## Early level 3 plan (term 4)

Planning notes
This plan is a starting point for planning a mathematics and statistics teaching programme for a term. The resources listed cover about 50% of your teaching time. Further resources need to be added to meet the specific learning needs of your class, to support your local curriculum and to provide sufficient teaching for a full term.

## Calibrating Clocks

Level Three
Geometry and Measurement
Units of Work
In this unit students construct several simple clocks including water clocks and pendulums. They investigate time as measured by these clocks and make comparisons with standard units of time.
• Measure time in periods of up to 15 minutes.
• Compare non standard and standard measurements for time.

## Multiplication and division symbols, expressions and relationships

Level Three
Number and Algebra
Units of Work
This unit develops an understanding of multiplication and division, including the inverse relationship between the two operations, and when and how to use them in problem solving situations. Students learn the conventions of how multiplication and division operations are represented as equations.
• Read, write and understand the multiplication and division symbols and the equals sign, and the language associated with these symbols.
• Write story contexts for given multiplication and division equations.
• Recognise that the operation of multiplication is commutative.
• Identify related...

## Paint It!

Level Three
Geometry and Measurement
Units of Work

In this unit we measure small quantities of paint accurately to produce our own colour range. We also think about the physics of colour and how our eyes see different colours.

• Accurately measure volumes up to 10 ml.
• Estimate amounts of constituent paint used in given mixtures, up to 10ml.
• Describe the effect of adding incremental amounts of white paint to an existing colour.

## Dare to compare

Level Three
Number and Algebra
Units of Work
This unit encourages students to compare quantities both additively and multiplicatively.
• Describe multiplicative comparison between two quantities using appropriate language, such as, “three times more/greater/bigger”, “three times less/smaller.”
• Record multiplicative comparisons using equations, such as, 4 x 3 = 12 (for 12 is four times greater than three).
• Find the unknown...

## Perspective on Picasso

Level Three
Geometry and Measurement
Units of Work
In this unit students develop a further understanding of two-dimensional representations of three-dimensional shapes through creating drawings and artworks that explore shapes from different viewpoints.
• List the features of common three-dimensional shapes.
• Recognise that the faces of 3D shapes are 2D plane shapes.
• Explore and understand the defining features of a prism.
• Draw (freehand) common 3D shapes from different viewpoints.
• Represent different viewpoints of the constructions with...
Source URL: https://nzmaths.co.nz/user/75803/planning-space/early-level-3-plan-term-4

## Calibrating Clocks

Purpose

In this unit students construct several simple clocks including water clocks and pendulums. They investigate time as measured by these clocks and make comparisons with standard units of time.

Achievement Objectives
GM3-1: Use linear scales and whole numbers of metric units for length, area, volume and capacity, weight (mass), angle, temperature, and time.
Specific Learning Outcomes
• Measure time in periods of up to 15 minutes.
• Compare non standard and standard measurements for time.
Description of Mathematics

Students' experiences with time throughout the learning sequence has two aspects:
duration and telling time.

Telling time must enable them to:

• develop an understanding of the size of the units of time. This includes being able to estimate and measure using units of time;
• read and tell the time using both analogue and digital displays.

Standard units
When students can measure the duration of events using non-standard units, they are ready to move to the use of standard units. The motivation for moving to this stage often follows from experiences where the students have used different non-standard units for the same event. They can then appreciate that consistency in the units used would allow for the easier and more accurate communication of duration.

Measuring with standard units involves the introduction of minutes, hours and seconds in addition to reading time on analogue and digital clocks.

Duration
The minute is often introduced first because it is small enough to measure common events. The duration of a minute can be established by watching the second hand on a clock or by constructing a minute sand-timer. An appreciation for the size of a minute can be built up through lots of experience in measuring everyday events. For example, how many minutes does the song play for? How long is morning break? How long does it take to walk around the school building? How many times can you hop in a minute? How many linked cubes can you join in a minute?

As the students become familiar with the size of a minute they should be given opportunities to estimate before measuring. Minutes need to be linked to the movement of the minute hand on the analogue clock and the digits on digital displays.

An understanding of the size of a second can be developed by investigating the relationship between seconds and minutes. This can be done by watching the digital displays on some watches, on stopwatches and on video-recorders. The students should be encouraged to develop their own reference for a second, for example, a counting pattern "one – banana – two – banana – three etc".

The underlying number skills should be mastered before teaching students the skill of telling and reading time.

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:

• providing support on how to use the timing devices (stopwatches or clocks)
• pairing the students so that they can be supported at the station tasks
• reducing the number of tasks that need to be completed at the stations.

The context for this unit can be adapted to suit the interests and experiences of your students. While the focus is on creating “clocks” that measure time, the tasks can be made more engaging by providing a purpose for each clock that connects to the everyday experiences of your students.  The clocks could then be used by your students for that purpose.  For example:

• a clock that can be used to time whether an egg is soft or hard boiled
• a clock that could be used to measure the time allowed for a school's performance at Polyfest
• a clock that can be used to measure the time spent cleaning teeth or washing hands.

Giving students the opportunity to suggest their own purposes for the clocks and then adapting the stations so that they can tailor the clocks to those purposes is also likely to increase their engagement in the task.

Required Resource Materials
• Getting started: Candles (enough for one per group or pair), matches, large dish of sand for candles to stand in.
• Station One: Plastic containers (e.g. milk bottles, yoghurt pottles), nails to make holes in plastic (variety of sizes), wooden sticks, water, trays or buckets for water collection, water, stopwatches or clocks with second hands.
• Station Two: Pairs of plastic bottles with the same size openings (e.g. soft drink bottles, water bottles), small squares of cardboard,  nails to make holes in cardboard, salt or dry sand (needs to be dry to run through hole), stopwatches or clocks with second hands.
• Station Three: String, a variety of weights, place to attach string to (e.g. edge of desk, overhanging shelf in classroom, empty chart holder), stopwatches or clocks with second hands.
• Reflecting: Variety of timing devices including stop watches, clocks, watches and devices made in the three stations.
Activity

#### Getting started

This session is taken as a whole class activity with students working in groups or pairs to experiment with the time taken for different candles to burn. The teacher retains control of the matches and the burning candles.

Brainstorm with students the different kinds of clocks they know: watches, wall clocks, oven timers etc. Explain that this week they will be making and experimenting with a variety of different clocks.

1. Show the students a candle. Ask them to estimate how long it would take for the whole candle to melt. How much would be melted in 5 minutes?
2. Group or pair students and give each group a candle.
3. Students record the length of their candle, either by drawing a line to show how long it is, or measuring it in cm. They also draw a mark on the candle to show how much they think will be melted in 5 minutes.
4. Collect candles together (maybe in a dish of sand so they stand up easily) and burn all candles for five minutes. Students observe and watch to see the accuracy of their five minute marks.
5. Blow out the candles after five minutes and return them to the students for them to check the accuracy of their mark and record how long the candle is now (either by drawing a line to show how long it is, or measuring it in cm).
6. Students put three more marks on their candles: showing how much they think will be melted in 5, 10 and 15 minutes.
7. Collect all candles together and burn for 15 minutes, observing the accuracy of the marks as they burn.
8. At the conclusion of the activity return candles to students and discuss
How accurate was your candle clock?
How could you make your candle clock more accurate?
What are the limitations of a candle clock?

Over the next three days have students work at the three stations, rotating groups so all students have a chance to try all three activities.

#### Station One: Water Clocks

In this station students make and experiment with water clocks.

Student Instructions (Copymaster 1)

1. Use the materials provided to make a water clock like the one in the diagram. The size of the hole you make and the size of the container you use will affect the way your clock measures time.
2. Estimate the time it will take for the water level to drop from one mark to the next on your clock. Record your estimates in a table.
 water moves down estimated time taken measured time taken 1 mark 2 marks 3 marks 4 marks

1. Measure the time taken for the water level to drop from one mark to the next on your clock. Record your results in a table.
2. Experiment with your clock and the way it measures time.
What happens if you make the marks further apart?
What happens if you make the marks closer together?
Can you make your clock measure 1 minute accurately?
Can you make your clock measure 5 minutes accurately?

#### Station Two: Sand Timers

In this station students make and experiment with sand clocks similar to egg timers.

Student Instructions (Copymaster 2)

1. Use the materials provided to make a sand timer like the one in the diagram. The size of the hole you make will affect the way your clock measures time.
2. Estimate the time it will take for the sand level to drop from one mark to the next on your clock. Record your estimates in a table.
 sand moves down estimated time taken measured time taken 1 mark 2 marks 3 marks 4 marks

1. Measure the time taken for the sand level to drop from one mark to the next on your clock. Record your results in a table.
Were the times between the marks the same?
Why / Why not?
How could you make the time between each mark the same?
2. Experiment with your clock and the way it measures time.
What happens if you make the marks further apart?
What happens if you make the marks closer together?
Can you make each mark measure the same time period?
Can you make your clock measure 1 minute accurately?
Can you make your clock measure 5 minutes accurately?

#### Station Three: Pendulums

In this station students make and experiment with pendulums to tell the time.

Student Instructions (Copymaster 3)

1. Use the materials provided to construct a pendulum like the one in the diagram.
2. Start your pendulum swinging to measure time.
3. To help your pendulum measure the same each time you use it, you will need to
• make sure you hold the pendulum in the same place each time you release it
• make sure you let the weight go rather than push it
• count the number of swings carefully from the highest place in the swing each time.
4. Estimate the time it will take for the pendulum to swing 5, 10, 15 and 20 times. Record your estimates on a table.
 number swings estimated time taken measured time taken 5 10 15 20

1. Measure the time taken for the pendulum to swing 5, 10 15 and 20 times. Record your measurements on a table.
2. Experiment with your pendulum and the way it measures time.
What happens if you make the string longer?
What happens if you make the string shorter?
What happens if you use a heavier weight?
What happens if you use a lighter weight?
Can you make your pendulum measure 1 minute accurately?
Can you make your pendulum measure 5 minutes accurately?

#### Reflecting - Comparing Clocks

This session is taken as a whole class activity with students working with different timing devices to time the same event. Equivalent times are recorded and compared.

1. Explain to students that today they will be using a variety of timing devices to measure the length of set tasks.
2. Distribute timing devices amongst pairs of students. Use a variety of timing devices including stop watches, clocks, watches and the devices made in the three previous stations.
3. One student performs a task while all other students time the task.
4. Record equivalent measures on a chart.
 Task Equivalent measures for time taken Recite the alphabet 55 seconds 1/2 candle mark 2 water clock marks 3 sand clock marks 20 pendulum swings Count to 500 2 minutes, 33 seconds 3 1/4 candle marks 6 sand clock marks 52 pendulum swings

1. Continue for several tasks. When complete review and discuss results.
What are the advantages of the candle clock (and other devices)?
What are the disadvantages of the candle clock (and other devices)?
Which measure would be the most useful if you wanted to describe to someone in London how long a task took? Why
?

## Multiplication and division symbols, expressions and relationships

Purpose

This unit develops an understanding of multiplication and division, including the inverse relationship between the two operations, and when and how to use them in problem solving situations. Students learn the conventions of how multiplication and division operations are represented as equations.

Achievement Objectives
NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.
Specific Learning Outcomes
• Read, write and understand the multiplication and division symbols and the equals sign, and the language associated with these symbols.
• Write story contexts for given multiplication and division equations.
• Recognise that the operation of multiplication is commutative.
• Identify related multiplication and division facts (‘families of facts’).
• Recognise the inverse relationship of the operations of multiplication and division.
• Recognise that division is not commutative.
• Use the words ‘factor’ and ‘product’ appropriately and identify factors of given amounts.
Description of Mathematics

This sequence of lessons makes the connection between repeated addition and multiplication. It introduces division and explores the relationship between the operations of multiplication and division.

There are three main understandings being developed within these lessons.

1. Students need to understand the relationships among quantities that are represented by multiplication and division equations. For example, 4 x 5 = 20 might represent that “Four quantities of five are the same quantity as 20” or “20 is four times greater than 5.”

2. Students need to learn the vocabulary associated with multiplication and division, and the meaning of those words. Important vocabulary includes factors (numbers being multiplied), product (answer to multiplication), multiplied by (making one quantity x times greater), equality (sameness of quantity).

3. Multiplication can also be represented spatially. Arrays are a powerful way of showing the structure and pattern of multiple groups, and, in this case, strongly linking multiplication and division to measurement.

In exploring the structure and pattern of multiplication and division, the focus is also on developing an early understanding of number properties. The commutative property of multiplication is formally explored in these lessons. The distributive property, in which one or both factors are partitioned, (eg. 12 x 55 = 10 x 55 + 2 x 55), is foundational to calculation strategies, including written algorithms.

In exploring the behavior of the operations of multiplication and division, it is important that students make generalisations in which they can state ‘what always happens’ when certain actions are undertaken. For example, recognising that the ‘turn around’ rule (commutative) is always true for multiplication, but that is it not true for division.

This series of lessons focuses on single digit factors and divisors. It recognises that to build a sound understanding of how we use multiplication and division symbols and expressions to think mathematically, and to express relationships, students must have many opportunities to represent the operations for solving word problems. Students also need to be able to create contexts that an equation might express. Making the connections between language and symbols is essential to developing a sound understanding of mathematical ideas and concepts.

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:

• Provide physical materials so that students can anticipate actions and justify their solutions. Use materials like cubes and square tiles to model situations and connect strategies used by the students to the quantities that are represented. Progress to creating array diagrams on squared paper.
• Connect symbols and mathematical vocabulary, especially the symbols for multiplication and division (x, ÷) and for equality (=). Explicitly model the correct use of equations and algorithms and discuss the meaning of the symbols in context.
• Alter the complexity of the numbers that are used. Multiplication with factors such as two, four, five, ten, and division with the same divisors, tend to be easier than factors such as three, six, seven, eight and nine.
• Encourage students to collaborate in small groups and to share, and justify, their ideas.
• Use technology, especially calculators, in predictive, pattern-based ways to estimate products and quotients, e.g. If the answer to 4 x 8 = 32,  will the answer to 32 ÷ 5 by more or less than 8? How do you know? Allow use of calculators where you want students to focus more on the process of getting a reasonable answer, or on noticing patterns, than on practising calculation skills.

The context used for this unit are quilts and tapa cloth. You may wish to change the contexts to situations more relevant to your students’ everyday lives, interests, or cultural identities. Arrays are common across cultures and can be found in tiling patterns, textiles, packaging arrangements, piles for houses, and playing boards for games. Encourage students to be creative by accepting a variety of strategies from others and asking students to create their own problems for others to solve, in contexts that are meaningful.

Required Resource Materials
• At least two rectangular quilts  or tapa cloth
• Coloured plastic square tiles (or small squares of different coloured card)
• Squared paper
• Calculators
• Unifix cubes
• Playing cards
• PowerPoints One and Two
• Copymasters One, Two, and Three

Activity

Session 1

Activity 1

1. Show the students two different rectangular patchwork quilts. Alternatively, use PowerPoint One to show pictures of suitable quilts or tapa cloth. For example:

Suppose that the class is going to make a quilt or tapa cloth for the children’s ward at the local hospital (or hospice).
Engage students in a quilt discussion, establishing how the designs are created.
2. Ask, “ What mathematics is there in these quilts?” (eg. the 3 x 3 quilt)
Record the students’ ideas on the class chart. (These may include number, geometry, measurement statements: for example, 3 + 3 + 3 = 9, 3 x 3 = 9, 9 squares, one big square, sides the same length, 9 divided by 3 etc.). Compare the number of squares in different examples.
3. Highlight the operation and relationship symbols (or words) that have been recorded. For example:
4. Write each symbol on a separate A4 sheet of paper. Ask student pairs to take one sheet (one symbol) and each take turns to record, in 2 minutes, using words and pictures/diagrams, a brainstorm of all they know about this symbol (or word). Ask students to give an example of where their symbol might be used.
5. Ask students to return to the mat, sitting in separate two groups: a group with operation symbols (+ - x ÷) and a group with relationship symbols (< > =). Have selected student pairs explain why they are sitting where they are, and what ideas they have recorded for their symbols.
In this discussion, emphasise the language used, develop the understanding of what a number operation is (a mathematical process that changes an number or amount), and review the meaning of the equals sign.
Keep the brainstorm sheets for future reference.

Activity 2

1. Prepare bags of 12, 18, 20, 24, and 30 plastic tiles, small coloured small squares of card, or fabric squares. Make these, pencils and paper, available to student pairs.

Pose the problem. “Show, using diagrams and equations, how many different ways can you arrange these patches to make a ‘mini patchwork quilt’?”
Have students work in pairs to record their ideas.
2. Have students pair share their ideas with a pair who had the same number of tiles, and record any arrangements they had not thought of.
3. As a class, share ideas, exploring and recording key understandings on the class chart. Keep this student work for Session 2.
For example: From the bag of 18 ‘patches’ (tiles).

In the discussion, build on ideas shared in Activity 1 (above), highlighting and recording in words, these ideas:
• The ‘patch’ arrangements can be recorded using different operations.
• Multiplication, using the x symbol, can show the same idea as repeated addition (of equal amounts), using the + symbol.
• The symbol for division, or sharing into equal groups, is ÷. It’s called the division symbol.
• This arrangement with equal rows and columns is called an array.
4. Pose and record: “ 9 + 9 = 6 x 3. Do you agree or disagree.” Ask student pairs to discuss this claim and be prepared justify their position (explain why they agree or disagree, and how they know that they are right).
Record student justification, highlighting the relationship of equivalence (both are equal to 18, there are 18 patches altogether in both arrays). Highlight multiplicative views such as "9 equals 3 x 3 so 9 + 9 equals 6 x 3".

Activity 3

Write two equations on the class chart, one multiplication and one division.
For example: 6 x 5 = 30      28 ÷ 4 = 7. Read them together.Have each student draw a diagram of a quilt or tapa cloth that represents the equation. Ask them to write a description in words of how the quilt/cloth represents the equation.

Activity 4

Conclude the session by reviewing the operation and relationship symbols and their meanings.

Session 2

Activity 1

1. Begin by having at least two students share their quilt/cloth diagrams from the previous session. Ask the other students to record equations that are represented in the diagram. Highlight the fact that real life maths can be represented using diagrams, words and symbols.
2. Brainstorm on a class chart other situations in our lives where we see and use multiplication or division. As students share ideas, require them to suggest specific numbers. Record these stories using diagrams, and words.
For example: We see multiplication when:
• 12 packets of 20 raisins are wrapped together in a bigger pack – four packets in a row and three rows.
• You buy three packets of chewing gum with ten pieces in each packet
• We make four teams of six for a sports game in PE.
3. Read the stories again together. Ask students to use symbols to record the equations for each of the stories in their books/on whiteboard/paper. Those students who finish quickly can develop more context stories.
Have students pair share their equations. If students have recorded using repeated addition, have them also record multiplication equations.

Activity 2

1. Review symbol information from Session 1, highlighting the operation symbols, + - x ÷ , and the relationship symbols, equals (=), greater than (>) and less than (<) relationship symbols.
Ask students to work in pairs using the situations from the previous activity. Students should discuss the situations, and see how many equations, or inequations they can write.For example:
3 x 4 = 4 x 3
3 x 4 < 2 x 10
4 x 6 > 2 x 10 > 4 x 3
They should use diagrams to show how they know that they are correct.
2. Have students pair share their work. As they do, they should take turns to read aloud what they have written.

Activity 3

1. Return to the quilts/tapa cloth (pictures). Explain that some young children like alphabet quilts in which each patch shows a picture of something starting with a different letter of the alphabet. Talk about what some of these might be. For example: A might show an apple, B a butterfly, C a cat and so on.

2. Make paper, pencils and felt pens available to the students.
Pose the problem: You are going to make an alphabet quilt/tapa cloth for a young child. You have till the end of the session today to plan your design and how you will arrange your 'square patches’. There might be a challenge somewhere in the problem. You decide the best way to solve this for your quilt design.
How many letters are in the alphabet? (26)
Why might making a quilt with 26 squares be a problem?
3. Let students experiment with 26 squares. They might draw the possible arrangements of use square tiles or cubes.
(26 will only make 1 x 26 or 2 x 13 arrays, which is undesirable for a quilt of this kind. Students will encounter ‘a remainder’ (6 x 4 + 2, 5 x 5 + 1) or they’ll find they are some ‘patches’ short (7 x 4). Accept realistic solutions for the context. (e.g. 5 x 5 quilt: put 2 letters on one patch, 6 x 4 quilt: make it 7 x 4 and include 2 novel or blank patches.)
4. Suggest: If we add patches for each of the digits 0-9, how many patches will we have then? (26 + 10 = 36)
See what quilts you could make then.
Look for students to find all the possible arrangements:
1 x 36        2 x 18        3 x 12        4 x 9        6 x 6
Which array is the best quilt/tapa cloth? Why?

Session 3

Activity 1

1. Have students share their alphabet quilt designs for 36 patches.Discuss the ‘remainder issue’ and commend creative solutions.
Why was a quilt with five patches in a row not possible?
Record 36 ÷5 = 7 r 1 and ask students what r 1 means (remainder of 1).
Point out that often division problems don’t work out evenly. We call what is left over, a remainder
2. Imagine we had 26 patches and tried to put six patches in each row. One way we could record this problem is 26 ÷ 6 = 4 r2.
3. On the class chart quickly draw the arrays that were developed for 26 patches.

4. Discuss the ‘dimensions’ of the array, introducing the words factors and product. Model with an example:

Have each student record below their quilt designs, what is in the box above, adjusting the numbers for their own design.

Activity 2

1. Write 4  36   9 on the board.
Here are three other numbers that are connected by multiplication and division.
Record a set of multiplication and division equations using these numbers.

2. Have student work in pairs to develop equations, and to create an array that represents all four equations. Students must be prepared to justify their position (explaining how they know they are right).
4 x 9 = 36     9 x 4 = 36     36 ÷ 4 = 9     36 ÷ 9 = 4
Relate each equation to the 9 x 4 array which students should recognise from the design a quilt activity previously. Pay particular attention to division. For example, 36 ÷ 4 = 9 gives the number of rows created from 36 patches (the area) if each row is made from four patches.

3. Summarise findings on a class chart. For example:
• There are four related facts only (family of facts) and no more.
4 x 7 = 28 7 x 4 = 28 28 ÷ 4 = 7 28 ÷ 7 = 4
• Multiplication is a ‘turn around’ operation. You can change the order of the factors without changing the product. (It’s like addition.)
We say that multiplication (and addition) are commutative.
4 x 7 = 7 x 4 = 28
• Division isn’t commutative, e.g.  36 ÷ 4 = 9 but  4  ÷ 36 = 0.1111… (1/9). The divisions do not have the same quotient (answer).
We say that division (and subtraction) are not commutative.

Activity 3

1. Ask students to play the game Multiply, Draw and Write in pairs.
They need playing cards (with numbers 2 to 9), pencil and paper.
The winner is the person who, after ten rounds, has the most pairs of cards with products that are the same, but which are made with different factors.
For example: 6 x 4 = 8 x 3 = 24,    or 4 x 4 = 2 x 8 = 16
How to play:
The cards are shuffled and placed face down in a pile between both players.
Players take turns to turn over three cards from the pile. These are the factors. The player returns one card to the bottom of the pile. The player must write the multiplication fact/s for the two cards. They might also draw an array and write the family of facts.
For example:
2. Students conclude the session by writing word scenarios for their equation sets (family of facts). These do not have to be quilt scenarios.
For example: “There were three bags with five apples in each. Fifteen shared among three bags is five. If these fifteen apples were put into five bags, there would be three in each. That would be five lots of three.”

Session 4

Activity 1

Show an alternative quilt or tapa cloth array (PowerPoint Two). For example:

Have four students record one of each of the related facts.
(6 x 5 = 30, 5 x 6 = 30, 30 ÷ 5 = 6, 30 ÷ 6 = 5) and explain each fact with reference to the quilt, including demonstrating the commutative (turn around) property of multiplication. Turn the quilt to demonstrate this.

Activity 2

1. Make linking cubes (or coloured counters) available to the students. Have pairs of students each take 48 cubes. Ask what factors could make 48. Record possibilities using multiplication; 1 x 48, 2 x 24, 3 x 16, etc.
2. Have one student of each student pair model 4 x 12, linking the cubes. Then have their partner use the same cubes to model 12 x 4. Discuss what happens. (They needed to regroup these). Repeat with 6 x 8 and 8 x 6. Emphasise that the commutative property involves the same factors and product, but requires a different way to look at the array(i.e. either the rows or the column form the equal sets).
3. Put the multiplication facts for 48 onto cards (Copymaster One). Hold back 5 x ? and 7 x ? Match up the pairs of equations that show the commutative property.
Do you think this is all the multiplication facts with a product of 48? (You could arrange the cards by order of the first factor.)
Why is there no 5 x and 7 x facts? (Use the cards. Students should realise that 48 is not in the set of multiples of 5 and 7. 48 is not divisible by 5 and 7).
Use a calculator to show that 48 ÷ 5 = 9.6 and 48 ÷ 7 = 6.857142857…
What do you think the decimal part of the product shows? (A remainder so 48 is not divisible by 5 and 7)
4. Ask students to explore multiplication facts with a different number of cubes, using the language of same factors and product, focusing on regrouping. There investigation might show that some numbers only have two factors, such as 17 and 31. These numbers are primes.

Activity 3

1. Record one familiar multiplication equation on the class chart. For example, 6 x 2 = 12. Have one of students in each pair model this, by making 6 groups of 2 and linking the cubes together into one line of 12.

Record 12 ÷ 6 = 2. Have the other student in the pair enact this with the cubes.

Have students describe what has happened and record ideas such as: it’s the opposite, division undid multiplication, it’s the reverse, we’re back where we started.
Ask, Is this always true? How can we find out? Accept student ideas. These should include the students exploring more examples.
2. Conclude that it is impossible to try all multiplication and division facts. Say that the ‘undoing’ idea means that multiplication and division are inverse operations, just like turning a light switch on and off. Undoing each other is just the way that multiplication and division behave.

3. Write inverse relationship on the class chart. Discuss words similar to inverse e.g. reverse, undo, put back, return, and their meaning. Make the connection to the inverse relationship between addition and subtraction. Highlight that in each operation pair, one operation or action undoes the other.

4. Return to the quilt in Activity 1 (above) and to the equations recorded:
(6 x 6 = 30, 5 x 6 = 30, 30 ÷ 5 = 6, 30 ÷ 6 = 5)
Ask students to explain the ‘undoing’ (the inverse relationship again, with reference to the quilt. (This is a little harder to see because this array cannot physically be ‘undone.’ However, you might create rows of six cubes and show 5 x 6 by arranging five rows vertically. How many patches do I have? What happens if I now divide by five?)

5. Write on the class chart:
Knowing that multiplication and division are inverse operations is helpful because……..
Have students suggest reasons and record these, including:
We can use multiplication to help us solve division problems.
We can check division operations using multiplication. (How?)

Activity 4

Distribute Copymaster Two for students to work from. Emphasise the inverse operations, and the need for students to show or explain how multiplication helps to solve division problems.

Session 5

Activity 1

Review key learning from Session 4. Have students work in pairs to share their solutions to the quilt problems from Session 4, Activity 4. Encourage them to question each other.

Activity 2

1. Display some quilt or tapa cloth examples using PowerPoint One:

2. Write on the class chart:
One quilt of sixteen patches:
One quilt of thirty patches:
One quilt of forty-five patches:
If I rearranged the patches into a single row, what would the quilt look like? (More like a long scarf)
3. Ask students to record multiplication equations for each of these statements.
One quilt of sixteen patches: 1 x 16 = 16
One quilt of nine patches: 1 x 30 = 30
One quilt of thirty patches: 1 x 45 = 45
If your equations are correct what are the answers to 16 ÷ 1 = ☐ , 30 ÷ 1 = ☐ , 45 ÷ 1 = ☐?
4. Ask students to discuss their ideas, then explain and justify their thinking. Do they relate the division to “How many columns of one patch make a total of 16, 30, or 45 patches?”
If your equations are correct what are the answers to 16 ÷ 16 = ☐ , 30 ÷ 30 = ☐ , 45 ÷ 45 = ☐?
Do they relate the division to “How many rows of 16 patches make a total of 16 patches, etc?
5. Pose other examples of dividing a number by one and itself. Calculators can be used to check answers.

Activity 3

Have students work in groups of 2 to 4 on Is it a fact? (Copymaster Three (Purpose: To discriminate between correct and incorrect multiplication and division equations and expressions, and be able to explain why, justifying their decision)

Students take turns to choose a statement and explain to others in the group, if and why the statement is a fact, or if and why it is incorrect (true or false).

Ask students to create their own facts or non-facts that involve multiplication and division, e.g. 8 x 9 = 72 so 72 ÷ 18 = 4. Exchange the facts and non-facts among students.

Activity 4

Conclude this session by reviewing the learning that has happened over the five sessions.

Attachments

## Paint It!

Purpose

In this unit we measure small quantities of paint accurately to produce our own colour range. We also think about the physics of colour and how our eyes see different colours.

Achievement Objectives
GM3-1: Use linear scales and whole numbers of metric units for length, area, volume and capacity, weight (mass), angle, temperature, and time.
Specific Learning Outcomes
• Accurately measure volumes up to 10 ml.
• Estimate amounts of constituent paint used in given mixtures, up to 10ml.
• Describe the effect of adding incremental amounts of white paint to an existing colour.
Description of Mathematics

In this unit students reads scales, and develop an understanding of the size of a millilitre by working with measurements. They compare volumes using phrases such as “less than” greater than” and “equal to”.

#### Associated Achievement Objective

Science, Physical World AO1: Explore, describe, and represent patterns and trends for everyday examples of physical phenomena such as movement, forces, electricity and magnetism, light, sound, waves and heat.

The concepts covered in this unit are outlined more fully in "Seeing Colours, The Spectrum, the Eye and the Brain", number 11 in the Building Science Concepts series.

The main idea covered is that the colours of objects come from their reflection and absorption of different parts of the spectrum of colours in light.

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

• providing physical materials, paint and measurement tools, so students can experiment, and have real experiences of colour mixing
• explicit modelling of ways to record paint recipes, including how measurements such as millilitres (mL) and litres (L) can be written, and the symbols used in ratios
• allowing access to calculators to reduce the calculation demands for students who need that support.

Tasks can be varied in many ways including:

• manipulating the complexity of the colours students work with. For example, start with just yellow and blue as primary colours
• limiting the number of drops allowable in a paint recipe
• supporting students to organise their recipe creation as an algorithm (sequence of steps). Provide a chart of steps if necessary for some students.

As an extension, students could be encouraged to research the difference between the RYB and CMYK models of colour mixing.

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Colours often have cultural sigificance. For example, green is associated with pounamu in Māori culture, Red is associated with birth, coming into being, and Papatuanuku, Earth Mother. Black represents the heavens and white the coming into light. Be sensitive to the cultural significance of colours for students in your class. In Chinese culture, red represents luck and fertility, but in some African cultures it symbolises death. Green is a sacred colour is Islamic cultures.  Ask your students what ideas they associate with different colours.

Required Resource Materials
• Acrylic paint in primary colours: red, blue, yellow and white.
• Plastic droppers or syringes to measure small amounts up to 5mL (These tools often have scales to measure in mL)
• Paint brushes
• Paper
• Samples of paint charts
Activity

#### Introduction, Session 1

1. Brainstorm students’ ideas about colour.
What can you tell me about colour and how we see it?
How is it that we are able to see colour? (light)
Why do colours appear different in different lights?
How can we create colours?
2. Explain that this week the students will become paint designers and technicians, creating their own new colour range of paints.
3. Allow the students time to experiment with mixing colours of paint to create new colours, focusing on the variety of combinations possible. Start with two colour blends of the primary colours.
4. Let students experiment with blending primary colours. Use droppers or syringes that are repeatedly cleaned to reduce waste. Use paint trays to hold small amounts of the primary colours and black art paper as the painting surface.
What will you need to record so you are able to reproduce these new colours?
Students might recognise that some record of the comparative amount of each primary colour will be useful. They might decide on a drop as a unit of amount (volume) and the use of ratio notation, e.g. 3 yellow: 1 blue. (20 drops is approximately equal to 1 millilitre).
5. Have students share some of the colours they have created and describe how they achieved the effect. Do they record the colours as ratios?
6. Discuss how colours might be altered to make them darker or lighter, more/less blue or red, etc.
7. Show the samples of paint charts.
Why do paint shops produce these charts?
If a customer chooses a colour how do the paint people make a whole can of it?
8. Look online for a video that shows how tint is added to a base in a strict recipe. Even with this precision there is always a disclaimer that colours vary slightly from the charts.
9. Discuss the need for consistency.
If customers were going to purchase our paints they would need to be the same every time.
How will we achieve this?

What units of measurement will we need to use?
10. Discuss how to use the droppers or syringes consistently, as a way to accurately measure small quantities of paint.
Are all drops the same?
What causes the drops to vary?
How might we be more consistent?
11. Outline the unit of one mL as one thousandth of a litre. Showing students a one litre container and a cylinder that has millilitre gradations will help students appreciate the small size of 1mL. A small place value cube is the volume of 1mL. A large place value cube is the volume of 1000 mL or 1 litre.
12. Ask: Suppose the recipe at the paint shop said 1 red: 3 yellow. What colour would that give?
Will all oranges made with this recipe be the same? (Maybe not, but close)
13. Allow students time to practise measuring small amounts, reinforcing the importance of being accurate and reading the scale carefully. Students can work in teams, taking the recipe of another pair and trying to reproduce exactly the same colour. (Note that wet paint changes colour as it dries. It usually lightens.)

#### Exploring, Sessions 2-4

1. Over the next few sessions have students develop 2-3 base colours and record the “secret formula” for these colours. For example, Groovy Green is made from 2.5mL of blue and 4mL of yellow. As they develop base colours they can present these in a paint chart type presentation, developing lighter hues by adding a constant amount of white and also naming these colours. For example, Avocado can be made from 2.5mL blue, 4ml yellow and 1 ml white, Apple can be made from 2.5mL blue, 4 mL yellow and 2 mL white.
2. As students work, reinforce the importance of accurate measurement. Have students reconstruct the colours others have created, using the secret formulae.
Are the colours the same? Why? Why not?
3. Discuss the new colours being created:
What new colours have we made?
Does it make a difference if the colours we mix are light or dark to start with?
If so, how are the resulting colours affected?
(Dark colours have a more pronounced effect on the final colour than light colours)
4. Explain that light is made up of lots of different colours. Most surfaces around us absorb light but not all surfaces absorb all the colours of light. What we see is the part of white light that is not absorbed by the surface. That light is reflected back into our eyes.
What colours make up white light?
What colours are reflected when we see a red object?
What colours are absorbed?

What colours are reflected when we see a blue object?
What colours are absorbed?
5. Have students calculate the formula for larger quantities of paint. For example:
If I wanted to order 2 litres of paint for my kitchen what would the secret formula be?
What if I wanted 6 litres for my dining and living area?

To scale up paint recipes students will need to convert between measures and apply multiplication. For example, Punk Purple is made with a ratio of 2 red: 3 blue : 1 white. How many millilitres of each colour are needed to make a 1 litre pot?
Since 1L = 1000 mL each part will need to be calculated as a fraction. There are 6 mLs per unit ratio, since 2 + 3 + 1 = 6. 2/6 or 1/3 of the mix is red, 3/6 or ½ is blue and 1/6 is while. The amount of each colour is:
1/3 of 1000mL = 333.33 mL, ½ of 1000mL = 500mL, 1/6 of 1000ml = 166.67 mL

#### Reflecting, Session 5

1. Present the students with a colour you have created and named, for example Awesome Orange.
2. Explain that this is one of the colours another paint company has created and they need to discover the secret formula as one of our loyal customers has requested it.
3. Allow the students time to experiment with paints to find the formula. As they work compare the results and the mixtures they have used. For example:
Why is your orange lighter/darker than Awesome Orange?
What do you need to change? Why?
What amounts of paint will you try now?
4. As a conclusion share the secret formula for Awesome Orange and have students share their finished paint charts with the group.

## Dare to compare

Purpose

This unit encourages students to compare quantities both additively and multiplicatively.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
Specific Learning Outcomes
• Describe multiplicative comparison between two quantities using appropriate language, such as, “three times more/greater/bigger”, “three times less/smaller.”
• Record multiplicative comparisons using equations, such as, 4 x 3 = 12 (for 12 is four times greater than three).
• Find the unknown scalar between two quantities using multiplication and division, such as, the multiplier between 6 and 24 is 4 because 24 ÷ 6 = 4.
Description of Mathematics

Multiplicative comparison between two quantities is significant to the future success of students, particularly for their understanding of proportions. Consider two quantities, such as five and fifteen. How can the two quantities be compared?

Here are some things that could be said:

• 15 is 10 more than 5
• 5 is 10 less than 15
• 15 is three times as much as 5
• 5 is one third as much as 15

The top two statements are additive comparisons that can be made about the difference between 5 and 15. The bottom two statements are multiplicative comparisons that can be made about the scalar between 5 and 15. You can think of the scalar as a multiplier.

Finding the scalar is simple with numbers like 5 and 15 as facts like 3 x 5 = 15 are commonly known. When the numbers are less friendly then students need an algorithm to find the scalar. If you ask, “How many times does 5 fit into 15?” you establish that the scalar between 5 and 15 is 3.

You could write the algorithm as 15 ÷ 5 = □ to find □ x 5 = 15.

The scalar between 5 and 15 is a more difficult relationship since it is a fraction. Let’s apply the same algorithm as above. If you ask, “How many times does 15 fit into 5?” you realise that the scalar must be a fraction less than one. Since five is one third of 15, the scalar is one third.

You could write the algorithm as 5 ÷ 15 = □ to find □ x 15 = 5. Note that a common misconception students hold about division is that the divisor is always less than the dividend. In this case the divisor, 15, is greater than the dividend, 5.

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:

1. Provide physical materials so that students can anticipate actions and justify their solutions. Linear models, like connecting cubes, are preferred. Use colours to show how many times the measurement length fits into the target length.
2. Connect symbols and mathematical vocabulary, especially the symbols for multiplication and division (x, ÷) and for equality (=). Explicitly model the correct use of equations and algorithms, and discuss the meaning of the symbols in context.
3. Alter the complexity of the numbers that are used. Scalars (multipliers) that are whole numbers are much easier than fractions, e.g. □ x 9 = 36 is much easier than □ x 9 = 6.
4. Encourage students to collaborate in small groups and to share, and justify, their ideas.
5. Use technology, especially calculators, in predictive, pattern-based ways to generalise how scalars can be found using division. Use of calculators broadens the range of numbers accessible to all students.

The context used for this unit involve lengths of tuataras, scores in a top student competition, ages, fish, and price inflation. Using a range of situations helps to engage students and helps them to see application of mathematics to the real world. Vary the contexts to suit the cultures and interests of your students while retaining the mathematical focus of the unit. You might compare improvements in kapa haka scores, changes in points scored in sport, number of whānau members at a marae gathering, or distance of houses from school. Ask students to choose the objects that appeal to them, in comparing 1991 and today’s prices.

Required Resource Materials
Activity

#### Session 1

1. Begin with a growth problem (see PowerPoint One).
Tuati and Tera are tuatara lizards. They are small reminders of a time when dinosaurs ruled the Earth.
Nowadays scientists measure tuatara to check that they are growing well, as they are an endangered species.
In 2000 Tuati was 10 centimetres long and Tera was 16 centimetres long.
This year Tuati is 30 centimetres long and Tera is 40 centimetres long.
Which tuatara grew the most between 2000 and today, Tuati or Tera?

2. You may like to make four models of the tuatara using linking cubes and arrange them in this way:
3. Let your students work in small groups to answer the question. Encourage them to rehearse the strategy they used so they are ready to share it with the whole class. After an appropriate time bring the class together for discussion.
Some students, possibly most, will use additive comparison. They will find the number of centimetres each tuatara grew in the time. Highlight that they have found the differences between the today and year 2000 lengths. Difference can be found either by ‘adding on’, or subtracting. For example:
I added on from 16 until I got to 40. That was 24 centimetres growth for Tera.
I subtracted 10 from 30 and got 20 centimetres for Tuati.

4. You may choose to use empty number lines to record the calculations.
Finding difference is a legitimate way to compare the growth of the two tuatara. That will lead a conclusion that Tera grew more than Tuati. However, a multiplicative comparison leads to another conclusion.
How many times as long is Tuati today than he was in the year 2000?
How many times as long is Tera today than she was in the year 2000?

The term ‘times as long’ needs explanation and representation. Use PowerPoint One to illustrate the concept. Tuati is three times as long as he was in 2000, and Tera is two and one half times longer than she was. The comparison of 40 with 16 requires students to recognise 16 as the unit of comparison, and express the eight cubes in the third iteration (copy) as one half of 16. You might record the relationships using equations;
3 x 10 = 30                           2½ x 16 = 40

5. Provide the students with Copymaster One which contains a variety of tuatara growths between 2000 and today. Student can work together to establish the ‘times as long’ relationship for each tuatara. If needed, provide linking cubes so students can form the two lengths and map the year 2000 length into the today length.

6. After sufficient time bring the class together to share the answers. Pay attention, particularly to:
• ​Anticipation of the relationship by thinking about “How many times does x fit into y?”
• Noticing the need for fractions where the 2000 length does not map into the today length a whole number of times
• Situations where the tuatara has reduced length between 2000 and today so the multiplier is a fraction less than one.

#### Session 2

Multiplicative comparison sometimes requires a multiplier that is not a whole number. In this session students explore applying a whole number multiplier then extending their understanding to fraction multipliers.

1. Start with simple problems where the multiplier is known. PowerPoint Two provides a scenario of students getting points in a ‘Top of the class’ competition for taking care of each other. Each odd numbered slide is a problem to solve. Use ‘think, pair, share” to give students the opportunity to solve each problem before turning to the even numbered slide that follows. Even slides illustrate how the ‘times as many’ relationship works using a linear cube model.

2. Give each pair of students a small set of cubes. Ask them to build a model of each situation below:
Amy has three cubes. Ben has three times as many. How many cubes does Ben have?
Zane has two cubes. Xena has eight times as many. How many cubes does Xena have?
Kayla has some cubes. Mike has 15 cubes, that’s three times as many as Kayla. How many cubes does Kayla have?
Sam has some cubes. Lia has 20 cubes, that’s two and one half times as many as Sam. How many cubes does Sam have?

Note that the last two examples involve an inverse operation. For example, Kayla and Mike’s situation might be recorded as 3 x □ = 15. Look to see if your students understand that the unit of comparison is mapped onto the other quantity a certain number of times. In Kayla and Sam’s situation they need to find a unit that maps into 15 three times.

3. You might extend the problems to include fractional multipliers, depending on how students are progressing.
Gina has nine cubes. Fran has two thirds as many. How many cubes does Fran have?
Ena has some cubes. Dan has 15 cubes, that’s one and one half times as many as Ena.
How many cubes does Dan have?

4. Record each problem as an equation. For example, the Ena and Dan situation is represented by 1½ x □ = 15.

5. Finish the session with a complex problem that involves organising a lot of ‘times as many’ relationships. Copymaster Two contains the problem. Let your students attempt the problem in small groups or pairs. Useful prompts might be:
• How can you organise the information to help take a load off your brain?
• Look at clue 3. What ages could Dad be? Why?
• Can you use that thinking to find other possible ages?
The solution is: Rex is four, Dave is six, Kate is nine, Mum is 36, Dad is 40, Nana is 66 and Pop is 72.

6. Students might make up their own family ages problem for others to solve. You could make a book of ages problem to put in the mathematics corner.

#### Session 3

In this session students identify the multiplier connecting two quantities. They learn that the multiplier can be found using division and use that knowledge to solve more complex problems than they can solve mentally. Calculators are used to increase the range of numbers accessible to the students.

1. Use PowerPoint Three. Students should recognise that there are three times as many orange fish as blue fish on Slide 1.
What equation can you write for this problem?

2. Let your students try to write an equation then share. Most will write a multiplication problem like this: □ x 4 = 12 or 3 x 4 = 12
Some may reverse the order of the factors, i.e. 4 x 3 = 12. In English speaking countries the multiplier is conventionally written as the first factor. Discuss what the 3 and 4 refer to in the equation. Four is the unit of comparison, the known set. Three is the multiplier that tells how many times 4 maps into 12.
3. How did you work out that the first factor was 3?
Some students may have skip counted or scanned their four times multiplication facts to find the missing factor. It is possible some students used division, asking themselves “How many times does 4 fit into 12?”

4. Write an equation for “How many fours are in 12?”

5. Record the multiplication and division equations for the fish scenario: □ x 4 = 12 and 12 ÷ 4 = □
Note that 12 ÷ 4 = □ refers to measurement division (quotative) meaning “How many fours measure 12?” Use calculators to verify that 12 ÷ 4 = gives the multiplier, three.

6. Move to Slides 2, 3 and 4 of PowerPoint Three that increase in complexity. As the numbers become more difficult students should realise that using division has strategic advantage. Allow use of calculators after students attempt the problems mentally.
Look for students to recognise that if 3 x 4 = 12 then 9 x 4 = 36. Also encourage them to use known facts to solve complex problems. For example (Slide 3):
If there were blue fish, what would the ‘times as many’ factor be? (6 x 8 = 48)
How can you use that fact to work out □ x 3 = 48? (Twice as many threes as sixes measure 48)

7. For each problem write multiplication and division equations to represent the solution.

8. Complete the lesson with Copymaster Three that contains calculator change problems. For example, "What single multiplication operation changes 6 into 48?"
Since 48 ÷ 6 = 8 the multiplier is eight and 8 x 6 = 48.

9. After students complete Copymaster Three share their strategies. Do they use division to find the unknown multiplier. To extend their thinking you might pose change problems that have fraction multipliers. For example: 6 → 57, or 24 → 18

#### Session 4

1. Remind the students of the problems they investigated to date (tuatara lengths, top student points, tropical fish, calculator change). Ask them to write down features that are the same about all four problem sets. Encourage them to use examples of problems to illustrate their ideas.

2. Discuss the comparison nature of all the problems. One quantity has been measured with another to see how many times it fits in. Expect students to also see that all the situations can be represented as multiplication and division equations.

3. To see if you understand the ideas in this unit, I have another problem for you to solve. Have you heard of inflation? What does that mean?
You may need to explain that inflation is the increase in the price of things, from food to electronic devices to rent on houses. Statistics New Zealand uses a mixture of things to work out how much prices are increasing or decreasing.

4. In 1990, about when your parents were born, the average rent for a house in Auckland was about \$200 per week. Today, to rent the same house would cost at least \$500 per week.
How much has the rent increased since 1990?

5. Students might express the relationship between 200 and 500 as an increase of \$300 or as “two and one half times as much.” Reiterate that the unit is about multiplicative comparison, “times as many” or “times as much.” Ask for the equations that represent the comparison (2½ x 200 = 500 or 500 ÷ 200 = 2½).

6. Copymaster Four has average prices for items in 1990 (source New Zealand Yearbook, 1991). Tell your students that their task is to work out how many times dearer each item costs today compared to the price in 1990. To help them provide junk mail flyers from supermarkets, and appliance stores, and/or access to online. Search for the latest New Zealand Yearbook. Students will also need calculators as the numbers are not always friendly.

7. Let your students work in small teams to find the multipliers. Watch for students using consistent units. Comparing prices may require them to convert dollar amounts into cents or vice versa. Do your students apply division to find the multipliers?

8. After a suitable period discuss what they have found out.
Are the increases consistent?
Why might some things increase more than others?
Which things increased the most? Which things increased by the least?

Students might notice that the stove and computer cost about the same number of dollars as they did back in 1990. Food items tend to be two to three times dearer, except for bananas that are about the same. Students at Level 3 are just beginning their learning of decimals so encountering numbers like 2.38 is an opportunity to discuss rounding, possibly with the support of a number line.

## Perspective on Picasso

Purpose

In this unit students develop a further understanding of two-dimensional representations of three-dimensional shapes through creating drawings and artworks that explore shapes from different viewpoints.

Achievement Objectives
GM3-3: Classify plane shapes and prisms by their spatial features.
GM3-4: Represent objects with drawings and models.
Specific Learning Outcomes
• List the features of common three-dimensional shapes.
• Recognise that the faces of 3D shapes are 2D plane shapes.
• Explore and understand the defining features of a prism.
• Draw (freehand) common 3D shapes from different viewpoints.
• Represent different viewpoints of the constructions with drawings on isometric paper.
• Create cube constructions from viewpoint drawings made by another student.
• Become familiar with and describe the features of a Picasso portrait and of cubism.
• Create a portrait in the Picasso style.
• Appreciate and write about their own artwork and that of one other student, recognising the influence of shape and viewpoint (or perspective).
Description of Mathematics

In level two, students have been learning what is meant by ‘dimension, and coming to understand that three-dimensional shapes, called polyhedra, are bounded by flat (planar) shapes that have two dimensions. The language associated with the features of 2D and 3D shapes is becoming familiar, such as side, vertex (corner), edge, face.

At level three, students are developing more depth in their understanding of the properties of 2D and 3D shapes, including prisms. Prisms are three dimensional solids with a constant cross-section, and are named after that cross-section. For example, a triangular prism can be cut into parallel cross-sections that are triangles. Other classes of solids, such as pyramids or curved surfaces, have variable cross sections.

As they consolidate their understanding of polyhedra, students recognise that the faces are polygons (bounded planar shapes), and that the properties of those faces contribute to the properties of the polyhedron. Although dynamic computer software enables more helpful representations of three dimensional shapes, the images are still flat, that is in two dimensions. Such flat representations depict limited information about the three dimensional shape they represent. Through representing three dimensional shapes from different perspectives, student learn to focus on different attributes of the shapes and recognise that representations of the same shape can look very different.

In this unit of work, these understandings combine to inform the student’s appreciation at a simple level, of some of the features of Picasso’s portraits and of his defining art style, cubism. The development of cubism marked a pronounced shift in artists' interpretation of real-world objects. Picasso, and Braque, attempted to capture the characteristics of three dimensional shapes in a single two-dimensional image.

Associated Achievement Objectives

Art
Visual Arts

• Explore some art-making conventions, applying knowledge of elements and selected principles through the use of materials and processes.
• Develop and revisit visual ideas, in response to a variety of motivations, observation, and imagination, supported by the study of artists’ works.
• Describe the ideas their own and other’s objects and images communicate.

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

• providing physical materials, such as models of three dimensional solids, so students can move themselves in relation to those objects
• providing access to sculpting materials, such as plasticine, so students can create their own three dimensional models, and cut cross sections
• orientating cube models so the viewpoint matches the orientation of isometric paper
• explicitly modelling drawing from various viewpoints, including the use of digital cameras, so students appreciate the minimal requirements of plan views.

Tasks can be varied in many ways including:

• manipulating the complexity of the three dimensional objects the students represent
• scaffolding the process of representational drawing into a sequence of steps.

The contexts for this unit can be adapted to suit the interests and cultural backgrounds of your students. Capitalise on the interests of your students. Cubism is one form of artistic representation. Other forms might be more appropriate to the cultural background and interests of your students. Modern art marries traditional aspects of Māori and Polynesian art with modern techniques and media. Students may want to study how modern New Zealand artists represent their subjects to convey meaning and feelings.

Required Resource Materials
• Three-dimensional solid shapes: sphere, cylinder, cone, cube, cuboid, hexagonal prism, triangular prism, square based pyramid
• Small plain cubes
• Play dough
• Plastic knives
• Paper
• Pencils
• Copymasters One, Two and Three
• Chart paper
• Art paper
• Pastels
Activity

Whilst this unit is presented as sequence of five sessions, more sessions than this may be required. It is also expected that any session may extend beyond one teaching period.

#### Session 1

This session is about students becoming familiar with the defining features of common three-dimensional shapes and representing these shapes with freehand drawings from different viewpoints.

SLOs:

• List the features of common three-dimensional shapes.
• Recognise that the faces of 3D shapes are 2D plane shapes.
• Explore and understand the defining features of a prism.
• Draw (freehand) common 3D shapes from different viewpoints.

Activity 1

1. Make available pencils and paper.
Draw a diagram of a circle.
Give student pairs 1 minute to write some of the objects that this circle could represent.
Have student pairs share their ideas with the class.

2. Display solid sphere, cylinder and cone shapes.

Ask, “Could this (circle) be a drawing of a sphere or a cone or a cylinder?” Have students discuss this and reach agreement that this could be a bird’s eye view of a sphere or a cylinder or a view of a cone looking at it from underneath, (or perhaps from the top if the point were very fine).
Write ‘viewpoint’ on the class chart, agree on a definition that a viewpoint is a perspective of an object from a specific location.  Discuss and identify other possible drawings of the three objects. Ask students to be specific about the viewpoint each picture is taken from. For example, a cone appears as a triangle from side on, and a cylinder looks like a rectangle from side on.
3. Does a sphere always look like a circle, no matter the viewpoint?
4. Note that many students want to behave like cubist artists, and convey the curvature of surfaces in their drawings. For example, they might draw a side view of a cone with a curved base. Taking and displaying digital photographs of the solids can support students to accept the loss of information in a two dimensional drawing.

Activity 2

1. Add to the solid shape display, a cube, a cuboid, a hexagonal prism, a triangular prism, and a square based pyramid.
2. Create a chart, writing on it the names of each of the shapes and the headings, faces, edges and vertices. Define the meaning of each feature of the shapes. For example, a face is a polygon that is part of the boundary of a solid.  Model and record the features of at least 2 of the shapes. Note that a sphere is a curved surface and has no faces. A cone also has a curved surface.
 Shape Faces Edges Vertices Sphere 0 0 0 Cube 6 12 8 Cylinder Cone Cuboid Triangular prism
3. If you do not have enough models of the shapes available, provide play dough and plastic knives so students can sculpt the shapes.
4. Ask students to create and complete their own chart (individually or in pairs), making models of each shape with play dough as required, to explore and confirm their features.
5. Discuss the strategies the students used to count the number of faces, edges and vertices for each solid. Their strategies typically include counting top and bottom of a solid then counting the remaining faces, edges, or vertices.
6. Discuss what a cylinder, cone and sphere have no edges or vertices. That is because edges and vertices are formed where faces, polygons, meet.
How many faces meet at an edge? (Two – this is an important generalisation)
How many corners meet at a vertex? (That number depends on the number of faces surrounding each vertex. For a cuboid the number is three)
7. What do students notice about the number of faces, edges and vertices of a cuboid? (the same)
Why does that happen? (A cube is a type of cuboid)

Activity 3

Have each student choose at least two of the shapes that they have made, and make freehand drawings of each, from at least two different viewpoints. Have students share their drawings with a partner, and talk about what they have done, using the language of faces, edges and vertices.

In the follow up discussion ask one student at a time to show just two viewpoints of the same solid.
Can you identify which solid these viewpoints represent?
Do you need more viewpoints or less to make a specific identification? Why?

Activity 4

1. Invite students to visualise straight (planar) cuts of a square based pyramid and a cuboid (rectangular prism), parallel to the base. You might use the analogy of slicing loaves of bread.
What shape will the cross sections be? (squares for the pyramid, rectangles for the prism)
Will the cross sections be exactly the same? Students should notice that cross sections of the pyramid vary in size, while the prism has constant cross section.
Alternatively students can make a pyramid and rectangular prism from play dough, plasticine, or cheap vegetables such as potatoes, pumpkin, or carrot (Make a vegetable soup for lunch afterwards). They can make cuts through each shape.

2. Discuss the results, eliciting the explanation that the faces of the cuts of the pyramid change, whilst the prism faces remain unchanged.
Does the property of constant cross section also apply to other types of prism? (Yes)

3. You might explore the idea by slicing other prisms. Agree on a definition of a prism.

Activity 5

Conclude the session by having students draw any prism from at least two viewpoints. Ask them to record their work, using the language of faces, edges and vertices, and to write their own definition of a prism.

Is a prism easy to identify by its viewpoints? Why?

#### Session 2

This session is about exploring cube constructions, representing those constructions from different viewpoints on isometric paper, and interpreting the drawings of others.

SLOs:

• Make simple constructions using interlocking cubes.
• Represent different viewpoints of the constructions with drawings on isometric paper.
• Create cube constructions from viewpoint drawings made by another student.

Activity 1

1. Begin by having students share their work from Session 1, Activity 5.
2. Explain that they will use isometric paper to to reduce the need to measure, and to make their drawings more precise.
3. Introduce the task by showing a cube, selecting a viewpoint, and modelling how to represent a single cube on isometric paper.

Notice that the cube has a leading edge, marked by the central vertical line. To faces that meet at the edges are drawn as rhombi rather than as squares. The top face is also drawn as a rhombus.
4. Make cubes and isometric paper (Copymaster 1) available to each student.
5. Walk students through the steps of drawing a single cube then try joining additional cubes back, and below, from it.
The drawing above shows a large cube. Discuss any challenges that arise.
What is the smallest cube you could draw?

Activity 2

1. Make available isometric paper and interlocking cubes, sufficient for each student to have nine. Have students join the cubes to make a shape of interest, then draw this shape from at least two viewpoints. It is important that the faces of the cubes meet exactly.
2. If students have difficulty drawing the shapes hold art class.
• First, get students to tilt their model so there is a leading edge, and the top faces are visible from their viewpoint as rhombi.
• Second, draw the leading edge vertically on the isometric paper. That edge might be several little cube edges joined.
• Third, draw the lateral faces coming back from that edge.
• Finally, complete the whole shape, including the top.

Activity 3

Ask students to display their models on a table top, and pass in their isometric drawings. Mix-up the drawings and redistribute them randomly amongst the students. Each student must then locate the model that matches the drawing that they have been given. Confirm their matches.

How do you know that this is the correct model? (Viewpoint matches)
What strategies did you use to locate the model?
Is it possible to have different isometric drawings of the same model? Why?
Is it possible to have several models that match a single drawing? Why?

This discussion helps students appreciate the loss of information about a three dimensional object when it is represented in a two dimensional drawing.

Activity 4

Have students dismantle their shapes into single cubes. Collect in the drawings once more, shuffle these and redistribute them. Have students construct the model shape from another student’s isometric drawing.

Activity 5

Conclude the session by discussing the process, having students talk about the representation of shapes from different viewpoints and highlighting the geometric language, e.g. faces, edges, vertices.

#### Session 3

This session is about exploring a selection of Picasso portraits, identifying some distinguishing stylistic characteristics.

SLOs:

• Become familiar with and describe the characterising features of a Picasso portrait.
• Understand some of the defining features of cubism.
• Write a description of the influence of geometry in a Picasso portrait.

Activity 1

1. Write Pablo Picasso on the class chart. Brainstorm and record student knowledge of the artist. Show his picture, and share the information in Copymaster 2, or set a short time limit and have the students research pertinent information about the artist online.
I wonder why his style is known as cubism?
2. Discuss how Picasso tried to depict three dimensional features of his subject on a two dimensional surface. Isometric drawing belongs in the perspective methods of realist artists before Picasso.

Activity 2

1. Make available to students in pairs, one of the pictures in Copymaster 3, paper and pencils. Have students discuss their Picasso portrait, and record all the things that they feel and notice about it.
What was Picasso trying to represent about his subject?
How did he alter what can really be seen from one viewpoint?
2. Have students pair share their feelings, and observations and agree on common features.
3. As a class, discuss their feelings about the artworks. Record these responses, (like because….., dislike because….., puzzled because…., etc.)
4. Discuss and record on the class chart the features that they notice about the artworks.
These are likely to include:
• You can see the person from a front viewpoint and a side viewpoint (profile) in the same picture
• There are geometric shapes in the pictures
• There are many light and dark contrasts (chiaroscuro is the arrangement of light and dark elements in a pictorial work of art, also known as positive and negative space)
• Black is used to outline shapes and to draw features
• The pictures are not realistic – they are not how people really look
• The pictures appear to be fragmented, or broken up into parts
• They are a combination of two-dimensional flat plane shapes and some realistic rounded and shaded three-dimensional images.

Activity 3

1. Write ‘cubism’ on the class chart and have students share their knowledge of this art form. Agree that many of the features they have identified in 2 (above) are defining features of cubism. Explain that Pablo Picasso and Georges Braque began their innovative, new art movement, cubism, in the early 20th century (the 1900s).
What did art look like before that time?
(Some students may know about impressionists such as Van Gogh and Monet who were instrumental in the movement towards conveying feelings about a subject, rather than just capturing a realistic image).
2. Make a copy of one of the pictures from Copymaster 3 available to each student. Have them write a description on the picture and the artist, synthesising the information that has been presented so far about both and including references to geometry and viewpoints.

Activity 4

1. Display both photographs of Picasso (Copymaster 2).
2. Explain that in the next session, each student will be choosing one of the photographs and creating their own portrait of Picasso based upon the photograph. The portrait will be in the cubist style.
3. Have students begin a draft design of their Picasso portrait.

Activity 5

Conclude the session by having students share their work.
Ask each student to bring to the next session a photograph of a familiar person or pet that they would like to use as the subject of an artwork.

#### Session 4

This session is about using the style of cubism to create a portrait of Picasso, and begin an artwork of a person or pet significant to the student.

SLOs:

• Create at least one portrait in the Picasso style.
• Give feedback to other students on the cubist influence evident in their artworks.

Activity 1

1. Begin the session by reviewing the features of cubism listed in Session 3, Activity 2, Step 4.
2. Display both pictures of Picasso (Copymaster 2).
3. Students  select one of the pictures of Pablo Picasso, and create a draft cubist portrait.
4. Have students share the progress on their draft portraits of Picasso.
What viewpoints did you imagine?
How did you represent how Picasso would look from each viewpoint?
What geometric shapes did you use? Why?

Activity 2

1. Make available art paper and pastels. Students choose their own subject. Picasso tended to paint people (women mainly), though animals were another common subject.
2. Let students begin their Picasso works in a pastel medium. Insist they sketch the work first in pencil. Art takes careful planning, like other forms of design. Stop at significant points to review progress, checking against the listed cubist criteria.

Activity 3

As students complete their works, have them self-check and peer evaluate the works, referring to the cubist features.

Activity 4

Conclude the session with a gallery walk for the students to appreciate and give feedback on each other’s Picasso portrait artworks.
As a class, recognise those artworks that successfully and strongly show the influence of cubism. Look, in particular for representations from different viewpoints and the use of shapes in the fragmented (or disjointed) forms. Have each student identify improvements they might make to their own artwork.

#### Session 5

This session is about using self-evaluation and peer feedback to inform a second artwork in the cubist style.

SLO:

• Appreciate and write about their own artwork and that of one other student, recognising the influence of shape and viewpoint (or perspective).

Activity 1

1. Begin this session by having students review feedback and self-evaluation, using this to inform their design of their second artwork of the person or pet that is significant to them personally.
2. Stop at significant points to review progress, checking against the listed criteria and against their identified intended improvements. As students complete their works, have them self-check and peer evaluate the works.

Activity 2

Upon completion of the second artwork, have each student write a statement about the way in which the work has been created with reference to the cubist features, and whether the final work has captured the character of the special subject (the person or the pet), and if so, how.

Activity 3

1. Conclude the session by discussing the ‘geometry’ in their art and in the art of Picasso, referring to the learning in Sessions 1 to 3.
2. Discuss how artists like Picasso might represent a triangular prism or square based pyramid as a single two dimensional drawing. Let students choose a solid model and ‘Picasso’ it.
Can you capture the ‘feel’ of the solid as well as it’s appearance?
From this cubist drawing can you tell what solid has been used?
Would you ever see a picture like this is real life?