Stage 4 - algebra - number patterns
Continue a number pattern to match a table of values; determine the rule which involves more than one operation to match a number pattern
Strategy
Students can:
- continue a number pattern to match a table of values
- determine the rule which involves more than one operation to match a number pattern
Activities to support the strategy
Activity 1
Provide students with a collection of matchsticks. Students build a sequence of squares with different side lengths using the matchsticks.
Model a sequence of squares with sides measuring 1 matchstick, 2 matchsticks, 3 matchsticks.
Students then make the next two squares of the pattern. The perimeter of a square is the distance all the way around. Create a table showing the perimeter of each of the squares.
| Length of one side of square (in matchsticks) | Perimeter of square (in matchsticks) |
|---|---|
| 1 | 4 |
| 2 | enter answer |
| 3 | enter answer |
| 4 | enter answer |
| 5 | enter answer |
Predict the perimeter of a square with sides of six matchsticks.
Discuss
- If the square has sides 8 matchsticks long, what is the perimeter? 10 matchsticks long? and so on.
- Students determine the rule to describe their matchstick pattern.
2. Use coloured counters to show various patterns for students to identify and describe.
Ask students to
- look at the pattern of counters and draw what they think the fifth and sixth shapes in the pattern would look like
- describe the patterns they can see
- develop an expression to show the number of counters needed for the nth shape
Activity 2
A variable is a symbol or letter which represents a number in an expression or equation. For example, "b" is a variable in the expression 3b + 5. This means, "b" can be equal to any number in this expression.
In an equation, variables can be independent or dependent. For example, in the equation c = 3b + 5, b is the independent variable (can be equal to any number) and c is the dependent variable (the value of c is determined once we know the value of b).
For example, anthropologists have developed a formula to determine the height from femur length. In cm, a man's height is given as
- height = 2.59 x femur length + 66.4
Using pronumerals, we can use f to stand for femur length and h to stand for the man's height. The formula may then be written as
- h = 2.59f + 66.4
The man's height depends on the length of the femur, so we say that f is the independent variable and h is the dependent variable. The formula is written with the dependent variable (h) as the subject.
1. Ask students to work through the following questions:
| No | Equation | Independent variable | Dependent variable |
|---|---|---|---|
| 1 | y = 3x | enter answer | enter answer |
| 2 | h = 3k + 2 | enter answer | enter answer |
| 3 | v = u + 5t | enter answer | enter answer |
| 4 | p = 3q - 2 | enter answer | enter answer |
| 5 | t = 3x2 + 2x - 5 | enter answer | enter answer |
| 6 | x + 3y = 4 | enter answer | enter answer |
| 7 | 3x + 2y = 9 | enter answer | enter answer |
| 8 | 3x + 2y - 9 = 0 | enter answer | enter answer |
| 9 | x - 3y - 5 = 0 | enter answer | enter answer |
| 10 | 12m = 4n + 6 | enter answer | enter answer |
Identifying the Independent and Dependent variables in an equation (PDF 104.74KB)
Discuss the answers as a class, with particular emphasis on questions 6 -10.
2. As a class, determine the rule giving the relationship between x and y in each of the following and write the corresponding equation.
| x | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| y | -2 | 0 | 2 | 4 | 6 | 8 |
The rule is y is always double x.
The equation is y = 2x
| a | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| b | -4 | 0 | 4 | 8 | 12 | 16 |
The rule is __________________________
The equation is ______________________
3. Working in pairs, students determine the rule and write the equation for each of the following relationships.
| x | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| y | 3 | 5 | 7 | 9 | 11 | 13 |
- When x = 0, y = 5 = 0 + 5 = 2 x 0 + 5
- When x = 1, y = 7 = 2 + 5 = 2 x 1 + 5
- When x = 2, y = 9 = 4 + 5 = 2 x 2 + 5
- So the equation is y = 2 x + 5
| a | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| b | -1 | 3 | 7 | 11 | 15 | 19 |
- When a = ______ b = ______________
- When a = ______ b = ______________
- When a = ______ b = ______________
- So the equation is _________________
Determine the relationship and write the equation worksheet 1 (PDF 40.61KB)
Determine the rule and write the equation worksheet 2 (PDF 40.92KB)
References
Australian curriculum
ACMNA175: Introduce the concept of variables as a way of representing numbers using letters.
NSW syllabus
MA4-10NA: Uses algebraic techniques to solve simple linear and quadratic equations.