## Late level 1 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.

## Lonely Pig

Level One
Statistics
Units of Work
In this unit we have a first look at uncertainty and chance. We begin to develop an intuitive sense of what chance and possibilities are about through games that involve following rules, making predictions and seeing what happens.
• Use everyday language to talk about chance.
• List outcomes of simple events.

## The teddy bears’ fraction picnic

Level One
Number and Algebra
Units of Work
This unit supports students to equally partition objects and sets into fractional parts.
• Partition a length, area or volume into equal parts.
• Partition a set into equal parts and anticipate the result.
• Recognise that the numerator of a fraction is a count.
• Recognise that the bottom number of a fraction gives the size of the parts being counted.

Level One
Number and Algebra
Units of Work
In this unit students explore different ways to communicate and explain adding numbers within ten. The representations included are number lines, set diagrams, animal strips and tens frames.
• Draw representations to show simple addition equations.
• Write an equation/number sentence to match their representation.

## Taller/Wider/longer

Level One
Geometry and Measurement
Units of Work
This unit encourages students to use measurement language and counting to compare the attributes of length, width or height of objects in a variety of game situations. The transition from counting all to advanced counting is also supported.
• Use measuring language to compare length, width, and height.

## Inequality symbols and relationships

Level One
Number and Algebra
Units of Work
The purpose of this unit of three lessons is to develop understanding of how to recognise and record relationships of (equality and) inequality in mathematical situations.
• Understand the equals symbol as an expression of a relationship of equivalence, and explain this.
• Recognise situations of inequality and use the inequality (‘is not equal to’) symbol, ≠.
• Understand that < and > symbols can make equivalent statements.
• Use relationship symbols =, <, > in...
Source URL: https://nzmaths.co.nz/user/75803/planning-space/late-level-1-plan-term-4

## Lonely Pig

Purpose

In this unit we have a first look at uncertainty and chance. We begin to develop an intuitive sense of what chance and possibilities are about through games that involve following rules, making predictions and seeing what happens.

Achievement Objectives
S1-3: Investigate situations that involve elements of chance, acknowledging and anticipating possible outcomes.
Specific Learning Outcomes
• Use everyday language to talk about chance.
• List outcomes of simple events.
Description of Mathematics

Although students at this level won't be ready to fully grasp the idea of chance they will develop some valuable intuitive notions. Underlying the activities is the idea that some events can be influenced by information, whereas others rely largely on luck. This provides the basis for the two ways of determining probabilities: theoretically and experimentally. Later on we explore certain situations such as rolling dice, using spinners, choosing cards to determine precise values of probability. This is the theoretical aspect. Ideally you would always like to do this as you are then sure exactly what the probability is. However, in other situations we have to rely on a series of experiments and deduce the likely probability of an event through the long-term frequency of its occurrence. This applies to the kind of events that insurance companies are interested in such as the likelihood of an accident or the length of life of a 30 year old male.

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

• altering the numbers of each type of animal
• having two students play a demonstration game so that the whole class can see how the game works before playing independently
• change the number of sides with each colour on the dice for the Greedy Cat game.

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

• Use different cards instead of the provided animal cards. This could be any kind of collecting card, or cards that you make yourself with images to interest your students. Ensure that the numbers of each added type are the same as those in the original collection.
• Choose a different story for the Greedy Cat game, This could be an existing story about a hungry person or animal (for example Awarua, the taniwha of Porirua) or a made up story about a familiar local animal such as the school cat or the teacher's dog.
Required Resource Materials
• Resealable sandwich-size plastic bags to store cards
• Animal cards
Activity

#### Session One: Cats and Dogs

Today we make our own set of cards using pictures of cats and dogs.

1. Give each pair of students 20 cards. You might like to use the cats and dogs from Copymaster 1. Otherwise you will need to direct the students to draw cats on 10 of the cards and dogs on the other 10.
2. When the students have completed their pack of cards demonstrate a game of Cats and Dogs.
3. Rules
Shuffle the cards.
Put the cards face down in one pile.
Players decide who will collect same pairs and who will collect different pairs.
Take turns turning a card from the top of the pile.
Compare the two cards. If they are the same, the player collecting same pairs takes them. If they are different, the other player takes them.
The game continues until all cards are used.
The winner is the player who collects the most cards.
4. After the students have played a few games lead a discussion highlighting their observations about the chances of winning.
Was one of the players luckier than the other one? What made you think this?
How many times did you win?
How many times did your partner win?
Was it better to be a same or a different? Why?

(note: The probability of different pairs is slightly greater than that of same pairs.)

#### Session Two: Cats, Dogs and Mice

Today we add 6 mice to our pack of cat and dog cards and play the game again.

1. Give the pairs 6 blank cards to draw the mice or use the mice from Copymaster 1. Add the mice to the pack of 20 cats and dogs used in session one.
2. Play the same pairs and different pairs game from yesterday.
3. After the students have played a few games lead a discussion highlighting their observations about the chances of winning.
Was one of the players luckier than the other one? What made you think this?
How many times did you win?
How many times did your partner win?
Was it better to be a same or a different? Why?
Was it easier to win today? Why/Why not?

#### Session Three: The little lost pig

Today we add one pig to our pack of cats, dogs and mice and play a new game.

1. Give each pair one pig card to add to their set.
2. Explain that today we are using the cards to play animal memory.
Rules
Spread the cards out face down.
Players take turns to turn over two cards.
If the cards are the same the player keeps the pair of cards but does not have another turn.
If the cards are different the cards are turned back face down.
Continue taking turns until all the cards (except the pig) are collected.
3. As the students play the game ask questions that focus on the likelihood of finding pairs.
What card do you think you will turn up next? Why do you think that?
Which cards do you think will be last?
Which are the hardest pairs to find? Why?
Which are the easiest pairs to find?
What can you tell me about the pig?
Why do you think the game is called The little lost pig?
Can you think of another name for our game?

#### Session Four: Greedy Cat

Resources:
• Wooden cubes with two of the faces red, two blue and two green (dot stickers work well)
• Pictures of 3 cats from Copymaster 2
• Pictures of fish cut outs from Copymaster 3 (30 per page)
• Felt pens (colours to match the dots)

In this activity the students roll a dice to feed fish to our three cats. The students will investigate the chance of giving a fish to their cat.

1. Give groups of 3 students the three cat pictures, a prepared dice and 30 fish.
2. Each student chooses one of the three colours for their cat and colours it in (single colour only).
3. The students play Greedy cat
Rules
The students take turns rolling the dice.
The student whose colour shows gives their cat a fish.
The game continues with players taking turns until all the fish are eaten.
4. After allowing the students to play the game(s), discuss:
Which coloured cat got the most fish?
Was there a lucky colour in your group?
Was it lucky in all the groups? Why/Why not?

#### Session Five: Feeding Greedy Cat

Resources:
• Dice 1: 5 blue faces representing the greedy cat, one red face for the second cat.
• Dice 2: 3 blue faces, 3 red faces.
• Copymaster 3 of Fish Cut Outs (30 per page)
• Copymaster 4 of blue greedy cat and red greedy cat

In this game the students experiment with different dice.

2. Discuss Greedy Cat’s need for food.
3. Give pairs the two cat pictures, the prepared dice and 20 fish.
Rules
The students take turns first selecting and then rolling one of the dice.
They give a fish to the cat whose colour is shown.
The game continues with players taking turns until all the fish are eaten.
4. After allowing the students to play the game(s), discuss:
Which cat got the most fish?
Was there a lucky colour in your group? Why?
Was it lucky in all the groups? Why/Why not?
What difference did the dice you chose make?
5. Now pose a problem for the students.
In the game we want to stop Greedy Cat from yowling for food. What should we do? (Check that the students understand that they need to feed Greedy Cat to stop him yowling.)
6. Send the students away to explore the problem as they play the game.
7. After allowing the pairs time to explore the solution and share their thinking about the problem.
Attachments

## The teddy bears’ fraction picnic

Purpose

This unit supports students to equally partition objects and sets into fractional parts.

Achievement Objectives
NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
Specific Learning Outcomes
• Partition a length, area or volume into equal parts.
• Partition a set into equal parts and anticipate the result.
• Recognise that the numerator of a fraction is a count.
• Recognise that the bottom number of a fraction gives the size of the parts being counted.
Description of Mathematics

In this unit students explore equal partitioning of objects and sets, and how to name the parts they create using fractions. The easiest partitions are those related to halving since symmetry can be used. Halves come from partitioning a whole into two equal parts. Quarters come from partitioning a whole into four equal parts, etc.

An object is partitioned according to an attribute, or characteristic. For example, a banana might be partitioned by length, but a sandwich partitioned by area. Some attributes are more difficult to work with than others. Length is the easiest attribute but others such as area, volume (capacity), and time are more difficult.

The results of partitioning a set into fraction parts can be anticipated, using number knowledge and strategies. For example, one quarter of 20 could be found by:

• Dealing 20 objects onto four quarters one by one, or in collections of two.
• Halving 20 into 10, then halving again.
• Realising that putting one object on each quarter uses four each time, and repeatedly adding four until 20 is used up.

In this unit we apply three criteria when evaluating students’ capacity to partition and object or set into fractional parts:

• The correct number of parts is created
• The shares are equal
• The whole is exhausted (used up).

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

• encouraging students to work collaboratively in partnerships
• pausing to allow students to share their work with others
• restricting or extending the range of fractions students are asked to work with.

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

• The first slide of PowerPoint 1 could easily be replaced to change the whole context of the unit to be about any different characters you choose.
• You could add more types of food to the ones used to better reflect the food your students usually eat.
Required Resource Materials
• Teddy Bears (brought from home) or Teddy Bear counters
• Pattern blocks or Attribute blocks
• Play dough and plastic knives
• Counters, cubes, or beaNZ
• Blank wooden cubes (to make dice)
• Apple, doughnut, banana, bread (optional)
• Clear plastic drink bottle and plastic cups
• Scissors
• Copymasters One, Two, Three, Four, Five, and Six
• PowerPoints One, Two, and Three
Activity

#### Lesson One

1. Use PowerPoint One to tell the story of two teddy bears going on a picnic. Unfortunately, Papa Bear, who made the lunch, forgot to cut up the food items equally. What are Ben and Barbara Bear supposed to do?
Ben and Barbara are friends. They always share their food equally. How could they cut this sandwich? (Slide Two)
2. Discuss ways to cut the sandwich in two equal pieces. Having two teddy bears available allows for acting out of the equal sharing.
What does equal mean?
What do we call these pieces? (halves)
3. Write the words “one half” and the symbol 1/2.
Why do you think there is a 1 and the 2? (2 refers to the number of those parts that make one)
4. Look at Slides Three and Four of PowerPoint One.
Is the sandwich cut in half? Why? Why not?
5. Provide the students with three ‘sandwiches’ made from Copymaster One.
What other ways could Ben and Barbara cut their sandwich in half?
6. For students who complete the task quickly, give them another sandwich.
Cut this sandwich in two pieces that are not halves.
7. When each student has produced at least two halved sandwiches bring the class together to share their thinking. Make a collection of halves and ‘non-halves’ and discuss those collections.
What is true for the sandwiches that are cut in half?
What is the same or the sandwiches that are not cut in half?
The key idea of halving is that both Ben and Barbara get the same amount of sandwich. As the sandwich is not exactly symmetric through a horizontal mirror line you might get interesting conversation about the equality of shares. Slide Five may provide some stimulus for the discussion. Using the cut-out pieces of sandwich, show how halves map onto each other by reflection (flipping) or rotating (turning). With paper sandwiches you can fold to show that mapping of one half onto another.
8. Luckily Papa Bear prepared more than just one sandwich. Copymaster Two has some other food items that Papa packed. Ask your students to work in pairs to cut each item in half. They need to record where they will cut each item and justify why they put the line where they did.
9. After enough time bring the class together and discuss their answers. You might have real items to half. This is particularly important in the case of the bottle of fruit drink and the bag of 12 cashews. Note there are different ways to partition each object into two equal parts. Points to note are:
• Do your students create two parts, exhaust all of the objects or set, and check that the parts are equal in size?
• Do your students see halving as a balancing, symmetric process?
10. For example, get two clear plastic cups. Ask a student to pour half of the drink bottle into each cup, one for Ben and one for Barbara. Look to see that the student balances the level in each cup. Capacity is a perceptually difficult attribute for measurement.
11. Similarly, look to see it students equally share the cashews by one to one dealing.
12. Finish the lesson with the final two slides of PowerPoint One.
Can your students distinguish when a shape or set has been halved?

#### Lesson Two

In this lesson students explore the impact of other Teddy Bears joining in the picnic. What happens to the size of the portions as more shares are needed? How do we name the shares?

1. Remind the students about the previous lesson. Use a chart of halves and ‘not halves’ to focus attention on equal sharing and the words and symbols for one half.
Imagine that four Teddy Bears went on the picnic instead of two. Papa still packed the same amount of food. What would happen then?
2. Sit four teddy bears around in a circle. Give them names like Ben, Barbara, Bill and Bella. Use Copymaster One to make a few sandwiches.
How could we cut a sandwich into four equal parts? What will the parts be called?
3. Students should find quartering intuitive, as it involves halving halves. Symmetry is still useful. Let students share how they think a sandwich could be partitioned into four equal parts. Act out giving one part to each Teddy Bear.
Does each Teddy Bear get the same amount of sandwich?
Common solutions are:

Students often miss the option of dividing the area of the sandwich by length…

Or other possibilities…
4. Ask students: How big is one quarter, compared to one half? How many quarters is the same as one half?
5. Use pieces of’ ‘sandwich’ to check. It is good for students to see that two quarters make one half irrespective of the shape of the pieces. Challenge the students to find some ways to cut a sandwich into four pieces that are not quarters. Examples might be:

6. You may need to highlight the lack of equality of non-quarters by mapping the pieces on top of one another. Discuss that quarters are only quarters if:
• All the sandwich is used
• The pieces are all equal in size.
7. Provide the students with copies of Copymaster Two again.
This time the challenge is to share each picnic food into quarters.
8. Give the students enough time to record where they would partition the items before gathering as a class to discuss their ideas. Pay attention to the different ways that the foods are partitioned. Act out the partitioning either with real objects or imitations, e.g. banana and cheese made from playdough of plasticine. For example, the banana is a length, so the partitioning is most likely to be by length. Some students may attend to area (second example).
or
The shape of the sector of cheese places a restriction on how it can be partitioned. Likely responses are (left is correct, right is not):

9. Equally sharing the bottle of fruit juice among four glasses might be done in two main ways.
• Set out four plastic cups and keep pouring while balancing the water levels.
• Halve the bottle into two plastic cups. Pour half of each cup into another cup until the levels match.
10. Look to see that your students can equally share the 12 cashews by dealing one to one for each teddy bear. Highlight that the half share is six cashews and the quarter share is three cashews.
11. Record the words and symbol for one quarter. Create a chart of quarters for the food items.
12. Finish the session with the game called ‘Make a whole sandwich.” Students need copies of the first page of Copymaster Three (Halves and quarters) and a blank dice labelled, 1/2, 1/2, 14, 2/4, 3/4 ,choose. The aim of the game is for players to make as many whole sandwiches as they can.
Players take turns to:
• Roll the dice.
• Take the amount of sandwich shown on the dice, e.g. If 3/4 comes up, then the player takes three one quarter pieces of sandwich.
• If choose comes up the player chooses a single piece.
At anytime a player can rearrange their sandwich pieces to make as many whole sandwiches as possible. Time the game to finish in five minutes which is long enough to gather a lot of pieces.
13. Discuss what might be learned from the game. For example:
• Three quarters is three lots of one quarter.
• Two quarters is the same amount as one half.
• Two halves and four quarters both make one whole.
14. Record those findings symbolically and discuss the meaning of + and = as joining and “is the same amount as”, e.g. 3/4 = 1/4 + 1/4 + 1/4 or 2/4 = 1/2.

#### Lesson Three

In this lesson students explore the equal sharing of picnic foods among three Teddy Bears. Thirding an object or set is generally harder than quartering because symmetry is harder to use. Your students should observe that thirds of the same object or set are smaller than halves but larger than quarters.

1. Begin with a reminder of what you did in the previous lesson:
We explored how to cut sandwiches into two and four equal parts. What are those parts called? (halves and quarters)
2. Be aware to the non-generality of early fraction words in English. Halves should be twoths, and quarters should be fourths, to match what occurs with sixths and further equal partitions. In many other languages, such as Maori and Japanese, the word for a fraction indicates the number of those parts that make one. Maori for one fifth is haurima (rima means five) and in Japanese one fifth is daigo (go means five).
What parts will we get if we share the food among three Teddy Bears?
3. Point out that third is a special English word for one of three equal parts. Show the students one half and one quarter of a sandwich.
Which is bigger, one half or one quarter? Why?
How big do you think one third will be? Why?
4. Students might conjecture that thirds are smaller than halves but bigger than quarters. That is true on the assumption that the whole (one) remains the same. Give the students copies of four ‘sandwiches’ from Copymaster One. Set out three Teddy Bears.
Find different ways to share one sandwich equally among the three Teddy Bears.
5. PowerPoint Two shows two examples and three non-examples of cutting a sandwich into thirds. Note that the successful cuts are vertical and horizontal. Students should notice that the two non-examples produce unequal parts. On Slide Five ask:
Is this sandwich cut into thirds? (No, the parts are unequal)
6. Give the students a copy of Copymaster Two. Ask them to work in pairs to share each food item into thirds. After a suitable period bring the class together to discuss the equal sharing.
7. PowerPoint Two contains animations of partitioning some of the foods into thirds. You may want to act out model answers, certainly with pouring among three plastic glasses while keeping the levels balanced. Discuss issues that arise. For example:
Do the parts have to be the same shape to be equal? (Banana is a good example)
Why are thirds harder to make than haves and quarters? What other fractions would be hard to make? Why?
8. In the next part of the lesson students anticipate the result of cutting thirds and quarters in half. What are the new pieces called?
PowerPoint Two, Slides 9-11, shows how halving quarters produces eighths and halving thirds produces sixths. It is important for students to connect the naming of these equal parts to how many of those parts form one whole (sandwich).
9. Use parts of sandwiches made from Copymaster Three to build up students’ counting knowledge of fractions. Lay down and join the same sized pieces and ask students to name the fraction. For example:

One sixth                     Two sixths                   Three sixths                Four sixths
(or one third)               (or one half)                (or two thirds)
10. Count with the same sized parts past one whole and write the fraction symbols as you count.
How much of a sandwich is 2/3? 3/4? 5/8?  etc.
11. It is important that students understand that the top number of a fraction (the numerator) is a count of equal parts. The bottom number (the denominator) tells how many of those parts make one.
12. Change the game from Lesson Two called “Make a Whole sandwich.” Include all the pieces from Copymaster Three so that thirds, sixths and eighths are also available. Alter the dice to read 1/2, 1/4, 1/3, 1/6, 1/8, choose. Allow the students more time to play that game, say ten minutes.
13. Discuss some ways that the students found to make one whole sandwich. Record their suggestions using symbols, e.g. 1/2 + 1/3 + 1/6 or 3/4 + 2/8. Begin a chart of ways to make one that students can add to using pictures (using pieces from Copymaster Three) and symbols.

#### Lesson Four

In this lesson students work from part to whole. Usually students encounter problems where the whole is well-defined, and they are shown, or must create, the required fraction. As they progress to more complex tasks, it is also important that students can relate a fraction part back to the whole from which that part may have been created.

1. Use Copymaster Four to create a set of paper part foods. With each challenge below the aim is to draw the appearance of the whole, in schematic rather than detailed form. Place the fraction piece in the middle of an A3 sheet of photocopy paper to allow space to draw.
Suppose there are eight Teddy Bears at the picnic. This piece is one eighth of a sandwich, how can we find the size of the whole sandwich?
2. Check that students recognise that eight pieces of that size will make the complete sandwich. Ask a student to draw around the outside of the piece to show the whole sandwich.
How can we check [Name]’s estimate of the whole sandwich?
3. Iterating (copy and pasting) eight copies of the piece will give the area of the sandwich. Note that the whole could look differently as the eighths could be arranged on a line or as a rectangle or other formations. However, the context suggests a rectangular arrangement is best. Other challenges are:
This is one fifth of the banana. How long is the banana?
This is one third of the doughnut. How big is the doughnut?
This is one half of the bun. How big is the bun?
This is one sixth of the chocolate cake. How big is the cake?
This is one quarter of the packet of walnuts. How many walnuts are in the whole packet?
4. Build on the capacity example used earlier. Tell your students that a plastic cup filled to a certain level is one third, quarter, fifth, etc. of a whole bottle. Can they make the quantity of water that fills the bottle?

Independent work

1. Show your students a set of pattern blocks. Choose one shape as ‘the piece of food’, e.g. a rhombus.
I will make a chart with this pretend piece of food.
2. Develop the chart as you go rather than present it as complete.

Provide the students with a copy of Copymaster Five to reduce the writing load.

#### Lesson Five

In this lesson students connect fractions of regions and fractions of sets. Partitioning of a set into equal parts fluently requires multiplicative thinking that most young students do not possess yet. However, exposure to problems with equal partitioning provides opportunities to learn that develop additive part-whole thinking and the ‘sets of equal sets’ concept, that is fundamental to multiplication and division.

1. Use PowerPoint Three to introduce the context of the four Teddy Bears sharing a chocolate cake.
How many smarties do you think go on each quarter? Remember that the quarters must be equal.
2. Students need to use their number understanding to anticipate a result. Some students may ‘virtually’ share the smarties one at a time (look for eye, hand and head gestures). Tracking the number of ‘virtual’ smarties in each quarter puts a significant load on working memory. Estimates of three, four or five should be worked with.
Let’s imagine there are three smarties on each quarter. How many smarties would that be on the whole cake?
3. Working from part to whole in this way checks the estimate. Using actual objects, like counters, to enact the sharing, might follow once anticipations are made and justified.
4. Other students may use additive knowledge, such as 4 + 4 = 8 (one half), then 8 + 8 = 16 (whole cake). Students might adjust from an initial prediction, such as 20 smarties is four more than 16, so each quarter must get 4 + 1 = 5 smarties.
5. Slide Two makes a minor adjustment to the number of shares to see if students can anticipate the effect of more parts on the size of shares. With five parts (fifths) additive knowledge with fives is more likely to be used.
Last time the cake was in quarters. What fractions is it in now? (fifths)
Will the five Teddy Bears get more or less than four Teddy Bears got?
6. Check that students recognise that fifths are smaller than quarters, even in the sets of smarties context. Encourage use of additive thinking by animating one smarty on each fifth of the cake.
How many smarties are on the cake?
If I put two smarties on each fifth how many would be on the whole cake?
How many smarties can I put on each fifth?
7. Slides Three and Four present two other situations of equally sharing smarties onto a cake. Ask your students to name the fraction parts and anticipate the result of the equal sharing.
8. Let your students play the Birthday Cake Game in pairs or threes. The game is made from Copymaster Six. You will need blank dice and counters as well.

Purpose

In this unit students explore different ways to communicate and explain adding numbers within ten. The representations included are number lines, set diagrams, animal strips and tens frames.

Achievement Objectives
NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
NA1-4: Communicate and explain counting, grouping, and equal-sharing strategies, using words, numbers, and pictures.
Specific Learning Outcomes
• Draw representations to show simple addition equations.
• Write an equation/number sentence to match their representation.
Description of Mathematics

In this unit students are introduced to different ways to represent addition. Students will be able to communicate their thinking using the representations. Teachers may choose to spend more time on some representations depending on students’ familiarity with them. The unit is designed to help students transition from Counting All to Advanced Counting, using a counting on strategy.

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

• adjusting the size of numbers used as appropriate
• encouraging students to work collaboratively.

The contexts for this unit are strictly mathematical but the materials used can be adapted. Physical items that have significance to your students might be better used than standard mathematical equipment. For example, if you have a big set of shells for environmental studies you might use those shells as the materials for session 1.

Required Resource Materials
Activity

#### Session 1

In this session students are introduced to using a diagram or picture to communicate an addition equation.

1. Draw a circle and place 5 blue counters in the circle.
How many blue counters are there?
2. Place 2 red counters in the circle and ask:
How many counters are in the circle now?

Students at a Counting All stage of the Number Framework will count from one, i.e. 1, 2, 3, 4, 5, 6, 7.
Students at an Advanced Counting of the Number Framework will be able to count on from the five i.e. 5... 6, 7.
Using two different colours will help encourage students to count on.

3. Draw another circle and place 5 blue counters in it.
How many blue counters are there?
Count with me as I add more counters.
4. Add 3 counters and count as each counter is added "6","7","8".
How many counters are there in the circle now?
How many red counters did we put in?
5. Introduce the idea of recording what the diagram shows. Tell the students you are going to write the number sentence.
How many blue counters are there? (Write 5 in blue.)
And how many red counters are there? (Write 3 in red)
5 and 3 makes how many counters?

6.
7. Work with the students to draw set diagrams. Use coloured pens to draw circles instead of using counters (or use coloured round stickers). Use the coloured pens to write the number sentence beside the set diagram. Number sentences can focus on facts to five or facts to ten depending on the students’ knowledge.
8. Students can work in pairs or as individuals to practise using the set diagram.

#### Session 2

In this session students are introduced to tens frames as way to represent addition equations.

1. Show the students the tens frames.
How many dots are on the frame?
2. Students should be able to recognise the patterns and answer instantly, rather than counting each dot.
3. Show the students a tens frame. For example:

How many dots are on the frame?
How many dots will there be if I add another 2 dots?

Students may be able to image 2 more dots and then count all the dots, or they may need you draw the 2 dots before they can count all the dots.

4. To encourage students to use counting on strategies use a coloured pen to draw 2 more dots and count with the students as each dot is added "7", "8".
5. Write beside the tens frame the number sentence 6 and 2 makes 8. Again using coloured pens to match the coloured dots to help students connect the representation with the number sentence.
6. Make copies of the tens frames and ask pairs or individual students to complete addition equations. For example, add 3 dots. Encourage them to count on as they add the dots. Students then write the number sentence.
7. Number sentences can focus on facts to five or facts to ten depending on the students’ knowledge.

#### Session 3

In this session students are introduced to animal strips as a way to communicate addition equations.

1. Show the students the animal strips.
How many animals are on the strip?
2. Students should be able to recognise the patterns and answer instantly, rather than counting each animal. Show the students that the dotted line after the 5th animal can help them recognise the number.
3. Show the students an animal strip.
How many animals are on the strip?
4. Show the students another animal strip and ask the students:
How many animals are there altogether?
5. Encourage the students to count on. i.e. "8", "9".
6. Write under the animal strip the number sentence "7 and 2 makes 9”.
7. Make copies of the animal strips and ask pairs or individual students to make their own addition equations. Encourage them to count on as they add the animals. Then ask the students to write the number sentence underneath.
8. Give students animal strips appropriate for their knowledge stage, for example facts to 5, facts with 5, doubles, or facts to 10.

#### Session 4

In this session students are introduced to the number line as a way to communicate addition equations.

1. Draw a number line and mark a number by putting a coloured counter over the number 4.
If we count on 2, what number on the number line will we be at?
2. Count with students as you make the jumps, "5", "6" and slide the counter along to the new place.
3. Discuss with the students the number sentence 4 and 2 makes 6 and write it below the number line.
4. Give the students a copy of a number line. Ask the students to show on the number line 6 and 3 makes ?
5. Encourage the students to count on as they mark the jumps with their counter and their pencil.
6. Ask the students to write the number sentence under the number line.
7. Give the students copies of the number line and ask them to show addition equations.

#### Session 5

In this session students choose two or three addition facts and show each of them using the 4 different representations covered in this unit.

Attachments

## Taller/Wider/longer

Purpose

This unit encourages students to use measurement language and counting to compare the attributes of length, width or height of objects in a variety of game situations. The transition from counting all to advanced counting is also supported.

Achievement Objectives
GM1-1: Order and compare objects or events by length, area, volume and capacity, weight (mass), turn (angle), temperature, and time by direct comparison and/or counting whole numbers of units.
Specific Learning Outcomes
• Use measuring language to compare length, width, and height.
Description of Mathematics

Measuring is about making a comparison between what is being measured and a suitable measurement unit.The first step in the measuring process is understanding that objects have attributes that can be measured. Initial experiences are needed to develop awareness of the attribute and to introduce the necessary language, for example, long, longer, short, shorter, tall, taller, wide, wider, narrow, narrower.

The activities in this unit provide experience in using a range of measurement language.

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

• modelling each game for the class before they begin
• reducing the number of pieces added or removed each turn, or eliminating subtraction altogether.

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

• make your own templates for the games. You may prefer to use a different shape of tree for session 3 or fence pieces instead of brick walls for session 4.
Required Resource Materials
Activity

#### Session 1

In this session we introduce the language of comparison that will be used throughout the unit.

1. Ask students to put their hand up if they think they are tall.
2. Choose two of the taller students.  Get them to stand up.  Ask the rest of the class:
Are they tall?
Who is taller?

Compare the heights of the two students to see who is taller.
Ensure that all students understand that taller means more tall than.
3. Repeat with short and shorter.
4. Give each student a piece of string (ensure that the pieces are of a variety of lengths).
5. Allow students to compare in pairs:
Is my string longer?
Is my string shorter?
6. If you have a variety of toy cars or similar, you could use them to practice width comparison language in a similar way.

The following three sessions provide games in which students will practice the language of comparison.The games could be introduced in three separate sessions or all be introduced in one session and then played in groups rotating over several sessions. These games are also suitable to go into a general box for early finishers to use during other maths lessons.

#### Session 2: Wiggly Worms

Wiggly Worms is about the language of length. Students will be encouraged to use the words long, short, longer, shorter, longest, shortest.

1. Wiggly Worms is a game to play in pairs. Each pair needs a dice with the sides labelled +1, +2, +3, +4, -1, -2, and a set of Wiggly Worm pieces (Copymaster 1). Students can cut out and colour in their own worm pieces to make them more appealing.
2. Students start by each building a 5 piece worm as shown below.
3. Students take it in turns to roll the dice and follow the instructions on it.  If the dice says +2, they add two pieces to their worm, if it says -1, they remove one piece.
4. Each time a student has a turn they have to say whether their worm is longer or shorter than their partner’s or than their own before they rolled the dice.  They should check which is longer by counting the number of pieces in each worm.
5. As the worms get longer, students should be encouraged to keep track of how long their worm is and count on or back to find its new length.  For example My worm was 7 long and I’m adding 3 so it’s 8, 9, 10 long now, it’s longer than it was.
6. The game finishes when one student’s worm reaches 20 pieces long (or whatever other number you assign).  If a student rolls a minus which is greater than the number of pieces left in their worm they should ignore it.
7. The game can be easily modified to suit your students’ ability by changing the numbers on the dice (or simply using a counter with 1 on one side and 2 on the other and excluding subtraction altogether).

#### Session 3: Tremendous Trees

Tremendous Trees is about the language of height, students will be encouraged to use the words tall, short, taller, shorter, tallest, shortest.

1. Tremendous Trees is a game to play in pairs.  Each pair needs a dice with the sides labelled +1, +2, +3, +4, -1, -2, and a set of Tremendous Trees pieces (Copymaster 2).  Students can cut out and colour in their own tree pieces to make them more appealing.
2. Students start by each building a 5 piece tree on their trunk as shown below.
3. Students take it in turns to roll the dice and follow the instructions on it.  If the dice says +2, they add two pieces to their tree, if it says -1, they remove one piece.
4. Each time a student has a turn they have to say whether their tree is taller or shorter than their partner’s or than their own before they rolled the dice.  They should check which is taller by counting the number of pieces in each tree.
5. As the trees get taller, students should be encouraged to keep track of how tall their tree is and count on or back to find its new height.  For example My tree was 7 tall and I’m adding 3 so it’s 8, 9, 10 tall now, it’s taller than it was.

6. The game finishes when one student’s tree reaches 20 pieces tall (or whatever other number you assign).  If a student rolls a minus which is greater than the number of pieces left in their tree they should ignore it.
7. The game can be easily modified to suit your students’ ability by changing the numbers on the dice (or simply using a counter with 1 on one side and 2 on the other and excluding subtraction altogether).

#### Session 4:  Wonderful Walls

Wonderful Walls is about the language of width, students will be encouraged to use the words wide, narrow, wider, narrower, widest, narrowest.

1. Wonderful Walls is a game to play in pairs.  Each pair needs a dice with the sides labelled +1, +2, +3, +4, -1, -2, and a set of Wonderful Walls pieces (Copymaster 3).  Students can cut out and colour in their own wall pieces to make them more appealing.
2. Students start by each building a 5 piece wall as shown below.
3. Students take it in turns to roll the dice and follow the instructions on it.  If the dice says +2, they add two pieces to their wall, if it says -1, they remove one piece.
4. Each time a student has a turn they have to say whether their tree is wider or narrower than their partner’s or than their own before they rolled the dice.  They should check which is wider by counting the number of pieces in each wall.
5. As the walls get wider, students should be encouraged to keep track of how wide their wall is and count on or back to find its new width.  For example My wall was 7 wide and I’m adding 3 so it’s 8, 9, 10 wide now, it’s wider than it was.
6. The game finishes when one student’s wall reaches 20 pieces wide (or whatever other number you assign).  If a student rolls a minus which is greater than the number of pieces left in their wall they should ignore it.
7. The game can be easily modified to suit your students’ ability by changing the numbers on the dice (or simply using a counter with 1 on one side and 2 on the other and excluding subtraction altogether).

#### Session 5:  Reflection

1. In this session students could be given time to play their favourite game from the previous three sessions.  Alternatively, your class might want to create their own comparison game to play.  Ideas could include trains (adding carriages), skyscrapers (adding floors), fences (adding rails), or anything else their imagination provides.
2. As a final revision (and possible summative assessment) students could be given Copymaster 4 to work through.  In this Copymaster they are asked to use the words they have been practising all week in sentences.  Possibly the first couple could be done as a class experience to ensure that students understand the task, and then they could work independently on the remainder.

## Inequality symbols and relationships

Purpose

The purpose of this unit of three lessons is to develop understanding of how to recognise and record relationships of (equality and) inequality in mathematical situations.

Achievement Objectives
NA1-4: Communicate and explain counting, grouping, and equal-sharing strategies, using words, numbers, and pictures.
Specific Learning Outcomes
• Understand the equals symbol as an expression of a relationship of equivalence, and explain this.
• Recognise situations of inequality and use the inequality (‘is not equal to’) symbol, ≠.
• Understand that < and > symbols can make equivalent statements.
• Use relationship symbols =, <, > in equations and expressions to represent situations in story problems.
• Understand how to find and express the difference between unequal amounts.
Description of Mathematics

The first relationship symbol that most students encounter is the equals symbol, =, which communicates a relationship of equivalence between amounts. It is important for students to understand that symbols help us to express relationships between numbers and that equivalence is just one such relationship.

Inequality is the relationship that holds between two values when they are different. Their relative value is described with specific language including ‘is greater than’, ‘is more than’, ‘is bigger than’ or ‘is less than ‘ or ‘is fewer than’. These are expressed using the symbols, <, >, which are said to show ‘strict’ relationships of inequality. Whilst not introduced here, the symbols, ≤ , meaning ‘is less than or equal to’, and , ≥, meaning ‘is greater than or equal to’, are known as ‘not strict’. The notation ≠, meaning ‘is not equal to’ is briefly introduced here as it is a useful, if infrequently used, relationship symbol.

Algebra is the area of mathematics that uses letters and symbols to represent numbers, points and other objects, as well as the relationships between them. Through exploring both equality and inequality relationships, and the symbols used to express these, students develop an important and heightened awareness of the relational aspect of mathematics, rather than simply holding the computational view of mathematics that arises from the arithmetic emphasis that is dominant in many classrooms.

The activities suggested in this series of lessons can form the basis of independent practice tasks.

Counting all (Stages 2 and 3)

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

• encouraging students to work collaboratively in partnerships
• 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 for this unit can be adapted to suit the interests and experiences of your students. For example:

• instead of heights of buildings you might use heights of trees, or heights of people
• change the stories from the copymaster to more familiar contexts.
Required Resource Materials
• Unifix cubes
• Street map diagram (a simple, made up one), A1 or A2 size, for example:
• Small blank cards
Activity

The activities suggested in this series of lessons can form the basis of independent practice tasks.

Session 1

SLOs:

• Understand the equals symbol as an expression of a relationship of equivalence, and explain this.
• Recognise situations of inequality and use the ‘is not equal to’ symbol.
• Recognise and describe in words the relative size of amounts.

Activity 1

1. Begin by talking about buildings in your school, suburb, town, city, or a city nearby. List any tall buildings which are known by name. Ask if anyone knows any (familiar) buildings that might be the same height.
2. Show a skyline picture such as this and ask what features the students notice. (eg. ‘buildings are of different heights’.)

Elicit descriptive, comparative language: tall, taller, tallest, short, shorter, shortest, same).
Point out that we have been comparing and describing the buildings in relation to one another. Explain we will be investigating relationships between numbers.
3. Make unifix cubes available to the students, and tell them to think of their favourite number between (and including) one and ten. Ask them to take this number of cubes of one colour.
For example, one student takes seven pink cubes.
Place a simple city street map in front of the students, or create one with them.

Have the students join their cubes to make buildings for this city. When they made their ‘tower buildings’, have them locate them, standing up, in places of their choice between the streets, creating a ‘cityscape.’
4. Have students look carefully at their ‘city’ and identify any buildings that they think might be the same height. Select several students to test their idea by taking the two identified ‘towers’, standing them side-by-side and comparing their heights. If they are the same they should count the number of ‘storeys high’ they are (number of cubes) and, on the class chart, write an equation and words to show this. For example:
5 = 5      five is equal to five
Read the equation together, “Five is equal to five” and “Five is the same as five.”
The buildings are then returned to their place ‘in the city’.
5. Have students now identify towers that are not the same in height.

Have a student describe how these numbers (of storeys) are ‘related’: “six is more than four”, “four is less than six”, “six is not equal to four.” Ask, “How can you write this?” Record the students’ suggestions, accepting all ideas.
On the class chart write, and have students in pairs, read this (expression of inequality) to each other. Read this together.
6 ≠ 4, six is not equal to four
6. Have students identify more ‘unequal buildings’ and record these as inequality statements on the chart and read them.
If possible retain the class ‘cityscape’ for Session 2.

Activity 2

Make plain A4 paper, felt pens, and cubes available to students. Have them work in pairs to create their own small ‘city’ (with street map and cube buildings). On separate paper, each student is to write about the buildings in their ‘city’. They should draw at least four pairs of buildings and for these, write both equality and inequality statements in words and symbols, as modelled in Activity 1, Step 5 (above).
Have student pairs retain their maps for Session 2.

Activity 3

Conclude the session by sharing their recording and discussing how the symbols = and ≠ show us how numbers are related to each other.
Suggestion: Photograph class and pair ‘city models’ to display with student recording from 2. (above), and from further sessions.

Session 2

SLOs:

• Recognise situations of inequality and use the appropriate symbols, ≠, <, >, to express this.
• Understand that in using < and > symbols, we can make equivalent statements.

Activity 1

1. Place the class ‘city’ from Session 1, with its map and tower buildings, in front of the students. Review the symbols = and ≠ and ask the students, “What is common about these symbols?” (They both express a relationship between numbers.)
Explain that there are more relationship symbols, and that they will learn about two more in this session.
2. Ask for a volunteer to find two towers that match this number expression:
6 ≠ 4
Have student pairs discuss the towers,

then, as a class, record their observations, including ‘6 is more than 4’ and ‘4 is less than 6.’ Ask if anyone knows symbols that show each of these relationships.
3. Write these symbols on the class chart.
< >
Write the words, ‘is more than’, and ‘is greater than’, together on the class chart, and ‘is less than’ or ‘is fewer than’ together.
Have students discuss these in pairs and decide which symbol goes with which pair of phrases and why they think that.
4. Accept all ideas. Conclude, agree, model and record:

six is greater than/is more than 4

Activity 2

1. Make available to the students small pieces of card (the same size)and felt pens or pencils. Explain that they are now to work in pairs with their own ‘city’.
Each student is to write at least four inequality cards for pairs of ‘buildings’. For example:
2. Have the students then mix up their cards so that they don’t match the ‘buildings’, swap with another student pair, and correctly match their cards and ‘buildings’.

Activity 3

1. As a class, now discuss and conclude that the same relationships can also be expressed using the “is less than’ or ‘is fewer than’ symbol. Demonstrate with ‘buildings’ (cubes) from Activity 1, Step 4. (above):

four is less than/is fewer than 6
2. Have students return to their own displays from 2.i. (above) and write four more cards expressing ‘is less than’ relationships’.
Each student should now have written at least 4 pairs of cards, 16 cards in total for the pair.

Activity 4

1. Have students shuffle the cards they have made in Activity 3, Step 2 (above) and swap these with another student pair.
Each pair should play a short Memory game with these cards by spreading them out, face down in front of them, trying to find matching pairs of statements, for example:

2. Students who finish quickly can create towers to match some of the pairs of inequality statements.

Activity 5

Conclude the session by reviewing the four relationship symbols, one of equality and three of inequality, that have been used in Sessions 1 and 2.
=, ≠, <, >.
Retain student ‘cities’ and relationship cards for Session 3.

Session 3

SLOs:

• Use relationship symbols =, <, > in equations and expressions to represent situations in story problems.
• Understand how to find and express the difference between unequal amounts.

Activity 1

1. Begin asking, “Who walked to school this morning?” Say that you are going to read a short story (Attachment 1).
Explain that the students must listen very carefully to the story. As they do so, they should record relationship expressions, in order, for any numbers that they hear. Read the story once. Highlight an example (eg. 3 >2 weetbix) and read the story again.
2.  Have students compare their expressions and equation in pairs. Share and discuss these as a class, recording them on the class chart.

Activity 2

1. Write the word ‘difference’ on the class chart. Ask students to explain this, giving examples from their own life, and record their ideas. (for example: ‘There’s a difference between the number of people in my family and in Maia’s family. There’s five in my family and eight in Maia’s. They’re not the same.’)
2. Refer to the inequality expressions recorded on the class chart in Activity 1, Step 2 (above).
For each discuss and record the difference. For example:
Weetbix: 3 > 2, 2 < 3,
Three is one more than two. Two is one less than three.
The difference is one.
Age: 60 > 50, 50 < 60
Sixty is ten more than fifty. Fifty is ten less than sixty.
The difference is ten.
Cats: 6 > 0, 0 < 6
Six is six more than zero. Zero is six less than six.
The difference is six.
Dog: 1 = 1
One is the same as one. There is no difference.
The difference is zero.

Activity 3

1. Make available to the students, small pieces of card (the same size) and felt pens or pencils.
Display two ‘towers’ from the class ‘city’. Ask. “What is the difference between the two towers? How do you know?” For example:

Show:     and write
Elicit explanations such as: there are two more blue ones, there are two less/fewer green ones.
Write 6 – 4 = 2 on the class chart and on a card.
Highlight the fact that when we solve a subtraction problem, we are finding the difference.
2. Have student pairs go to their ‘cities’ and relationship cards from Session 2.
Explain that they are to write a difference card, and a subtraction equation card as shown in Activity 3, Step 1 for each of their inequality expression pairs. Have partners check each other’s cards.
For the pair, there are now 32 cards in total, 8 sets of four cards.

These can be put together in a bag, or combined with an elastic band.
3. Have student pairs exchange full sets of cards. Have pairs, or fours, play Fish for Four with one set of cards.
(Purpose: To recognise equivalent pairs of inequality expressions, and their matching subtraction equation and difference statements)
How to play:
Cards are shuffled. Five are dealt to each player. The spare cards are put in a pile, face down, handy to all players.
Players check to see if they have any complete sets in their hand. If so, these are displayed face up in front of them. Each player then privately identifies which set they will collect and they take turns to ask one other named player for a specific card to complete their set.
For example:
In hand:  and
At their turn, the player says, “ Name, do you have the card, four is less than six?”
If the named player has the card, they must forfeit it. The successful player can ask again till they are told, “No. Go fish.” That player then takes a card from the top of the upturned pile of spare cards. It is then the turn of the next player.
The winner is the player with the most complete sets when all cards are used.

Activity 4

Conclude this session by reviewing key learning from this series of three lessons. Cities can be dismantled. Sets of cards can be used as an independent consolidation task.