## Late level 1 plan (term 2)

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.

## I Like Trucks

Level One
Statistics
Units of Work
In this unit we explore ways to pose and answer investigative questions about our favourites by gathering and analysing data and discussing the results.
• Pose investigative questions with support from the teacher.
• With the teacher decide on how to collect the data to answer the investigative question.
• Sort objects into categories for display.
• Make a display of the data collected (pictograph).
• Make statements about data displays.

## Beetle Wheels

Level One
Number and Algebra
Units of Work
In this unit of work we link the development of skip-counting patterns to bars on a relationship graph. We also plot our skip-counting patterns on a hundreds board.
• Continue a skip-counting pattern.
• Describe skip-counting patterns.
• Use graphs to illustrate skip-counting patterns.

## Equality and equations

Level One
Number and Algebra
Units of Work
The purpose of this unit of five lessons is to develop the algebraic understanding that the equals symbol, = , indicates a relationship of equivalence between two amounts.
• Review number expressions involving the operations of addition and subtraction.
• Make and recognise combined amounts that have the same value.
• Write statements of equivalence in words.
• Solve addition and subtraction balance problems and...

## Making Patterns

Level One
Geometry and Measurement
Units of Work
In this unit students explore patterns involving transformations of shapes. Students make and identify patterns that involve translation, reflection, and rotation. They make their own patterns and pictures to show the transformations, and discuss with others how they included the different...
• Make patterns that involve translations, reflections, and rotations.
• Identify translations, reflections, or rotations in patterns.

## Matariki - Level 1

Level One
Integrated
Units of Work
This unit builds the learning of mathematics around celebrations of Matariki, the Māori New Year. The sessions provide meaningful contexts that highlight Māori culture and provide powerful learning opportunities that connect different strands of mathematics.
• Assemble parts of a shape to form the whole.
• Create symmetrical figures (reflection and rotation).
• Calculate the number of direct ancestors they have.
• Use fractions to create rhythmic percussion patterns.
• Order events.
• Describe the likelihood of outcomes using the language of chance.
• ...
Source URL: https://nzmaths.co.nz/user/75803/planning-space/late-level-1-plan-term-2

## I Like Trucks

Purpose

In this unit we explore ways to pose and answer investigative questions about our favourites by gathering and analysing data and discussing the results.

Achievement Objectives
S1-1: Conduct investigations using the statistical enquiry cycle: posing and answering questions; gathering, sorting and counting, and displaying category data; discussing the results.
Specific Learning Outcomes
• Pose investigative questions with support from the teacher.
• With the teacher decide on how to collect the data to answer the investigative question.
• Sort objects into categories for display.
• Make a display of the data collected (pictograph).
• Make statements about data displays.
Description of Mathematics

In this unit the students begin by brainstorming areas for investigation. Young students' areas of interest are likely to focus on themselves and the activities that they are engaged in. Using the students and their interests is a recurring technique used in junior classes. It provides students with contexts that are meaningful and motivating. With the teacher investigative questions are posed about categorical data.

In this unit we use favourites as the theme for the investigations. Much of the data collected at level one will be real objects. In this unit we begin by posing an investigative question about our favourite toys. Once the data (toys) are collected together they can be sorted into categories ready for display. It is important that the students are involved in deciding how to sort the objects. We then draw pictures of other favourites and use them to make displays.

Once more we stress the importance of letting the students decide how to sort and display the data. In this unit we photocopy the drawings so that each pair of students gets the opportunity to make decisions about how the data should be sorted and displayed. The follow-up discussion of the displays will involve the students making statements about the number of objects in each of the categories. In this unit we do not attempt to get the students to formalise their displays into pictographs. However the thinking carried out in this unit means that the students would be ready to use pictographs in future statistics units.

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:

• reducing the number of favourites in the data set for students who are beginning to count one-to-one when they are doing individual or small group work
• extending by recording two types of favourites on data cards so that they can sort in two ways, e.g. by favourite food and by favourite drink.

The context, about favourites, for this unit can be adapted to suit the interests and experiences of your students by selecting favourites of interest to your students.

Required Resource Materials
• Paper for drawing (cut A4 sheets into 8 pieces)
• Chart paper
• Scissors
• A favourite toy from home
Activity

#### Getting Started

1. The previous day or previous week ask the students to bring their favourite toy to school.
2. We begin the week by looking at all the toys the students have brought to school. Ask the students, seated in a circle, to introduce their toy to the class.
Let’s try to find out about our favourite toys. The investigative question we are exploring is "What are the favourite toys of the children in our class?"
3. Write the investigative question up on the board or a chart.
4. Ask a student to put their toy in the centre of the circle.
Do any of you have toys that could belong with this one? Is there some way that your toy is like this one? How does your toy belong? Who has one that doesn’t belong? Why not?
5. Continue until all the toys are sorted into categories. Together count the toys in each group.
Four of us brought dolls or action men. Three of us brought balls to kick. Six of us brought toys with wheels. Two of us brought soft animals etc
6. Record statements on the board or a chart, for example, "Four of us like dolls the best."
Let’s think about some other favourites that we can investigative with the class this week.
7. With the students, discuss different objects and items that we could explore to see what our favourites are.  Collect the ideas on the board or chart.
8. Choose three favourites to explore with the class this week.
Possible favourites include: food, colour, drink, number, animal.
9. With the students pose investigative questions to explore. For example: What are the favourite foods of the children in our class? What colours do Room 30 students like best? What are our class’s favourite numbers? What animals do the children in Room 2 have as favourites?
10. Discuss with the students that we need to collect data from each of them to answer the three investigative questions.  With the students pose survey questions for each of the investigative questions.  For example: What is your favourite food? What colour do you like the best? What number is your favourite? What is your favourite animal?
11. Write each of the three survey questions on an envelope. Pin the envelopes up where everyone can reach them.
12. Ask the students to draw an answer for each of the survey questions onto the prepared pieces of paper (A4 cut into 8 pieces). If they are able ask them to write their answer beside the picture. Circulate and help those who are unable to write their answers beside the picture.
13. Put the named pictures (answers) in the appropriate envelope.
14. Send home the family and whānau letter regarding answering the investigative question "What are the favourite fruits of our class and their families?"

#### Exploring

In preparation for the next three days, make a set of picture sheets for each pair by photocopying the answers (8 per sheet of A4). It is worth taking the time to make copies as it gives everyone the opportunity to sort and display the data.

1. Each day select one of the investigative questions to explore.  Take the envelope that contains the student responses to the associated survey question. Spread the pictures out on the mat for the students to look at.
2. Can you see your drawing?
3. Do you see any that are like yours?
4. Which ones are different to yours?
5. Ask the students for ideas for sorting the pictures.
6. Sort the pictures according to one of the suggestions, for example, sort foods with others of the same type, for example, ice-creams, fruit, cakes, chips, fish
7. Together count the pictures in each collection.
8. Have the students return to their seats to work with their partners. Give each pair a set of the photocopied answers prepared for the day’s investigative question.
9. Each pair needs to cut the pictures apart and then decide how to sort them.
10. As the students work ask questions that focus on the approach they are taking to sort the pictures:
How are you sorting the pictures?
How many categories or groups have you got?
Is it easy to decide where to put the pictures? Why/Why not?
11. Once the pictures are sorted get the students to glue their pictures onto chart paper that have the investigative question already written at the top. Encourage students to make statements about what their displays shows. Help them write statements about their display if they are unable to write their own statements.
12. At the end of each session ask the students to share their posters. Ask questions, such as in point 10 above, and 2, 3, and 4 below, and offer models of appropriate language to support the students to effectively talk about the process of making their display and what is presented on it.

#### Reflecting

We begin today’s session by getting the students to select one of their favourites investigations to display on the classroom walls.

1. Let's look at all the great statistical investigations that we did this week.
2. Spend some time looking at the displays and asking the students to tell you what the display is saying about the investigative question.  For example, if the investigative question was "What colours do Room 30 students like best?", then the teacher would say Can you tell us what your display is saying about the colours that Room 30 students like best.
How many chose that favourite (e.g. colour)?
Which things are favourites? How do you know? How does your display show that?
3. Conclude the session by exploring the investigative question "What are the favourite fruits of our class and their families?"  Gather together the fruit pictures that families drew.
4. Talk about ways to sort the fruit pictures. Choose four or five types and then discuss the use of an "other fruit" category.
5. When you have agreed on ways to sort the fruit let the students add their families’ pictures to categories. Glue the pictures onto a chart to make a display.
6. Leave the activity open as you challenge the students to write statements to answer the investigative question "What are the favourite fruits of our class and their families?" They can add these to the display during their own time in the next week or so.

## Beetle Wheels

Purpose

In this unit of work we link the development of skip-counting patterns to bars on a relationship graph. We also plot our skip-counting patterns on a hundreds board.

Achievement Objectives
NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
NA1-6: Create and continue sequential patterns.
Specific Learning Outcomes
• Continue a skip-counting pattern.
• Describe skip-counting patterns.
• Use graphs to illustrate skip-counting patterns.
Description of Mathematics

In this unit we look at skip-counting patterns. These are patterns obtained by adding the same, constant, number to make the next number every time. So the difference between any two terms in a skip-counting pattern is the same. This is a good exercise to help reinforce the various concepts relating to pattern. In particular, it helps us to understand the idea of a recurrence relation between consecutive terms.

Skip-counting patterns are also called arithmetic progressions. In secondary school, expressions for both the general term of an arithmetic progression and the sum of all of the numbers in the progression are found.

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 the tasks include:

• Varying the difficulty of the skip.  For example, reduce the complexity by using skips of 2, 5 or 10.  Increase the complexity by using odd numbers or numbers larger than 5.
• Increase the difficulty by asking the students to extend the number of skips in the pattern.  For example, in the Exploring part of the unit ask the students to extend the pattern to 10 or more skips.

The contexts for this skip patterns used in this unit can be adapted to suit the interests and experiences of your students. For example in the Exploring part of the unit:

• involve the students in finding or suggesting the skip counting images
• use images of spiders (8 legs), lizards (4 legs), frogs (4 legs) and butterflies (2 wings) that are native to New Zealand
• use images of animals, plants and insects that are found in the local area.
Required Resource Materials
• Counters
• Cubes
• Squared paper for graphing
• Picture of VW beetle
• Pictures of objects for exploration
Activity

#### Getting Started

Today we explore the pattern of 4s by counting the number of wheels on cars.  We then use this information to build a relationship graph.

1. Ask: How many wheels does a beetle have?
2. Share ideas. Hopefully someone will link the beetle to the Volkswagen car rather than the insect or you may have to give a few more hints. Show students a picture of a VW beetle and discuss why it got this nickname (it is shaped like a beetle).
3. Using counters begin to develop a chart of the number of wheels to the number of cars.
4. Ask: How many wheels are there on 2 beetles?
How did you work that out?
5. It is useful for the students to listen to the strategies that others use. More advanced Level 1 students will be able to count on from 4 to find the answer and many may have 4 + 4 as a known fact.
6. Repeat the process with 3 and 4 VW beetles. Each time continue to add the information to the chart.
7. Ask the students to work out how many wheels there would be on 6 beetles. If some of the students find the answer quickly, ask them to find the answer using another strategy.
8. Share solutions. These may include:
• skip-counting with or without the calculator
• counting on using a number line or hundred’s board
• using counters to find 6 groups of 4.
9. Ask the class to complete the chart up to 6 cars.
Share ideas. Encourage the students to focus on the relationship between the number of cars and the number of wheels.
11. Ask the students how they could record this information using grid paper.

#### Exploring

Over the next 2-3 days, the students work in pairs to explore the number patterns of other skip-counts. At the end of each session the students share their charts with the rest of the class.

1. Place pictures of items that the students are to investigate in a “hat”. Ask each pair to draw one out and then investigate the pattern up to at least 6. Encourage the more able to students to extend the pattern beyond 6.
2. Pictures could include:
• tricycles (3 wheels)
• bicycles (2 wheels)
• hands (5 fingers)
• spiders (8 legs)
• glasses (2 lenses)
• frog (4 limbs)
• stool (3 legs)
3. Remind the students that they are to record their explorations on a chart.
4. At the end of each session share and discuss charts and number patterns.  Ask the students to identify the patterns that are the same.

#### Reflecting

In today’s session we use calculators to extend our skip-counting into the hundreds.  We record our patterns on a hundreds chart.

1.  As a class look at the chart to show hands (5 fingers). Skip count together in 5s, shading the counts on a hundreds chart.
2. As the chart is shaded ask questions which encourage the students to look for patterns in the numbers as they make their predictions.
Which number will be next?
How do you know?
3. Give the students (in pairs) a hundred’s chart and ask them to shade in one of the skip counting patterns that they had charted on the previous days.
4. Display, share and discuss at the end of the session.

## Equality and equations

Purpose

The purpose of this unit of five lessons is to develop the algebraic understanding that the equals symbol, = , indicates a relationship of equivalence between two amounts.

Achievement Objectives
NA1-4: Communicate and explain counting, grouping, and equal-sharing strategies, using words, numbers, and pictures.
Specific Learning Outcomes
• Review number expressions involving the operations of addition and subtraction.
• Make and recognise combined amounts that have the same value.
• Write statements of equivalence in words.
• Solve addition and subtraction balance problems and explain the solutions, using the language of equivalence.
• Recognise expressions that are equal in value.
Description of Mathematics

This sequence of lessons lays a fundamental and important foundation for students to be able to read, write and understand an equation.

The essence of an equation is that it is a statement of a relationship between two amounts. This relationship is a significant one of equivalence. The understanding that the amounts on either side of the equals sign are equal in value, is essential if students are to experience success in algebra in particular, and in mathematics in general, into the future.

The most common misunderstanding is when students develop a process view of an equation as a procedure to follow to get an answer, rather than a structural or relational view of equivalence.

Students should be immersed in a range of experiences that support them to explore the concept of equivalence and balance. During these experiences, the teacher must carefully choose the language they use and model. As equations are introduced, recorded, read and interpreted, words and phrases such as ‘has the same value as’, ‘is the same as’, ‘is equal to’ and ‘ is equivalent to’, rather than ‘makes’, or ‘gives an answer of ’, become very important. It is interesting to note that the word ‘equals’, on its own, has subtly become more synonymous with ‘makes’ or ‘gives an answer’, rather than giving the message of equivalence that it should.

When posing problems that position the unknown amount at the beginning or in the middle of an equation, we are challenging the student to explore the relationship statement and the operations from a different perspective. This also occurs when students are asked to find ‘different names’ for the same amount.

Students should have opportunities to both read and respond to equations, and to record them, having interpreted a number problem expressed in words. In developing the ‘balance’ view of an equation, students will understand the equality relationship expressed in an equation such as 6 = 6, rather than being perplexed by the fact that there is no number problem to ‘answer’. Students will also readily understand relationships expressed in equations such as 4 + 2 = 1 + 5, rather than developing an expectation that a single ‘answer’ will follow the = symbol. Instead of expressing solutions in the arithmetic ‘voice’ of ‘problem, calculation and answer’, it is important in early algebra work, for students to explain their solutions in words that make the equivalence relationship explicit.

Counting all (Stages 2 – 3)
Counting on (Stage 4)

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 the tasks include:

• Varying the complexity of the numbers used in the problem to match the number understanding of students in your class.  For example, increase the complexity by using larger numbers for students who are able to count-on to solve problems.

The contexts used in the word problems in this unit can be adapted to suit the interests and experiences of your students. For example:

• In session 1, change Jack and the beans to characters that are popular with your students (e.g. dinosaurs, unicorns or kiwis).
• In session 3, activity 2 replace the beans with objects that are of interest to your students (e.g. lego figures).  Remember that the use of the balance scale means that the objects need to be the same weight.
Required Resource Materials
• Class set of balance scales
• Plastic coloured beans
• Plain envelopes
• Jack and the Beanstalk story
• Plastic containers
• Plastic pegs
• Cardboard (labels)
Activity

These learning experiences use numbers in the range from 1 to 20, however the numbers in the problems and the learning experiences should be adapted, as appropriate, for the students.

#### Session 1

SLOs:

• Review number expressions involving the operations of addition and subtraction.
• Make and recognise combined amounts that have the same value.
• Write statements of equivalence in words.
• Write and read equations, using the language of equivalence.
• Understand the word ‘equation’.

Activity 1

1. Introduce the story of Jack and the Beanstalk. Ask who has planted or picked beans. Read the story. Explain that when the beanstalk is chopped to the ground, Jack picks handfuls of beans from it, some of which are bright green and others dark green. Unfortunately, they are no longer ‘magic’.
2. Draw on the class chart, the combinations of beans in Jack’s handfuls. Have students record beside them, in words and number expressions, what they see. For example:
three and four beans (3 + 4)
two plus five beans (2 + 5)
Pose subtraction scenarios and have students record their number expressions.
For example:
“Jack has eight beans and drops four.” (8 - 4)
“Jack has 6 beans and drops 1.” (6 - 1)

Activity 2

1. Make available to the students pencils, envelopes, and sets of two different coloured beans. Have students work in pairs.
Pose the problem:
“Jack wants to give away some packets of beans. He decides he’ll put six in each packet. He puts some beans of each colour into each packet and writes on the outside of the packet how many there are of each colour."
Write 6 on the class chart.
Demonstrate. For example:
Put 2 bright green and four dark green beans into one envelope and write 2 + 4 in pencil on the outside.
Tell the students that they should take turns to put the beans into the packets and to write on the outside.
2. Have student pairs share their packets and discuss if they have the same combinations recorded. Have them investigate any anomalies. (They may have put more or fewer than six in a packet).
3. Have student pairs return to the mat with their bean packets, which they place in front of them. On the class chart record:
6 is the same amount as:
Have students take turns to record their number expressions beside this.
6 is the same as: 5 + 1, 4 + 2, 3 + 3, 2 + 4, 1 + 5
Read these together using the language of, “is the same as.”
Ask whether it would be fair for Jack to give these to his friends. (Yes, because they would be getting the same amount. They would be getting an equal amount.)

Activity 3

1. Write the word ‘equal’ on the chart.
Have students tell you what ‘equal’ means. Brainstorm ideas and record these.
2. Add to the recording in Activity 2, Step 3.
6 is the same (amount) as: 5 + 1, 4 + 2, 3 + 3, 2 + 4, 1 + 5
6 is equal to: 5 + 1, 4 + 2, 3 + 3, 2 + 4, 1 + 5
Ask if students know how to write “is equal to” using a symbol. Introduce =.
Model writing 6 = 5 + 1.
3. Have student pairs share the task of writing the complete equation on each of their packets.
4. Have students place their “6” packets into a class container labelled with a 6 to be used in a later session.

Activity 4

1. Place containers labelled 3, 4, 5, and 7 in front of the students.
Explain that Jack needs packets with these different amounts. Demonstrate, using a ‘six packet’, that each envelope must have the “number equals story” on it.
2. Ask what is the correct word for a “number equals story”. Elicit and record the word equation, highlighting that 6 = 5 + 1 (for example) is called an equation because it uses the = sign to show that both amounts are the same. Ask if they can see part of the word ‘equals’ in the word equation.
3. Have students make up packets, as before, this time choosing 3, 4, 5, or 7 as their total, and recording a full equation on each packet. For example, 3 = 2 + 1, or 4 = 2 + 2.
4. Students should pair share and check their packets and equations before they are placed in the appropriate containers.

Activity 5

Conclude the session by reviewing =, equals and its meaning and the meaning of the word equation. Have students explain these, and record what they say.

#### Session 2

SLO:

Activity 1

1. Review the words, equal, equation and the symbol =, recorded on the class chart in Session 1.
Record a ‘six’ equation and read it in different ways together. For example:
6 = 5 + 1, “six is equal to five plus one”, “six is the same as five plus one”.
Highlight the fact that each of the packets in the 6 container have an equal or same amount.
2. Make available to the students, pencil and paper.
Have students in pairs choose one of the containers (you may need to make multiples of each container depending on class size).
Students begin by taking turns to read aloud to their partner, in the two ways modeled in Step 1 (above), an equation on an envelope selected from the container. They should return these once read.
3. Explain that in shops, staff do ‘stocktaking’ to check the amount of items they have. Students are to “stock take’ the beans by checking each packet to see that the equation on the outside matches the beans inside.
They should take two packets at a time, check that they have exactly the same amount and record what they find on their “stocktaking sheet” like this:

Students with containers 3 and 4 in particular, will accomplish this quickly.

Activity 2

1. Have each student pair join one other pair in this way: two pairs of seven and three, two pairs of six and four and one pair of five and five.
2. Refer to Jack and the Beanstalk.
Place in front of the students the cardboard ‘tickets’ and the plastic pegs.
“Jack is going to have a bean stall. He needs ‘pegged pairs’ with ten beans altogether in each. We are going to help him. We need to make labels to show the contents, or what's inside."
Elicit from the students that by using one packet from each of their containers, they will have ten beans. If necessary, students can explore this idea and check, using their fingers, showing, for example: 10 = 7 fingers (up) and 3 fingers (down).
3. Demonstrate that the two packets can be pegged together to make one “pegged pair of ten.” Model on the class chart, how labels should show the content in 3 ways. For example:
1. Write an equation using the number on each of the containers.
We say "7 beans plus 3 beans equals 10 beans" and we write 7 + 3 = 10
2. Write each of the expressions written on each envelope. (The number of each colour in each envelope)
We say: "This envelope has 5 dark and 2 light and this envelope has 1 dark and 2 light. Altogether that equals 10. " We write:  5 + 2 + 1 + 2 = 10
3. Tip out the beans and write the number of each of the colours.
We say" There are 6 dark beans and 4 light beans and that is 10 beans altogether." We write: 6 + 4 = 10
4. Conclude the session by having the students read aloud some of the tickets they have made for their pegged pairs.
Review the words, equal, equation and the symbol =, recorded on the class chart in Step 1, highlighting the language of ‘is equal to’ and ‘is the same as’ and that all the equations written are different names for ten.

#### Session 3

SLOs:

• Solve addition and subtraction balance problems and explain the solutions using the language of equivalence.

Activity 1

1. Introduce balance scales. Brainstorm and record on the class chart, students’ ideas about ‘how balance scales work’, eliciting language of ‘same, level, equal, balance, not tipped.’
2. Place one envelope pair (10) in one pan and ask what could be placed in the other to achieve balance. (Another pegged envelope pair.)
Again, record and ‘test’ student ideas, trying different combinations of pegged pairs. For example:
5 + 5 = 6 + 4
6 + 4 = 7 + 3
Ask why the results are recorded using =.
Elicit reasons such as ,”equals shows that they are the same”, “equals shows that they balance”, “equals shows that both amounts have the same value (10)” , “equals means is the same as”.
3. Record, 10 = 10 and discuss why this has been written and why it makes sense.

Activity 2

1. Model 5 + 5 = 6 + 4 using the scales.
Remove the packet of 4 beans, leaving 6 only on one side. Discuss the tipped scales and how to record the removal of the 4 beans.
Record suggestions. For example:
5 + 5 is not the same as 10 – 4
5 + 5 is not equal to 10 – 4
10 is not equal to 6
2. Ask what can be done to restore the balance.
Accept, ‘put 4 back in again’, but work to elicit, ‘take 4 away from the other side.
Have a student remove 4 beans from one of the 5 bean envelopes (example above), saying how many are remaining in the envelope (1). Return it to the scales.
Record suggestions that describe what has happened now the balance is restored. For example:
5 + 5 - 4 is equal to 10 – 4
10 - 4 is the same as 10 – 4
10 – 4 = 10 – 4
6 = 6
As equations are recorded, have students explain or demonstrate, using the materials, exactly what is happening. Together reach the conclusion: if you take away the same amount from each ‘side’ or pan, the scales will still balance.
3. Make available to the students, fresh envelopes (or erasers to clear used envelopes), and pegged bundles of ten from Session 2.
Have student pairs combine the beans from the pegged pairs into single envelopes of ten beans, writing 10 on each.
Have students work in pairs with envelopes of ten beans, some spare beans, paper to record equations and a set of balance scales.
Have students undertake the following tasks.
1. Student One removes a number of beans from one envelope, unseen by the other student, and returns the envelope to the scales. This student ‘secretly’ records the equation. For example: 10 – 3 = 7.
Student Two guesses how many were removed, removes this number from the other envelope, ‘secretly’ records the equation, for example 10 – 5 = 5, and returns it to the scales. They look carefully to check to see if the scales balance. If the scales do not balance, Student Two repeats their turn with another amount. When the scales do balance, both students share their final equations and check the amount in each envelope. Both students finally record the balance, for example, 7 = 7.
The students reverse rolls.
2. Students make teen numbers and record equations.
Student One places one ten envelope and a mixture of both colours of beans into one pan to make a number between ten and twenty. The student records the equation: for example, 10 + 2 + 3 = 15.
Student Two places one ten envelope and a mixture of both colours of beans into the other pan. The two-bean mix must be a different combination, but the total must balance the scales (in this case must equal 15). This student records their equation: for example, 10 + 1 + 4 = 15.
Both students then record what they can see in both pans.
10 + 2 + 3 = 10 + 1 + 4
15 = 15
4. Conclude this session with some students sharing their equations from tasks A. and B. Record a selection on the class chart and discuss these.
It is important to highlight the balanced nature of the equations. Elicit from the students, ‘an equation is like a thing that balances.’

#### Session 4

SLOs:

• Interpret addition and subtraction word problems that involve start unknown, change unknown and result unknown amounts.
• Write addition and subtraction equations from word problems.

Activity 1

1. Review conclusions from Session 3, Activity 2, Step 3, referring to the balance scales.
2. Make available to the students: balance scales, packets of beans, spare beans, and a pencil.
Explain that Jack, of Jack and the Beanstalk fame, has some problems for the students to solve and that they may want to use the equipment to help them.
Distribute a copy of Attachment 1 to each student. Read through the problems together.
Highlight that each student will be writing equations for each problem.
Students should choose whether to work on the problems alone or with a partner however, each student should complete their own recording sheet.
3. As students complete the recording task, have them compare and discuss their equations and solutions. They can then write some problems for their partner to solve.

#### Session 5

SLOs:

• Identify true (correct) from false (incorrect) equations and justify the choice.
• Recognise expressions that are equal in value.

Activity 1

1. Students will play two games in the session. Make available beans and balance scales.
Introduce the True/False game. (Attachment 2)
(Purpose: To recognise when amounts are equal or not equal.)
Model a ‘true’ equation such as 1 + 3 = 2 + 2, highlighting the fact that the amount on both sides are the same or equal to each other. Each expression is equal to 4. Model a ‘false’ equation such as 1 + 3 = 3 + 2, highlighting the fact that both sides are not the same and not equal to each other. 4 is not equal to 5. This is false (not true).
2. How to play:
Students play in pairs. They shuffle the playing cards and deal 10 to each player. The remainder of cards is placed in a pile, face down, handy to both players.
The aim of the game is to be the first person to have an equal number of true and false equations (five of each).
As each player turns over their cards, they sort them into true and false groups, face up in front of themselves. If they have more of one group than the other, they continue to take cards from the top of the pile, till the number of their true and false cards is equal.
The first player to have equal numbers of true and false cards calls, “Stop!”
This caller must explain to their partner, for each of their decisions, how they know they are correct in their true/false decisions. They can use beans to support their explanation.
The game begins again. The winner is the person who wins the most of three games.

Activity 2

Students play Same Name snap, using cards from Attachment 3.
(Purpose: To recognise when amounts are equivalent (or not equivalent) and to give the ‘number name’ for the ‘same name’ expressions.)
How to play:
Student pairs shuffle the cards and deal all cards so each student has an equal number of cards. These are placed in a pile, face down in front of each student. Student One turns over the top card and places it, face up, between both students. Student Two does the same, placing their card on top of their partner’s card. If the two expressions have equal value, either student calls Same Name, states the number that the expression represents, and the correct equation using either ‘is equal to’ or ‘is the same as’. For example:
2 + 3 is placed on top of 4 + 1.
“Same name! Five! Two plus three is equal to four plus one.” or
“Two plus three is the same as four plus one.”
The caller collects the card pile, records the equation, 5 = 2 + 3 = 4 + 1 on their scoring paper, and the game begins again, with the winner of this round placing the first card.
The student who does not call, can challenge the caller if they believe the “name” is not true for either or both expressions. If they are correct, they collect the pile and record the correct equation. The original caller must erase the incorrect equation.
The game finishes when one student has all the cards, or when one student has recorded ten ‘same name’ equations.

Activity 3

Conclude this session by discussing learning from the games, and reviewing ideas recorded on the class chart over five sessions.

## Making Patterns

Purpose

In this unit students explore patterns involving transformations of shapes. Students make and identify patterns that involve translation, reflection, and rotation. They make their own patterns and pictures to show the transformations, and discuss with others how they included the different transformation elements.

Achievement Objectives
GM1-5: Communicate and record the results of translations, reflections, and rotations on plane shapes.
Specific Learning Outcomes
• Make patterns that involve translations, reflections, and rotations.
• Identify translations, reflections, or rotations in patterns.
Description of Mathematics

Translations (slides), reflections (flips), and rotations (turns) are explored in this unit.

Translations are slides or shifts of a shape along a line.

Reflections are flips of a shape to make an image as though it is reflected in a mirror.

Rotations are turns, so when an shape is turned about a point, either inside or outside of itself, the image is a rotation of the original shape. This unit uses examples where the rotation happens around the centre point of an shape.

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

• In session 1, show students how to make the first translation of an object before asking them to independently create patterns using translation.  You could also begin with translating a real object and drawing around it before introducing a stamp pad.
• In session 3, focus on two or three examples of rotations using objects before introducing pictures that show rotations.

The objects and pictures used in this unit can be changes to suit the interests and experiences of your students.  For example, local flowers and pictures of native insects and butterflies are likely to be more engaging than generic pictures of flowers and butterflies. You could also incorporate an outdoor walk where students look for examples of translations, reflections and rotations.

Required Resource Materials
• Wallpaper or wrapping paper with translation pattern
• White and blue A4 paper
• Pictures that show reflection patterns
• Mirror
• Scissors, glue, crayons
• Copymaster
Activity

#### Session 1

In this session students make patterns that show translations.

1. Show the students a piece of wallpaper, wall frieze or wrapping paper that shows a translation pattern.
2. Ask the students: what picture is repeated on the paper to make a pattern?
Are the pictures the same each time? (Yes)
How are the pictures the same? (shape, size, orientation, colour)
3. Using A4 paper and stamps students are to make their own translation pattern on the page. Ensure that the students keep the stamp orientated the same way as they make repeated stamps on the page.

#### Session 2

In this session students make patterns that show reflections.

1. Show the students pictures that show reflections, for example scenery reflections in lakes, butterfly wings.
2. Explain to the students that a reflection picture looks like it could be folded in half so the two sides match. Using a mirror on the fold to show students that the reflection is the same as the other side.
3. Colouring pictures of butterflies so the wings show a reflection pattern is a common and popular activity. Another idea is to make a reflection patterns on the wings of paper planes.

#### Session 3

In this session students make patterns that show rotations.

1. Show the students pictures that show rotations, for example, star fish arms, flower petals, windmill blades, propeller blades, bike spokes.
2. Explain to the students that in these types of examples part of the object has been turned around a centre point. Ask them to identify the part that has been rotated. For example, if you take one spoke on the bike wheel, leave one end at the centre and turn the other end it will rotate on to the position of the next spoke.
3. Show the students pictures where the object itself has been rotated, for example:
4. Using the stamps and inkpads students can show a rotation pattern where the whole object is rotated.
5. Students can make patterns where part of the object is rotated. For example, drawing a flower by cutting out multiple petal shapes and gluing them around the centre or an aircraft with nose and wing propellers that show blade rotation.

#### Sessions 4 and 5

In these sessions students make an underwater sea picture that shows translation, reflection and rotation.

1. Discuss with students the patterns they have been making over the last 3 sessions. Explain that they will be making a picture that shows all 3 transformations.
2. Give the students the picture page. Explain that they will need to make the other half of the octopus. Name the animals in the picture and explain that sea stars (or star fish) are usually found at the bottom of the sea. They will need to cut out the pictures but they don’t need to use them all in their underwater sea picture.
Which animal shows reflection? (Octopus)
How else could you show reflection in your picture? (Put two fish nose to nose.)
How could you show translation in your picture? (Use 2 or more of the same animal and orientate them the same way.)
How could you show rotation in your picture? (Use 2 of more sea stars and rotate each one.)
4. Help the students complete the octopus. Ask the students to cut around the box, fold it in half on the dots, cut around the shape, and open it out.
5. Students cut out the other animals, colour them and glue them on to blue paper. Remind the students the finished picture needs to show translation, reflection, and rotation.
6. Students share their pictures with each other. Individuals look for the translation, reflection and rotation elements in each others’ pictures.
Attachments

## Matariki - Level 1

Purpose

This unit builds the learning of mathematics around celebrations of Matariki, the Māori New Year. The sessions provide meaningful contexts that highlight Māori culture and provide powerful learning opportunities that connect different strands of mathematics.

Specific Learning Outcomes
• Assemble parts of a shape to form the whole.
• Create symmetrical figures (reflection and rotation).
• Calculate the number of direct ancestors they have.
• Use fractions to create rhythmic percussion patterns.
• Order events.
• Describe the likelihood of outcomes using the language of chance.
• Measures quantities to follow a recipe.
Description of Mathematics

In this unit the students will apply many different mathematical and statistical ideas (see specific learning outcomes). They apply the properties of symmetry, that is the features of a shape that change and remain invariant under translation, reflection and rotation.

Students also apply simple probability. For example, given this set of cards, what is the chance of getting a bright star if you choose one card at random? Random means that each card has the same chance of selection.

The set of all possible outcomes contains four possibilities. Two of those possible outcomes are selecting a bright star card. The chance of getting a bright star are two out of four or one half. There is a one quarter chance of getting a fuzzy star and the same chance of getting a rainy cloud.

Required Resource Materials
• Mirrors
• Ingredients and baking equipment for making Rawena bread (see Lesson Five)
• Copymasters One, Two, Three, Four, Five, and Six
• PowerPoints One, Two, Three, and Four
Activity

#### Lesson One

1. Introduce Matariki, The Māori New Year, to your class. Matariki begins with the rising of the star cluster, (Pleiades in Greek), in late May or June. For the previous three months the Matariki cluster is below the horizon so it cannot be seen. The rising signals the turn of the seasons and sets the calendar for the rest of the year.
2. Show your students the first few slides on PowerPoint One. Discuss what stars are and how our Sun is an example.
What shape is a star?
Today we are going to make some stars to display using shapes.
3. For each slide discuss how the left-hand star is made then built with to form the right star. Encourage your students to use correct names for the composing shapes, such as triangle, square, hexagon, trapezium. If you have access to sets of pattern blocks students can make the stars on Slides Three and Four themselves by copying the pattern.
4. Many learning objects are available online that provide a virtual set of pattern blocks. Your students might use the objects to recreate the star patterns. Also encourage students to create their own stars, using the blocks. Slide six is a blank canvas of pattern block. With the slide in edit rather than display mode, you can use the blocks to form other patterns.
5. Copymaster One provides two different puzzles for your students. They cut out the pieces to form a star. Glue sticks can be used to fasten the parts in place (tricky for young fingers).
6. Other Activities with Stars
• Most star designs have mirror symmetry. That means that a mirror can be placed within the star, so the star appears complete. The reflection provides the missing half of the star. Demonstrate to your students how that works. Copymaster Two provides four different star patterns. Give students small mirrors and ask them to find the places where a mirror can go so the whole star is seen. Note that Star Four has no mirror symmetry so it is a non-example. Star Four does have rotational symmetry so it can map onto itself by rotation.
• Copymaster Two also has half stars on page two. Ask your students to complete the whole star. Be aware that attending to symmetry is harder when the mirror line is not vertical or horizontal. Can your students attend to perpendicular (at right angle) distance from the mirror line in recreating the other half?
7. Creating stars by envelopes
• The diagonals of some polygons create beautiful star patterns. The most famous pattern is the Mystic Pentagram that is created within a regular pentagon. Video One shows how to get started and leaves students to complete the pattern. The exercise is good for their motor skills as well as their attendance to pattern and structure. Copymaster Three has other shapes to draw the diagonals inside. Note that a diagonal need not be to the corner directly opposite, it can also go to any corner that it does not share a side with.
• Nice questions to ask are:
How do you know that you have got all the diagonals? (Students might notice that the same number of diagonals come from each corner)
Does the star have mirror lines? How do you know?

#### Lesson Two

In this lesson your students explore family trees, working out the number of people in their direct whakapapa. Thinking about our relatives who are no longer with us, or have just arrived, is a traditional part of Matariki, the Māori New Year. According to legend Matariki is the time when Taramainuku, captain of Te Waka of Rangi, and gatherer of souls, releases the souls of the departed from the great net. The souls ascend into the sky to become stars.

1. Begin by playing a video or reading a book about Ranginui (Sky Father) and Papatuanuku (Earth Mother), the mother and father of Māori Gods.
2. Ask: How many parents (matua) do you have?
That question needs to be treated sensitively but the focus is on biological parents, usually a father and mother. You might personalise the answer by telling your students the names of your mother and father. Draw a diagram like this:
My mother and father had parents too. What are your parent’s parents called?
How many grandparents (koroua) do you have?
Students may have different ways to establish the number of grandparents, such as just knowing, visualising the tree and counting in ones, or doubling (double two),
4. Extend the whakapapa tree further.
5. Ask: What do we call your grandparent’s parents? (Great grandparents, koroua rangatira)
Nowadays, many students will still have living great grandparents. You might personalise the idea using your whakapapa.
What do we call the parents of your Great Grandparents?
Now I want you to solve this problem. How many Great Great Grandparents do you have?
6. Encourage students to work in small groups. Provide materials like counters or cubes to support students. Ask students to draw the whakapapa diagram to four layers and record their strategy as much as they can. After an appropriate time, share strategies.
7. Discuss the efficiency of counting based strategies, counting by ones and skip counting in twos. Highlight more efficient methods such as doubling, e.g. 2 + 2 = 4, 4 + 4 = 8, 8 + 8 = 16.
8. Ask: Now let’s just think about our parents. If three children in this class invited their parents along, how many parents would that be?
Use the context as a vehicle for introducing even numbers (multiples of two), Act out three children getting their parents (other students) and bringing them to school. Change the number of students and work out the total number of parents. Find a way to highlight the numbers that come up, such as shading the numbers on a virtual Hundreds Board.
9. Ask: Can anyone see a pattern in the numbers of parents?
Students might notice that even numbers occur in the 0, 2, 4, 6, 8 columns of the hundreds board.
10. Introduce a challenge: If everyone in this class brought their parents along, how many parents would that be?
11. Ask the students to work in small teams with materials. Providing transparent counters and individual Hundred Boards will be useful but provide a range of options. Watch for your students to:
• Structure their model as “two parents for every one student.”
• Deploy materials in an organised way to represent the whole class.
• Use efficient counting strategies, to systematically find the number of parents.
• Use recording to organise their thinking, particularly the use of number symbols.
12. Share the strategies students used with a focus on the points above. Depending on the current achievement of your students you might extend the problems. For example:
• Ten parents came along. How many students brought them?
• How big would our class be if 100 parents came along?
• If every student in another class brought two parents along, could there be 54 parents? What about 53 parents? Etc.
• If students in our class brought all their Grandparents along, how many Grandparents would that be?
• If we had to provide Matariki celebration for our Grandparents, what food and drink would we need?
• How much of each food and drink would we need?

#### Lesson Three

Matariki is a time for cultural activities, such as story-telling, music, and games. Titi Rakau is a traditional game that involves hitting and throwing sticks, usually to a rhythmic chant. It was used to enhance the hand-eye co-ordination of children and warriors. Rakau can be used as a vehicle for fractions and musical notation, as well as physical co-ordination. You can make the tasks below as simple or as difficult as you like.

1. Look at slide one of PowerPoint Two.
What fractions has the bar been broken into? (Quarters)
Each of these notes (crotchet) is one quarter of a bar in this music.
2. Find a piece of music online with a clear 4/4  time signature. That means there are four crochet (quarter) beats to the bar. This timing is very common in popular music. “Tahi” released by Moana and the Moa Hunters in 1994 is a good example that is easily found online. Ask your students to clap in steady 1, 2, 3, 4, … time with a consistent time between claps as the music plays.
3. Introduce rakau, made from rolled up magazines taped together with duct tape. Students might practise hitting the ground with the ends of the sticks on every beat of the 4/4  time. This can be changed to beat one on the ground, and beat two ‘clapping’ the sticks together in the air, beat three on the ground, and beat four in the air, etc.
4. Introduce the rest symbol using slide two of PowerPoint Two. In 4/4  time the rest is for one beat. So the rhythm is ‘clap, clap, clap, rest, clap, clap, clap, rest,…” as is used in “We will rock you,” by Queen. Slide three has a bar with two rests. See if the students can maintain that rhythm.
5. Slides four and five introduce the quaver which is a one-eighth note in 4/4  time. See if students can manage the two different rhythms, including the beamed (joined) quavers. Copymaster Four can be made into cards, or cut out as is, to create different rhythmic bars in 4/4  time. Note that the semibreve (circular note) denotes the whole of four beats, and the minim (stemmed hollow note) denotes one half of a bar. A minim is equivalent to two crochets.
6. Let your students make up a single bar using the cards. Encourage them to experiment with possible rhythms by trailing them with Rakau. Rests are usually part of Rakau to allow movement of the sticks from one position, e.g. floor, to another, e.g. chest. The rhythm a student creates can be played by another using Rakau.
7. Look for your students to:
• Apply their knowledge of fractions, such as one half and two quarters make one whole (bar)
• Recognise equivalence, such as two quarters make one whole or two eighths make one quarter.
8. Share the bar patterns that students create and play them with Rakau. Rhythms can also be checked by finding an online music composer for children and entering the notes. The software usually has playback.
9. Natural extensions of the task are:
• Explore different time signatures. Many Māori action songs are in Waltz time (3/4) meaning that there are three crocket beats to a bar. A crochet is one third of a bar in that time signature and a quaver is one sixth of a bar.
The popular chant associated with Ti Rakau is E Papa Waiari is in 6/8  time meaning there are six quaver beats to a bar. If you watch a video of a performance with Rakau the sticks are often hit on the ground on the first and fourth beats, or clicked together on the second, third, fifth and sixth beats.
• Try to work out and record the rhythm of pieces of music, using the cards. Choose a difficulty that suits your students. For example, E rere taku poi, is in 4/4 time and is the tune to “My Girl.”  Kiri Te Kanawa’s recording of Te Tarahiki in 1999 features a wonderful six quaver rhythm in 3/4  time.

#### Lesson Four

The rising of Matariki, in late May or June, signals to Māori that it is the start of a new year. It is appropriate for students to reflect on the passage of time. For young students there are important landmarks in the development of time, including:

• Recalling and sequencing events that occurred in their past.
• Anticipating events that might occur in the future.
• Recognising that time is independent of events, it progresses no matter what is occurring.

Cooking in a hangi

In the first part of the lesson students work with the first two ideas, recalling the past and anticipating the future.

1. Show students a video about preparing and cooking a hangi. There are many examples online. Before viewing the video prompt your students:
Watch carefully. At the end of the video I will ask you about how to make a hangi.
2. At times pause the video to discuss what might be occurring. Use the pause as an opportunity to introduce important language, like hangi stones, kai (food), prepare, cover, serve, etc.
3. Give each pair or trio of students a copy of the first six pictures of Copymaster Five.
I want you to put the pictures in the order that they happened. Put them in a line. Be ready to explain why you put the pictures in that order.
4. You might allow groups to send out a ‘spy’ to check the order that other groups are using. After a suitable time let the groups ‘tour’ the lines that other groups have created and change their own line if they want to. Bring the class together to discuss the order of events.
Why does this happen before this?
Why does this happen after this?
5. Do your students recognise the consequential effect of order? e.g. The fire cannot be lit until the hole is dug and there is somewhere to put it.
6. Can the students recognise what events occurred between two events? e.g. Covering the food with soil and waiting four hours occurred between putting the food and stones in the hole and taking the cooked food out.
7. Discuss:
What might have happened before the hole was dug?
What might have happened after the food was served?
8. Ask students to draw and caption an event that occurred before the sequence of pictures, and another event that happened after. For example, the food must be prepared before or while the hole is dug. It must be bought or gathered before it can be prepared. After the food is served it will be eaten. Copymaster Five, pictures 7 and 8 are before and after pictures.
9. Add students’ before and after pictures to the collection from Copymaster Five. You might create a wall display. Some before and after pictures might need to be sequenced. Picture 9 is an event (uncovering the hole) that occurs between two of the six events. Where does it go? Why?

Chances of a good year

In former times Tohunga, wise people of the village, looked at the sky before dawn to watch the rising of Matariki. They used the clarity of the stars to predict what the new year would bring. A clear sky with the stars of Matariki shining brightly signalled a good season for weather and the growing and harvesting of crops. A cloudy sky signalled bad luck.

1. At the rising of Matariki, some stars shine brightly while others do not. Each star has a special job. Use PowerPoint Three to introduce the stars and their jobs. Play a game with Copymaster Six. The first page is a game board and use the second page to make a set of 12 cards (bright stars, fuzzy stars, and clouds). Note the second page can be used to make three sets of the cards.
2. Put the gameboard down and spread the cards face down on the floor. Mix the cards up while students close their eyes. Students select cards one at a time to cover each of the seven stars. For example, Matariki might be covered by the card for a fuzzy star. Slide Two of PowerPoint Three shows a completed gameboard (Click through it to place the cards).
If you saw this what would you predict?
3. Students should make comments like:
There will be plenty of rain but not too much, and the crops will grow well.
It is going to be a bit windy.
There will be lots of food in the rivers, lakes and sea.
4. Let students play their own game of predicting the upcoming year. Look to see whether students consider what is on the set of cards in predicting what card might come next.
5. After playing the game discuss:
• Is it possible to have a year where every star shines brightly? (No. There are six bright star cards and seven stars of Matariki)
• What is the worst year you can have? (All two clouds and four fuzzy stars come up)
• How likely is it that you will have a good year? (Quite likely since half the cards are bright stars and one third of the cards are fuzzy stars)

#### Lesson Five

This lesson involves making rewena paraoa (potato) bread. The process of making it takes three stages; preparing the ‘bug’, mixing and baking, then serving. Therefore, it is not a continuous lesson. Preparation and serving food are important activities for Matariki celebrations.

1. Explain: In the next three days we are going to make rewena bread from potatoes (paraoa). Why is the bread you buy at the supermarket so light and fluffy?
2. Some students may have made bread with their parents or grandparents and can talk about yeast as the ‘leavening’ agent. You can explore the science of breadmaking using this resource. Play an online video of breadmaking and discuss what each ingredient contributes.
3. In rewena bread the natural yeast from potatoes is used to raise the dough. The best potatoes are older taewa (Māori potatoes) which are small and knobbly, but any medium sized aged potato will do. One medium sized potato is need per recipe (for three students).
4. Weighing the potatoes on kitchen scales is a good opportunity to introduce the students to the gram as a unit of mass. Can you students predict the weight of each potato? You might have a potato peeling competition, using proper peelers (not knives). The student who gets the longest peel wins. Naturally, you will need to measure the lengths of the peels and come up with a class winner.
5. To make one batch of ‘the bug’ cut up the potatoes into smaller bits and boil them in clean water (no salt) until they are soft. You might time how long that takes. Let the potatoes cool and don’t drain the water. Mash the potatoes, water included. Add in (for each recipe):
2 cups of flour
1 teaspoon of sugar
Up to one cup of luke-warm water (as needed to maintain a paste-like consistency)
6. After you have made a bulk lot of ‘the bug’ put it into clean glass jars to ferment. Fill each jar to one third as the mixture will expand. Cover the jar with greaseproof paper and fix it with a rubber band. Over three of four days the mixture will ferment. Feed it daily with a mix of one teaspoon of sugar dissolved in half a cup of potato water. Your students will be intrigued by the foaming concoction that develops.
7. After ‘the bug’ has developed let your students create their own batch of rewena bread by following the recipe (PowerPoint Four). This is a good exercise in interpreting procedural language. Read the instructions to the class if necessary or use your most competent readers.
8. Once the bread is made it needs to be cut into slices.
How many slices should we make?
How thick will the slices be?
How many cuts will we make?
9. You might explore sharing slices equally among different numbers of students. Naming the equal parts will introduce fractions. You might explore the different ways to cut a slice in half or quarters.