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.
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Session One: Getting started
- We begin the week with the ‘Orchard Problem’. A picture book about gardens, such as Nana's Veggie Garden - Te Māra Kai a Kui by Marie Munro, could be used to ignite interest in this context.
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 1 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. One child could draw their picture on the whiteboard. This could then be referred back to throughout the rest of the lesson.
Look to see if they attend to the rows and columns layout even if the numbers of trees have errors. Discuss the layout.
- 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. - 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).
- 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?
- 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 …. Tahi, rua, toru, whā, rima, ono, whitu, waru, iwa, tekau, tekau mā tahi… | 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?
- Show students PowerPoint 2. The PowerPoint encourages students to disembed 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.
- Provide your students with Copymaster 1. 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.
- As a class, share the different ways that students used to solve the Orchard Problems. You might model on the Copymaster to show how various students partitioned the arrays.
Sessions Two and Three: Exploring through work stations
The picture book Hooray! Arrays! by Jason Powe could be used to ignite interest in this learning. In the next two sessions students work in pairs or threes to solve the problems on Copymaster 2. Consider choosing these pairs to encourage tuakana teina through the pairing of more knowledgeable and less knowledgeable students. Enlarge the problem cards and place them at each station. Provide students with access to copies of Copymaster 3 and Copymaster 4 (arrays students can draw on), and physical equipment such as counters, cubes, and the Slavonic Abacus.
Read the problems from Copymaster 2 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 solutions. 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 or 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 coordinate 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 consists 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 gives 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 for 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 3 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 5 provides students with further examples of visualising the masked array.
The Kapa Haka problem is designed around the array structure of seating arrangements for Kapa Haka performances at school.
Begin by role playing the Kapa Haka 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 audience members could be seated altogether?
- If the performance needed 24 seats what could they do?
Use different arrangements of columns and rows.
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.
- Tell the students that they are to pretend to be kūmara growers. They decide how many rows of kūmara plants they want in each row and how many rows they will have altogether. As part of this learning, you could look into how early Maori people grew kūmara. This plant arrived in New Zealand with Polynesian settlers in the 13th Century. However, the climate here was much colder than the Polynesian islands. As a result, the kūmara had to be stored until the weather was warm enough for it to grow. The kūmara plant became even more important once settlers discovered that some of their other food plants would not grow at all in New Zealand’s climate. These kūmara were different to the ones we eat today - which came to us from North America. The books Haumia and his Kumara: A Story of Manukau by Ron Bacon, and Kumara Mash Forever by Calico McClintock could be used to engage students in this context.
- Then they challenge their partner to see if the partner can work out how many kūmara plants they will have altogether.
- 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.
- 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.
Parties and favourites
In this unit we conduct a number of investigations using a party or favourites as a theme. Ākonga count, compare, organise, analyse, display and interpret data.
At Level 2 you can expect ākonga to be posing (with teacher support) a greater range of questions, including investigative questions and survey questions. They will also begin to understand some of the issues involved in conducting surveys and learn new methods for collecting data. While at Level 1 ākonga 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 ākonga own displays and the displays of others.
Investigative questions
At Level 2 ākonga should be generating broad ideas to investigate. The teacher supports ākonga 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 development of investigative questions is led by the teacher. Questioning of ākonga, leads to the identification of the variable of interest, and the group the investigative question pertains to. Ultimately, the teacher forms the investigative question with ākonga 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 ākonga in our class like?” a corresponding survey question might be “What is your favourite ice cream flavour?”
As with the investigative question, the development of a survey question is led by the teacher. Purposeful questioning of ākonga leads to the collaborative development of suitable survey questions.
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? What is the least common? How many of a certain category? What is the highest value (for numerical data)? What is the 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.
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.
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 ākonga 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. The fifth stroke is drawn across the previous four. This process is continued until all the required data has been collected. Tallying enables the number of objects to be recorded and counted quickly and efficiently (i.e. by skip counting in fives)
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 and Google sheets can be used to enter data into spreadsheets, analyse data, and create graphs. CODAP – Common Online Data Analysis Platform, is an online statistical tool that is accessible from a young age.
The learning opportunities in this unit can be differentiated by providing or removing support to ākonga and by varying the task requirements. Ways to support ākonga include:
The context for this unit can be adapted to suit the favourites of your ākonga. Other possible contexts for this learning might be:
Te reo Māori vocabulary terms such as tuhuru (investigate) and pātai (question) could be introduced in this unit and used throughout other mathematical learning.
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 ākonga in our class like?”
How many ākonga liked long wiggly balloons?
How many ākonga liked long straight balloons?
How many ākonga liked long balloons altogether?
How can you add the numbers together?
How many ākonga liked balloons that were not long?
How many more ākonga liked long wiggly balloons than long straight balloons? All ākonga counting methods should be valused in this activity. However, it may be appropriate for you to (model and reinforce the use of subtraction or addition, rather than counting on or back.)
Try to find analysis questions that will allow ākonga 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) for the statistical enquiry cycle. You can find out more about the PPDAC cycle at Census At School New Zealand.
If the birthday party context is not suitable for your ākonga, choose another context (e.g. Diwali, matariki). The process described here will work for other contexts.
PROBLEM: Generating ideas for statistical investigation and developing investigative questions
Using the starter “I wonder…” Ask the ākonga what they wonder about birthday parties. Record their ideas. For example:
I wonder…
The amount of work needed to tidy up the investigative questions will depend on the responses of your ākonga in the brainstorming session. 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 ākonga to identify the information needed e.g. variable, group and with this information develops the investigative question.
For the favourite kai at a birthday party example some possible questions are:
For each of the ideas generated in part 1, possible investigative questions are:
Each group selects one of the investigative questions to explore.
PLAN: Planning to collect data to answer our investigative questions
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:
Possible survey questions are:
DATA: Collecting and organising data
ANALYSIS: Making and describing displays
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 graph and their “I notice…” statements.
For example: What are some favourite birthday cake flavours for children in Room 30?
Answer: The most popular birthday cake flavour for Room 30 is chocolate cake. 15 ākonga 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: 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 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.
What was the most common number of pieces of popcorn?
What was the least common number of pieces of popcorn?
How many more pieces of popcorn were there in the cup with the most, than the one with the least?
How many pieces of popcorn did each table group count?
Discuss strategies for adding the numbers together (for example: combine the numbers that add to 'tidy' numbers; use place value; use doubles or near doubles).
Session 4. 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 ākonga different things that they have a favourite of. You might 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 ākonga to explore. A tuakana/teina model could be used here.
Investigative questions might be:
PLAN
As ākonga have had some practice with planning previously, allow them some freedom, as appropriate, to plan their data collection. Check in on the survey questions they are planning to ask. Encourage ākonga to use the tuakana/teina model to support their learning journey.
DATA
Ākonga collect the data that they need to answer their investigative question. Be prepared for some potentially inefficient methods. Use any resulting errors or problems to improve their data collection methods.
ANALYSIS
Ākonga can display the data to answer their investigative question. They may use a pictograph or a bar graph. Remind them to label using 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.
Dear parents and whānau,
In math this week we are doing a statistics study on favourites. Can you please help us by filling in the survey form your child has produced to get some data about favourite things? If there is more than one person filling out this survey, please support your child to make space for this data collection.
Thank you.
Name:
(You could adapt this list to meet with what your class is interested in)
Figure it Out Links
There are a wide range of activities in the Figure It Out series based around statistical investigations. Look in the front half of the Figure It Out! Statistics book from the relevant level to find one which is suitable.
There are also activities around the idea of statistical investigations in the Theme books.
Arrays hooray
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.
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 researchers in mathematics education because they model the binary (two factors) nature of multiplication. Therefore, this unit can be used as an introduction to 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.
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.
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 contexts in this unit can be adapted to recognise diversity and student interests to encourage engagement. For example:
Consider how these contexts could provide links to other areas of current learning (e.g. about community gardens, school trips, planting out the school garden).
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.
Session One: Getting started
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 1 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. One child could draw their picture on the whiteboard. This could then be referred back to throughout the rest of the lesson.
Look to see if they attend to the rows and columns layout even if the numbers of trees have errors. Discuss the layout.
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.
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:
Can you think of a more efficient way to work out how many trees there are?
How many trees are there in one row?
Do you know what 6 + 6 =? Or 3 + 3 = ?
Can that knowledge help you solve this problem more efficiently?
You used addition to work that out.
Do you know any multiplication facts that could help?
2 x 6 = 12;
12 + 12 + 12 = 36
The shared strategies can be put into similar groups.
Who used a strategy like this one?
Sessions Two and Three: Exploring through work stations
The picture book Hooray! Arrays! by Jason Powe could be used to ignite interest in this learning. In the next two sessions students work in pairs or threes to solve the problems on Copymaster 2. Consider choosing these pairs to encourage tuakana teina through the pairing of more knowledgeable and less knowledgeable students. Enlarge the problem cards and place them at each station. Provide students with access to copies of Copymaster 3 and Copymaster 4 (arrays students can draw on), and physical equipment such as counters, cubes, and the Slavonic Abacus.
Read the problems from Copymaster 2 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 solutions. 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:
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 coordinate 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 consists 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 gives 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 for 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 3 provides some examples of partially revealed chocolate blocks. For each block ask:
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 5 provides students with further examples of visualising the masked array.
The Kapa Haka problem is designed around the array structure of seating arrangements for Kapa Haka performances at school.
Begin by role playing the Kapa Haka 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:
Use different arrangements of columns and rows.
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.
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.
Dear family and whānau,
This week we have been looking at arrays in class. Arrays have rows and columns like orchards. We have found it easy to figure out the number of things that are in arrays.
Ask your child to find some arrays in your home or neighbourhood. These could include a supermarket car park or an orchard, or an egg box, or the bus seats. Use these arrays to count the number of things in those arrays. You can also help your child to draw a sketch of the arrays showing the columns and rows.
Figure it Out Links
Some links from the Figure It Out series which you may find useful are:
Supermarket displays
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.
Patterns are an important part of mathematics. It is valuable to be able to recognise the relationships between things. This enhances understanding of how things are interrelated and allows us to make predictions.
Patterns also provide an introduction to algebra. The rules for simple patterns can be discovered in words and then written using more concise algebraic notation. There are two useful rules that we concentrate on here.
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.
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 in this unit can be adapted to recognise diversity and student interests to encourage engagement. For example:
Getting Started
Today we look at the number patterns in a tower of tins (tini).
Show the students the arrangement:
How many tins are in this arrangement?
How many tins will be in the next row (kapa)?
Then how many tins will there be altogether?
How did you work that out?
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.]
Show the students the next arrangement of tins. They can check that their predictions were correct.
How many tins will be in the next row?
Then how many tins will there be altogether?
How did you work that out?
“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."
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.
“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]
How many tins will you need altogether?
How are you keeping track of the numbers?
Do you know how many tins will be on the bottom row? How do you know?
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?
Exploring
Over the next 2-3 sessions the students work with a partner to investigate the patterns in other stacking problems. Consider pairing together students with mixed mathematical abilities (tuakana/teina). We suggest the following introduction to each problem.
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?
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?
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?
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?
Reflecting
In this session the students create their own “growth” pattern for others to solve.
Dear parents and whānau,
In maths this week we have been looking at patterns. Patterns are an important part of mathematics. It is always valuable to be able to recognise the relationships between things to help us see how things are interrelated and allow us to make predictions.
The patterns below are to do with buildings. We have been learning about how patterns like these can be continued. An important part of this has been learning to use tables to keep track of the pattern and the relationships between terms.
Ask your child if they can continue the pattern below and say what patterns they notice in the numbers. Can they draw or fill out a table to show how the pattern would progress? Can you work out how many crosses would be in the triangle with 15 crosses along the bottom?
Number of crosses high
Outlining area
This unit of work explores the measurement of area. Ākonga estimate and measure area using square centimetres.
When ākonga 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 ākonga 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.
The measurement experiences of ākonga must enable them to:
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. centimetre cubes 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. Ākonga should be encouraged to develop their own reference for a centimetre, for example, a fingernail or a pebble.
As ākonga become familiar with the size of the square centimetre they should be given many opportunities to estimate before using precise measurement. This could include using centimetre-squared paper to create different shapes of the same area.
Consider integrating real-world and cross-curricular examples, such as the area of grass seed or bark needed for a school garden, or the area of space available on a tapa cloth, or the area of new rakau sticks. You might also draw on the books you have been reading as a class to frame the learning in a new context (e.g. estimating and measuring the height of different characters or items from the story).
Te reo Māori vocabulary terms such as mehua (measure), horahunga (area) and mitarau pūrua (square centimetre) could be introduced in this unit and used throughout other mathematical learning.
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 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, shells following a trip to the beach, or pebbles found at your local awa. For the activities to work there needs to be a collection of objects, all with a range of areas around 120cm.
Session 1
We start this unit with a guessing game which introduces the idea of estimation. Consider how the mystery object you choose might reflect the learning interests and cultural make-up of your class.
What do you think that this could be the outline of?
How many cubes do you think I would need to cover this shape?
Give each student a 1cm 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
Lillie-Moana
Show the class a shape made with 5cm cubes, for example a rākau stick
Ask ākonga to record the shape on cm squared paper.
What is the area of this shape?
Why do you think it is called a square centimetre?
What is the area of this shape? (5 square centimetres or 5 square cm)
(These shapes are called pentominoes and there are 12 distinct shapes that can be made. Some ākonga may wish to explore this concept further).
Session 2
Session 3
Session 4
Session 5
Today we use the measurement skills we've been working on to find out who has the smallest and largest footprint in our class.
How could we find out?
About how many square centimetres do you think it would be? Why do you think that?
Dear family and whānau,
This week at school we have been measuring the area of objects using square centimetres.
At home this week your child is asked to figure out the area of largest and smallest footprints in your family using the attached square centimetre paper. They can draw and outline and then figure out the area. Encourage your child to estimate the area in square centimetres before working it out. We will use these to make a classroom display of big and little footprints.
Figure It Out Links
Some links from the Figure It Out series which you may find useful are:
Matariki - Level 2
This unit consists of mathematical learning, at Level 2 of the New Zealand Curriculum, focused 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 should anticipate how many counters remain. Students should use place value to calculate, as opposed to counting back. The use of a linear model for representing the calculations is recommended. A bead string is ideal and can be mounted along the edge of a whiteboard. Jumps can then 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 this 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:
In this diagram, 93 – 7 is calculated in two steps. First, the seven in 17 is imaged as 10 and 7. Three is subtracted from 93 to get to 90 and then four is taken away to get 86. Thus, the seven from 17 has been subtracted, leaving only the 10 to be taken away next. This is a ‘back through ten’ strategy, meaning it uses a decade number (i.e. 90) as a benchmark. Alternatively, a student might take away the ten in 17 first, and then the 7.
Session three involves dealing with "like" measures, e.g. dividing or multiplying weights. 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 need to be ordered for the hāngī. If possible, bring a few kūmara along so students can experiment and 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 in total for the hāngi. If 24 kūmara are needed and four kūmara weigh one kilogram then 26 (24 ÷ 4) kilograms of kūmara 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 shapes 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.
The learning opportunities in this unit can be differentiated by providing or removing support to students, or by varying the task requirements. Ways to support students include:
Tasks can be varied in many ways including:
Although the context of Matariki should be engaging, and relevant, for the majority of your learners, it may be appropriate to frame the learning in these sessions around another significant time of year (e.g. Chinese New Year, Samoan language week). This context offers opportunities to make links between home and school. Consider asking family and community members to help with the different lessons. For example, members of your local marae may be able to share local stories and traditions of matariki with your class.
Te reo Māori language is embedded throughout this unit. Relevant mathematical vocabulary that could be introduced in this unit and used throughout other learning include āhua (shape), shape names (e.g. whetū - star), tātai (calculate, calculation), tāpiri (add, addition), ine (measure), taumaha (weight), huri (rotate, rotation), whakaata (reflect, reflection), and neke (translate, translation, move)
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 at least 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 hāngī 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 the year are exhausting. It is said the steam of the first hāngī 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 hāngī.
“Preparing for the hāngī” is a Level 3 activity from the Figure It Out series.
“Hanging out for hāngī” is a unit at Level 3 that develops a statistical investigation around deciding which foods to cook.
The notes below are an adaptation more suitable for Level 2 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. 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 is commonly used to form lines for brickwork so is available at most hardware stores in a variety of colours. Nylon string is usually 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, depending on the version of the game that is played. A digital version of this game is available online - search for “Mū Tōrere - HEIHEI Games”.
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 off. 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.
Dear parents and whānau,
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 hāngī, 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.