Math 7 · Patterns and expressions
From a pattern, to a rule you can use
A pattern that grows by the same amount at every step can be written as a rule with a variable in it. The rule, a table of values and a graph are three views of the same thing. This unit moves between them, and sorts out the difference between an expression, which you work out, and an equation, which you solve.
- 1. The words, first
- 2. From a pattern to a rule
- 3. Tables of values and graphs
- 4. Expressions, equations and evaluating
- 5. What costs marks
The words, first
The idea: Expression and equation sound alike. The difference is one symbol: the equals sign.
| Word | What it means |
|---|---|
| Variable | A letter standing for a number that can change, like n for the number of tables. |
| Expression | Numbers, variables and operations with no equals sign, such as 2n + 2. |
| Equation | Two expressions with an equals sign between them, such as 2n + 2 = 30. It is only true for some values of n. |
| Coefficient | The number multiplying a variable: 2 in 2n. 2n means 2 × n. |
| Constant term | A number on its own, with no variable: the last 2 in 2n + 2. |
| Linear relation | A rule where the output changes by the same amount each time the input goes up by 1. Its graph is a straight line. |
| Table of values | Matching inputs and outputs in two rows or columns. |
| Ordered pair | An input and its output as a point: (1, 20) means 1 across, 20 up. Across comes first. |
| Evaluate | Replace the variable with a number and work out the value. |
From a pattern to a rule
The idea: How much the pattern grows at each step is the number that multiplies the variable. Then add or subtract whatever makes the first case come out right.
Worked example. Square tables are pushed together in a row, with one person at each open side.
| Tables, n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| People, p | 4 | 6 | 8 | 10 |
- Find the step. Each new table adds 2 people, one on each long side. So the rule contains 2n.
- Fix the start. For 1 table, 2n is 2, but 4 people fit. Add 2: p = 2n + 2. The extra 2 are the people at the ends, who are there however long the row gets.
- Check another case. n = 3 gives 2 × 3 + 2 = 8 ✓.
Use it. 15 tables seat 2 × 15 + 2 = 32 people, with no drawing.
From words. A pool charges $4 to get in and $2 for each waterslide ride: c = 2r + 4. “For each” points to the coefficient; a one-time charge is the constant term.
Tables of values and graphs
The idea: Make a table from the rule, plot each pair as a point, and check that the points line up. Then the graph answers questions the table never listed.
Worked example. Leo has $15 saved and adds $5 every week, so after w weeks he has m = 5w + 15 dollars.
| Weeks, w | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Money, m ($) | 15 | 20 | 25 | 30 | 35 |
- Plot it: weeks across the horizontal axis, money up the vertical one. Each column is an ordered pair: (0, 15), (1, 20), and so on.
- The points make a straight line, because every week adds the same $5.
- Read beyond the table. When will he have $50? From 50 on the vertical axis, go across to the line, then down: 7 weeks. Check: 5 × 7 + 15 = 50 ✓.
Is a table linear? Look at the steps. Outputs of 3, 6, 9, 12 go up 3 every time, so yes. Outputs of 1, 2, 4, 8 go up 1, then 2, then 4, so no — those points would curve.
Expressions, equations and evaluating
The idea: An expression is a recipe for a number. An equation says two things are equal. To evaluate an expression, substitute the value, then follow the order of operations.
| In words | As an expression |
|---|---|
| a number plus 6 | n + 6 |
| 7 less than a number | n − 7 |
| triple a number | 3n |
| a number divided by 4 | n/4 |
| 5 more than twice a number | 2n + 5 |
“Less than” turns the order round. 7 less than a number starts with the number and takes 7 away: n − 7, not 7 − n.
Worked example. Evaluate 4x − 3 when x = 5. 4x means 4 × x, so it is 4 × 5 − 3 = 20 − 3 = 17. When x = 2.5 it is 4 × 2.5 − 3 = 10 − 3 = 7. Writing the 5 straight after the 4 and reading 45 is the classic slip.
Expression or equation? 2n + 5 has no single value until someone chooses n. 2n + 5 = 17 makes a claim, and it is only true when n = 6: 2 × 6 + 5 = 17 ✓. You evaluate an expression; you solve an equation.
What costs marks
The idea: Testing a rule on the first case catches the first one every time.
- Forgetting the constant term. p = 2n says one table seats 2 people. It seats 4.
- Swapping an ordered pair. (1, 20) is 1 across, then 20 up.
- Reading 4x as 45 when x = 5. It is 4 × 5 = 20.
- Writing “7 less than n” as 7 − n.
- Calling 2n + 5 an equation. With no equals sign, it is an expression.