Integers are needed to meet the demands of situations where a larger whole number is subtracted from a smaller whole number. For example, if a person has $5 available in cash but owes $8 then their net situation is 5 – 8 = ^{-}3. The origins of integers historically lie in the algebra of subtraction of whole numbers. Integers, as quantities, usually reflect a state of balance between directional forces, such as cash being and asset and debt a liability. Furthermore, integers are sometime called directional numbers because they represent a magnitude (size) and a direction, e.g. ^{-}3 represents a magnitude of 3 units in a negative direction from zero. The directional nature of integers is important to real world applications such as transmission factors of gears and pulleys, and to enlargement (dilation).

A common use of integers in real life is to label and quantify points on a scale, such as temperature and height above sea level. In both cases the location of zero is important to the attribute being measured. For example, both the height above normal sea level (0 m) of a spring tide and the temperature below the freezing point of water (0°C) have significant consequences to the severity of the situation. Zero acts as an important benchmark indicating normality or balance. This is also true in sport or games like Bridge where negative numbers reflect a state relative to expectation, e.g. ^{-}6 in golf means six under par when a player has taken six fewer shots than the expected norm.

#### Specific Teaching Points

Integers are an extension of the whole number system. Therefore, the properties of Integers under the four operations should be the same as those for whole numbers. With addition and subtraction four main properties hold:

**The commutative property of addition**

The order of the addends does not affect the sum. If ^{-}3 + 4 = 1 then 4 + ^{-}3 = 1. Note that the commutative property does not hold for subtraction. For example, 4 - ^{-}3 = 7 but ^{-}3 – 4 = ^{-}7.

**The distributive property of addition **

This property is really about the partitioning of addends and recombining those addends. For example, if 5 = ^{-}1 + 6, then ^{-}2 + 5 = (^{-}2 + ^{-}1) + 6. This property does not hold for subtraction.

**The associative property of addition**

This property is about ‘associating’ pairs of addends one pair at a time. For example, (^{-}4 + 3) + ^{-}1 = ^{-}4 + (3+ ^{-}1). This property does not hold for subtraction.

**Inverse operations**

Addition and subtraction are inverse operations so one operation undoes the other. For example, ^{-}2 + ^{-}3 = ^{-}5 so^{ -}5 - ^{-}3 = ^{-}2.

It is the need for these number laws to hold that establishes the effect of operations, such as subtracting a negative integer has the same effect as adding a positive integer.

This unit combines two of Hans Freudenthal’s (1983) models for operations on integers, the annihilation and vector models. In the annihilation model positive and negatives cancel each other, so +1 and -1 pairs equal zero. The act of creating or removing one positive and one negative pairs that equal zero does not alter the quantity being represented. The vector model presents integers as magnitudes with direction. +1 is represented by a vector of length one in a positive direction and -1 as a vector of length one in a negative direction. Freudenthal cautioned that a quantity of +1 or -1 was easily confused with the operation of adding or subtracting one and teaching needed to make that difference explicit.

## Integers

This unit helps students to develop procedural fluency with integers and have conceptual understanding of integers in the real world.

Integers are needed to meet the demands of situations where a larger whole number is subtracted from a smaller whole number. For example, if a person has $5 available in cash but owes $8 then their net situation is 5 – 8 =

^{-}3. The origins of integers historically lie in the algebra of subtraction of whole numbers. Integers, as quantities, usually reflect a state of balance between directional forces, such as cash being and asset and debt a liability. Furthermore, integers are sometime called directional numbers because they represent a magnitude (size) and a direction, e.g.^{-}3 represents a magnitude of 3 units in a negative direction from zero. The directional nature of integers is important to real world applications such as transmission factors of gears and pulleys, and to enlargement (dilation).A common use of integers in real life is to label and quantify points on a scale, such as temperature and height above sea level. In both cases the location of zero is important to the attribute being measured. For example, both the height above normal sea level (0 m) of a spring tide and the temperature below the freezing point of water (0°C) have significant consequences to the severity of the situation. Zero acts as an important benchmark indicating normality or balance. This is also true in sport or games like Bridge where negative numbers reflect a state relative to expectation, e.g.

^{-}6 in golf means six under par when a player has taken six fewer shots than the expected norm.## Specific Teaching Points

Integers are an extension of the whole number system. Therefore, the properties of Integers under the four operations should be the same as those for whole numbers. With addition and subtraction four main properties hold:

The commutative property of additionThe order of the addends does not affect the sum. If

^{-}3 + 4 = 1 then 4 +^{-}3 = 1. Note that the commutative property does not hold for subtraction. For example, 4 -^{-}3 = 7 but^{-}3 – 4 =^{-}7.The distributive property of additionThis property is really about the partitioning of addends and recombining those addends. For example, if 5 =

^{-}1 + 6, then^{-}2 + 5 = (^{-}2 +^{-}1) + 6. This property does not hold for subtraction.The associative property of additionThis property is about ‘associating’ pairs of addends one pair at a time. For example, (

^{-}4 + 3) +^{-}1 =^{-}4 + (3+^{-}1). This property does not hold for subtraction.Inverse operationsAddition and subtraction are inverse operations so one operation undoes the other. For example,

^{-}2 +^{-}3 =^{-}5 so^{ -}5 -^{-}3 =^{-}2.It is the need for these number laws to hold that establishes the effect of operations, such as subtracting a negative integer has the same effect as adding a positive integer.

This unit combines two of Hans Freudenthal’s (1983) models for operations on integers, the annihilation and vector models. In the annihilation model positive and negatives cancel each other, so +1 and -1 pairs equal zero. The act of creating or removing one positive and one negative pairs that equal zero does not alter the quantity being represented. The vector model presents integers as magnitudes with direction. +1 is represented by a vector of length one in a positive direction and -1 as a vector of length one in a negative direction. Freudenthal cautioned that a quantity of +1 or -1 was easily confused with the operation of adding or subtracting one and teaching needed to make that difference explicit.

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 unit uses the contexts of money, positive and negative spaces, and scale. Other contexts might better suit the interests and cultural backgrounds of your students. Students interested in sports might enjoy golf as a context, and those who enjoy computer find points schemes interesting. Some students may enjoy the context of comparing temperatures from locations around the world, and finding out which locations have the most and least variation in a full year.

## Session One

This session introduces negative integers in real life and presents integers as vectors.

^{-}8 to 8. Mention that usually we do not write^{+}8 for 8 though we could.Ask one student to act out the addition of integers as vectors. Addition is the operation of combining quantities. Ask him or her to begin on zero and act out the cards they are given. Draw a set of three cards randomly (made from Copymaster 1). The

^{+}1 card represents one unit in a positive (right) direction and^{-}1 represents one unit in a negative (left) direction. As the student walks out the movement given by the cards explain to the whole class why they are moving in that direction and the distance of that move.Act out several combinations of cards then ask the students if it is possible to predict the finishing location if the cards are known. Choose a set of five cards and invite predictions.

What do you notice about what happens when^{ +}1 and^{-}1 cards combine?Is that also true of^{-}1 and^{+}1?How can the fact that positive and negative cards form zeros be used to predict the finishing location?^{+}1 and^{-}1. Each turn the cards are shuffled and the top four cards turned overone at a time. With a counter the player acts out the moves, starting at zero. They finish each play by leaving the counter at the finishing number on the board. Play continues like that for 16 turns.The ‘house’ gets any counters on

^{-}2 or^{+}2 and the team gets any counters that are left.Is the game fair?

What cards will give^{+}4? What about^{ -}4? (You might record^{-}1 +^{-}1 +^{-}1 +^{-}1 =^{-}4 or 4 x^{-}1 =^{-}4.)What cards will give you^{+}3 or^{-}3? (Impossible? Why?)What cards will give you^{+}2? (^{+}1,^{+}1,^{+}1,^{-}1) How many different orders can the cards come in?How can you use the number of ways to get^{+}2 to find the ways to get^{-}2?What other finishing numbers are possible? (Only zero – Why?)How many orders of cards will give zero?This table shows all the possible outcomes for the four card game:

Card One

Card Two

Card Three

Card Four

Total

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4

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2

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2

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2

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2

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0

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0

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0

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0

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0

←

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→

→

0

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←

←

^{-}2←

→

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^{-}2←

←

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←

^{-}2←

←

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→

^{-}2←

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^{-}4Session Two

In this session students explore the ‘Hills and Dales’ context for application integers. The context was used in the Oscar nominated film “Stand and Deliver” about Jaime Escalante, an American teacher working with disadvantaged students in Los Angeles. A short video of him teaching algebra using the ‘Hills and Dales’ model is easily accessed online. The video finishes with Escalante asking his students why a negative number multiplied by a negative number gives a positive answer. Good question!

In this unit the context is about road builders. In real life one of the largest costs of new roads is relocation of earth, particularly if earth must be brought in from off-site. Roads are best flat and both hills and dales present potential costs unless a hill can be used to fill a dale.

^{+}1) might fill a dale (^{-}1) to create flat land (0). With each slide ask the students two questions:What will happen in this situation?What is the equation for this situation?For example, scenario one is three hills (

^{+}3) and two dales (^{-}2). You might connect that scenario to the arrow cards by getting three^{+}1 cards and two^{ -}1 cards. Students should recognise that you are combining integers and offer the equation 3 +^{-}2 = 1 or^{-}2 + 3 = 1. In fact, noting the commutative property holds with Integers is important. Work through the four scenarios before giving the students Copymaster 3 to complete independently.The foreman notices that there are three more dales than hills.Draw some landscapes where that would be true.What is the same aboutall the landscapesyou could draw?## Session Three

In this session students explore the addition of Integers in the context of dollars and debts. The net financial position of a person is the sum of the money they have available and the debts they owe.

Layla has $7 in her bank account. She owes her parents $3. How well off is Layla?^{ +}1 representing $1 and^{-}1 representing a bill for $1.Layla has $7 in her bank account. She owes her parents $3. Her parents say they will not need her to repay $2 of the debt. How well off is she now?This story can be represented by the equation 4 -

^{-}2 = 6. Originally Layla’s position was $4, accounting for her money and debt. Removal of a two dollar debt is represented by -^{-}2.Is Layla better or worse off by her parents’ decision?Layla has $8 in her bank account. She owes her parents $13. Her parents say they will not need her to repay $5 of the debt. How well off is she now? (^{-}5 –^{-}5 = 0)^{-}5## Session Four

In this session the vector model is connected to the Hills and Dales and Dollars and Debts models. The aim is to generalise addition and subtraction of Integers. It is important to distinguish the vectors that represent positive and negative numbers and the addition and subtraction as operations, addition as movement to the right and subtraction as movement to the left.

What do these models have in common?Invite students to give the balance of

^{-}2. Ask:Where is^{-}2 is in each model?If we start with a balance of^{-}2 and subtract 3, what does that look like in each model?^{-}2 – 3 =^{-}5^{-}2 – 3 =^{-}5^{-}2 – 2 =^{-}4^{-}2 – 1 =^{-}3^{-}2 – 0 =^{--}2What happens if -1 is subtracted (removed)?^{ -}1. Continue the pattern to^{-}2 –^{-}3 =^{-}1. Discus the positive effect of removing a negative amount, that is the balance is greater than before (There is more earth and more money).^{+}3 – 2 =^{-}_^{+}3 – 1 =^{-}_^{+}3 – 0 =^{-}_^{+}3 –^{-}1 = _^{+}3 –^{-}2 = _Ask them to justify their answers using one of the models that have been used.

## Session Five

In this session the vector model is developed into a number line model which highlights the direction of change when Integers are added and subtracted.

When two numbers are added the sum is always greater than the number you start with.True or False or Sometimes TrueWhen a number is subtracted from another the difference (answer) is always less than the starting number.True or False or Sometimes True## Getting partial to percentages

This unit supports students to recognise percentages as equivalent fractions, and to carry out simple calculations involving finding percentages of amounts.

In this unit we build on work by Prediger and Pohler (2015) to develop students’ concept of percentages using the metaphor of a data download. The model is linear which has been effective in the teaching of fractions and decimals.

Percentages are a specific type of equivalent fraction with a denominator of 100. The symbol %, is derived from the vinculum of a fraction /, and the two zeros from 100. So, a percentage literally means “for every hundred.”

In many situations a percentage describes a part-whole relationship. For example, Mani lands 24 out of 30 shots in her netball game. What is her shooting percentage? The calculation of her percentage is 24/30 = ?/100. There is an assumption of

homogeneity, that Mani will shoot at the same rate.Percentages are very useful to establish a common base for comparison. Most situations in which part-whole relationships need to be compared, the whole (or bases) are not the same. For example, Mani takes 30 shots, but Alicia takes 28 shots. Some situations involve comparison of different wholes or different parts. In those situations, percentages greater than 100% are possible. For example, there are 12 girls and 18 boys. The number of boys is 150% of the number of girls.

In other situations, percentages are used as a fractional operator, e.g. 30% of $45. That calculation is equivalent to 30/100 x 45 = ?. Common examples of percentages as operators are discounts in shops, and interest on loans.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. The difficulty of tasks can be varied in many ways including:

The contexts used in the unit can also be adapted to cater for the cultural backgrounds and interests of students. Choose situations that are likely to be familiar to your students. The unit uses situations around shopping, download bars, and popular travel destinations, which will appeal to many students. Other situations such as sharing collected food, a walk or cycle between two local points, battery charge remaining in a computer game, or sports points, might be more motivating to your students.

## Session One

When we go to a 25% off sale, how much is taken off the normal price?(one quarter)How much of the normal price do we pay?(75% or 3/4)What does it mean when you pay 75% of the normal price, but the item does not normally cost $100? Say it costs $60.00?What number replaces the question mark to show how much you pay for the item?Students should use their knowledge of equivalent fractions, and recognise that multiplying both the numerator and denominator of three quarters by 15 gives 45/60. So, the item costs $45 at the sale.

Students must be able to justify the percentage to fraction equivalence. For example:

_{1}: 25 multiplied by four equals 100 so 25% is one quarter of 100%_{2}: 50% equals one half, so 25% equals one quarter since half of 50 is 25.## Session Two

In the next two sessions, students explore using the percentage bar as a model for solving a variety of part-whole percentage problems. The language of rate, amount, and base are introduced. Calculation steps are derived from the strategies that students use when solving problems with the percentage bar.

Put your hand up when (fraction) of the data is downloaded.Justify why most of you chose this spot (pointing to download bar).Vary the fractions to include quarters, thirds, fifths and tenths.

Basemeans the unit of measurement or comparison. The percentage is calculated as a fraction of the base. Sometimes the fraction is more than one, so the percentage is greater than 100%.Ratemeans a relationship between two measures. The relationship is multiplicative not additive. The word per is used to mean ‘for every’. For example, 60 km/h means 60 kilometres pass for every hour. 75% means 75, of the amount, for every 100 of the base.Amountmeans the result of applying the rate to the base. For example, if the base is 40 and the rate is 25% then the amount is 25% of 40 = 10.## Session Three

Here is a percentage problem for you to solve.What is the base? (24 people)What is the amount? (from 9, 5, 4, and 6)What is missing for you to find out? (the rate or percentage)Ask them to use the marks on the wheel to create a pie chart of the holiday destinations from Slide One.

What fraction of the circle have you given to Fiji? Explain why you did that.Samoa was chosen by 25% of students. What fraction is that?Session FourThis session extends application of the percentage bar to more complex percentage problems. The previous sessions were about part-whole situations. This session introduces the use of percentages to compare two different whole or two different parts.

Begin with this problem:In this story we have two files to download at the same time. One file is 12 kilobytes in size and the other is 18 kilobytes. Let’s look at what happens if the files download at the same rate, in bytes per second.Slide One of PowerPoint Four has an animation of the respective downloads. You may want to play it twice or three times.

What did you notice about the percentages?Students might notice that the percentage for the smaller file is one and one half times the percentage for the larger file. Note that 1 ½ = 3/2. The percentage for the larger file is two thirds the percentage for the smaller file. Note that 2/3 is the reciprocal of 3/2, meaning the numerators and denominators are swapped.

What will happen if the animation bars keep growing until the large file is 100% loaded?What will the percentage be for the smaller file?Is the percentage more or less than 100%? How do you know?Expect students to recognise that if the base is greater than the amount then the percentage is less than 100%. If the base is less than the amount, the percentage is greater than 100%.

Family and whānau,

This week we have been learning to solve percentage problems. We have used a diagram called the percentage bar to visualise the problems and to make good estimates of the answer.

We have used file downloads as a context for our work. Ask us about what we have learned. You might use this context:

I scored 24 points in my basketball game. Henry scored 16 points.

What percentage is my score of Henry’s score?

## Solid Understanding

In this unit students make and investigate various solids, including regular and semi-regular polyhedra, and cylinders and cones. They look for patterns in the numbers of faces, edges and vertices they see whether they can “discover” Euler’s famous formula. By truncating the vertices of the Platonic solids, they find a class of semi-regular polyhedra. They also generalise how to know if arrangements of polygons will create a closed polyhedron and the common properties of nets for prisms and pyramids.

A polyhedron (singular) is a three-dimensional solid object which consists of a collection of polygons that bound a space. That means that the space is fully enclosed by the polygons. The simplest polyhedra (plural) are created by joining regular polygons, such as equilateral triangles, squares and regular pentagons. This family of polyhedra are known as the Platonic solids, named after the Greek mathematician Plato (though actually proved by Euclid).

There are 5 platonic solids, the cube (6 squares, 3 meeting at each vertex), the tetrahedron (4 triangles, 3 meeting at each vertex), the octahedron (8 triangles, 4 meeting at each vertex), the dodecahedron (12 pentagons, 3 meeting at each vertex), and the icosahedron (20 triangles, 5 meeting at each vertex).

Terms commonly used to describe the attributes of polyhedra include:

In the 1750’s Leonhard Euler discovered a famous relationship between these three properties. The number of vertices, plus the number of faces take away two equals the number of edges.

E = V + F - 2

This unit involves exploration with three dimensional shapes and would be ideal as a lead in to the unit

Building with Triangles, a level four unit which goes on to look at a group of polyhedra, the Platonic solids, in more detail.This unit builds on the Level Three unit called Polyhedra (3D Shapes). It extends students’ knowledge of polyhedra to include Euler’s theorem and the properties of shapes around a vertex for semi-regular solids.

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

This unit is focussed on the construction of specific geometric shapes and as such is not set in a real world context. There are ways that it could be adapted to appeal to the interests and experiences of your students. For example, students could be given the opportunity to decorate a model of their favourite polyhedron in a style of their choosing for a class display. This could range from cultural motifs to favourite colours, patterns or images.

Session OneWhat do you notice about these solids?What shapes make up each solid? (Platonic solids are made from one type of regular polygon)What are the flat surfaces (polygons) that make up the solid called? (faces)How many faces does each solid have? How did you count the faces systematically?Look for edges and corners (vertices – singular is vertex). How many edges and vertices has each solid got?How many triangles will there be? (Twenty since icosa- is the prefix for twenty)How many corners (vertices) will the twenty triangles have in total? (20 x 3 = 60)How many of the sixty corners form each vertex of the icosahedron? (Refer your students to the model and systematically count the five triangles that surround each vertex)If five triangle corners make one vertex, how many vertices must the icosahedron have? (60÷5 = 12)Could the same kind of thinking help us to work out the number of edges? (The twenty triangles have 20 x 3 = 60 sides. Two sides are needed to make one edge. Since 60÷2 = 30 the icosahedron must have 30 edges.)## Session Two

In this session students work with truncations of the Platonic Solids. A truncation occurs when the vertices are ‘cut off’. PowerPoint Two begins with the simplest truncation, that of a cube.

These two solids are related yet they look so different. How are they related?Students might notice if the corners of a cube are cut off the result is the right-hand solid, called a truncated cube. You can demonstrate truncation using a cube cut from a large potato or kumara. Where vertices are truncated new triangular faces are formed.

How many faces, edges and vertices will the truncated cube have?What shapes are the faces? Why are the faces those shapes.Expect your students to use structure as well as visualisation to answer the question. For example, there are eight vertices on a cube so changing each vertex into a triangular face adds eight new faces. That gives a total of 6 + 8 = 14 faces. The faces made from a vertex are triangles because three squares meet at each vertex of a cube. That also means that 8 x 3 = 24 new edges are added making 12 + 24 = 36 edges in total. Each of the eight vertices of a cube are replaced by three new vertices. That makes 8 x 3 = 24 vertices.

A truncated cube has 14 faces, 24 vertices and 36 edges. Does it still fit Euler’s theorem?Show the students slide two of PowerPoint Two. Ask the students to name the Platonic Solids (Tetrahedron and Octahedron) and recall the properties of the solids, including numbers of faces, vertices and edges.

Here is the challenge. Use the connecting shapes to build the truncations of these solids. Before you start, visualise what the truncated solid will look like. Then build it.Give your students ample time to construct the solids. Look for the following:

Do your students apply structure to anticipate the result of truncation? For example, a tetrahedron has four vertices so four new faces will be formed by truncating each vertex. For the octahedron six new faces will be formed.

Do your students predict the shape of the new faces from the number of triangles around each vertex of the original solid? The tetrahedron has three triangles around each vertex but an octahedron has four. Therefore, the new faces will be triangles for a tetrahedron and squares for an octahedron.

Can your students anticipate how the original faces will be changed? Cutting off the vertices of triangles results in hexagons.

Students can check their models against the picture on slide three of PowerPoint Two. Draw their attention to the arrangement of shapes around each vertex. For the truncated tetrahedron, one triangle and two hexagons surround a vertex. For the truncated octahedron, the arrangement is one square and two hexagons.

Challenge your students further.

Here are the dodecahedron and icosahedron. Imagine these solids are truncated.What shapes would the faces of the truncated solid be?How many faces would there be?Let your students solve the problem in small groups. Access to the models they built in Session One will be helpful. To build a truncated dodecahedron requires 20 triangles and 12 decagons (ten sided polygons). Therefore, a model cannot be built with a normal set of connecting shapes. However, a truncated icosahedron can be built from 12 pentagons and 20 hexagons.

Slide four of PowerPoint Two has images of both truncated solids. Ask students to name the arrangement of shapes around each vertex. That arrangement is consistent for the whole solid.

It is hard to visualise how many edges and vertices each solid has. You do have Euler’s theorem to help. Use your knowledge of the faces to work the numbers out.Look for your students to use the properties of shapes meeting at vertices and edges to solve the problem. They could organise the data in a table:

Solid

Number of faces

Number of edges

Number of vertices

Truncated Dodecahedron

20 triangles

+ 12 decagons = 32

(20 x 3) + (12 x 10) ÷ 2 = 90

20 x 3 = 60

Truncated Icosahedron

12 pentagons

+ 20 hexagons = 32

(12 x 5) + (20 x 6) ÷ 2

= 90

12 x 5 = 60

Does Euler’s theorem hold for both solids?## Session Three

In this session students create nets for the Platonic solids and possibly their truncations.

Use protractors, rulers and scissor to make and cut out an equilateral triangle, a square, a regular pentagon, a regular hexagon, a regular octagon and a regular decagon. Every side must be 5cm long. Use light cardboard.The purpose of making the shapes is to create templates to form nets with. Some students will need support with creating the polygons. Ideally students will use the sum of internal angles to work out the angle measures. That is a nice investigation but may interfere with the flow of this unit.

Number of sides

Name of polygon

Sum of internal angles

Each internal angle

3

Equilateral triangle

180°

60°

4

Square

360°

90°

5

Regular pentagon

540°

108°

6

Regular hexagon

720°

120°

7

Regular heptagon

900°

128.57°

8

Regular octagon

1080°

135°

9

Regular nonagon

1260°

140°

10

Regular decagon

1440°

144°

The halves can be joined to form the full net.

## Session Four

In this session students find the properties of shapes surrounding a vertex that can be used to predict whether a polyhedron will be formed.

What is the angle defect, that is the angle that is missing from a full turn of 360°?How many vertices does the octahedron have? (six)There are six vertices where the angle defect is 120°.What is the total of six defects of 120 degrees? (6 x 120 = 720°)Maybe that is just co-incidence. Check out the total of the angle defects of the other Platonic solids.The students should notice that the sum of the angle defects is always 720°.

How could we use this theorem to see if an arrangement of shapes will make a closed solid?Could we then know how many vertices the solid will have?The truncated cube has an angle defect of 30°, and 24 x 30 = 720, so the solid will have 24 vertices.

The truncated octahedron also has an angle defect of 30°, and 24 x 30 = 720, so the solid will have 24 vertices.

## Session Five

In this session students explore the classes of solids known as prisms and pyramids. Assigning solids to these classes allows the students to generalise the structure of nets and volume formulae. In the process a cylinder can be regarded as the ‘limiting case’ of a prism and a cone as the ‘limiting case’ of a pyramid.

What is the same about each solid?What differences are there among the solids?How do the similarities among the prisms show in their nets?How do the differences show?Students should note that all three nets have rectangular faces, and the number of those faces matches the number of sides of the end faces. The endpoint faces are those that give the prism its cross-section.

Is this solid a type of prism?Most definitions of a cylinder classify it as a curved surface. A prism is a type of polyhedron which means it is bounded (enclosed) by flat polygons. However, it is advantageous to consider the similarity of a cylinder to a type of prism. A cylinder has consistent cross-section (a circle) so the volume is worked out identically to any prism. Ask the students to sketch the net for a cylinder. The standard net looks like this:

The net has much in common with the other nets for prisms. The two end faces are circles and there are an infinite number of infinitely small rectangular faces which form a continuous whole. The length of the whole rectangle is equal to the circumference of the circle cross-section.

How do you find the volume of a cuboid? (rectangular prism)Students are likely to suggest length x width x height. Rebuild the cuboid model and ask how many 1 cm

^{3 }place value blocks will fit in. Highlight that multiplying two dimensions is like finding the area of the cross-section. Multiplying by the other dimension layers the cross-section.How do you think we could find the volume of the triangular and hexagonal prisms, and the cylinder?The same method applies. Find the area of an end face (triangle, hexagon, and circle) and multiply that area by the other dimension. Students may know the formula for the area of a circle a = πr

^{2}, though that knowledge is not expected at Level Four. That means the volume of a cylinder involves the area of the circular face multiplied by height (v = πr^{2 }x h). You might test the formula out with a cylinder, e.g. a tin can. Measure the radius and height in centimetres and calculate the volume in cubic centimetres. Fill the container with water and measure to see if the capacity in millilitres matches the volume, since 1 cm^{3}= 1 mL.Expects students to notice a base and triangular faces converging to an apex. Students should also note that the bases are different. Like prisms, pyramids are named by their bases, e.g. square based pyramid.

^{2 }x h so the volume of a cone is given by v = ⅓πr^{2 }x h.If you are ambitious, you might test the volume formula out by making a cone with card, lining the cone with plastic wrap and filling it with water.

Dear family and whānau,

This week we have been exploring polyhedra, which are 3-dimensional shapes made from 2-dimensional shapes. Ask your child to explain how these solid shapes have faces, edges and vertices. For homework your child has been asked to either:

## Getting partial: Multiplying decimals

This unit requires students to apply their number sense about the size of decimals to estimate and calculate the product of decimal fractions. In doing so they generalise about the effect of multiplying and dividing by ten and one hundred.

In this unit the students will develop number sense related to multiplying decimals. Making sensible estimates for the products of decimals requires a flexible connection of place number understanding with whole numbers, the meaning of multiplication, and multiplication with fractions.

In the age of calculators, it is a dubious use of instruction time to concentrate extensively on pencil and paper algorithms. Rather, emphasis should go on giving students strategies to recognise the reasonableness of answers and to estimate the products of decimals.

The decimal system includes a restricted set of equivalent fractions for situations where one units are inadequate for purpose. Commonly situations where tenths, hundredths, thousandths, etc. of units are needed involve measurement. The prefixes deci (tenth), centi (hundredth) and milli (thousandth) are applied to base units, such as metres and litres, to obtain a necessary degree of precision. Tenths, hundredths, thousandths, etc. are the result of division by ten of the previous larger unit. Therefore, understanding the effect of multiplying and dividing a decimal amount by ten, one hundred, and one thousand is foundational to estimating the products of decimals. The effect of division by ten is to make each unit one tenths of its previous size, represented as a shift of the digits one place to the right. For example, 3.9 ÷ 10 = 0.39, 3.9 ÷ 100 = 0.039, 3.9 ÷ 1000 = 0.0039.

Applying multiplication to measurement situations often involves a multiplier and a rate. A rate is a relationship between two measures, such as 60 kilometres per hour (velocity), 456 kilograms per cubic metre (density), or 30 people per square kilometre (also density). The multiplier is applied to the rate resulting in a measure, e.g. 3 kilograms of meat at $12.50 per kilogram costs $37.50.

Decimals might more correctly be termed decimal fractions. Deci is the prefix meaning one tenth, to indicate that decimal fractions are powers of one tenth, or negative powers of ten, e.g. 1/1000 = (1/10)³ or 10⁻³. Estimation of the size of products requires understanding of the multiplication of decimal fractions. Knowing 1/10 x 1/10 = 1/100 and 1/10 x 1/100 = 1/1000, etc., makes it possible to know the size of the products for factors like 0.6 x 0.4 and 0.3 x 0.08. Expressing both multiplications as fractions gives 6/10 x 4/10= 24/100 and 3/10 x 8/100 = 24/1000, so the products are 0.24 and 0.024 respectively. A sound understanding of multiplication of fractions is therefore a prerequisite for multiplying decimals with a sense of the size of product.

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:

confirm predictionsabout the results of multiplying decimals. Where differences between predictions and outcomes occur, ask students to work out where their predictions went awry. This strategy helps to cause cognitive conflict and address common misconceptions. The area model is particularly good in identifying where the size of units is misinterpreted.The contexts for this unit are about rates, which is unavoidable with multiplication. Situations involve electricity and fuel consumption, density, and cost. Adapt the contexts if needed to meet the interests and backgrounds of your students. For example, using lengths of fabric at a decimal amount per metre, the cost of petrol per litre, or costs of vegetables, seafood, or other ingredient per kilogram, might be more familiar to your students. For students who have whānau in other nations, such as India or Samoa, currency exchange rates might be of interest.

## Session One

In this session students are introduced to multiplying decimals in everyday contexts. They refresh their understanding of multiplying fractions and the size of decimals using an area model. Students should work through the units Getting Partial to Fractions and Getting Partial to Decimals before attempting this unit.

What operation do we need to carry out to solve this problem?

Treat the significant digits as whole numbers to calculate the product:

451 x 100 = 45 100 45 x 208 = 9 360 15 x 537 = 8 055

What does 0.4 mean?(four tenths)What does 0.6 mean?(six tenths)What does x mean?(of)What does equals mean?(the same amount as)How big is the area that represents 0.4 x 0.6?(24 hundredths)What is the answer to 0.4 x 0.6?(0.24)## Session Two

In the next two sessions student learn more about the place value structure of decimals. They connect their knowledge of multiplication with fractions to finding decimals place value units, especially tenths of tenths, tenths of hundredths, and hundredths of hundredths. The learn about the effect of multiplying and dividing a number by ten and one hundred.

Who can mark one tenth of one tenth on this square metre?(You might use a place value block flat to mark the area)What will you call one tenth of one tenth?(one hundredth of a square metre(m²))Note that one hundredth of a square metre is a square decimetre (dm²) though the unit is rarely used in New Zealand.

How might we record finding one tenth of one tenth?(1/10 x 1/10 = 1/100 or 0.1 x 0.1 = 0.01)How do I write that calculation using fractions?(6/10 x 5/10 = 30/100 )Why is the answer not 0.030?(Thirty hundredths equals three tenths)What is the effect of multiplying/dividing by ten?3.7 x 10 = 3.7 x 100 = 3.7 ÷ 10 = 3.7 ÷ 10 = 1/10 x 3.7= 1/100 x 3.7=

Be aware that students may not see that 3.7 divided by ten has the same result as one tenth of 3.7. That is an important connection if they are to use the effect to estimate the results of decimal multiplication.

## Session Three

## Session Four

This session is devoted to applying multiplication of decimals. The beginning explicitly confronts common misconceptions about decimals as a window to the understandings that students have developed so far. The final part of the session involves a game where multiplication of decimals is required.

To PlayPlayers take turns to:

Family and whānau,

This week we have been exploring multiplication of decimals. We have used an area model to show how the answer to a decimal multiplication can be estimated using our fraction knowledge. A lot of everyday situations involve decimals. We have explored exchange rates, electricity and fuel consumption, and density of firewood.

## Cubic Conundrums

This unit provides a set of learning tasks that integrate across the strands of the Mathematics and Statistics learning area of the New Zealand Curriculum and provide opportunities for assessment of student achievement across those strands. Each session may involve more than one lesson, especially if students need more time to become proficient at the tasks.

Session OneUse systematic approaches to find all the possible outcomes, e.g. tree diagrams, organised lists.

Session TwoUse tables, graphs, and word rules to represent growing patterns.

Session ThreeDraw cube models using plan views.

Session FourDraw cube models using isometric projections.

Session FiveFind the volume of cuboids given edge lengths.

The mathematics in this unit spans all three strands of the Mathematics and Statistics learning area of the New Zealand Curriculum. Key ideas are:

## Specific Teaching Points

Observations of students during this unit can be used to inform judgments in relation to the Learning Progression Frameworks. Click for tables of guidelines.

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:

Activities can be adapted to meet the interests and cultures of your students. Where necessary use internet resources to support students to understand contexts. For example, students may not be familiar with apartment complexes, and architectural drawings. Find resources that broaden their world view. As much as possible, give students the opportunity to be creative mathematicians, for example, create their own cube structure to draw. Foster a climate of problem posing as well as problem solving. Contexts can also be altered, for example a townhouse of two colours might become a tātua, traditional woven belt, and a cube structure might be reframed as the contours of pa site. Arrangments of foods for storage, e.g. kumara in pits, might provide a nice context for volume.

## Session One: Two Colour Townhouses

In this session students explore combinations with two or more colours in a probability situation. They link the combinations to powers of two and to possible outcomes.

How many possibilities might we expect for the four-storey townhouse?What are the chances that the buyer will get a townhouse of only one colour? (two out of 16 or one-eighth)

What are the chances that the buyer will get an alternate pattern?, i.e. dark-light-dark-light or light-dark-light-dark. (Also two out of 16 or one-eighth)

What are the chances of the buyer getting two storeys of each colour? (six out of 16 or three eighths)

Students might make up other possible colour options and work out the probability of that option occurring. They might act out the drawing cubes scenario to see what colour schemes they actually get with 16 attempts. Does every possibility come out? Why is this unlikely? High achievers might investigate the connection between the possible townhouses and Pascal’s triangle.

## Session Two: Square Doughnuts

In this session students use connecting cubes to create growing patterns and attempt to find general rules for those patterns. They also graph the growing patterns using spreadsheets and look for patterns in the ordered pairs.

Imagine someone gave you 96 cubes to build a model in this pattern. How long would the side length of your model be in cubes?Why would the sequence going up in fours give you a rule that uses ‘multiply by four’?Would the problem be easier if you had 400 cubes instead? Why? What would you do to solve it?

The problem can be solved by auto-filling but encourage students to suggest formulas that might be used. One possibility is to put the formula =A5+1 into cell A6 and =B5+4 into cell B6. These are recursive formulas that work out members of the sequence from the one before. Copying down the formulas into columns A and B will continue the pattern.

## Session Three: Spylights

In this session students explore plan views to represent three dimensional models made with cubes. They interpret the plan diagrams and create their own diagrams for others to build from.

What does each square represent?(A single face of a cube)How many cubes across is the building?(Four as shown by both the front view and Satellite image)What is height of the tallest part of the building?(Four cubes high as verified by both the front and left plans)How tall is this part of the building(pointing to the front square of the satellite image)? (Two cubes high at it is the only stack of cubes visible on the right of the left image).## Session Four

In this session students learn to represent models made with connecting cubes using isometric dot paper (orthogonal diagrams or projections). Students will need isometric dot paper (Copymaster 2) and connecting cubes.

## Session Five

In this session students learn to find the volume of a cuboid (rectangular prism) by visualising the arrangement of cubes that will fill it. They also connect multiplication of length, width and depth to the arrangement of cubes.

How many cm³ fit into one of the connecting cubes?(2 x 2 x 2 = 8). So if the number of cubes is multiplied by 8 then that gives the volume in cubic centimetres.How many different cuboids can be made with the cubes that have a volume of sixty 2 x 2 x 2 = 8cm³ cubes? Sixty cubes is a good target volume since 60 has many factors. Slides 6-8 give one possible answer of 3 x 4 x 5 = 60 cubes. Uniqueness is an issue. For example, in this investigation 4 x 3 x 5 should be considered as the same cuboid as 3 x 4 x 5.

Do the students approach the task systematically?

How will you know when you have found all the possibilities?One approach is to close off the first factor, like this:

1 x Possibilities: 1 x 1 x 60, 1 x 2 x 30, 1 x 3 x 20, 1 x 4 x 15, 1 x 5 x 12, 1 x 6 x 10 (all done – How do you know?

2 x Possibilities: 2 x 2 x 15, 2 x 3 x 10, 2 x 5 x 6 (Why we not include any ones as factors)

3 x Possibilities: 3 x 4 x 5 (All done – How do you know?)

Are there other possibilities?(No. Why not?)Parents and caregivers

This week our unit of work will be based around connecting cubes. We will explore how many different four storey towers can be made using just two colours, how to generalise sequential patterns made with the cubes, representing models made with the cubes as plans and isometric drawings, and finding the volume of boxes using the cubes as a measure.