Early level 2 plan (term 2)

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

In this unit we conduct a number of investigations using a party or favourites theme. Students count, compare, organise, analyse, display and interpret data and at the same time, apply early additive strategies for combining numbers.

  • Pose investigative questions.
  • Plan for and collect category data.
  • Display data in tally charts, pictographs and bar graphs.
  • Make statements about data displays.
  • Answer the investigative question.
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Level Two
Number and Algebra
Units of Work
In this unit students are given the opportunity to explore multiplication concepts using arrays. The use of multiple strategies and the sharing of strategies is encouraged, in group and whole class situations.
  • Solve multiplication problems by using skip counting or additive strategies.
  • Interpret and solve multiplication story problems.
Resource logo
Level Two
Number and Algebra
Units of Work
In this unit students explore the number patterns created when tins are stacked in different arrangements and keep track of the numbers involved by drawing up a table of values.
  • Identify patterns in number sequences.
  • Systematically “count” to establish rules for sequential patterns.
  • Use rules to make predictions.
Resource logo
Level Two
Geometry and Measurement
Units of Work
This unit of work explores the measurement of area. Students estimate and measure area using square centimetres.
  • Recognise the need for a standard unit of area.
  • Measure surfaces using square centimetres.
  • Estimate the measure of surfaces using square centimetres.
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Level Two
Integrated
Units of Work
This unit builds the learning of mathematics around celebrations of Matariki, the Māori New Year. The sessions provide meaningful contexts that highlight Māori culture and provide powerful learning opportunities that connect different strands of mathematics.

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.
  • Calcul...
Source URL: https://nzmaths.co.nz/user/75803/planning-space/early-level-2-plan-term-2

Parties and favourites

Purpose

In this unit we conduct a number of investigations using a party or favourites theme. Students count, compare, organise, analyse, display and interpret data and at the same time, apply early additive strategies for combining numbers.

Achievement Objectives
S2-1: Conduct investigations using the statistical enquiry cycle: posing and answering questions; gathering, sorting, and displaying category and whole-number data; communicating findings based on the data.
Specific Learning Outcomes
  • Pose investigative questions.
  • Plan for and collect category data.
  • Display data in tally charts, pictographs and bar graphs.
  • Make statements about data displays.
  • Answer the investigative question.
Description of Mathematics

At Level 2 you can expect students to be posing (with teacher support) a greater range of questions, including investigative questions and survey questions. They will also be helped to understand some of the issues involved in conducting surveys and learn new methods for collecting data. While at Level 1 students collected data and chose their own ways to display their findings, at Level 2 they will be introduced to pictographs, tally charts and bar graphs. More emphasis here will also be placed on describing the data and the making of sensible statements from both the student’s own displays and the displays of others.

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.

Analysis questions

Analysis questions are questions we ask of displays of data as we start to describe it.  Questions such as: what is the most common? the least common? how many of a certain category? what is the highest value (for numerical data)? lowest value (for numerical data)?

Pictograph

In a pictograph the pictures are drawn on uniform pieces of paper. This means that the number of objects in each category now bears a direct relationship to the size of each category on the display. An example is shown in the diagram below. 

pictograph.

In a further development the pictures can be displayed on a chart with axes and titles. The vertical axis can be numbered to match the pictures.

Bar Graph

In a bar graph equal-width rectangles (bars) represent each category or value for the variable. The height of these bars tells how many of that object there are.  The bars can be vertical, as shown in the example, or horizontal.

  graph

The example above shows the types of shoes worn in the class on a particular day. There are three types of shoes: jandals, sneakers, and boots. The height of the corresponding bars shows that there are six lots of jandals, 15 lots of sneakers and three lots of boots. It should be noted that the numbers label the points on the vertical axis, not the spaces between them. Notice too, in a convention used for discrete data (category and whole number data), there are gaps between the bars. 

Tally Chart

A tally chart provides a quick method of recording data as events happen. If the students are counting different coloured cars as they pass the school, a tally chart would be an appropriate means of recording the data. Note that it is usual to put down vertical strokes until there are four. Then the fifth stroke is drawn across the previous four. This process is continued until all the required data has been collected. The advantage of this method of tallying is that it enables the number of objects to be counted quickly and easily at the end.

tally chart.

In the example above, in the time that we were recording cars, there were 11 red cars, four yellow cars, 18 white cars and five black ones and 22 cars of other colours. Microsoft Excel is a program available on most types of computers that allows data to be entered onto a spreadsheet and then analysed and graphed very easily.  There are a number of freely available tools for graphing data, for example CODAP – Common Online Data Analysis Platform, is an online statistical tool that is accessible from a young age.

Opportunities for Adaptation and Differentiation

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:

  • Direct students to collect category data or whole number data – whole number is harder
  • Give students summarised data to graph rather than them having to collect it and collate it
  • Give students a graph of the display and ask them to “notice” from the graph rather than having them draw the graph
  • Write starter statements that students can fill in the blanks to describe a statistical graph e.g. I notice that the most common XXXX is ________, More students chose _______ than chose _______.

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

  • Planning for a class party
  • Planning for a special occasion – e.g. deciding on the type of food, the activities to include, venue
    • Matariki
    • Diwali
    • Kai festivals e.g. Motueka Kai Fest, Hokitika Wildfoods Festival, Kāwhia Kai Festival
  • The favourite example can be adapted to explore any favourites
Required Resource Materials
  • Packet of balloons – different shapes and colours if possible
  • Sheets of A4 cut into eighths
  • Prepared bar graph outlines
  • Multi packs of chips (popcorn and cups)
  • Party props: hats, candles, cards, sweets, blind fold
Activity

Session 1: Balloons investigation

Today we will make a pictograph of our favourite balloon shapes. We are going to answer the investigative question “What different balloon shapes do the students in our class like?”

  1. Take a bag of balloons and spread out. Discuss shapes. Suggest the investigative question “What colour balloons do the students in our class like?”
  2. tudents choose favourite shape (or colour if different shaped balloons are not available) and draw it on a piece of paper (one eighth of an A4).
  3. As a class, discuss ways to display the data. If matching pictures in 1:1 lines (pictograph) is not suggested, teacher will need to direct them to this.
  4. Students each attach their drawing to the class chart.
  5. Ask the students what they notice about the  information shown on pictograph. Use the prompt “I notice…” to start the discussion. These “noticings” could be recorded as "speech" bubbles around the chart.
  6. Talk about the need to label the axes and give the chart a title so that others could make sense of the display. A good idea is to write the investigative question as the chart title.
  7. Ask analysis questions, to extend the noticing, about the results that require students to combine sets:
    How many students liked long wiggly balloons?
    How many students
     liked long straight balloons?
    How many students
     liked long balloons altogether?
    How can you add the numbers together?
    How many students
     liked balloons that were not long?
    How many more students
     liked long wiggly balloons than long straight balloons? (Model and reinforce the use of subtraction or addition rather than counting on or back to solve this type of question.)
    Try to find analysis questions that will allow students to use strategies such as near doubles and adding to make 10s.

Session 2: Birthday Party investigation

This birthday party investigation is described in full as a possible model for teaching and developing ideas for each of the stages of the statistical enquiry cycle at Level 2.  In New Zealand we use the PPDAC cycle (problem, plan, data, analysis, conclusion).  You can find out more about the PPDAC cycle on Census At School New Zealand.

Check in: if the birthday party context is not suitable for your students you can find another context, the process described here will work for other contexts.

PROBLEM: Generating ideas for statistical investigation and developing investigative questions

  1. Ask the students to think about the topic of birthday parties.  Explain that we will be going to collect some information to answer different investigative questions about birthday parties that we are going to pose.
    Using the starter “I wonder…” ask the students what they wonder about birthday parties. On the board or on a chart record their ideas. For example:
    I wonder…
    1. What are our favourite birthday cakes?
    2. What birthday games we like?
    3. How many birthday parties kids have gone to? (check in on this, will it be a tricky question to answer, could it cause distress?)
    4. Where kids like to have their birthday parties?
    5. What birthday presents we want?
    6. What types of food we like?
    7. How many people we want at our birthday party? (this too might cause distress, as may other questions, as the teacher you will have a reasonable idea of the questions that will be ok and those that will not).
      Using the “I wonder” prompt helps with generating investigative questions, questions we ask of the data.
       
  2. Depending on what the students give you in the brainstorming session depends on how much work is needed to tidy up the investigative questions.  New Zealand based research has identified six criteria to support the development of and/or critiquing of investigative questions. These criteria are used in the example below.  The teacher asks questions of the students to identify the information needed e.g. variable, group and with this information develops the investigative question.
    • The intent of the investigative question is clear – we need to pose summary investigative questions (about category or whole number data)
    • The variable of interest is clear e.g. favourite birthday cake flavour, number of people at our party
    • The group we are interested in is clear e.g. our class, Room 30, Kauri class
    • We can collect data to answer our investigative question
    • The investigative question considers the whole group e.g. How many birthday parties have the children in Room 30 been to this year? – considers the whole group; whereas What is the most number of birthday parties that a child in Room 30 has been to this year? – does not and is not an investigative question (it is an example of an analysis question and asks about an individual)
    • The investigative question is interesting and/or purposeful – in this case if the ideas are generated by the students then we would expect them to be interesting to the students.
       

    For the birthday cakes example some possible questions are:

    • What will we wanting to find out about? (Favourite birthday cakes)
    • Who are we going to ask? (Our class)
    • Do you think we could find this out by asking our class? (Yes)
    • Are you interested in knowing about favourite birthday cakes? (Yes (if no, then ditch the question).)
       

    For each of the ideas generated in part 1, possible investigative questions are:

    1. What are favourite birthday cakes for children in Room 30?
    2. What are Room 30’s favourite birthday games?
    3. How many birthday parties have the children in Room 30 been to this year?
    4. Where do Room 30 children want to have their birthday parties?
    5. What presents do Room 30 children want for their birthday?
    6. What are favourite birthday foods for the children in Room 30?
    7. How many people do Room 30 children want at their birthday parties?
       
  3. Each group selects one of the investigative questions to explore.

     

PLAN: Planning to collect data to answer our investigative questions

  1. Explain to the students that they need to think about what question or questions they will ask to collect the information they need to answer their investigative question.
     
  2. Explain that these questions are called survey questions and they are the questions we ask to get the data. Work with groups to generate survey questions. For example:
    • If the investigative question is: “What are the favourite birthday cakes for children in Room 30?”, ask the students how they could collect the data. 
    • A possible response is to ask the other students “What is your favourite birthday cake?”
    • This might lead to a discussion about whether they mean the flavour and/or the style (a possible way to extend this for some students), e.g. I might respond my favourite birthday cake is a dinosaur, when they might have been meaning the flavour e.g. chocolate, banana etc.
    • This might also mean that the investigative question needs adjusting to: What are favourite birthday cake flavours for children in Room 30?
    • Also, the students might want to ask, “What is your favourite birthday cake out of chocolate, banana, ice-cream or sponge?” 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 your favourite birthday cake flavour?
    • What is your favourite game to play on your birthday?
    • How many birthday parties have you been to this year?
    • Where do you want to have your birthday party?
    • What presents do you want for your birthday? (this could give multiple answers, may want to change to what is the present you most want…)
    • What is your favourite birthday food?
    • How many people do you want to have at your birthday party?

    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 favourite birthday cakes for the children in Room 30? – overall about the class data) rather than a survey question (What is your favourite birthday cake flavour? – asking the individual).
     

  3. Get the students to think about how they will record the information they get. Options may include:
    • Tally chart
    • Writing down names and choices
    • Using pre-determined options
    • Using a class list to record responses
       
  4. Let them try any of the options they suggest.  They are likely to encounter problems, but this will provide further learning opportunities as they reflect on the difficulties and how they can improve them.

DATA: Collecting and organising data

  1. Students collect data from the rest of the class using their planned method. Expect a bit of chaos. Possible issues that lead to useful teaching opportunities include:
    • Pre-determined options
      • What happens for students whose choice is not in the pre-determined options?
      • What if nobody likes the options given and they end up with a whole lot of people choosing other? They only have tally marks so they cannot regroup to new categories.
    • Using tally marks only
      • The discussed issue above about the “other” category
      • Have fewer tally marks than the number of students in the class
        • and they think they have surveyed everyone
        • or they do not know who they have not surveyed yet
      • Have more tally marks than the number of students in the class
    • Possible solutions to the above issues could be (generated by the students please)
      • Recording the name of the student and their response and then tallying from the list
      • Giving everyone a piece of paper to write their response on, then collecting all the papers in and tallying from the papers
         
  2. Regardless of the process of data collection we are aiming for a collated summary of the results.
    table

ANALYSIS: Making and describing displays

  1. Taking their summarised information the students make a pictograph to help to answer their investigative question. As for the balloon activity we want to have uniform pieces. Provide:
    • Squares of paper all of the same size for students to create their own pictures
    • Chart paper
       
  2. Students give the chart a title – a good option is the investigative question.
     
  3. Students make the pictograph by gluing enough pictures to represent the data they collected.
     
  4. Teacher roams questioning for understanding and ensuring that students can correctly construct a pictograph.
     
  5. Once students have completed their pictograph they should write at least three “I notice…” statements about their pictograph. They can write the “I notice…” statements onto the chart paper as well.
     
  6. Teachers can prompt further statements by asking questions such as:
    • What do you notice about how many students liked cakes that were not chocolate?
    • What do you notice about the number of birthday parties attended? Did you notice the greatest number of birthday parties? The least number of birthday parties?
    • Emphasise questions that require students to operate with the numbers in their displays.
       
  7. Check the “I notice…” statements for the variable and reference to the class.  For example: “I notice that the most favourite birthday cake flavour for Room 30 children is carrot cake.” This statement includes the variable (favourite birthday cake flavour) and the class (Room 30 children). Support students to write statements that include the variable and the group.
     
  8. Get students to leave their charts on their desks.  Hand out post it notes to the students and get them to wander around the class and to look at all the other graphs.  Encourage them to add “I notice…” statements to the graphs of others by using the post it notes.

CONCLUSION: Answering the investigative question

At the end of the session get each group to share their chart. They should state their investigative question and then the answer to the investigative question. The answer should draw on the evidence from their pictograph and their “I notice…” statements.

For example: What are favourite birthday cakes for children in Room 30?

Answer: The most popular birthday cake flavour for Room 30 is chocolate cake. 15 students in our class had chocolate as their choice. The other flavours that were liked included carrot cake, banana cake and ice-cream cake.  Carrot cake was the least popular cake flavour for Room 30.

Extending: If I (the teacher) was to make a cake for the class what flavour should I make?

Session 3: Chips or popcorn

The previous session involved the full PPDAC cycle.  In this session today we are going to look at using tally marks to record the number of chips in a snack bag or the number of pieces of popcorn in a small cup and a bar graph to display the data.  We are focusing on the data collection and analysis phases.

  1. Display a snack bag of chips (or a small cup of popcorn) and ask the students to guess how many chips (popcorn) they think are in the bag (cup).
  2. Pose the investigative question: How many chips are in XXX brand snack bags? (How many popcorns are in the small cups?
  3. We are going to collect data to answer our investigative question by counting how many chips are in each of the snack bags I have here (count the number of popcorns in the small cup).
  4. How should we do that? Elicit ideas including counting them all. Ask how we could count them and keep a track? Accept all ideas including using tally marks to keep a track.
  5. Teacher models using tally marks to track how many chips (popcorns) she/he eats.
  6. Distribute individual bags of chips to small groups.
  7. Students eat chips and use tally marks to record the number of chips in each bag by adding the total of the tally marks each student in the group recorded. (Record the number of popcorns in the cup, but don’t combine, will use each students cup count as a data point – this gives more data points than the chips do, unless you give each child their own bag of chips.)
  8. Gather the total tallies on the board or a chart.
  9. Using a prepared bar graph outline the teacher constructs a bar graph with the information from the individual total tallies.
  10. Discuss features of the graph and summarise the information shown.
    What was the most common number of chips (popcorns)?
    What was the least common number of chips (popcorns)?

    How many more chips (popcorns) were there in the packet (cup) with the most than there were in the one with the least?
  11. As a class challenge, try to work out how many chips (popcorns) the class ate altogether.
    How many chips (popcorns) did the boys eat?
    How many chips (popcorns) did the girls eat?
    Discuss strategies for adding the numbers together 
    (for example: combine the numbers that add to 'tidy' numbers; add the tens and then the ones; use doubles or near doubles).

Session 4. Party props.

Today we will plan our own investigation from a party prop display.

  1. Use "party props" to generate discussion about parties.
  2. Brainstorm possible investigative questions using the prompt “I wonder…”. You may need to model possible investigative questions as you display party props. For example:
    Here are some balloons. Gosh I’m hopeless at blowing up balloons. It probably takes me 12 or more breaths to blow it up. How many breaths does it take you?

    I wonder how many breaths it takes our class to blow up a balloon.
    Generate a few ideas using this process and following the ideas from session 2.
  3. In pairs students select an investigative question to explore and plan how they are going to collect the data. Use the ideas from session 2 to support students to plan for data collection.
  4. Once the data is collected the pairs need to display the data using either a pictograph or a bar graph. Ask them to write a couple of ”I notice…” statements to accompany their graph.
  5. Share survey results. Students can be challenged to answer the questions on each other's data displays.

Session 5. Favourites

In this session we will undertake a statistical investigation using the idea of favourites as our starting point.  The big ideas for the investigation are detailed in session 2.  Ideas to support the specific context are given here.

PROBLEM

Brainstorm with the students different things that they have a favourite of.  You might also use the starter “I wonder what are favourite _________ for our class?”

Using the ideas developed previously identify 10-15 favourites to be explored and develop investigative questions for pairs of students to explore.

Investigative questions might be:

  • What are favourite sports that the children in our class play?
  • What are our class’s favourite waiata?
  • What are Room 30’s favourite kai?

PLAN

As the students have had some practice with planning previously allow them some freedom to plan for their data collection.  Check in on the survey questions they are planning to ask.

DATA

Students collect the data that they need to answer their investigative question, remember that they will have potentially inefficient methods they employ to do this.  Be prepared for some chaos. Use any resulting errors or problems to improve their data collection methods.

ANALYSIS

Get the students to display the data to answer their investigative question.  They may use a pictograph or a bar graph.  Remind them to label with the investigative question and to write “I notice…” statements about what the data shows.

CONCLUSION

Allow time for pairs to present their findings by giving their investigative question and then answering it using evidence from their displays and noticings. 

Arrays hooray

Purpose

In this unit students are given the opportunity to explore multiplication concepts using arrays. The use of multiple strategies and the sharing of strategies is encouraged, in group and whole class situations.

Achievement Objectives
NA2-1: Use simple additive strategies with whole numbers and fractions.
NA2-2: Know forward and backward counting sequences with whole numbers to at least 1000.
Specific Learning Outcomes
  • Solve multiplication problems by using skip counting or additive strategies.
  • Interpret and solve multiplication story problems.
Description of Mathematics

In this unit the students use arrays to solve multiplication problems. Arrays are an arrangement of objects in rows and columns. For example, chocolate blocks are made up of an array of smaller pieces. The block below has two rows and five columns.

Orchards’ layouts are also arrays where the fruit trees are grown in rows and columns to make them easier to look after and easier to pick fruit from.

Arrays are strongly advocated by many researchers in Mathematics Education because they model the binary (two factors) nature of multiplication. The number of rows and columns gives the factors by which the total number can be found, e.g. 2 x 5 or 5 x 2 for the chocolate block above. Arrays are also used extensively in the measurement of area, in finding all outcomes of a probability situation (Cartesian product), in grid systems on maps, and in spreadsheets and other digital tools. Recognising the multiplicative structure of arrays can be challenging for students, especially those who have little experience with equal sets. This unit is an introduction to multiplication.

Estimation is also an important component of this unit. Students are encouraged to use their number knowledge to anticipate approximate products when given two factors.

Opportunities for Adaptation and Differentiation

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:

  • Support students to solve problems by counting if that is their most sophisticated strategy. Encourage the use of skip-counting where possible.
  • Model effective use of the arrays on Copymasters 3 and 4, showing students how to partition the arrays using known number facts.
  • Reduce the numbers in the problems so that students are working with smaller arrays.

The contexts in this unit can be adapted to recognise diversity and student interests to encourage engagement. For example, lines of students in kapa haka groups, groups of people travelling in a bus or van, or planting seedlings in rows can be used as alternative contexts for arrays.

Required Resource Materials
  • Large pieces of paper for recording
  • Scissors
  • Counters and cubes
  • Copymasters One, Two, Three, Four, and Five (Copymaster Two is made into cards for group station work)
  • PowerPoints One Two and Three
Activity

This series of lessons provides different contexts to explore multiplication concepts using arrays such as the one below. This array has 5 rows and 10 columns.

 array. array.

Getting started

  1. We begin the week with the ‘Orchard Problem’.
    Jack the apple tree grower has to prune his apple trees in the autumn. He has 6 rows of apple trees and in every row there are 6 trees. How many apple trees does Jack have to prune altogether?

The start of PowerPoint One shows the whole array. Show the complete array. Ask your students to open their eyes and take a mind picture of what they see. Click once to remove all the trees and ask your students to draw what their mind picture looks like.

Look to see if they attend to the rows and columns layout even if the numbers of trees have errors. Discuss the layout.

  1. Have a pile of counters in the middle of the mat. Ask a volunteer to come and show what the first row of trees might look like. Or get 6 individuals to come forward and act like trees and organise themselves into what they think a row is.
    Alternatively click again in the PowerPoint so it’s easy for all to see what the first row of apple trees will look like. Ask your students to improve their picture if they can.
    What will the second row look like? 
    It’s important for students to understand what a row is so they can make sense of the problem. It is also important for them to notice that all rows have the same number of trees.
  2. Arrange the class into small mixed ability groups with 3 or 4 students in each. Give each group a large sheet of paper. Ask them to fold their piece of paper so it makes 4 boxes (fold in half one way and then in half the other way).
  3. Allow some time for each group to see if they can come up with different ways to solve the Orchard Problem and record their methods in the four boxes. Tell them that you are looking for efficient strategies, those that take the least work.
    Allow students to use equipment if they think it will help them solve the problem. 
    Rove around the class and challenge their thinking with questions like:
    • How could you count the trees in groups rather than one at a time?
    • What facts do you know that might help you?
    • What sets of numbers do you know that might help you?
    • What is the most efficient way of working out the total number of trees?
  4. Ask the groups to cut up the 4 boxes on their large sheet of paper and then come to the mat. Gather the class in a circle and ask the groups to share what they think is their most interesting strategy. Place each group’s strategy in the middle of the circle as they are being shared. Once each group has contributed, ask the students to offer strategies that no one has shared yet. 

    Likely strategies

    Possible teacher responses

    1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ….

    Can you think of a more efficient way to work out how many trees there are?

    How many trees are there in one row?

    6, 12, 18, 24, 30, 36

    Do you know what 6 + 6 =? Or 3 + 3 = ? 

    Can that knowledge help you solve this problem more efficiently?

    6 x 6 = 36

    What if Jake had 6 rows of trees and there were 7 trees in each row?

    6 + 6 = 12; 12 + 12 = 24; 24 + 12 = 36

    You used addition to work that out. 

    Do you know any multiplication facts that could help?

    2 x 6 = 12;

    12 + 12 + 12 = 36

    If 2 x 6 = 12, what does 4 x 6 =? How could you work out 6 x 6 from this?

    3 x 6 = 18 and then doubled it

    That is very efficient. Could you work out 9 rows of 6 for me using 6 x 6 = 36?

    5 x 6 = 30; and 6 more = 36

    The shared strategies can be put into similar groups.
    Who used a strategy like this one?

  5. Show students PowerPoint Two. The PowerPoint encourages students to dis-embed a given smaller array of trees from within a larger array. They are also asked to use their knowledge of the smaller array to work out the total number of trees in the larger array. This is a significant ability for finding the totals of arrays using the distributive property of multiplication.
  6. Provide your students with Copymaster One. The challenge is to find the total number of trees in each orchard. Challenge your students to find efficient strategies that do not involve counting by ones.
  7. As a class share the different ways that students used to solve the Orchard Problems. You might draw on the Copymaster shown on an Interactive Whiteboard to show how various students partitioned the arrays.

Sessions Two and Three: Exploring through work stations

In the next two sessions students work in pairs or threes to solve the problems on Copymaster Two. Enlarge the problem cards and place them at each station. Provide students with access to copies of Copymasters Three and Four (arrays students can draw on), and physical equipment such as counters, cubes, and the Slavonic Abacus.

Read the problems from Copymaster Two to the class one at a time to clarify the wording. You may need to revisit the meaning of rows and columns by creating simple examples.
As students work on a station activity, ask them to create a record of their thinking and solution/s. The record might be a recording sheet or in their workbook. Note that Part 2 of each problem is open and requires a longer period of investigation.

As the students work watch for the following:

  • Can they interpret the problem wording either as a physical representation of as symbolic equations?
  • Do they create arrays of equal rows and columns?
  • Are they able to use skip counting, additive or multiplicative strategies to find the total number of trees?
  • Do they begin to see properties of whole numbers under multiplication? (for example, Apple Orchard Part 2 deals with the commutative property)

At times during both sessions you might bring the class together to discuss confusions or misconceptions, clarify language and share efficient strategies and ways of representing the problems.

Below are specific details related to each problem set.

Orange Orchard

Orange Orchard (Part 1) involves 6 x 8 (or 8 x 6). Students might use their knowledge of 6 x 6 = 36 and add on 12 more (two columns of six). That would indicate a strong understanding of the multiplicative structure of arrays.

Most students will use strategies that involve visualising the array and partitioning the array into manageable chunks (dis-embedding). For example, they might split rows of eight into two fours (6 x 8 = 6 x 4 + 6 x 4), or into fives and threes (6 x 8 = 6 x 5 + 6 x 3). Other students will use less sophisticated strategies such as counting in twos and fives, or a combination of skip counting and counting by ones.

Part 2 is an open task which requires students to identify the factor pairs of 24.

Encourage capable students to be systematic in finding all the possibilities (1 x 24, 2 x 12, 3 x 8, 4 x 6).

Orange Orchard (Part 1)

Tame has an orange orchard with 6 rows of trees.

In each row there are 8 trees. 

How many trees does Tame have altogether?

Your prediction:

Your answer:

Orange Orchard (Part 2)

Tame wants to plant another orchard with oranges. 

He gets 24 trees. 

Find different ways Tame can plant 24 trees in rows and columns. 

Show all the different ways.

 

Kiwifruit Orchard

Part 1 requires students to co-ordinate three factors as the problem can be written as 3 x (4 x 5). Multiplication is a binary operation so only two factors can be multiplied at once. Do your student recognise the structure of a single orchard (4 x 5) and realise that the total is comprised of three arrays of that size?

Similarly, in Part 2 students must restructure 36 plants into two sets. Do they partition 36 into two numbers, preferably that have many factors? The problem does not say that the two orchards must contain the same number of plants though 18 and 18 is a nice first solution. Once the two sets of plants are formed can your students find appropriate numbers of rows and columns that equal the parts of 36?

Kiwifruit Orchard (Part 1)

Lana has three kiwifruit orchards that are the same. 

In each orchard she has 5 kiwi fruit plants in every row.

There are 4 rows.

How many kiwi fruit plants does Lana have altogether?

Your prediction:

Your answer:

Kiwifruit Orchard (Part 2)

Lana’s son, Bruce, buys 36 plants to start two Kiwi fruit orchards. 

How can Bruce arrange the plants into rows and columns?

Show different ways.

Remember that he must share the 36 plants between two orchards.

 

 

Strawberry Patch

Part 1 is a single array (5 x 12). Students might use the distributive property and solve the problem or 5 x 10 + 5 x 2 (partitioning 12) or 5 x 6 + 5 x 6. Some may re-unitise two fives as ten to create 6 x 10. These strategies are strongly multiplicative. Most students will use smaller units such as fives or two and apply a combination of repeated addition (5 + 5 = 10, 10 + 10 = 20, etc.) or skip counting (2, 4, 6, 8, …).

Part 2 is about factors that have the same product (24). This gives students a chance to recognise that some numbers have many factors and the expressions of those factors have patterns. For example, 6 x 4 and 3 x 8 are related by doubling and halving. The logic behind the relationship may be accessible for some students. If the rows are halved in length, then twice as many rows can be made with the same number of plants.

Strawberry Patch (Part 1)

Hera has a strawberry patch. 

There are 5 rows with 12 strawberry plants in each row. 

How many strawberry plants does Hera have altogether?

Your prediction:

Your answer:

Strawberry Patch (Part 2)

Sam, Kim and Toni also have strawberry patches.

Sam has 6 rows with 4 plants in each row.

Kim has 3 rows with 8 plants in each row.

Toni has 2 rows with 12 plants in each row.

Who has the most strawberry plants, Sam, Kim or Toni?

 

Apple Orchard

Part 1 give students a chance to ‘discover’ the commutative property, the order of factors does not affect the product. In this case 5 x 10 = 10 x 5.

Part 2 applies the distributive property of multiplication though many students will physically solve the problem with objects. Look of some students to notice that 12 extra trees shared among six rows results in two extra per row. So, the number of rows stays the same, but the rows increase in length to six trees. Similarly, if more rows are made the 12 trees are formed into three rows of four. The number of rows would then be 9. 6 x 6 and 9 x 4 are the possible options.

Apple Orchard (Part 1)

Fatu’s apple orchard has ten trees in each row. There are five rows.

Min’s apple orchard has 5 trees in each row. There are ten rows.

Who has more apple trees, Fatu or Min?

Your prediction:

Your answer:

Apple Orchard (Part 2)

Besma has six rows of apple trees.

Each row has four trees.

If she plants 12 more trees, how many rows might she have then.

How many trees will Besma have in each row?

There are two answers.

Show both answers.

 

Sessions Four and Five

Sessions Four and Five give students an opportunity to recognise the application of arrays in other contexts.

The chocolate block problem involves visualising the total number of pieces in a block even though the wrapping is only partially removed. PowerPoint Three provides some examples of partially revealed chocolate blocks. For each block ask:

  • How many pieces are in this block?
  • How do you know?

Look for students to apply two types of strategies, both of which are important in measurement:

Iteration: That is when they take one column or row and see how many times it maps into the whole block.

Partitioning: That is when they imagine the lines that cut up the block, particularly halving lines. They look to find a partitioning that fits the row or column that is given.

Copymaster Five provides students with further examples of visualising the masked array.

The Air New Zealand problem is designed around the array structure of seating arrangements on airplanes.

Begin by role playing the Air New Zealand problem. Use chairs to make a simulated arrangement of seats. You might like to include grid references used to locate specific seats.

Try questions like:

  • How many rows are there? How many columns are there?
  • How many passengers could be seated altogether?
  • If the airline needed 24 seats what could they do?

Use different arrangements of columns and rows. Students might know that airplanes tend to be long and narrow, so some arrangements are very unlikely.

Give the students counters, cubes or square grid paper to design possible seat layouts with 40 seats. Encourage them to be systematic and to look for patterns in the arrangements. Some students will find efficient ways to record the arrangements such as:

2 rows of 20 seats                4 rows of 10 seats                5 rows of 8 seats

Record these possibilities as multiplication expressions on rectangles of card. Put pairs of cards together to see if students notice patterns, like doubling and halving.

It is important to also note what length rows do not work.

  • Could we make rows of 11 sets? 9 seats? Why not? (40 is not divisible by 11 or 9 as there would be remaining seats left over)

If students show competence with finding factors, you could challenge them to find seating arrangements with a prime number of seats such as 17 or 23. They should find that only one arrangement works; 1 x 17 and 1 x 23 respectively.

Reflecting

As a final task for the unit ask the students to make up their own array-based multiplication problems for their partner to solve.

  1. Tell the students that they are to pretend to be tomato growers. They decide how many rows of tomato plants they want in each row and how many rows they will have altogether. 
  2. Then they challenge their partner to see if the partner can work out how many tomato plants they will have altogether.
  3. Tell the students to create a record of their problem with the solution on the back. The problems could be made into a book and other students could write other solution strategies on the back of each problem page.
  4. Conclude the session by talking about the types of problems we have explored and solved over the week. Tell them that the problems were based on arrays. Let them know that there are many ways of solving these problems, tough multiplication is the most efficient method. Ask students where else in daily life they might find arrays.

Supermarket displays

Purpose

In this unit students explore the number patterns created when tins are stacked in different arrangements and keep track of the numbers involved by drawing up a table of values. 

Achievement Objectives
NA2-8: Find rules for the next member in a sequential pattern.
Specific Learning Outcomes
  • Identify patterns in number sequences.
  • Systematically “count” to establish rules for sequential patterns.
  • Use rules to make predictions.
Description of Mathematics

Patterns are an important part of mathematics. It is always valuable to be able to recognise the relations between things in order to understand how they inter-relate and predict what will happen.

Patterns also provide an introduction to algebra proper as the rules for simple patterns can be first discovered in words and then be written using the much more concise algebraic notation. There are two useful rules that we concentrate on here.

  • The recurrence rule tells the way that a pattern is increasing. It tells us the difference between two successive terms. So if we think of the pattern 5, 8, 11, 14, 17, … we can see that this pattern increases by 3 each time. So here the recurrence rule says that the number at any stage in the pattern is 3 more than the previous number.
  • The general rule tells us about the value of any number in the pattern. So for the pattern above, the general rule is that the number connected to any term of the sequence is 2 plus 3 times the number of the term. For instance, the third number in the sequence above is 2 plus 3 x 3, which equals 11. And the sixth number is 2 plus 3 x 6 = 20. To see why this general rule works it is useful to write the initial term (5) in terms of the increase (3). So 5 = 2 + 3.

It should be noted that there are many rules operating in these more complicated patterns. Encourage students to look for any relation between the numbers involved.

In this unit we ask students to construct tables so that they can keep track of the numbers in the patterns. The tables will also make it easier for the students to look for patterns.

In addition to the algebraic focus of the unit there are many opportunities to extend the students computational strategies. By encouraging the students to explain their calculating strategies we can see where the students are in terms of the Number Framework. As the numbers become larger expect the students to use a range of part-whole strategies in combination with their knowledge of the basic number facts.

Opportunities for Adaptation and Differentiation

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:

  • provide students with additional time to explore the patterns by drawing and counting tins, before expecting them to continue the patterns using only numbers.
  • work in small groups with students who need additional support, solving problems together.

The context in this unit can be adapted to recognise diversity and student interests to encourage engagement. For example, growing number patterns could be explored in the context of tukutuku panels in the wharenui, or the layout of seedlings for a community garden.

Required Resource Materials
Activity

Getting Started

Today we look at the number patterns in a tower of tins.

  1. Tell the students that today we will stack tins for a supermarket display.

  2. Show the students the arrangement:

    tin arrangement.

    How many tins are in this arrangement?
    How many tins will be in the next row?
    Then how many tins will there be altogether?
    How did you work that out?

  3. Encourage the students to share the strategy they used to work out the number of tins. “I can see 4 tins and know that you need 5 more on the bottom. 4 + 5 = 9”

    “I know that 1 + 3 + 5 = 9 because 5+3= 8 and 1 more is 9.”
    [These strategies illustrate the student’s knowledge of basic addition facts.]

  4. Show the students the next arrangement of tins. They can check that their predictions were correct.

    tin arrangement.

    How many tins will be in the next row?
    Then how many tins will there be altogether?
    How did you work that out?
  5. Encourage the students to share the strategies they used to work out the number of tins.
    “I know that we need to add 7 to 9 which is 16.” [knowledge of basic facts]
    “I know that 7+ 9 = 16 because 7 + 10 = 17 and this is one less." [early part-whole reasoning]
    “I know that we are adding on odd numbers each time. 1+3+5+7 = 16 because 7+3 is 10 + 5 + 1 = 16."
  6. Add seven tins to the arrangement and ask the same questions. As the numbers are becoming larger expect the range of strategies used to be more varied.

    tin arrangement.

    “16 + 9 = 25. I counted on from 16.” [advanced counting strategy]
    “16 + 10 = 26 so it is one less which is 25.” [part-whole strategy]

  7. Tell the students that the supermarket has asked for the display to be 10 rows high.
    How many tins will you need altogether?
  8. Ask the students to work in small groups to find out how many tins are needed. As the students work circulate asking:
    How are you keeping track of the numbers?
    Do you know how many tins will be on the bottom row? How do you know?
  9. Gather the students back together as a class to share solutions.
  10. Discuss the methods that the groups have used to keep track of the number of tins.
  11. Work with students to make a table showing number of rows and total number of tins. Complete the first couple of rows together.
  12. Ask the small groups to complete their own copy of the table on Copymaster 1. As they complete the chart ask:
    Can you spot any patterns?
    Write down what you notice?
    Can you predict how many tins would be needed when there are 15 in the bottom row?
  13. Encourage the students to explain their strategies for “counting” the numbers of tins.
  14. As a class share the patterns noted.

Exploring

Over the next 2-3 sessions the students work with a partner to investigate the patterns in other stacking problems. We suggest the following introduction to each problem.

  1. Pose the problem to the class and ask the students to think about how they might solve it.  In particular encourage them to think about the table of values that they would construct to keep track of the numbers.
  2. Share tables.
  3. Ask the students to work with their partner to construct and complete their own table.
  4. Write the following questions on the board for the students to consider as they solve the problem.

    How many tins are in the first row?
    How many are in the second row?
    By how much is the number of tins changing as the rows increase?
    What patterns do you notice?
    Can you predict how many tins would be needed for the bottom row if the stack was 15 rows high?
    Explain the strategy you are using to count the tins to your partner?
    Did you use the same strategy?
    Which strategy do you find the easiest?

  5. As the students complete the tables and solve the problem circulate asking them to explain the strategies that they are using to “count” the numbers of tins in the design.
  6. Share solutions as a class.

Problem 1:

Copymaster 1

A supermarket assistant was asked to make a display of sauce tins. The display has to be 10 rows high.
How many tins are needed altogether?
What patterns do you notice?

tin display.

Problem 2:

Copymaster 2

A supermarket assistant was asked to make a display of sauce tins. The display has to be 10 rows high.
How many tins are needed altogether?
What patterns do you notice?

tin display.

Problem 3:

Copymaster 3

A food demonstrator likes her products displayed using a cross pattern. The display has to be 10 products wide.
How many products are needed altogether?
What patterns do you notice?

cross pattern display.

Reflecting

In this session the students create their own “growth” pattern for others to solve.

  1. Display the growth patterns investigated over the previous sessions.
  2. Gather the students as a class and tell them that their task for the day is to invent a pattern for the supermarket to use to display objects.
  3. Ask the students in small group to decide on a pattern and the way that it will grow. (A supply of counters may be helpful for some students.)
  4. The students need next to construct a table to keep track of their pattern (up to the 10th model).
  5. Once they have constructed the table ask them to record the any patterns that they spot in the numbers. Ask them also to make predictions about the 15th and 20th model.
  6. Swap problems with another group.  When the problem has been solved compare solutions with each other.

Outlining area

Purpose

This unit of work explores the measurement of area. Students estimate and measure area using square centimetres. 

Achievement Objectives
GM2-1: Create and use appropriate units and devices to measure length, area, volume and capacity, weight (mass), turn (angle), temperature, and time.
Specific Learning Outcomes
  • Recognise the need for a standard unit of area.
  • Measure surfaces using square centimetres.
  • Estimate the measure of surfaces using square centimetres.
Description of Mathematics

When students can measure areas 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 area and have realised that consistency in the units used would allow for the easier and more accurate communication of area measures.

Students’ measurement experiences must enable them to:

  1. develop an understanding of the size of a square metre and a square centimetre;
  2. estimate and measure using square metres and square centimetres.

The usual sequence used in primary school is to introduce the square centimetre and then the square metre.

The square centimetre is introduced first, because it is small enough to measure common objects. The size of the square centimetre can be established by constructing it, for example by cutting 1-centimetre pieces of paper. Most primary classrooms also have a supply of 1-cm cubes that can be used to measure the area of objects. An appreciation of the size of the unit can be built up through lots of experience in measuring everyday objects. The students should be encouraged to develop their own reference for a centimetre, for example, a fingernail or a small button.

As the students become familiar with the size of the square centimetre they should be given many opportunities to estimate before using precise measurement. They can also be given the task of using centimetre-squared paper to create different shapes of the same area.

Opportunities for Adaptation and Differentiation

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:

  • provide smaller shapes for students to work with that have an area of a whole number of square centimetres
  • model how to visualise the first row and column of a grid and use this to estimate area.

The context in this unit can be adapted to recognise diversity and student interests to encourage engagement. For example, the activities could focus on measuring familiar objects such as leaves in autumn, or shells flowing a trip to the beach. For the activities to work there needs to be a collection of objects, all with a range of areas around 120cm.

Required Resource Materials
Activity

Session 1

We start this unit with a guessing game which introduces the idea of estimation.

  1. Show the students the outline of an object, for example; a small book, a pebbles packet or a calculator.
    What do you think that this could be the outline of?
    How many cubes do you think I would need to cover this shape?
  2. Give each student a cube and ask them to write their guess on a piece of paper. Introduce the idea that an estimate is a thoughtful guess. 

    I think the area of the mystery object is ......cubes

    James T

    Show the class a shape made with 5 cm cubes. 
    cubes

    Ask the students to record the shape on cm squared paper.
    What is the area of this shape?

  3. If the students say 5 squares tell them that the unit square is called a square centimetre.
    Why do you think it is called a square centimetre?
  4. Ask a volunteer to make a different shape with the 5 cubes. Tell them that the shape must be flat and the whole sides of the squares must touch.
    What is the area of this shape? ( 5 square centimetres or 5 square cm )
  5. Give each student 5 cm cubes and challenge them to find other shapes that can be made with the cubes. Ask them to record the shapes on the cm grid paper.
    Shapes to make box's 

    (These shapes are called pentominoes and there are 12 distinct shapes that can be made)

  6. Share shapes. Check again that the students understand that each has an area of 5 square cm.

Session 2

  1. Look at the outline of the mystery object from yesterday.
 How can we work out whose guess was closest to the area of the object? 
  2. Give each pair of students an outline of the mystery object and ask them to work out its area in square centimetres. Have cm cubes and squared paper available and support students to make decisions about how they will measure the area. Share areas and approaches used.
  3. Talk about how to handle part squares.
  4. Ask the students to write what they think the object is, and their measurement for its area, on the object’s outline. Display the outlines on a Mystery Object chart.

Session 3

  1. Pose the question: What objects do you think have about the same area as our Mystery Object? Note that students will need to use their estimation skills to accurately identify objects of a similar area and discuss possible estimation strategies. 
  2. Brainstorm ideas for objects that have about the same area as the Mystery Object. Write the names of these objects on strips of paper and put them into a hat.
  3. Working with a partner the students take a strip, and find the object it names. They then make an outline of the object, calculate its area, and write the name of the object and its area on the outline. 
  4. At the end of the session work together to order the objects measured from smallest area to largest area, and identify objects with a similar area to the Mystery Object.

Session 4

  1. Establish a challenge: Today we’re going to challenge ourselves to identify objects with a specific area. We’ll need to use our estimation skills.  
  2. Before the session fill the hat with strips of paper. Each strip needs to have the measurement of an area written on it. Include several strips of the same measurement.
  3. Students work in pairs to take a strip with the measurement of an area, and draw or find five objects with that area.
  4. As a class, review the task together and find out how successful students were at estimating area. Discuss useful estimation strategies.
  5. Students who have been working with the same measurement compare results and discuss any differences, checking each other’s measurements.

Session 5

Today we use the measurement skills we've been working on to find out who has the largest foot.

  1. How could we find out?
    About how many square cm do you think it would be? Why do you think that?
  2. Ask small groups of students to think about a way of measuring feet to find out whose is the largest.
  3. When the outline is made the students need to work out the area of their foot.
  4. Share outlines and measurements. Display from smallest to largest.
Attachments

Matariki - Level 2

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

  1. 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. 
  2. 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. An animated version of story is available at this link: https://www.youtube.com/watch?v=bEmcv8IbPSg.
  3. 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).
  4. 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.
  5. 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.
  6. 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.
  7. 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. 
  8. 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?
  9. 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. 
  10. Ask: What might Tamarereti scratch into the side of his waka to keep track of his number of stones?
  11. 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.
  12. 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)
  13. You might use the work samples students produce as evidence of their additive thinking.

Session Two

  1. 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.
  2. 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. 
  3. 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.
  4. 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.
  5. 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? 
  6. 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.
  7. 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.
  8. 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?
  9. 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.

  1. 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.
  2. Your investigation question is “What hāngi foods do people in our class like to eat?”
  3. 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.
  4. 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.
  5. 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. 
  6. 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.
  7. Ask: How might we organise these data so we can order food for the hāngi?
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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.

  1. 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. 
  2. 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.
  3. 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: 
  4. 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. 
  5. 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. 
  6. 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:
  7. 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.
  8. For an interesting geometry challenge consider looking for different shapes within a whai pattern. For example the gate pattern looks like this
  9. Ask your students what shapes they can see in the figure. Here are a few possibilities:
  10. 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.

  1. 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. 
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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?

Printed from https://nzmaths.co.nz/user/75803/planning-space/early-level-2-plan-term-2 at 12:17pm on the 21st January 2022