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.
solve multiplication problems by using strategies other than counting all (counting both sets from one in an addition or subtraction problem);
interpret and solve multiplication story problems.
In this unit the students use arrays to solve multiplication problems. First of all arrays are just a way of neatly lining up objects in rows and columns. Orchards are usually examples of arrays where the fruit trees are grown in rows and columns to make them easier to look after and easier to pick the fruit.
Arrays provide a quick and efficient way to count things. For example, this can be done by adding the numbers in each row together. However, it is quickest to determine the number of objects in an array by multiplying the number of rows by the number of columns. So this current unit is an introduction to multiplication.
But this is not the only use for arrays. Spreadsheets are becoming ever more common these days. Spreadsheets are just arrays. So arrays provide a convenient and easily understood way to present data. In parts of geometry, where we want to see the effect of certain transformations on objects, ways have been devised of multiplying arrays. These arrays are called matrices and are very powerful tools.
As an important side issue to this unit we ask the students to be involved in guessing and estimating. These are both useful skills that take time to develop. This unit provides some practice in this area.
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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Getting started
- 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 apples does Jack have to prune altogether? - 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 use an overhead projector with counters so it’s easy for all to see what the first row of apple trees will look like.
What will the second row then look like?
It’s important for students to understand what a row is so they can make sense of the problem. - 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 like this. (Fold in half one way and then in half the other way.)
- Then allow some time for each group to see if they can come up with 4 different ways to solve the Orchard Problem and record their methods in the 4 boxes.
Allow students to use equipment if they think it will help them solve the problem.
The teacher roves round the class and challenges their thinking.
Are there quicker ways of working out how many trees there are without having to count every one? - 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.
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Likely strategies |
Possible teacher responses |
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 …. |
Can you think of a quicker way to work out how many trees there are? How many trees are there in one row? |
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6,12,18,24,30,36 |
Do you know what 6 + 6 =? Can that help you solve this problem quicker? |
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6 x 6 = 36 |
What if Jake had 9 rows of trees and there were 12 trees in each row? |
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6 + 6 = 12; 12 + 12 = 24; 24 + 12 = 36 |
You used adding to work that out. How could you have used your times tables? |
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2 x 6 = 12; 12 + 12 + 12 = 36 |
If 2 x 6 = 12, what does 3 x 6 =? How could you work out 6 x 6 from this? |
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3 x 6 = 18 and then doubled it |
Awesome, clever you, could you work out 9 rows of 6 for me? |
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5 x 6 = 30; and 6 more = 36 |
The shared strategies can be put into similar groups.
Who used a strategy like this one?
- The teacher then shows students copymaster 1 – an array of trees with 5 groupings. How do you think we could use this to solve the Orchard Problem? Send groups off to experiment with Copymaster 1. Again move round the class and observe what the groups are doing.
- As a class share the different ways that students used the array to solve the Orchard Problem. These can be modelled on a large array or on an OHT.
Challenge the students to use the array in the same way as another group did and pose the following problems.
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Jake plants 2 more rows of 6 trees. How many rows does he have now? How many trees is that altogether?
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Jake’s sister Mary has an orchard exactly the same size with the same amount of trees. How many trees do they have altogether?
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Exploring
Before the students work on the stations of Copymaster 3, you will need to prepare the material from Copymasters 1, 2 and 4.
The next three days will involve working on similar problems. Allow students the opportunity to opt into ‘clinics’ if they are finding the problems challenging and are unable to get started independently. This is an opportunity for the teacher to go over more examples and clarify any misunderstandings.
Begin by role playing the Air New Zealand 737 problem from copymaster 3 to get the students started.
Ask 6 students to pretend to be the passengers on the plane. Ask them to bring their chairs to the front and sit in a row. Leave a gap in the middle to show where the aisle is. Then choose a second lot of six students to make a second row behind the first row of 6 students.
Who thinks they can come up with a fast way to work out how many students there are altogether, without having to count them all?
Ask students to think about:
How many passengers there would be if there were 4 rows of 6?
What about if there were 9 rows?
Encourage students to share and demonstrate their strategies.
Enlarge the problem cards from copymaster 3 and have them placed at each station with copymasters 1 and 2 and equipment including counters and the 5’s abacus.
Read the problems to the class one at a time to clarify any misunderstandings. |
Explain the recording sheet (copymaster 4) and how it works e.g. that as students have finished a station activity they complete the tracking box and continue to the next problem. Encourage students to show their working and solutions on this sheet.
Then set the class off in pairs to one of the 4 stations shown in copymaster 3 - Orange Orchard, Kiwi fruit, Strawberry patch, Flying 737.
Then spend the session roving around the stations and questioning students.
How many did you think there would be to start with?
Why did you predict that to start with?
Can you think of another way to use the array to solve that problem?
Can you think of a way to solve the problem without using the array?
Which way do you think is faster, why?
Station 1
Have counters available for students to use and the arrays of copymaster 1 and 2.
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Orange Orchard Frederick has an orange orchard with 6 trees in each row and there are 8 rows of trees. How many trees does he have altogether Prediction Solution |
Orange Orchard Frederick wants to plant another field of orange trees. He can get 24 trees at a cheap price. How many different ways can he plant this trees. How many rows should he plant? How many trees should be in each row? Show all the different ways Frederick could plant his trees in rows by making your own arrays on a blank piece of paper. |
Station 2
Students may need to use equipment to do the second part of the Kiwifruit problem. Allow students to manipulate the counters to make arrays.
For Part 2 of the Kiwifruit problem, give students the opportunity to take a piece of chalk outside and draw on the pavement to represent Toby’s son’s choices for arranging his kiwifruit plants.
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Kiwifruit (Part 1) Toby has a kiwifruit orchard. He has 4 kiwi fruit plants in each row and 4 rows. He has a total of 3 fields the same size. How many kiwi fruit plants are there on his orchard? Prediction Solution |
Kiwifruit (Part 2) Toby’s son has just brought the plants to start his own Kiwi fruit orchard. He has purchased 36 plants and wants some help on how to arrange the plants into rows. What suggestions would you give Toby’s son? Draw all the possibilities. |
Station 3
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Strawberry Patch (Part 1) Hera has a strawberry patch. In each row she has 12 strawberry plants and there are 4 rows. How many strawberry plants does she have altogether. Prediction Solution |
Strawberry Patch (Part 2) Hera wants to double the size of her strawberry patch each year. How many plants will she have altogether afte 1 year 2 years 3 years Record your strategies on an array |
Station 4
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Let’s fly away (Part 1) Air New Zealand’s 737 plane has 6 seats in each row. The seat row numbers go up to 21. How many passengers can the 737 plane hold? Prediction Solution
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Lets fly away (Part 2) Air New Zealand’s 737 planes make 2 trips to Wellington each day from Dunedin. If the plane was full each time. How many passengers would Air New Zealand take to Wellington each day? Prediction Solution
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At the end of each session allow for a sharing time to discuss what students were finding interesting and challenging. Share some strategies used.
Reflecting
On the final day of the unit ask the students to make up their own multiplication problems for their partner to solve.
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Tell the students that they are to pretend to be tomato growers. They are to decide how many rows of tomato plants they want in each row and how many rows they will have altogether.
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Now they challenge their partner to see if the partner can work out how many tomato plants they will have altogether.
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Tell the students to make up 2 more problems for their partner. Suggest that they solve the problem using an array Copymaster.
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Conclude the session by talking about the types of problems we have explored and solved over the week. Tell them that the problems were multiplication problems. Let them know that there are many ways of solving these problems and that arrays are just one of these ways.
Dear Family
This week we have been looking at arrays in class. Arrays have rows and columns like orchards. We have found it easy to count things that are in arrays.
Find some arrays in a supermarket car park or in an orchard near you. Use these arrays to count the number of cars or trees in those arrays.
| Attachment | Size |
|---|---|
| ArraysHoorayCM1.pdf | 59.12 KB |
| ArraysHoorayCM3.pdf | 91.61 KB |
| ArraysHoorayCM4.pdf | 66.31 KB |
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