In this unit we work on word problems about cars and people, fish and fish bowls, and tables and table legs. We learn about different types of problems and modelling in a variety of ways. We link these to the commutative properties of multiplication (though we don’t expect the students to use these words for them).
AO elaboration and other teaching resources
- pose different types of word problems
- explain their mathematical thinking in solving problems
- use a variety of equipment to model their solutions
The basic concept of multiplication is an important one because of its practicality (how much do 4 ice creams cost at $2 each) and efficiency (it is quicker to determine 4 x 2 than 2 + 2 + 2 + 2). But multiplication is used in a number of situations. Here we think about multiplication as repeated addition and in rate problems, comparison problems and array problems. This latter type of problem helps students to see the commutative property of numbers, namely that 4 x 3 = 3 x 4. In other words, the order of the factors does not affect the outcome in the multiplication of two or more numbers.
As well as thinking about multiplication in a variety of situations, we also encourage students to use a variety of materials to solve the various problems. Both of these give the students a broader experience and hence aid their overall understanding.
At this Level, the emphasis of course is on the multiplication of number. However, further on, in Algebra, students will find a use for multiplying numbers and unknowns (things like 3p) and even unknowns and unknowns (things like t x t = t2). These turn out to be invaluable in solving a great range of problems.
But there are other things that can be ‘multiplied’. Matrices are an example of this. They provide quite a different concept of multiplication. Despite the very abstract nature of this multiplication, it does have important applications in areas such as 3-and higher-dimensional geometry.
- We introduce the session by asking the students to work through several equivalent group [set] problems first and then ask them to pose their own problems. For example:
There are 3 cars. Each one has 2 people in it. How many people are there altogether?
There are 6 fish bowls. Each bowl contains 4 goldfish. How many goldfish are there altogether?
There are 7 tables. Each table has 4 legs. How many legs altogether?
- The students can model these and similar types of problems with:
- towers of interlocking cubes
- threading beads
- jumps on the number line
- interlocking cubes on a number track
- draw a picture to show the number of tables and the corresponding number of legs
- Note: It is important to link the idea of environmental examples (where possible) of equivalent sets with the idea of multiplication as repeated addition. As well as modelling with equipment, students should write the same equation using repeated addition and using multiplication. For example: 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28 or 7 x 4 = 28.
- Now ask the students to make up word problems using the problem structure above with different answers. For example write a problem with an answer of 24.
- Now use several sets of ice-cream containers (all with the same number of items in them) with the contents of each covered except for one. Ask the students to write problems for each example. What strategy did the students use to solve this problem? For example did they try to count the contents of each ice-cream container; that is those that are visible and those that are concealed?
Over the next 3 days the students pose a number of different types of story problems. They are encouraged to model the problems using different types of equipment and explain their answers to others. They will begin to think about the most efficient ways of solving the problems. It is important that students are provided with opportunities to build up multiplication facts to 10 and then to 20. Some students may solve these problems without equipment.
- On the first day work through several rate problems. These might be equivalent to some of the problems in the previous lesson but expressed as rate or ratio problems. Do a few problems of this type with the students first and then ask them to pose their own problems.
If you need 1 car for 2 people, how many cars will you need for 6 people?
If you need 1 fish bowl for 4 goldfish, how many bowls will you need for 24 goldfish?
If each table has 4 legs, how many tables are there if there are 28 legs?
- The students can model these and similar types of problems with pictures
- Now ask the students to make up word problems using the problem structure above and to pose these problems to each other. Encourage them to explain their answers to each other.
- On the second day work through several comparison problems, where possible using a context familiar to the students. For example:
Susie has 4 crackers in her lunchbox, and Rima has 3 times as many crackers as Susie. How many crackers does Rima have in her lunchbox?
- On the third day work through several array problems based on situations in which there are equivalent groups. Teams with the same number of members in each are often used during the school day. Pose problems such as:
The students are lined up in 3 teams. Each team has 6 members. How many students are there altogether?
Encourage the students to draw problems like this using three rows (one for each team) and six columns (one for each team member).
- Note: When modelling arrays it may be helpful to talk about the lines across as ‘rows’ and the lines up and down as ‘columns’. Arrays are useful for illustrating the commutative property of multiplication. There are many everyday objects that are examples of arrays. These include egg cartons, bars of chocolate, and buttons on clothing such as double breasted jackets. These might also provide opportunities to count in twos and practice the doubling strategy.
- The students can model these and other array problems with
- pegboards: pegs on a pegboard can be used to illustrate arrays in multiplication.
For the problem above this could be talked about as 3 rows of 6 pegs or 3 sixes,
or 3 rows of 6, or 3 x 6 = 18
By turning the pegboard a quarter turn, the array still has a total of 18 pegs;
This could be talked about as 6 columns of 3 pegs or 3 rows of 6 pegs or 6 threes
or 6 columns of 3 or 6 X 3 + 18.
- interlocking cubes;
- threading beads;
- jumps on the number line;
- interlocking cubes on a number track.
On the final day of the unit we play a game in which the students use graph paper to plot arrays.
For this activity you will need
- a sheet of graph paper for each player;
- 2 ten sided dice - each side has a different digit on it
- different coloured felts
How to play
roll both dice, for example 8 and 2;
mark out a rectangle of that size, for example 8 rows of 2, or 2 columns of 8;
write the multiplication basic fact in the rectangle, for example 8 x 2 = 16 or 2 x 8 = 16;
roll both dice again and mark out a new rectangle in a different colour according to the 2 numbers shown on the dice;
keep doing this until you fill the graph paper.