That will not work as a strategy. Students will need to estimate what the multiplier is, possibly from two pairs, then check to see if the multiplier applies to the other pairs.

If division is in the rule, how can you tell?Students usually say that division, like subtraction, makes the output number smaller. That generalisation is naïve since division with a fraction between zero and one makes the output number larger. However, provided the divisor is larger than one the claim that “division makes smaller” holds. As with multiplication, the divisor will need to be estimated first then tried on the input-output pairs.

Challenge students to come up with their own tables of values. You might set up a spreadsheet of tables as a template, with hidden rules. Alternatively create a paper template as seen in Copymaster 1.

## Session 5

These activities involve students in working out the number of counters or cubes that are in each cup of a given colour. Several clues are provided, and students must combine these clues to find a solution. Use opaque coloured plastic cups, that are readily available in supermarkets, warehouses, and dollar stores. For each problem all students should solve it, then record their reasoning, before the solution is “revealed.”

Encourage students to check their solutions by putting the values they assign to each cup back into the original clues. Do all the clues work? If not, try again!

Examples of cups and counters problems follow. The first sentence is the instruction for the teacher and gives the answer. It does not appear on (Copymaster 2) as students use the clues to work out how many cubes are in each coloured cup.

Clues:

14 cubes in total 16 cubes in total

Clues:

Four yellow cups have the same number of cubes as two blue cups.

Three yellow cups and one blue cup have 15 cubes in total.

Clues:

Three yellow cups have the same number of cubes as five blue cups.

One yellow cup and two blue cups have 11 cubes in total

Clues:

Six yellow cups have the same number of cubes as three blue cubes that have the same number of cubes as two red cups.

Two blue cups and one red cup have 14 cubes in total

Clues:

Five yellow cups have the same number of cubes as seven blue cups.

Five red cups have the same number of cubes as seven blue cups.

One red cup and one blue cup contain 12 cubes.

Is there more than one possible answer?(y = 6, r = 6, b = 6 also works)Dear parents and whānau,

This week we have been looking at equations with unknowns. We are learning to express relationships using the language of algebra, particularly using letters to represent amounts that we do not know.

Try the following with your child:

Write 8 on a small piece of paper and 4 on another piece. Hold the numbers in different hands so that your student can’t see them. Say, “One number is twice the other, and both numbers add to 12. What numbers are in my hands.”

Get your child to make a problem like this for you. He or she might share the problems they are solving in class.

Enjoy challenging each other!

## Figure it Out Links

Some links from the Figure It Out series which you may find useful are:

Level 4, Algebra, Book Two: Straw Chains, page 2; Number Crunching, page 4; Number Puzzles, page 22.

## Getting partial to fractions

In this unit students partition a length model into equal parts, to create unit fractions. Students form non-unit fractions (e.g. 3/4 and 7/8) and develop strategies to find different names for the same fraction (equivalent fractions). Fractions are added and compared to find the difference and a fraction of a length is determined. Finally, a length model is used to find a fraction of a whole number amount.

Fractions are an extension of whole numbers and integers. Fractions are needed when wholes (ones) are not adequate for a task. Mathematically the need for rational numbers comes from division. Division often requires equal partitioning of ones. Sharing two chocolate bars equally among five people requires that the bars be cut into smaller equal parts. The operation might be recorded as 2 ÷ 5 = 2/5. Note that the number two fifths, is composed of two units of one fifth. In practical terms the equal share can occur by dividing each of the two bars into fifths, then giving each person one fifth from each bar.

If the bar was made up of ten pieces then each person might be given two tenths from each bar, giving them four tenths in total. Four tenths are the same quantity of chocolate as two fifths. Any fraction can be expressed as an infinite number of equivalent fractions that represent the same quantity.

Fractions are very important to measurement. The presence of fractions is often masked by the fact that most measurements are expressed as decimals. Suppose your height is 1.78 metres. Metre lengths are the whole (ones) in this representation of quantity. Whole metres are inadequate for most length-related measurement tasks. So equal parts of metres are used to achieve greater precision.

The decimal system uses repeated equal division into ten parts to create smaller units. Dividing one metre into ten equal parts creates deci-metres, a unit that is used in Europe but seldom in New Zealand. A decimetre is one tenth of a metre. If one decimetre is cut into ten equal parts, the parts are called centimetres. That is because 100 centimetres compose one metre (one tenth of one tenth equals one hundredth). A height of 1.78 metres is a combination of 1 whole metre, 7 tenths of one metre, and 8 hundredths of one metre.

Addition and subtraction require quantities that are expressed in the same unit. With whole numbers, quantities are always expressed using units of one. With fractions the units can be different, e.g. 2/5 + 1/2 = ?. Two fifths are made up of two units of one fifth, that are different in size to halves. A standard method is to rename the fractions so they share a common denominator. Two fifths equal four tenths (2/5 = 4/10) and one half equals five tenths (1/2 = 5/10) Since four tenths and five tenths share the same unit they can be combined to make nine tenths (4/10 + 5/10 = 9/10).

Fractions are numbers that behave in the same way as whole numbers, albeit with more complexity. The properties of addition and multiplication that hold for whole numbers also hold for fractions. To find a fraction of another amount is to treat the fraction as an operator. For example, three quarters of 24 is represented symbolically as 3/4 x 24, with three quarters operating on 24 as a multiplier or scalar.

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:

Adaptation involves changing the contexts used for problems to meet the interests and cultural backgrounds of your students. Where contexts such as food and computer games may not be appropriate for your students, find other situations likely to engage them. For example, points earned on a computer could be reframed as points earned in a Kapa Haka performance or talent quest. You might find contexts familiar to your students, such as fractions of journeys in familiar locations (Copymaster 3).

## Session one

This session begins with a lesson created by Thomas Kieren, a distinguished Canadian mathematics educator. You will need many strips of paper that are the same length as a standard ruler (30cm). Also have a strip that is 48cm long but has the same width.

How do I know that these parts are halves?(Equal parts, all the strip is used, two parts)What is true if the parts are truly halves?Look for students to say that one is cut into

two equal parts. Fold a 48cm strip into halves.Are these parts halves? Why are they a different size to the other halves?The important point is that the unit of one determines the size of the halves. Use the half strips to draw two different number lines to represent the relationship of one half to one (whole).

Where would the fraction 2/2 (or two halves) be on the number line?Where would fraction 5/2 (or five halves) be on the number line?The important point is that non-unit fractions are located by repeatedly joining unit fractions. Five halves are made of five units of one half.

You all know how to fold a strip of paper in two equal parts and call those parts halves. How can you fold one strip into three equal parts? How can you fold one strip into five equal parts?Is it possible to fold the parts without measuring?)Why are thirds smaller than halves?)If we can fold into halves, thirds and fifths what other fractions can we make by combining these folds?What fraction parts have I made?If you halve first, then third, what fraction parts do you make?(sixths)Does it matter if you change the order of folding to thirding then halving?(No)How can you record your folding?(e.g. 1/3 x 1/2 = 1/6) Note that x means "of" as in “one third of one half".)Session twoWhat do you think the word equivalent means?From everyday life, students may know that equivalents are the same, e.g. equivalent amounts, equivalent methods, equivalent ingredients or medications.

Do your students recognise that the fractions are all names for one (whole)?

You may need to make each fraction using a student’s strips from the previous session.

What other fractions also equal one?What is the same about all fractions that equal one?(The numerator and denominator are the same)What equivalent fractions can you see on this page?Students should notice that there are three fractions equivalent to one half. Record their observations as an equality, 1/2 = 2/4 = 3/6 = 4/8

.Imagine that tenths and twelfths were on the sheet. What equivalent fractions for one half could you find?Students might notice other equivalents to one third and two thirds (1/3 = 2/6 and 2/3 = 4/6) and one quarter and three quarters (1/4 = 2/8 and 3/4 = 6/8).

The fifths do not seem useful to find equivalent fractions. Why is that?Do you notice any patterns in the equations?Students are likely to notice doubling of both numerator and denominator without fully appreciating why that occurs, e.g. 1/2 = 2/4 = 4/8.

Find as many examples of equivalent fractions as you can using this Copymaster. You may need to cut out some strips so you can move them around.Record what you find as equations and drawings.

Is there a way to anticipate if fractions are equivalent, without using fraction pieces?When four is doubled to give eight, what does this mean? (There are twice as many parts in eight tenths, than there are in four fifths)When five is doubled to give ten, what does this mean? (There are twice as many tenths in one as there are fifths. Tenths are half the size of fifths)What fraction part can you see?How many of those parts are shaded?Which two fractions have you found to be equivalent?If you fold the fifths/thirds in half, what sized parts do we make?How many of the pieces are shaded?How do you write an equation to represent what we’ve found?What happens if you fold the fifths/thirds into other numbers of parts? What equivalent fractions can you make?Can you make fifths into hundredths? How? What about thirds?Put these fractions in order from smallest to largest.

Ask students to solve the problem first then check their answer using fraction strips. Have them explain their strategy to a classmate.

Session threeIn this session students explore ordering fractions and finding the difference between two fractions. Difference can be found either by subtraction or adding on. Students should come to understand that both strategies yield the same result. The length model is used again.

Which is more, one third or one half of the same submarine sandwich?How much more?Why is the difference one sixth?Students should notice that 2 x 3 = 6, the connection between the denominators. However, the important understanding is that sixths work because both fractions can be renamed into equivalent fractions with that common denominator.

How many sixths is one half equivalent to?How many sixths is one third equivalent to?Could we use twelfths instead? What would be the difference then?

The difference can be one sixth or two twelfths. Is it strange to get different answers?Students should notice that one sixth and two twelfths are equivalent fractions so both answers are different names for the same number. Slide three shows different ways the problem can be expressed as an equation. Discuss what each equation means, e.g. “One third and how much more equals one half?”

I have a new problem.What is the difference between two thirds and five eighths of the same submarine sandwich?Where should I start?Students should suggest that a common denominator is needed because the parts are different sizes.

What common denominator will work?Hopefully, students suggest 24 using 3 x 8.

Can two thirds be renamed as so many twenty fourths? How many?Can five eighths be renamed as so many twenty fourths? How many?Session fourIn this session students explore how to add fractions. They learn to recognise when one or both fractions in an addition operation need to be renamed.

Tieri’s parents pay for two credits each week so she can play her favourite computer game.She is careful not to use all her credits up in one day.On Saturday Tieri used up three quarters of a credit. She used another three quarters of a credit on Sunday.How much credit does she have left for the rest of the week?What was it about the fractions in this problem that made it reasonably easy to solve?Students might note that only quarters were involved. Change Tieri’s computer credits scenario to five eighths plus one quarter (slide three of PowerPoint 2). Let your students solve the problem in pairs. Watch out for incorrect strategies such as adding numerators and denominators, e.g. 5/8 + 2/8 = 7/16. Use a strip model to represent the problem and use that model to address problems your students have.

How many credits does Tieri have left?(nine eighths, which equals one and one eighth)Knowing that fractions can only be added if they refer to the same sized parts is very important. The denominators must be the same.

Imagine a week in which Tieri uses up two thirds of a credit on Saturday and one half of a credit on Sunday.How much credit does she have left for the rest of the week?In a week of the school holidays Tieri uses up four fifths of a credit on Saturday and three quarters of a credit on Sunday.How much credit does she have left for the rest of the week?Session FiveIn this session students learn to distinguish between problems in which fractions are treated as measures (numbers), and problems in which the fraction operates on another number. Rational numbers (which fractions are a subset of) conform to the same properties as whole numbers when operated with (added, subtracted, multiplied and divided). Recognition of that conformity is a critical transfer for students at Level Four.

How is this possible?Who gets more pocket money each week, Elise or Harry? How do you know?Organise the solution data in a table, such as:

Elise’s weekly pocket money (e)

Harry’s weekly pocket money (h)

$18

$16

$81

$72

$45

$40

$9

$8

etc.

etc.

Ask: Do you see any patterns in the table?Students might notice that Elise’s amounts are multiples of nine and Harry’s amounts are multiples of eight.

What operations change Elise’s amounts into Harry’s amounts?Try dividing Elise’s amounts by nine. What do you notice?Students might see that dividing Elise’s amounts by nine then multiplying the result by eight gives Harry’s amount. High achieving students can be encouraged to express that relationship algebraically (8/9 e = h or e = 9/8 h).

Kia ora family and whānau,

This week we are exploring fractions as numbers and as operators. We will be using lengths to find equivalent fractions, add and subtract fractions, and find a fraction of an amount. You might look for examples in real life where fractions of a length are found, such as cutting a length of wood into equal parts or breaking a long trip up into equal parts.

Talk to your child about what they are learning. They could show you what they have learnt about fractions using paper strips at home.

## Oranges L4

In this unit we focus on selecting appropriate units for measurement in practical situations. Students are required to justify the instrument they have used in relation to the degree of accuracy required in their measurements.

Measuring is about quantifying a feature, or attribute, of an object. Examples of these attributes include length, mass, and temperature. Measurement involves making a comparison between the size of the attribute being measured and a suitable measurement unit. For example, if the peel of an orange is measured for length, a measure of 34cm means that 34 units of 1 centimetre fit into the orange peel with no gaps or overlaps.

Central to the development of students’ measurement skills and processes is ample practical measuring experience. Also important is the reality that measurement is never exact. As measurement involves continuous quantities, even the most careful measurements are only approximations. For example, the length of the orange peel might be measured as 34cm or 338mm, depending on the precision needed for the measurement purpose. This unit gives students the opportunity to carry out practical measuring tasks and emphasises the fact that there are many attributes of objects that can be measured.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

^{2}as 45 square metres; 45g as 45 grams, and 45mL as 45 millilitresTask can be varied in many ways including:

The context for this unit can be adapted to suit the interests and cultural backgrounds of your students. If use of fruit, or any food, is a cultural issue for your students, then choose an item that is more appropriate. Note that in this unity no food is wasted. You might integrate the investigation of

The Orangewith a unit on healthy eating or food production. Students may be curious about the amount of vitamin C in an orange (about 70mg). Students might explore the concept of a milligram, in connection with their learning around a millimetre.Students might explore fruits and vegetables in relation to diet. How many of each fruit or vegetable do they eat in one year?

The same types of activities used with an orange might be adapted for another consumable product, such as a tube of toothpaste or a roll of paper towels. Consider how this learning could reflect the cultural diversity of your students. For example, could you compare food items from different countries?

Te reo Māori vocabulary terms such as mehua (measure), karamumano (milligram), karamu (gram), mitarau (centimetre), mitamano (millimetre), rūri (ruler), āwhata (scale of a measuring instrument) and ine-taumaha (scale for measuring weight) could be introduced in this unit and used throughout other mathematical learning.

Session 1In this session we discuss the attributes of an orange that could be measured.

How can you measure an orange?You might go online to look up “fascinating facts about oranges” that might yield interesting attributes to measure.

How can we make our measurement as precise as possible?Students might mention that oranges vary a bit so multiple oranges might be needed. An average might be used.

How can we make sure we are consistent in the way we measure?Session 2In small groups we attempt to measure one of the attributes of an orange identified in Session 1. Our measurements will be compiled into a class report on

TheOrange.What units will you use?(cubic centimetres (cm^{3}) or millilitres (mL) for volume, centimetres or millimetres for length, grams for mass)How will you ensure that your measurements are accurate?(repeat where possible, double check when reading scales, read the scale correctly, take three different measurements then average, etc.)Session 3In this session students complete the write up of their Orange investigations, that are then compiled into a class report on

The Orange. Students then select a different attribute which they will investigate as a group over the next two sessions.What are each group’s results?What could each group have done differently?How accurate are each group’s results?How could they have been more accurate?What attributes were easiest/hardest to measure?Why?What were successful/unsuccessful approaches to measuring each attribute?Why were the approaches successful or unsuccessful?The Orange.Session 4This session is given to students to work in their groups measuring their oranges.

Circulate and provide assistance to students, where required, and ensure that students are using good measurement techniques. Can your students?

Where necessary provide mini-tutorials for groups of students about using measurement tools.

What can you do to get your results to be at a sensible level of accuracy?Why might results be different? (Different oranges, different choice of tools and units, measurement error, etc.

Session 5In this session students complete their measurements of their oranges, and write up their results to share with the class.

What have you learned about your orange?Were any of the results surprising?What have you learned about measurement?Were there attributes that you measured that you didn’t previously know how to?What was the most interesting thing that anyone in the class measured? What made it interesting?Who might be interested in our results? Why would they be interested?(e.g., Marmalade makers might be interested in mass of oranges)What other objects could we measure?ko wai au? (Who am I?)and be used to develop connections between students. It could also be used as a context to look at how different items have changed over time. Some examples include:Family and whānau,

This week we have been working on getting accurate measurements of everyday objects, both large and small. We spent a lot of time finding out about oranges which was very appealing! Each student has been asked to identify a common household object and present a brief report on the detailed measurements of the object's features (attributes). Ask your child to explain their project to you and how they are making sure their measurements are as accurate as possible.

## Figure it out

Some links from the Figure It Out series which you may find useful are:

## Weighty Problems

This unit comprises six problems for students to apply and interpret measurement of mass. Students are also introduced to the concepts of net and gross mass.

Mass is the force created by gravity acting of on an object. Mass is felt as weight, a force that pulls the object towards the centre of the Earth. Mass is measured in units based on grams, and tonnes. Larger or smaller units are created by combining or equally partitioning these units. One kilogram is a combination of 1000 grams (kilo means 1000). One milligram is 1/1000 of a gram and one microgram is 1/ 1 000 000 of a gram. The units for mass come from the mass of water. One cubic metre of water has a mass of 1 tonne, or 1000 kilograms. One millilitre of water has a mass of one gram.

Note that in the New Zealand Curriculum document, “weight” and “mass” are used interchangeably. In a science context, the definition of “force created by gravity acting on an object” would often be equated with weight, not mass. Consider the scientific knowledge of your students (e.g. are they studying forces in science). It may be more appropriate to define mass as the amount of matter in an object (measured in kilograms) and weight using the adorementioned definition (measured in Newtons, N).

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

Tasks can be varied in many ways including:

The context for this unit can be adapted to suit the interests and cultural backgrounds of your students. Use the interests of your students to create contexts that will engage them. Students may be interested in the mass of rugby players. Students from large whānau, or who prepare food for large numbers of people, may relate to measuring quantities to scale up recipes. Carrying heavy objects was a major problem for pre-European Māori. How did they carry heavy loads, or move waka? Counting on Frank by Rod Clement may inspire some students to look for eccentric ways to apply measurement to their daily lives. For example, the human body is 60% water, by mass. How much water is in their body?

Te reo Māori vocabulary terms such as maihea (weight / mass), karamu (gram), manokaramu (kilogram), andtana (tonne) could be introduced in this unit and used throughout other mathematical learning.

The following six stations provide a range of problems for students to apply and interpret measurement of mass. Consider what would be the most effective method for introducing these to your class. You could work on them as a whole class and provide support to groups of students. Alternatively, you could use another relevant igniting activity to introduce the context for learning, before directing students to work on one or more of these stations independently, or in small groups. These stations could serve as the basis for learning in different sessions, or could be used as one session. At the conclusion of these stations, students draw on the problems presented and create their own stations to be used in other lessons.

Station 1: A kilo of coinsYou have won a prize which can be just one of the following:

What is your choice?

Answers:1 kg of $1 coins (1000 ÷ 8 = 125 coins, so $125)

1.5 metre of $2 coins (1500 ÷ 26.5 = 57 coins, so $114)

0.5 metre stack of 50c coins (500 ÷ 1.7 = 294 coins, so $147)

Station 2: Largest LasagneThis problem could be adapted to reflect food that is meaningful to your students (e.g. the largest tray of pani popo).

The world’s largest lasagne was made in 2012 at a restaurant in Wieliczka, Poland.

It weighed 4865 kg and measured 25 m x 2.5 m.

The ingredients were:

2500kg of pasta, 800kg of mince, 400kg of mozzarella cheese, 100kg of peas, 100kg of carrots, and equal amounts of white sauce and tomato sauce.

Show how you arrived at your estimate.

Answers:500L of each sauce was used. Does that sound right?

Station 3: Weighing TonnesKonsihiki was the largest active sumo wrestler in the world with a mass of 287 kg. Now he is retired.

How many Konishikis weigh as much as 1 tonne?

Make a table of tonne weights using objects in the classroom. Remember that 1000 kg is a tonne.

Answers:The number of Konshihikis in 1 tonne equals 3.48, about 3 and ½ of him.

To find how many of any object make 1 tonne, divided 1 000 by the weight of the object in kilograms. For example, if a schoolbag weighs 5kg then 1 000 ÷ 5 = 200 make 1 tonne.

Station 4: Jumbo factsFind out facts about the mass of very large animals and make a report about these animals for the class. To get you started here are some facts about the Blue Whale, which can be seen in New Zealand waters.

The blue whale is the largest animal living on Earth. It can reach up to nearly 30 metres in length and weigh up to 180 tonnes (t). Their tongues alone can weigh as much as some elephants and their hearts are huge, weighing a whopping 180kg. They have the largest babies on Earth. When they are first born they can be 8 metres (m) in length and weigh 4000kg. Imagine a jet engine that registers at 140 decibels. A blue whale, when it calls, registers at 188 decibels. Compare the facts about the Blue Whale with the large African elephant

The African elephant is the biggest animal on land. Fully grown the male can be 7 metres long, 3.2 metres tall at the shoulder and have a mass of 6500kg. Its tusks can weigh as much as 100kg each. The largest pair of tusks on record are in the British Museum and weigh 133kg each.

What combination of animals could be equal to the elephant's weight?

For example, it takes 6500 ÷ 5 = 1300 big domestic cats to weigh 1 elephant or 130 big dogs.

How many rhinoceroses, lions, giraffes, or hippopotamuses weigh the same as an elephant?

Answers:Answers will vary depending on what other animals your students research.

Station 5: Mass of waterMeasure out one litre (l) of water.

If 1L = 1000ml, what is the mass of 1mL of water?

For each container, estimate the capacity of the container, measure it to check, estimate the mass of water when the container is full, and find the mass of the water using scales.

Record your results like this:

Answers:Answers depend on the size of the containers. Here is an example:

Station 6: Frank’s armsCounting on Frankby Rod Clement (1990; Harper Collins Publishers: Sydney) has some great ideas for measurement investigations. You can view readings of the book on YouTube if you cannot source a copy of the book. One of the ideas introduced in the story is about Frank carrying a trolley load of cans to the supermarket.What do those measures mean?

How did you work that out?

Here is another task based on Counting on Frank that you may choose to use.

Answers:Dear family and whānau,

In class we have talked a lot about weight this week. In particular, we discovered that Konshiki, the largest Sumo wrestler, weighs 226kg, a Blue Whale weighs 180 tonnes and an elephant weighs 6500kg. What do you think the Blue Whale weighs in kg?

Your child has been asked to look for other facts associated with weight by reading the newspaper or doing some reading on the internet. They should record what they find out. Please encourage them to discuss their findings with you.

You could support them further by weighing items at home or cooking with them.

## Figure It Out

Some links from the Figure It Out series which you may find useful are:

## Map it

This unit explores the mathematics of maps, including scale, coordinates, and bearings.

This unit investigates three mathematical concepts in the context of maps.

A map is a reduced version of a real landscape. Scale compares the size of lengths on a map to those in the real landscape.

With real maps the coordinates often refer to latitude and longitude.

Longitude is the number of degrees 'around the world' from the Prime Meridian, and latitude is measured in degrees north or south from the equator.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

The context of mapping is very realistic for all students. Adaptation of the context should include locations that students are less familiar with, after competence is developed in working with local surroundings. Students will enjoy using digital tools to plan a magic carpet ride to locations within New Zealand and overseas. “The guidance system on your magic carpet requests instructions to travel to Samoa. Give coordinates for the destination, and bearing and distances for the journey.”

Students might be interested in traditional forms of navigation. Tipuna (Ancestors) had clever ways to navigate in the absence of the sophisticated GPS systems used nowadays. Polynesian navigators used ocean currents, the sun and stars, and migration of whales and birds to find their way around the Pacific Ocean.

Te reo Māori vocabulary terms such as ahopae (latitude), ahopou (longitude), mahere (map), tuhinga āwhata (scale drawing), āwhata (scale), taunga tukutuku (cartesian coordinates), raki (north), runga (south), rāwhiti (east), and rātō (west) could be introduced in this unit and used throughout other mathematical learning.

## Session 1: Introduction to the mathematics of maps

This lesson provides an introduction to the context of the mathematics of maps, and the idea of reading maps. You could start this session with investigation into the traditional navigation methods used by Māori and Pasifika peoples. Consider inviting local iwi members to your classroom to discuss where the local iwi heralded from. These locations, and the locations of other iwi relevant to your students, could be identified on a map.

## Session 2: Scale

In this session we look at the scale of maps, what they mean, and how to interpret and apply them. We draw a scale map of the classroom.

How far is it from …?Model calculating this first, and then support students to independently figure out the distances. Calculators can be used to help students make accurate calculations with larger numbers.## Session 3: Coordinates

In this session we will draw a grid on the map of the classroom and use it to identify places within the class. Note that there are two ways in which location is represented on maps, grid references and coordinates. In both cases the representation is an ordered pair, e.g. F4 as a grid reference, and (6,4) as a pair of coordinates. The convention is to state the horizontal reference first, and the vertical reference second.

In the grid system (left image) the location of the person is an area. In the coordinate system (right image) the location of the person is a specific point at the intersection of the lines.

What are the vertical and horizontal lines on the map there for?What do the numbers at the tick marks (ends of the lines) represent?How do they work?Map Mysteriesto practise finding coordinates in a pirate context. If you have a large world map with coordinates on the x-axis and y-axis, then you could use this as an opportunity for students to identify countries they whakapapa to. In turn, this could help establish connections and cultural understanding between your students. If your students are ready to be extended in their use of coordinates, they could complete this activity independently using a printed world map, and compare their maps to those completed by their peers. They could also investigate the places their ancestors travelled from to arrive in New Zealand (e.g. from Samoa) and record this information using coordinates.## Session 4: Polar Coordinates

In this lesson we draw a north arrow on our maps, and introduce polar coordinates.

What is north of the teacher’s desk?What is east of the door?What is north-east of the mat?What is south-west of the bookshelf?What is north-north-west of the middle of the whiteboard?What is west-south-west of the doorway?What object is 5m on a bearing of 130^{0}from where I am standing?## Session 5: Guest speaker

In this session we invite a member of a local orienteering club, or possibly a local surveyor to come and talk to the class about how they use maps.

Dear family and whānau,

This week in maths we are learning about maps. Ask your child to tell you about how learning about maps is an important part of learning about mathematics. If you have any maps at home, ask your child to show you the important parts and show you how to describe where things are on the map.

## Figure it Out Links

Some links from the Figure It Out series which you may find useful are: