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
Session One
- Use place value based strategies to subtract single and two digit numbers.
Session Two
- Interpret a calendar to make decisions about dates.
- Add two digit numbers.
Session Three
- Gather and sort data to make decisions about quantities of food to order.
- Calculate with measures, including money.
Session Four
- Recognise shapes in a figure.
- Follow a set of instructions for movement.
Session Five
- Use symmetry to recognise when winning positions are the same.
Description of Mathematics
Specific Teaching Points
Session one involves subtracting single digit and two digit numbers starting at 200. As students take handfuls or counters from their waka they will need to anticipate how many counters remain. Students will use need to use place value to calculate in ways other than counting back. The session notes recommend using a linear model for representing the calculations. A bead string is ideal and can be mounted along the edge of a whiteboard so jumps can be recorded on the whiteboard.
The session notes recommend linking two strings end on end to form a line of 200 beads. An important strategy in the activity is ‘back through ten’. For example, a student has 93 counters left and removes a handful of 17 counters. How many do they have left?
On the bead string this calculation can be modelled like this:
93 – 7 can be carried out in two steps, take away three to get to 90 then take away four to get to 86. This is a ‘back through ten’ strategy so it applies to using any decade number as a benchmark. Of course a student might take away a the ten of 17 first.
Session three involves dealing with like measures, e.g. dividing or multiplying weights. Actually measuring objects with devices like kitchen scales is important to the development of students’ understanding of the measurement system. For example, students will need to find out how many kilograms of kūmara will need to be ordered for the hāngi. If possible bring a few kūmara along so students can experiment to find out how many kūmara make up one kilogram in weight. They will then need to use division or multiplication to calculate how many kūmara they need. So if 24 kūmara are needed and four kūmara weigh one kilogram then 24 ÷ 4 = 6 kilograms will need to be purchased.
Session four develops important geometry ideas out of whai (string figures). A common issue with the learning of geometry is that students form prototypical views of shapes. For example, they might consider an equilateral triangle to be the only shape that is a triangle. All of the shapes below are triangles:
The issue of prototypical ideas will also apply to other polygons such as hexagons and octagons.
These three shapes are all hexagons. Note that the bottom hexagon is concave as it has two internal angles greater than 180°. It is important to discuss the defining characteristics of a class of shapes like hexagons. The only required property is that the shape is closed by six sides.
Session five also involves an important mathematical idea, distinctness. Rotating or reflecting a shape does not change its properties, except orientation (direction it is facing). The idea is fundamental to determining if given shapes are similar or different. For example, all of the quadrilaterals below are similar even though they look different. They can all be mapped onto each other using translation (shifting), reflection (flipping), and rotation (turning).
Similarity is applied in Session Five by looking for different winning positions. If the positions are reflections or rotations of one another then they are not considered to be distinct.
Required Resource Materials
Activity
Prior Experience
The activities are mostly open ended so they cater for a range of achievement levels. It is expected that students have some experience with naming and classifying basic geometric shapes, with measuring weight in kilograms, and with translating, reflecting and rotating shapes. They should also have place value knowledge to 200.
Session One
- The Māori New Year is celebrated at a different time each year. That is because the date depends on two events, the rising of the star cluster Matariki and the arrival of a new moon. In June Matariki and the other six or eight stars of the cluster become visible in the Eastern Sky about 30 minutes before dawn. This is known as the rising of Matariki as for the month prior to Matariki it is below the horizon. After the rising of Matariki, Māori look for the next new moon to signal the New Year. The week prior to the new moon, excluding the night of no moon, is when Matariki is celebrated. Slide one of the PowerPoint shows the best known seven stars of Matariki. Slide two shows how to find Matariki should you want to organise a pre-dawn star spotting expedition.
- The Tūwharetoa legend of Tamarereti has connection to Matariki. Versions of the story vary but all name him as responsible for creating the stars in the night sky. Slides 3-8 tell the legend in abbreviated form. Yo umay like to show an animated version of the story.
- So Tamarereti cast the shining stones into the heavens on his journey across Lake Taupō. The stones stuck in the dark night sky to become the stars. Ranganui, the sky father, put Tamarereti’s waka into the sky in honour of his deeds, and the waka appears today as the Milky Way. The Southern Cross and Pointers make up the anchor and rope of this great canoe (see slide eight).
- Show the students slide nine which shows a satellite picture of Lake Taupō. Click to discuss where Tamarereti drifted to while asleep, cooked his fish, then set off from to return to his village.
- Select a student to act out the next part of the lesson.
What the legend does not tell you is that Tamarereti collected 200 bright shiny stones and put them at the bottom of the waka.
Have a ‘waka’ with 200 counters ready for the student to act out the story. Any narrow container will make a good waka.
- Tell the student to start rowing then grab a small handful of stones to throw into the sky. Remember that the stones have to last the whole journey so Tamarereti cannot use them up all at once. Ask the student to cast the counters onto a sheet of art paper so the whole class can see.
- Ask: Is there a way to group these stones to count them easily?
Look for students to suggest ways to group the counters. Combinations that add to ten are especially useful.
- Tell students: Tamarereti is being careful not to use all of his stones up because the Taniwha will eat him if he is unable to see. He wants to know how many stones he has left. How could he work that out?
- Let the students work out the remaining number in pairs. Then share the different ways the answer could be achieved. Look for part-whole strategies rather than counting back. For example if 15 stones are thrown into the sky subtracting ten then five is a good strategy. If 19 stones are thrown then subtracting 20 and adding one is effective.
- Ask: What might Tamarereti scratch into the side of his waka to keep track of his number of stones?
- Invite suggestions from the students about how to record the number of stones. Write on a map of Tamarereti’s journey the first stone toss or use the animation on Slide Five to show how the journey might be recorded. Let the students work in pairs to act out and record Tamarereti’s journey across Lake Taupō. Each pair will need a container, 200 counters and a copy of Copymaster 1. The number of counters can be lessened or increased to vary the challenge. Expect students to manage the distance to go on the map and the number of stones left. They should record the results of their calculations as each handful of stones is cast into the heavens. The grid system on Copymaster 1 could be used to create co-ordinates so students can indicate the position of Tamarereti’s waka each time he casts out shining stones.
- After a suitable period, bring the class back together to discuss the strategies they used to calculate the remaining stones. Use an empty number line or two connected hundred bead strings to illustrate strategies as students suggest them.
- Back through ten (173 – 16 = 157)
- Tidy numbers and compensation (145 – 29 = 116)
- Standard place value (156 – 34 = 122)
- You might use the work samples students produce as evidence of their additive thinking.
Session Two
- Slide eleven of the PowerPoint shows the phases of the moon. Ask the students why the moon (marama) changes appearance. Some may know that the change is caused by the moon’s orbit around the Earth and the extent to which the half of the moon lightened by the sun’s rays is visible. You can demonstrate this with a ball and a lamp.
- The phases of the moon are important to Māori as they indicate which days are best for traditional food gathering, particularly fishing. Slide Twelve shows a page from Mathematics Across Cultures (1992). Ask the students to interpret the calendar.
- Why does the month only have 30 days? That is the length of one moon orbit of the Earth (actually 29.5 days).
- When are the best days to fish in the lunar month? (The red days which are days 18, 24, and 25)
- When are the worst days of the month to fish? (The first 2 days of the new moon, days 6-7, 10, 16, 20-22, and the last two days of the old moon).
- So what are the best days to fish during Matariki? Matariki is celebrated in the last quarter of the lunar cycle but not on the day of the new moon.
- Use the timeanddate.com website to capture the lunar calendar for the current month. Give each student a copy of the calendar and ask them to make a puzzle for a classmate. They do that by cutting the calendar up into jigsaw pieces. Set the maximum number of pieces to eight and tell the students to use the straight lines of the calendar to cut along. They can cut vertically or horizontally so shapes like an L or a Z are encouraged.
- Once they have cut up their calendar, students give their pieces to a partner to reassemble. Look to students to attend to the progression of days at the top of the calendar, the maximum of seven days in each row, and the sequence of whole numbers to put the puzzle back together. Have the students glue their completed calendar into their mathematics book.
- With their calendar intact students can answer these questions:
How do we find out the date of the full moon from this calendar?
So when will the last quarter start?
When are the good days for fishing?
When will the New Moon appear?
So when does the New Year start?
- Tell the students that in honour of Matariki they are going fishing. If the day is not a good fishing day, wish them luck. If it is a good day for fishing say you are expecting a lot of success. The fishing game can be played in two ways:
- Cut the fish out (see Copymaster 2) and attach a paper clip to each fish. Make a fishing rod using a stick, a piece of string and a magnet (magnetic strip is a relatively cheap way to do this). Students capture a fish by getting it to stick to the magnet.
- Cut out the fish cards and turn them upside down. Players take turns to choose a fish.
- The game can be played in pairs or threes. The object of the game is for each player to gather fish that add to 100. They do that as often as they can. At any time players can trade fish with each other to make 100.
- Once the students have played the game on Copymaster 2, gather the class to share another legend. Māui was known as a trickster. It was Matariki, the New Year, and it was very cold outside. Māui’s brothers were getting bored (again) so he decided to play a trick on them. He made up the second set of fishing cards (see page 2 of Copymaster 2). The brothers tried for a long time to make 100 with the fish. They could not. Can you?
- Let the students try the second fishing game to see if they can do better than Māui’s brothers. It is actually impossible to make 100 with the cards but see if your students can figure out why. They may need to take the game home to their whanau to see if anyone can explain how Māui’s clever trick works. All of the numbers on the fish are answers to the nine times table so the totals must always be in the nine times table (multiples of nine). 100 is not a multiple of nine.
Session Three
Matariki is a time of cultural pursuits and feasting to celebrate the New Year ahead. The hāngi or earth oven has particular significance at the time of the new moon after the rise of Matariki in the Eastern pre-dawn sky. Matariki is the star at the bow of Te Waka o Rangi and her travels around the sky for eleven months of year are exhausting. It is said the steam of the first hangi in the New Year rises into the sky and replenishes the strength of Matariki. From the offerings she gathers strength to lead the giant canoe for another year. Without Matariki at the bow the canoe cannot travel and Taramainuku cannot cast his net to gather the souls of the departed. At the New Year the names of the dead are called out so the souls of the departed may be cast into the heavens as stars.
There are many resources already available about hāngi.
“Preparing for the hāngi” is a Level 3 activity from the Figure It Out series.
“Hanging out for hāngi” is a unit at Level Three that develops a statistical investigation around deciding which foods to cook.
The notes below are an adaptation more suitable for Level Two students.
- Tell your students about the types of food that are usually cooked in a hāngi. Chicken, pork and lamb are the most common meats used and the vegetables tend to be root crops like kūmara, potato and pumpkin. Stuffing is also popular. Before your class can plan the hāngi you will need to find out what people like to eat.
- Your investigation question is “What hāngi foods do people in our class like to eat?”
- Copymaster 3 has a photocopy sheet of ‘choice squares’. Put a container such as a shoebox or 2L plastic icecream container in the centre of the room. That is where the data will be placed. Show the students the first page of the Copymaster.
- Ask: If you want to eat any of these foods at our hāngi you need to cut out that square and put it into the box. Should there be some restrictions on what you can eat?
Students might mention that people should not eat every meat and every vegetable. Agree on some restrictions like one or two meats and up to three vegetables. Point out that stuffing is a yes or no choice.
- Explain that the data will be used to order the food. “If someone chooses two meats while another person chooses only one meat, how will we deal with that?” Students might suggest that a person choosing two meats can put in one half of each square while a person choosing one meat might put in the whole square.
- Give the students time to make their choices and put the squares of the food they choose into the container. It is important that they cut out squares rather than the food within the squares as scale is important for of possible data displays. Once you have brought the class together in a circle on the mat empty the container of squares.
- Ask: How might we organise these data so we can order food for the hāngi?
- Students should suggest putting the squares into categories so get a few students to sort the data into piles. Ask, “How might we show the data so the number of squares for each food is easier to see?” After some discussion you should end up with a picture graph made with the squares. Managing the half squares should provoke a discussion about how large fractions such as five halves are. You might glue the squares in place on a large sheet of paper and add labels and scale to the axes. The graph might also be given a title.
- Once the data display is complete put the students into small groups to discuss “How might we use these data to order food for the hāngi?” After a suitable time gather the class to share ideas. Expect students to consider the idea of a portion, that is how much of a food is reasonable as part of a meal. For example, one pumpkin is too much for a single portion so a fraction such as one eighth or one tenth is more sensible.
- Share the information about meat (see PowerPoint slide 13) for a poster about this information). The poster has some questions for the children to consider. To support their thinking have a set of kitchen scales available so you can find objects around the room that weigh the same as a lamb chop or a size 14 chicken. You might use the scales to count in lots of 100 grams to find out how many portions are in one kilogram of meat.
- Ask the students to work with a partner to decide how much of each food to buy. Look for them to consider the data on preferences you have collected, the information about portions of meat and their estimates of how much of each vegetable is required for each portion. You may decide to pool the data across several classes to make the task more challenging and avoid having a lot of pork left over! The students should produce a shopping list with clear working about how they decided on each amount.
- Share the shopping lists and agree on suitable amounts of each food. The amounts of vegetables are likely to be expressed as numbers of whole vegetables, e.g. two pumpkins, which will add interest to the next part of the lesson – working out the cost per person. Copymaster 3 has a fictitious flyer from the local butcher and fruiterer so that the students can create a budget for the hāngi food (see also Slide 13 of the PowerPoint). Allow students to use a calculator if they need to. Some may like to use a spreadsheet to keep track of their budget. Students will need to convert from numbers of vegetables into kilograms by estimating. For example, four or five good sized potatoes weight 1 kilogram. Students may realise that they need a recipe for stuffing so they can calculate how much bread to order. Let them search for a stuffing recipe. Onions are an important ingredient in stuffing.
- The final part of the budget is to work out a cost per person. This is a sharing context. The total cost, say $75, is divided equally among all the people in the class. Look for students to realise that the operation needed is division. You may need to link to simpler sharing problems so they connect the equal sharing to division and can write an equation for the solution, e.g. 75 ÷ 25 = 3. Talk about the meaning of the numbers in the equation, e.g. 3 represents $3 per person.
Session Four
Matariki was a time when food was already stored, and it was cold outside. So whānau (families) spent time together engaging in cultural pursuits such as storytelling, arts and games. Whai (string games) were popular with tamariki (children) and adults alike, especially when they involved co-operation. Whai has a long history and is common to many indigenous cultures around the world, including the indigenous tribes of North America. So if instructions tell you to “Navajo your thumbs” that means a common move that is attributed to a tribe of indigenous Americans. Traditionally whai was played with twine made from flax. The best man-made fibre to use for whai is nylon since it slides and flexes, and is soft on your hands. It commonly used to form lines for brickwork so is available at most hardware stores in a variety of colours. Nylon string is also available in craft shops.
- Ask your students to make a tau waru (Number 8) loop by wrapping the string loosely around their palms eight times, cutting the string, knotting it with minimal wastage, and trimming any loose ends.
- Whai relies on algorithms that are standard procedures. Algorithms are an important part of mathematics. Processes that initially take some time to master become standardised routines. The more complicated whai rely on the some basic moves being well known by the maker. That is where you should start with your students.
- Dasha Emery has created an excellent series of videos that give clear instructions about making well known whai. A good first move for students to learn is called “Opening A” which is a standard algorithm. This opening is the start of many whai, particularly those that end in diamond shapes. Play the video at the link below which talks students through Opening A. Dasha refers to this pattern as kotahi taimana (one diamond). Let students practise the opening until they have it mastered:
- Next follow the instructions in this video to create te kapu me te hoeha (cup and saucer):
- Cup and saucer (YouTube)
At 1:08 it is easier to think of going over two strings and ‘picking up the third string’ in that move. Note that the move where you use your mouth to shift the bottom of two strings over your thumbs (2:00 - ) is called ‘Navajoing your thumbs’ and is another algorithm common in whai.
- The next whai your students might make is ngā taimana e rua (two diamonds) which builds on Opening A. Tell your students to make the standard opening before the video starts. Alert them that somewhere in the video they will need to Navajo their thumbs. Work through the video, with students supporting each other to create the ngā taimana e rua pattern. Alternatively give the students copies of Copymaster 4 that has the instructions in graphic form. The written form will be much harder to interpret.
- Put the students into small groups of about three or four. Provide them with opportunity to become class experts in a particular whai pattern. Their job will be to teach the rest of the class how to make that pattern. Another YouTube chanel with many examples of string patterns is available at the link below:
- Once your students have practised their whai in small groups invite them to teach others how to make their pattern. You could do this as a whole class or create expert-novice pairs.
- For an interesting geometry challenge consider looking for different shapes within a whai pattern. For example the gate pattern looks like this
- Ask your students what shapes they can see in the figure. Here are a few possibilities:
- Some interesting points might arise such as:
- A three sided polygon is called a triangle, irrespective of the length of the sides, size of angles or orientation. The same is true of all five sided polygons being called pentagons and all six sided polygons being called hexagons.
- Non-regular means that all the sides and angles are not equal so a regular polygon, such as a square, must have equal sides and angles.
- The prefix ‘tapa’ means sides and the number name tells how many of those sides are in the shape, e.g. tapatoru means three sides (very helpful).
Session Five
In this session students learn to play the traditional Māori game Mū Tōrere which is like a form of draughts. The original game is sometimes referred to as the wheke (octopus) game or the whetū (star) game due to the shape of the board. It is appropriate that students learn to play the game at the time of Matariki, since the Māori New Year is a time of engaging in cultural pastimes. The board (see Copymaster 5) has been altered to include the nine or seven stars of Matariki, dependent on the version of the game that is played.
- Introduce the Mū Tōrere Ngāwari (easy) version first since it is simpler and is a good lead in to the more complex original game. The game is played in pairs with each player needing three counters for the easy game and four counters for the original game. Their counters should be of one colour. The rules are included on the gameboards.
- Let the students explore the easy game in pairs. Tell them that they need to record the winning position if one of them wins. Using black and grey for the counter colours can help to identify the arrangements that create wins. After they have played awhile bring the students together to share the winning positions. Create a set of diagrams. In these examples grey wins.
- While they may look different the winning arrangements are actually the same, and are just rotations or reflections of one another. That can be demonstrated by putting the patterns of card and turning them.
- Ask: What must be true for a player to win in the easy game?
The winner must occupy the centre circle, the opponent’s stones must be clustered together around the hexagon and the winner must have the ends of the cluster blocked of. You might try to find a winning arrangement by separating the loser’s stones into a group of two and one but there is no way for the other player to stop them moving.
- Transfer to the original game that has the same set of moves but more winning arrangements. Ask the students to create winning arrangements on their board prior to playing the game. Create a gallery so the students can look for similarities and differences. Here are winning positions for black. Notice how all four, three and one, and two and two configurations of grey can all result in a victory to black but the winner must always occupy the centre. Discuss the similarity of winning arrangements to students create as the diagrams are reflected or rotated.
- Is it possible to trap a player that has four ones, or two and two ones (as shown below)? Try colouring in four circles grey to achieve a trap. It is not possible.
- Once winning positions have been analysed let the students play the game. Competitive games go for over thirty moves so tell your students to be patient and think ahead. An interesting idea is that players can always create a draw if they know what they are doing. Is that true?
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Kia ora parents and caregivers
This week we are exploring some mathematical ideas with activities to celebrate Matariki, the Māori New Year. We will be learning the legend of Tamarereti and using that story to learn about numbers of bright stones. We will look at the phases of marama (the moon) to find the best days to go fishing in the New Year period, plan and cost a hāngi, and discover geometry in whai (string patterns). Lastly we will learn to play Mū Tōrere, a traditional Māori game that takes a lot of strategy to win.
These activities involve number, algebra, geometry, measurement and statistics so we will be very busy and doing a lot of mathematical thinking.
Popcorn
This unit is made of a number of popcorn investigations, which provide both a purposeful and enjoyable measuring context. The focus of the unit is introducing the students to the need for a standard unit for measuring volume.
When students can measure effectively using non-standard units, they are ready to move to the use of standard units. The motivation for moving to this stage, often follows from experiences where the students have used different non-standard units for the same volume. This allows them to appreciate that consistency in the units used allows for easier and more accurate communication.
The usual sequence used in primary school is to introduce the litre as a measurement of volume before using cubic centimetres and cubic metres.
Students’ measurement experiences must enable them to:
The standard units can be made meaningful by looking at the volumes of everyday objects. For example, the litre milk carton, the 2-litre ice-cream container and the 100-millilitre yoghurt pottle. Students should be able to use measuring jugs and to say what the measuring intervals on the scale represent.
This unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:
Including the process of making popcorn in a broader context within the class or school will encourage engagement. For example, the popcorn could be part of a shared lunch for the class, a popcorn stall could be set up as a fund-raiser or as part of a school fair, or the popcorn could be shared at a community event.
Session 1-2
In these sessions we investigate the volume of corn kernels before and after popping. The amount of popcorn to be made is based on the batch size using a popcorn maker, which is usually about 1/2 a cup of kernels. We use non-standard units to measure the volume of both kernels and popped corn, and think about why the measurements vary (because the units being used vary) and what could be done to improve the consistency of the measurements (use a standard unit).
Ask: Why are we coming up with different numbers of spoonfuls needed to fill the container?
Record students’ responses, encouraging them to compare the sizes of spoons.
Session 3
In this investigation the students revisit the results from the previous investigation, and the idea of measuring the popcorn using a litre measure is explored.
Session 4
In this investigation the students think about a standard serving size for the popcorn. The students will find out how many litres of popcorn will be needed for everyone in the class to get one serving.
Session 5
In the final day of the unit the students make cones to fit one serving of popcorn into. The batches of popcorn will be made and the students will be able to measure out their serving to eat.
Family and whānau,
This week we have been investigating the volume of popcorn. Do you know how many litres half a cup of corn makes when it is popped?
We designed containers at school to hold our popcorn. Ask your child to tell you about the one that they made. Together you could design another shaped container to hold the popcorn and make some at home.
Multiplication stories
In this unit students explore the different situation types to which multiplication can be applied. Particularly, they engage with rate, comparison and array problems.
The basic concept of multiplication is an important one because of its practicality (How much do 4 ice creams cost at $2 each) and efficiency (It is quicker to determine 4 x 2 than to calculate 2 + 2 + 2 + 2). Multiplication is used in many different situations. Here students think about multiplication as a short way to find the result of repeated addition of equal sets. They do so by solving rate problems, comparison problems and array problems.
A rate problem involves a statement of ‘so many of one quantity for so many of another quantity.’ All multiplication situations contain some form of rate but at this level, the problems are usually about equal sets or measurement. Take this example:
“Lena buys six bags of biscuits. Each bag contains four biscuits. How many biscuits does she buy altogether?”
This is an equal sets problem that contains the rate ‘four biscuits for every bag.’ A measurement rate problem is usually something like this:
“Hone’s kumara plant grows five centimetres each week after it sprouts. How long will his plant be after six weeks?”
The rate in Hone’s problem is “five centimetres for every week.” Comparison problems involve the relationship between two quantities. For example:
“Min’s apartment block has three floors. Anshul’s block has 12 floors. How much taller is Anshul’s block than Min’s?”
An additive answer is 12 – 3 = 9 floors. A multiplicative answer is 4 x 3 =12 so Anshul’s block is four times higher than Min’s. An array is a structure of rows and columns. For example, this chocolate block has two rows of five pieces (2 x 5 or 5 x 2).
An array is a structure of rows and columns. For example, this chocolate block has two rows of five pieces (2 x 5 or 5 x 2).
Array problems can help students to see the commutative property of multiplication, for example, that 5 x 2 = 2 x 5. In other words, the order of the factors does not affect the product (answer) in multiplication.
As well as thinking about multiplication in a variety of situations, students are encouraged to use a variety of materials to solve the problems. Using a variety of materials can help students see the multiplicative structure that is common to a variety of problems and assist them to transfer their understanding to situations which are new to them.
This unit can be differentiated by varying the scaffolding provided to make the learning opportunities accessible to a range of learners. For example:
Focus on familiar contexts which include multiplicative situations to appeal to students’ interests and experiences and encourage engagement. Examples may include:
Getting Started
Exploring
Over the next three days the students are exposed to a variety of different types of story problems. They are encouraged to model the problems using different equipment and explain their answers to others. They think about the most efficient ways of solving the problems. It is important that students are provided with opportunities to build up multiplication facts to 10 and then to 20. Some students may solve these problems without equipment, using the number knowledge they have available.
Rate problems
[Name] can write her name in 10 seconds. How long does he/she take to write his/her name four times?
You might select a student to role play writing their name and use an analogue clock (less battery) and/or stacks of cubes to track the time in seconds
[Name] drinks four cups of water each day. How many cups does he/she drink in one week?
Use plastic cups to build up the equal sets of four cups that are involved in this problem. Use another material to track the number of days.
[Name] sews five buttons on every shirt. How many buttons does he/she need for eight shirts?
[Name] puts three spoons of Milo in each cup. How many spoons of Milo does he/she need for 10 cups?
These problems are more accessible than the time related tasks. Shirts can be cut out of paper and buttons represented by counters. Cups and plastic spoons can be used to model the Milo problem. Both quantities in each rate are tangible.
The students can create similar types of problems with pictures and pose the problems to each other. Encourage them to explain their strategies to each other.
Discuss what is the same about all the problems you have just worked on.
Do students express the idea of a rate as being a “for every” relationship?
Now ask the students to make up similar word problems and to pose their problems to each other. Encourage them to explain their answers to each other.
Multiplicative comparison
How much taller is Jill’s apartment block than Jack’s apartment block?
S: Jill’s apartment block has 12 floors and Jack’s has four floors. Jill’s block is eight floors higher.
Students may not offer a ‘times as many’ multiplicative answer. If that occurs pose this problem:
Jill says that her apartment block is three time higher than Jack’s block. I wonder what she means?
Look for students to note the inverse relationships:
S: Jack’s apartment block is eight floors less (shorter) than Jill’s.
S: Jack’s apartment block is one third of the height of Jill’s.
Arrays
Pose problems such as:
The students are lined up in 3 teams for sport. Each team has 6 members. How many students are there altogether?
Encourage the students to draw representations of problems like this using three rows (one for each team) and six columns (one for each team member). Alternatively, use the students as the objects in the problem. If students draw the situation as three columns of six it opens discussion of the commutative property since 6 x 3 = 3 x 6.
For the problem above this could be talked about as 3 rows of 6 pegs or 3 sixes,
or 3 rows of 6, or 3 x 6 = 18
By turning the pegboard a quarter turn, the array still has a total of 18 pegs.
This could be talked about as 6 columns of 3 pegs or 3 rows of 6 pegs or 6 threes
or 6 columns of 3 or 6 X 3 + 18.
Reflecting
On the final day of the unit we play a game called Array Trap in which the students use graph paper to plot arrays.
For this activity you will need
Players take turns to:
Discuss strategies for the game such as:
Dear Family and Whānau,
At school this week we have been solving multiplication problems. Here is an example of one we have worked on:
There are 6 fish bowls. Each bowl contains 4 goldfish. How many goldfish are there altogether?
At home this week I would like your child to make up two more multiplication problems for us to solve in maths.
Figure it Out Links
Some links from the Figure It Out series which you may find useful are:
Planning a statistical investigation (Level 2)
In this unit students will identify how to plan and carry out a statistical investigation, looking at facts about their class as a context.
It is vital when planning statistical investigations that the students understand the importance of the way that they plan, collect, record and present their information. If they are not consistent in the way they carry out any of these steps, they could alter their findings, therefore making their investigation invalid.
In this unit the students will first look at choosing investigative questions to explore, making sure that the topic lends itself to being investigated statistically. They will then collect their data using structured recording methods. Once they have collected and recorded their data, they will present their findings using appropriate displays and making descriptive statements about their displays to answer the investigative question.
Dot plots
Dot plots are used to display the distribution of a numerical variable in which each dot represents a value of the variable. If a value occurs more than once, the dots are placed one above the other so that the height of the column of dots represents the frequency for that value. Sometimes the dot plot is drawn using crosses instead of dots.
Investigative questions
At Level 2 students should be generating broad ideas to investigate and the teacher works with the students to refine their ideas into an investigative question that can be answered with data. Investigative summary questions are about the class or other whole group. The variables are categorical or whole numbers. Investigative questions are the questions we ask of the data.
The investigative question development is led by the teacher, and through questioning of the students identifies the variable of interest and the group the investigative question is about. The teacher still forms the investigative question but with student input.
Survey questions
Survey questions are the questions we ask to collect the data to answer the investigative question. For example, if our investigative question was “What ice cream flavours do the students in our class like?” a corresponding survey question might be “What is your favourite ice cream flavour?”
As with the investigative question, survey question development is led by the teacher, and through questioning of the students, suitable survey questions are developed.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:
The context for this unit can be adapted to suit the interests and experiences of your students.
Although this unit is set out as five sessions, to cover the topic of statistical investigations in depth will likely take longer. Some of the sessions, especially sessions 4 and 5 dealing with data presentation and description, could easily be extended as a unit in themselves. Alternatively, this unit could follow on from a unit on data presentation to give students an appreciation of practical applications of data display.
Session 1
Session 1 provides an introduction to statistical investigations. The class will work together to answer the investigative question – How many brothers and sisters do people in our class have?
Explain that statistics concerns the collection, organisation, analysis and presentation of data in a way that other people can understand what it shows.
How many brothers and sisters do people in our class have?
Students might suggest that we can ask how many siblings, or they might suggest we ask how many brothers, how many sisters and how many altogether.
The idea of asking about brothers and sisters separately allows for a deeper exploration of the data and a more in depth answer to our investigative question.
Sticky notes could be a good way to collect this information from the students as it will allow rearrangements of the data quickly.
For example:
Pip records the following information about her brothers and sisters. She gives it to her partner. Her partner, Kaycee shares this information with another student. Kaycee says that Pip has three brothers and one sister. Altogether Pip has four siblings.
How can we use the pieces of paper (sticky notes) to show someone else how many brothers and sisters people in the class have?
How can we show the information so that people can easily understand what it is showing?
Hopefully, someone will suggest a more organised list, or counting the number of 0s, the number of 1s etc and writing sentences to explain how many there are of each.
Session 2
This session is ultimately about choosing an appropriate topic to investigate about the class. There will be a real need for discussion about measurable data and realistic topics that can be investigated in the given time frame. It would be a good idea to provide the students with a list of topics if they get stuck, but they should be encouraged to try and come up with something original where possible.
PROBLEM: Generating ideas for statistical investigation and developing investigative questions
If groups are having trouble thinking of ideas, you could try writing a list of suggestions on the board but limiting groups to using one of your ideas only, to encourage them to think of their own. Some ideas could be:
PLAN: Planning to collect data to answer our investigative questions
- If the investigative question is: “What are Room 30’s favourite pets?”, ask the students how they could collect the data.
- A possible response is to ask the other students “What is your favourite pet?”
- Also, the students might want to ask, “What is your favourite pet out of cat, dog, fish, bird?” You could challenge them as to if this would really answer the investigative question and suggest that possibly they might change the survey question to allow for other answers.
Possible survey questions are:- What is the colour of your eyes?
- How did you travel to school today?
- How many letters are in your first name?
- What month is your birthday?
In these examples you can see that the survey question and investigative question are very similar, but there are key differences that make it an investigative question (What are Room 30’s favourite pets? – overall about the class data) rather than a survey question (What is your favourite pet? – asking the individual).Session 3
Data collection is a vital part of the investigation process. In this session students will plan for their data collection, collect their data and record their data and summarise using a tally chart or similar for analysis in the following sessions.
PLAN continued: Planning to collect data to answer our investigative questions
DATA: Collecting and organising data
Session 4
In this session the students will work on creating data displays of the data collected in the previous session.
ANALYSIS: Making and describing displays
Numerical data – displaying count data e.g. “How many…” investigative questions
Categorical data – displaying data that has categories e.g. “What…” investigative questions
Session 5
Session 5 is a finishing off session. Students should be given time to complete their graphs if they have not already, and to write statements about what the graphs show.
Dear parents and whānau,
During the next week we will be working on statistical investigations in maths. Over this time, your child will be gathering data on the class and presenting it using data displays such as dot plots and bar graphs. If you know of any graphs or tables of information suitable to discuss with your child, either in the newspaper, or in a book, or perhaps on some advertising material, this week would be a good time to do so.
Figure it Out Links
Some links from the Figure It Out series which you may find useful are:
Pede patterns
This unit is about generating number patterns for certain ‘insects’ from the mythical planet of Elsinore. Each ‘Pede’ is made up of square parts and has a number of feet. The patterns range from counting by 2s and 3s to being the number of feet plus three.
Patterns are the basis of much mathematics. There is always a need to find a link between this variable and that variable. This unit provides an introduction to pattern in the context of ‘insects’. The students are given practice in finding the next insect in a sequence. This leads to the main aim, which is for the students to begin to see the link between the number of feet that certain insects have and the number of squares that make them up.
One of the things that is deliberately attempted here is for them to see the link above in both directions. So not only do they get practice in linking squares to feet but they also are asked to try to find the number of feet that an insect with a certain number of squares has.
This unit provides an opportunity to develop number knowledge in the area of Number Sequence and Order, in particular development of knowledge of the Forward Number Word Sequence and skip counting patterns.
This unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example:
The context of this unit can be adapted to address diversity, and appeal to students’ interests and experiences to encourage engagement. For example, students may like to colour their pede with their team's favourite colours, or decorate them with koru or other patterns.
As each pede is developed, help students focus on the number patterns involved by creating tables as below. Similar tables can be drawn for each type of pede.
Use of a hundreds chart will help students visualise the number patterns more easily and help them to predict which numbers will be part of the patterns.
The conclusion of each session is an ideal time to focus on the number patterns involved. Questions to develop number knowledge include:
Which number comes next in the number pattern for this pede? How do you know?
Which number will be before 20 in this pattern? (or another number as appropriate)
How do you know?
What is the largest number you can think of in this pattern? How do you know?
Could a pede with 20 squares be a Spotted Pede? Why / Why not?
Could a pede with 32 squares be a 2-pede? Why / Why not?
Are there any numbers that could be Spotted Pedes or Humped Back Pedes? What are they? How did you work that out?
Session 1
Here we explore a couple of number patterns related to the mythical insects that live on the planet Elsinore. The patterns involve skip counting by 2s.
On the planet Elsinore there live a strange collection of insects. There is the Humped-Back Pede. The Humped Back 1-pede looks like this. Can you see his eye? And the Humped Back 2-pede looks like this. He has an eye too. (Show them the pictures below.) Ask the students to work individually or in pairs to make a Humped Back 3-pede with the green tiles.
Can you work our how many squares a Humped Back 4-pede has?
Gather the students together to talk about the insects that they drew. Explore the number pattern of counting in twos that comes from the Humped-Back Pedes. Also ask them questions like:
Can you tell me how many green squares a Humped Back 5-pede will have?
Can you tell me how many green squares a Humped Back 7-pede will have?
Can you tell me how many green squares a Humped Back 10-pede will have?
How many feet has a Humped-Back Pede with 12 squares?
How many feet has a Humped-Back Pede with 18 squares?
How many feet has a Humped-Back Pede with 20 squares?
Can you tell me how to get the number of squares that a Humped-Back Pede with a particular number of feet has?
Can you tell me how to get the number of feet that a Humped-Back Pede with a particular number of squares has?
Session 2
Here we investigate some more of the mythical insects that live on the planet Elsinore. The patterns here involve skip counting by 3s.
How many feet would a 4-pede have? What about a 5-pede?
Can you tell me how many squares a 1-pede has?
How many squares does a 2-pede have?
What about a 3-pede?
(Put the numbers of squares beside the insects as the students answer the questions.)
How many feet will it have?
How many squares will it have?
Can someone make one for me with these square tiles?
Does everyone agree with that?
(Write 11 under the 4-pede.)
I wonder what sort of Pede comes next? (A 5-pede.)
How many squares does a 5-pede have? (14.)
I wonder what sort of Pede comes next? (A 6-pede.)
How many squares does a 6-pede have? (17)
Is there any pattern in the number of squares that Pedes have? (Add on 3 for each extra foot. Talk about skip counting by threes.)
Session 3
Here we explore patterns further. Here we are particularly interested in linking the number of squares on an insect and the number of feet it has.
How many red squares does a Spotted 4-pede have?
How many blue squares does a Spotted 4-pede have?
How many squares all together? How did you work that out?
Why are there more blue squares than red squares? How many more?
How many red squares does a Spotted 5-pede have?
Can you tell me how many blue squares a Spotted 5-pede has? Draw it.
How many red squares does a Spotted 6-pede have?
How many more blue squares does a Spotted 6-pede have? Draw it.
How many red squares does a Spotted 10-pede have?
How many blue squares does a Spotted 10-pede have?
What did you find out about the Spotted Pedes?
What patterns did you find?
(Try to get them to see that they have as many red squares as they have feet. This means that it is very easy to find out how many red squares they have.)
Session 4
How many yellow squares does a Big-Headed Pede 4-pede have?
How many yellow squares does a Big-Headed Pede 5-pede have?
How many yellow squares does a Big-Headed Pede 6-pede have?
Do you need to draw the insects to work out how many squares they have? Why or why not?
Could you work out how many yellow squares a Big-Headed Pede 10-pede would have?
What did you find out about the Big-Headed Pedes?
What patterns did you find?
(Try to get them to see that they have three more yellow squares than they have feet. This means that it is very easy to find out how many yellow squares they have.)
I saw a Big-Headed Pede with 16 yellow squares. How many feet did she have?
Session 5
Discuss the Pedes, the Humped-Back Pedes, the Spotted Pedes and the Big-Headed Pedes.
How many squares does one of your 5-pedes have?
Can you tell me the number of squares an X Pede with 10 feet (or some other relatively large number) has?
If I had 15 squares what is the Pede with the largest number of feet that I could draw?
Dear Parents and Whanau,
In maths this week we have been working on patterns. We have looked at some mythical insects from the planet Elsinore. We have seen the patterns that their numbers of squares have. Here is an animal from Elsinore. Perhaps together you could explore the links between its number of legs and the number of squares that make up its body.
Discuss with your child how many squares does a 2-legged animal have?
How many squares does a 3-legged animal have?
How many squares does a 4-legged animal have?
How many legs does an animal with 20 squares have? Talk together about how you know? What do you have to do to work this out?
We would also like you to help us come up with a name for our animal. Write your ideas here.
We hope that you enjoyed working together on this algebra patterning task.
Matariki - Level 2
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.
Session One
Session Two
Session Three
Session Four
Session Five
Specific Teaching Points
Session one involves subtracting single digit and two digit numbers starting at 200. As students take handfuls or counters from their waka they will need to anticipate how many counters remain. Students will use need to use place value to calculate in ways other than counting back. The session notes recommend using a linear model for representing the calculations. A bead string is ideal and can be mounted along the edge of a whiteboard so jumps can be recorded on the whiteboard.
The session notes recommend linking two strings end on end to form a line of 200 beads. An important strategy in the activity is ‘back through ten’. For example, a student has 93 counters left and removes a handful of 17 counters. How many do they have left?
On the bead string this calculation can be modelled like this:
93 – 7 can be carried out in two steps, take away three to get to 90 then take away four to get to 86. This is a ‘back through ten’ strategy so it applies to using any decade number as a benchmark. Of course a student might take away a the ten of 17 first.
Session three involves dealing with like measures, e.g. dividing or multiplying weights. Actually measuring objects with devices like kitchen scales is important to the development of students’ understanding of the measurement system. For example, students will need to find out how many kilograms of kūmara will need to be ordered for the hāngi. If possible bring a few kūmara along so students can experiment to find out how many kūmara make up one kilogram in weight. They will then need to use division or multiplication to calculate how many kūmara they need. So if 24 kūmara are needed and four kūmara weigh one kilogram then 24 ÷ 4 = 6 kilograms will need to be purchased.
Session four develops important geometry ideas out of whai (string figures). A common issue with the learning of geometry is that students form prototypical views of shapes. For example, they might consider an equilateral triangle to be the only shape that is a triangle. All of the shapes below are triangles:
The issue of prototypical ideas will also apply to other polygons such as hexagons and octagons.
These three shapes are all hexagons. Note that the bottom hexagon is concave as it has two internal angles greater than 180°. It is important to discuss the defining characteristics of a class of shapes like hexagons. The only required property is that the shape is closed by six sides.
Session five also involves an important mathematical idea, distinctness. Rotating or reflecting a shape does not change its properties, except orientation (direction it is facing). The idea is fundamental to determining if given shapes are similar or different. For example, all of the quadrilaterals below are similar even though they look different. They can all be mapped onto each other using translation (shifting), reflection (flipping), and rotation (turning).
Similarity is applied in Session Five by looking for different winning positions. If the positions are reflections or rotations of one another then they are not considered to be distinct.
Prior Experience
The activities are mostly open ended so they cater for a range of achievement levels. It is expected that students have some experience with naming and classifying basic geometric shapes, with measuring weight in kilograms, and with translating, reflecting and rotating shapes. They should also have place value knowledge to 200.
Session One
What the legend does not tell you is that Tamarereti collected 200 bright shiny stones and put them at the bottom of the waka.
Have a ‘waka’ with 200 counters ready for the student to act out the story. Any narrow container will make a good waka.
Look for students to suggest ways to group the counters. Combinations that add to ten are especially useful.
Session Two
How do we find out the date of the full moon from this calendar?
So when will the last quarter start?
When are the good days for fishing?
When will the New Moon appear?
So when does the New Year start?
Session Three
Matariki is a time of cultural pursuits and feasting to celebrate the New Year ahead. The hāngi or earth oven has particular significance at the time of the new moon after the rise of Matariki in the Eastern pre-dawn sky. Matariki is the star at the bow of Te Waka o Rangi and her travels around the sky for eleven months of year are exhausting. It is said the steam of the first hangi in the New Year rises into the sky and replenishes the strength of Matariki. From the offerings she gathers strength to lead the giant canoe for another year. Without Matariki at the bow the canoe cannot travel and Taramainuku cannot cast his net to gather the souls of the departed. At the New Year the names of the dead are called out so the souls of the departed may be cast into the heavens as stars.
There are many resources already available about hāngi.
“Preparing for the hāngi” is a Level 3 activity from the Figure It Out series.
“Hanging out for hāngi” is a unit at Level Three that develops a statistical investigation around deciding which foods to cook.
The notes below are an adaptation more suitable for Level Two students.
Students might mention that people should not eat every meat and every vegetable. Agree on some restrictions like one or two meats and up to three vegetables. Point out that stuffing is a yes or no choice.
Session Four
Matariki was a time when food was already stored, and it was cold outside. So whānau (families) spent time together engaging in cultural pursuits such as storytelling, arts and games. Whai (string games) were popular with tamariki (children) and adults alike, especially when they involved co-operation. Whai has a long history and is common to many indigenous cultures around the world, including the indigenous tribes of North America. So if instructions tell you to “Navajo your thumbs” that means a common move that is attributed to a tribe of indigenous Americans. Traditionally whai was played with twine made from flax. The best man-made fibre to use for whai is nylon since it slides and flexes, and is soft on your hands. It commonly used to form lines for brickwork so is available at most hardware stores in a variety of colours. Nylon string is also available in craft shops.
At 1:08 it is easier to think of going over two strings and ‘picking up the third string’ in that move. Note that the move where you use your mouth to shift the bottom of two strings over your thumbs (2:00 - ) is called ‘Navajoing your thumbs’ and is another algorithm common in whai.
Session Five
In this session students learn to play the traditional Māori game Mū Tōrere which is like a form of draughts. The original game is sometimes referred to as the wheke (octopus) game or the whetū (star) game due to the shape of the board. It is appropriate that students learn to play the game at the time of Matariki, since the Māori New Year is a time of engaging in cultural pastimes. The board (see Copymaster 5) has been altered to include the nine or seven stars of Matariki, dependent on the version of the game that is played.
The winner must occupy the centre circle, the opponent’s stones must be clustered together around the hexagon and the winner must have the ends of the cluster blocked of. You might try to find a winning arrangement by separating the loser’s stones into a group of two and one but there is no way for the other player to stop them moving.
Kia ora parents and caregivers
This week we are exploring some mathematical ideas with activities to celebrate Matariki, the Māori New Year. We will be learning the legend of Tamarereti and using that story to learn about numbers of bright stones. We will look at the phases of marama (the moon) to find the best days to go fishing in the New Year period, plan and cost a hāngi, and discover geometry in whai (string patterns). Lastly we will learn to play Mū Tōrere, a traditional Māori game that takes a lot of strategy to win.
These activities involve number, algebra, geometry, measurement and statistics so we will be very busy and doing a lot of mathematical thinking.