This activity gives students opportunities to use a range of multiplicative strategies when finding the capacity (volume) of prisms, in this case using non-standard units.
An important concept introduced in this activity is that to find the capacity of a prism, you can measure what will fit in one layer and then multiply this by the number of layers.
Although the word “prism” is not used in the activity, this is a good opportunity to introduce the term. This may be the first time that many of your students have met the word prism, at least in a mathematical context. If this is the case, discuss its meaning with them. Ask Apart from being full of buttons or shells, what do all the (5) containers pictured in questions 1 and 2 have in common?
(They all have completely regular cross sections. For example, if you “sliced” container 1c anywhere parallel to the base, you would reveal a triangle identical to the ones that form its top and bottom. It is this quality that makes the containers prisms.) If the students have trouble seeing where the question is leading, introduce some containers that are not prisms (for example, bottles that are spherical or tapering at the top) and ask how these containers are different from the ones pictured.
There are lots of opportunities in this activity to discuss the merits of knowing a range of different multiplicative strategies. See the notes for Bean Counters (page 21) for reasons.
Before the students start the activity, you could remind them of the multiplicative strategies that they already know. Ask How many different ways can you solve 5 x 28 =  ? Expect responses such as:
• doubling and halving: 5 x 28 = 10 x 14 = 140
• using a tidy number and compensating: 5 x 28 = (5 x 30) – (5 x 2) = 150 – 10 = 140
• using place value partitioning: 5 x 28 = (5 x 20) + (5 x 8) = 100 + 40 = 140.

Throughout the activity, ask questions such as:
• Before you start working out the problem, look at the numbers and decide which strategy is likely to be most efficient for this particular problem. Could you explain why to a classmate?
• Share your solution path with the group. Did someone else use a more efficient method? What is there about the problem that suggested the use of this strategy?
Extension investigation
How could you use Matilda’s method to measure the volume of a square or rectangular prism in square centimetres? Measure a box and explain what you did.

Answers to Activity

1. Answers will vary (you may count different numbers of buttons in each layer than other classmates).
Possible answers include:
a. About 100 buttons. (5 x 2 x 10)
b. About 300 buttons. (The bottle shape is triangular, with about 5 or 6 buttons on each of the front two sides. If this was a rectangle, this would be about 5 x 6 = 30 buttons. A triangle is half a rectangle, so 30 ÷ 2 = 15 for the top layer. There are about 20 layers, so 20 x 15 = 10 x 30 = 300.)
c. About 120 buttons. (There are about 6 x 4 x 5 buttons. 6 x 4 = 24. 24 x 5 = 12 x 10 = 120)
d. About 252 buttons. (There are about 8 buttons showing on the top layer, with probably 4 hidden at the back of the circle, and 21 layers.
12 x 21 = 10 x 21 + 2 x 21 = 210 + 42 = 252)
2. Yes, as long as the shells of different shapes and sizes are evenly mixed throughout the jar. 
 

The questions in this activity are all designed to help students develop their estimation skills using a range of data and taking into account various possibilities. Estimation strategies with respect to number include rounding up or down to reasonably close numbers that can be manipulated mentally. The estimates required in this activity involve multiplication and division. The activity is therefore suitable for students at the advanced additive stage or beyond of the Number Framework.
You could discuss with the students when and why they might estimate rather than calculate exactly.  Estimating is useful when an exact calculation isn’t required or, as in the case of the food for the school disco in question 1, when it’s not possible to calculate exactly because there is no exact data to work with. (The two students don’t know how many people will come to the disco or how much they will eat and drink.) Estimating is often quicker than calculating, especially when you don’t have a calculator handy. You could ask the students to compile a class list of situations in
which estimating is useful.
See the Answers section for the thinking behind the possible answers for question 1.
Question 2 is a problem that can be tackled initially in a similar way to finding the area of a rectangle, that is, by multiplying the number of rows (if you like, the length) by the approximate number of people in each row (if you like, the width). In this case, it means multiplying 15 by 18. This can be calculated mentally in several different ways, for example, (10 x 18) + 5 x (20 – 2), which is 180 + 100 – 10, giving 270. A different strategy is given in the Answers. Another possible strategy
is to use doubling and halving (15 x 18 = 30 x 9), then known multiplication facts (3 x 9) and multiplication by 10. Obviously, several valid estimates are possible for this problem, depending on how the data is interpreted.
The students could use a variety of advanced multiplicative part–whole strategies to estimate the number of words on a page in question 3. For example, if the page had 30 lines with an average of 17 words per line, they could use:
• standard place value: 30 x 17 = (30 x 10) + (30 x 7)
= 300 + 210
= 510
• tidy numbers: 30 x 17 = (30 x 20) – (30 x 3)
= 600 – 90
= 510
or 30 x 17 = (30 x 15) + (30 x 2)
= 450 + 60
= 510
 

Some possible estimation strategies for questions 4a–d are given in the Answers. The students could discuss other strategies that they think of with a classmate.
In question 5b, the students are asked to collect data and compare their own estimates with a classmate’s. There are various reasons why estimates can vary. For example, different students could well read the same School Journal story at different rates and thus arrive at different estimates, all of which could be valid.

Answers to Activity

1. Answers will vary. For example: 11 classes of 27 is approximately 10 classes of 30 students = 300. If 75% come, that’s 3/4 of 300; 1/2 would be 150, and 1/4 must be 75, so 3/4 would be 225. Most would have a drink and chips, so they’ll
need about 225 of each. 200 is 10 lots of 20, so 225 is 11 or 12 boxes of chips. 24 is about 25, so that’s 9 or 10 cartons of drinks. (It’s probably better to go for the higher figure in each case.)
2. Answers will vary. For example, 10 rows is about 180, and 5 rows is about 90, so that’s about 270. Add 30 for the one or more at the ends of rows, so that’s now about 300. Not all the rows were full, but there could have been 10 standing by
the door. So the total estimate could be between 300 and 315.
3. Answers will vary.
4. Answers will vary. For example:
a. Multiply the average number of people in each row by the number of rows.
b. Count the number of entries in a column. Multiply by the number of columns in a page and then multiply by the number of pages in the phone book.
c. Count the number of cars going past in 1 minute during the rush hour of 8 a.m. to
9 a.m. and multiply this number by 60 to get the number that could go by in the hour.
Then count the number of cars going past in 1 minute after 9 a.m. and extrapolate this
for the rest of the school day. The number of cars between 3 p.m. and 4 p.m. could
increase significantly, so you may decide to regard this period as another rush hour.
d. Time how long it takes to read 1 page with a reasonable number of words on it. Multiply this by the number of pages.
5. Answers will vary.

Using Materials

Provide the students with a copy of the dotty array. If you are using laminated copies, whiteboard pens are very practical.

Write 23 x 37 on the board or modelling book and ask the students to draw a border around the array that represents it. Tell them to partition the array in any way they  want that will make it easy to find the total number of dots.

Students are likely to use a variety of partitions including use of place value:

Discuss how the partitions relate to each other in that 3 x 37 = (3 x 30) + (3 x 7) and 20 x 37 = (20 x 30) + (20 x 7).

A key point is that the multiplication involves the idea of a cross-product. 23 x 37 can be calculated in this way, with each arrow representing a  separate multiplication.

Note that students may suggest valuable strategies for 23 x 37 that are efficient but not transferable to more complex examples, for example, 23 x 37 = (25 x 37) – (2 x 37) = 925 – 74 = 851, or 23 x 37 = (23 x 40) – (23 x 3) = 920 – 69 = 851.

Give the students other problems that can be modelled by partitioning the array.  Good examples are: 42 x 15 = 630,  26 x 24 = 624,  18 x 33 = 594,  45 x 35 = 1 575

Ensure the students have established a clear idea of the cross-product idea before proceeding to imaging.

Using Imaging and Number Properties

Provide further similar examples and ask the students to image the array and record the products that need to be calculated to ease memory load. Draw a border around the required array and hide it.

28 x 17 = 476,  19 x 43 = 817,  34 x 22 = 748,  40 x 34 = 1 360

Generalise the cross-product to include three-digit factors.

173 x 26 = (100 x 20) + (100 x 6) + (70 x 20) + (70 x 6) + (3 x 20) + (3 x 6) =
2 000 + 600 + 1 400 + 420 + 60 + 18
= 4 498

Prior knowledge

Background

This activity uses place value partitioning to solve multiplication problems. It uses the area model. This activity involves whole numbers and decimals in the problems. While the drawings involve using an area model to display the multiplications, (so the size of each box is in some form of proportion to the size of the numbers), it should also be noted that students can do the same problems using a table (or tabular form). In this form area is not considered.

For example: 14 x 51 can be done by using the following table (where all the boxes are the same size).

Comments on the Exercises

Exercise 1
Asks students to solve 2 digit by 1 digit multiplication using the area model to show place value multiplication. It is designed to follow on from a teaching episode in which the strategy is illustrated and used by the students. Once students have convinced themselves that the strategy works with a range of simple numbers they are ready to explore more difficult numbers including decimals.

Exercise 5
Asks students to solve multiplication problems with whole numbers between 10 and 100 for both factors.

Exercise 6
Asks students to solve multiplication problems where one of the factors is a one place decimal and the other factor is a whole number between 10 and 100.

Exercise 7
Asks students to solve multiplication problems where both factors are one decimal place decimals, using numbers less than 10 until the last question which is greater than 10.

Exercise 8
Asks students to find unknowns in problems that involve two decimals to 1dp.

Exercise 9
Asks students to solve word problems. Students may need to be encouraged to use units appropriately.

Exercise 10
Asks students to investigate the use of the strategy for numbers in the 100s

Exercise 11
This exercise introduces x as a length added to both sides of the rectangle and requires students to multiply the algebra and add like terms.

Prior knowledge.

Recall multiplication facts
Use the rounding and compensating strategy when multiplying whole numbers
Use the place value strategy to solve problems involving whole numbers

Background

in this activity students who can use place value and compensating with whole numbers learn to use these strategies when multiplying decimals.

Comments on the Exercises

Exercise 1
Asks students to solve a basic fact and then use compensating to solve a multiplication problem involving a one decimal place decimal.  For example, 3 x 5 = 15 so 3 x 4.9 = 14.7, 3 x 4.8 = 14.4, 3 x 5.1 = 15.3, 3 x 5.2 = 15.6

Exercise 2
Asks students to solve multiplication problems involving decimals by using compensating. For example 3 x 6.8 = (3 x 7 = 21) - (3 x 0.2) = 19.4

Exercise 3
Asks students to solve multiplication problems involving decimals by using place value strategies.  For example, 2 x 4.2 = 8.4.

Exercise 4
Asks students to solve multiplication problems involving decimals by using place value strategies but the problems involve adding the tenths to the whole number.  For example, 4 x 3.4 = 12 + 1.6 = 13.6.

Exercise 5
Asks students to solve multiplication problems involving decimals by choosing the most appropriate strategy.