I have been teaching mathematics in an Australian High School since 1982, and I am a contributing author to mathematics text books.
At school, students have access to specialised instructors and resources, all within an environment whose ambience promotes learning. The scenario, however, is different in a home setting. Mathematics homework is largely consolidation of what was covered in the classroom, usually via completion of prescribed exercises from the textbook.
To complement this learning regime, parents can make a significant contribution by accommodating learning using concrete aids. Outlined below are some important mathematics concepts children can explore as a group or by utilising items commonly found in the home.
Most people will be familiar with the 2-player game paper-scissors-rock. On the count of 1-2-3, each player uses their hand to represent paper (open palm), scissors (two fingers) or a rock (closed fist).
The object of the game is to win by showing a ‘winning hand’.
- Paper beats rock (paper can wrap a rock)
- Scissors beats paper (scissors can cut paper).
- Rock beats scissors (a rock can break scissors)
In the mathematics study of probability, the game is said to be fair or unbiased because each player has the same chance of winning. This can be shown by listing all possible outcomes of the game and calculating the probability of each winning combination.
There are nine different outcomes. Three of the nine outcomes favour player 1, so the chance of player 1 winning is 3/9 or 1/3. Similarly, three outcomes favour player 2, so the chance of player 2 winning is 3/9 or 1/3. Finally, there are three outcomes that lead to a draw, with probability 3/9 = 1/3. These are known as theoretical probabilities.
Play the Game
Play the game ten times with a parent, a brother or a sister. Before you start, choose who is to be player 1 and who will be player 2. As you play, fill in the table below.
When you have completed the table, calculate the probability of each player winning. This is known as the experimental probability. For example, if player 1 won 4 games, then his/her chance of winning is 4/10.
Now compare your results with the theoretical probabilities.
The game can be simulated using two dice.
Each player uses one die and assigns pairs of numbers to represent paper, scissors and rock.
player 1: 1/2 paper 3/4 scissors 5/6 rock player 2: 1/6 paper 2/4 scissors 3/5 rock
The dice are rolled simultaneously and the outcomes recorded. Thus, the following roll of the dice will represent a win for player 2.
player 1: 3 (scissors) player 2: 5 (rock)
Extension: 3 Players
The game can be modified to include 3 players, although it will be biased because each player does not have the same chance of winning. The rules for winning are as follows.
Player 1 wins if two hand signs are the same, player 2 wins if all three hand signs are different and player 3 wins if all three hand signs are the same.
It turns out that the chance of each player winning a game is:
Player 1: 18/27 Player 2: 6/27 Player 3: 3/27
This means that player 1 is the favourite to win. Their chance of winning is three times better than player 2 and six times better than player 3.
To obtain these theoretical probabilities, the parent can guide the players to complete a table that lists all 27 possible outcomes.
The game can also be simulated using three dice, with assignations as described in the two player simulation.
Activity: Create Square Numbers
A square number is the result of a whole number multiplied with itself. The first 10 square numbers are
Square numbers can be built up using the first square number, 1.
What You Need
Marbles, small lollies such as jelly beans, or counters of different colours (colours are optional). Alternatively, use small coins or draw and cut out small circles and colour them.
What to Do
Place one shape on the table. This represents a 1 x 1 square.
Now add shapes of another colour to create a 2 x 2 square, then add more shapes of another colour to create a 3 x 3 square, and so on to as many squares that you can create with the shapes you have.
The results will be similar to the following.
By noting how many shapes are added each time, a sequence is obtained.
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36
Parents, prompt your child to reach the generalisation that the sum of N odd numbers is N2.
Activity: Create Triangular Numbers
In a similar way to square numbers, a sequence of patterns can be created to illustrate triangular numbers.
What You Need
The same resource as that used to create square numbers.
What to Do
Determine the number of shapes required to add to each triangle to create the next triangle.
The sequence of numbers 1, 3, 6, 10, 15, 21, .. are triangular numbers.
Observe that the sequence of numbers formed by consecutive pairs of triangular numbers is 2, 3, 4, 5, 6, .. This allows more triangular numbers to be found without physically creating the shape. For example, the seventh triangular number is 21 + 7 = 28 and the eighth triangular number is 28 + 8 = 36.
Parents, please ask your child to verify that the number of shapes needed to make the Nth triangular number is N(N + 1)/2. For example, for the 6th triangular number, use N = 6 to get 6(6 + 1)/2 = 21.
Activity: Intersecting Lines
This activity involves a trial and error approach to maximise the number of vertices (points of intersection) obtained from intersecting straight lines.
What You Need
Bamboo skewers, long matchsticks or straws.
An interesting sequence of numbers is formed using the number of vertices created by intersecting lines. For instance, two intersecting lines produce one vertex and three intersecting lines can produce one, two or two vertices.
What to Do
Parents, please guide your child to complete table 1 below. The answers are given in table 2.
Note that using six lines is particularly challenging!
Have your child compare and comment on the number of vertices in column 2 with the triangular numbers found in the previous activity.
As a special challenge, ask for the solution using seven lines. The solution can be found by trial and error using seven straws or by noticing from the triangular numbers that 1 +2 = 3, 3 + 3 = 6, 6 + 4 = 10 and 10 + 5 = 15. This will mean for seven lines the number of vertices will be 15 + 6 = 21.
Activity: Area of a Circle
This activity assumes knowledge of the formula for the circumference of a circle.
What You Need
Paper, scissors, ruler, protractor (optional), colour pencils, compass or a flat round object to draw a circle.
What to Do
On the sheet of paper, use the compass or the base of your round object to draw a circle with radius approximately 2 inches (5 centimetres).
Use the protractor to divide the circle into 12 sectors. The angle between the two radii of each sector will be 300. Otherwise, the angle can be approximated without the use of a protractor.
Colour six sectors with one colour and the other six sectors with a different colour.
Cut out each sector and arrange them as follows.
The shape is approximately a rectangle. Ask for the dimensions (shown), in terms of r (the radius of the circle).
The reasoning is that arcs 1, 3, 5, 7, 9, 11 approximate to the length of the rectangle and are half the circumference of the circle. The width of the rectangle is approximately the radius of the circle.
Now ask how the formula for the area of a circle is obtained from the area of the rectangle. This should lead to the observation that the area of a circle is approximately the same as the area of the rectangle.
Since the area of a rectangle is length x width, then we have
Finally, ask how the approximation can be made more accurate. This will inspire the comment that using many more sectors will make the arranged shape look much more like a rectangle. This brings in the concept of the ‘infinite limit’. This means that the exact formula for the area of a circle is achieved when the number of sectors in the circle approaches infinity.
Demonstrating interest in your child’s mathematics education goes a long way to enhance their ability.
Reap a bountiful harvest with the above investigations!