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Algebra · Patterns and Relations · Lesson 2

Describing the Rule of a Relation in Words

A rule is only useful if you can tell someone else what it is. In this lesson you will learn the language mathematicians use to describe relations — one more than, twice, thrice, one more than twice — and use it to describe tables such as the cost of breakfast for workers on a trip or the fare for a taxi ride.

NaCCA B7.2.1.1.2 ⏱ about 45 min

By the end you will be able to

  • ✔Use phrases such as one more than, one less than, twice, thrice and one more than twice to describe a rule.
  • ✔Find the rule that links the input and output in a table of values and state it in words.
  • ✔Tell apart rules that sound alike, such as 'one more than twice x' and 'twice one more than x'.
  • ✔Use a rule in words to find a missing input or output in a real-life problem.

Key words

Rule
The instruction that tells you how to get the output from the input, e.g. 'three times the input'.
Input and output
The input is the number you start with (the domain); the output is the number you get after using the rule (the co-domain).
Twice / thrice
Twice means two times (×2); thrice means three times (×3).
One more than twice x
First double x, then add 1. In symbols: 2x + 1.
Twice one more than x
First add 1 to x, then double. In symbols: 2(x + 1).
Curriculum reference

Content standard: B7.2.1.1 Derive the rule for a set of points of a relation, draw a table of values to graph the relation in a number plane and make predictions about subsequent elements of the relation.

Indicator: Describe the rule for a given relation using mathematical language such as one more, one less, one more than twice, etc.

Starter

Breakfast for the field trip

Mr Asante is taking workers from his cocoa farm near Goaso on a field trip. One packed breakfast costs GH₵3. For 1 worker he pays GH₵3, for 2 workers GH₵6, for 3 workers GH₵9. At the end of the trip the receipt says GH₵120 — but he has forgotten how many workers went. If you can describe the rule in words, you can work it out in seconds.

1. The language of rules

When we describe a relation, we say what happens to the input to give the output. Mathematics has short, exact phrases for this. Learn them well — they appear in BECE questions every year.

Read each phrase slowly: the order of the words tells you the order of the operations.
Rule in wordsRule in symbolsOutputs for x = 1, 2, 3, 4
one more than xx + 12, 3, 4, 5
one less than xx − 10, 1, 2, 3
twice x (two times x)2x2, 4, 6, 8
thrice x (three times x)3x3, 6, 9, 12
one more than twice x2x + 13, 5, 7, 9
one less than thrice x3x − 12, 5, 8, 11
twice one more than x2(x + 1)4, 6, 8, 10
four times one less than x4(x − 1)0, 4, 8, 12

Look at the phrase one more than twice x. The words twice x are done first, and then you go one more than that answer. So for x = 5: twice 5 is 10, and one more than 10 is 11.

▶ Each figure has two dots for every term number plus one extra dot: one more than twice the term number (3, 5, 7, 9).

2. Finding the rule from a table

Here is the cost of breakfast for Mr Asante's workers:

Number of workers123456
Cost (GH₵)369121518
  1. 1Look at how the output changes when the input goes up by 1: 3, 6, 9, 12 … it goes up by 3 each time.
  2. 2That tells you the rule involves three times the input. Multiply each input by 3: 3, 6, 9, 12, 15, 18.
  3. 3Compare with the outputs. They are exactly the same, so nothing needs to be added or taken away.
  4. 4Say the rule in words: the cost is three times the number of workers.
  5. 5Test it on one more pair: 6 workers → 3 × 6 = 18 ✓

▶ A mapping diagram: every arrow follows the same rule, 'three times the number of workers'.

When multiplying is not enough, find what must be added or subtracted. For inputs 1, 2, 3, 4 and outputs 4, 7, 10, 13: the output goes up by 3, so try three times x: 3, 6, 9, 12. Each output is 1 more than that, so the rule is one more than thrice x (3x + 1).

3. Word order matters

Two rules can use the same words in a different order and give completely different answers. Compare them for x = 4:

Rule in wordsWhat you dox = 4 gives
one more than twice xdouble, then add 12 × 4 + 1 = 9
twice one more than xadd 1, then double2 × (4 + 1) = 10
one less than four times xmultiply by 4, then subtract 14 × 4 − 1 = 15
four times one less than xsubtract 1, then multiply by 44 × (4 − 1) = 12

4. Using a rule backwards

Once the rule is clear, you can answer questions in either direction. Mr Asante's rule is cost = three times the number of workers. To find the number of workers from the cost, undo the rule: divide by 3.

Receipt = GH₵120, so the number of workers = 120 ÷ 3 = 40 workers. Check: 3 × 40 = 120 ✓.

For a rule with two operations, undo them in the reverse order. If the output is one more than twice x and the output is 15: first take away the 1 (15 − 1 = 14), then halve (14 ÷ 2 = 7). So x = 7. Check: twice 7 is 14, and one more is 15 ✓.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

A 'times' rule

Inputs 1, 2, 3, 4 give outputs 5, 10, 15, 20. Describe the rule in words.
Show the solution
  1. Step 1The outputs go up by 5 each time, so try five times the input: 5, 10, 15, 20.
  2. Step 2These match the outputs exactly.

Answer: The output is five times the input (5x).

Example 2

A 'more than' rule

Inputs 1, 2, 3, 4 give outputs 4, 7, 10, 13. Describe the rule in words.
Show the solution
  1. Step 1The outputs go up by 3 each time, so try three times the input: 3, 6, 9, 12.
  2. Step 2Each output is 1 more than three times the input: 3 + 1 = 4, 6 + 1 = 7, 9 + 1 = 10, 12 + 1 = 13.

Answer: One more than thrice x (3x + 1).

Example 3

A rule with brackets

Inputs 1, 2, 3, 4 give outputs 0, 4, 8, 12. Describe the rule in words.
Show the solution
  1. Step 1The outputs go up by 4, so try four times the input: 4, 8, 12, 16.
  2. Step 2Each output is 4 less: 4 − 4 = 0, 8 − 4 = 4, … so the rule is four less than four times x.
  3. Step 3Another way: one less than x gives 0, 1, 2, 3, and four times these gives 0, 4, 8, 12.
  4. Step 4So 'four times one less than x', 4(x − 1), is the same rule as 4x − 4.

Answer: Four times one less than x, 4(x − 1) (the same as four less than four times x).

Example 4

The breakfast receipt

Breakfast costs GH₵3, GH₵6, GH₵9, GH₵12, GH₵15, GH₵18 for 1 to 6 workers. (i) Describe the rule. (ii) How many workers went on the trip if the cost was GH₵120?
Show the solution
  1. Step 1(i) The cost goes up by GH₵3 per worker, and 3 × 1 = 3, 3 × 2 = 6, … so the cost is three times the number of workers.
  2. Step 2(ii) Undo the rule: 120 ÷ 3 = 40.
  3. Step 3Check: 3 × 40 = 120 ✓

Answer: (i) Cost is three times the number of workers. (ii) 40 workers

Example 5

Taxi fare in Tamale

A taxi fare is GH₵7 for 1 km, GH₵9 for 2 km and GH₵11 for 3 km. Describe the rule in words, then find the fare for 12 km and the distance for a fare of GH₵45.
Show the solution
  1. Step 1The fare goes up by GH₵2 per km, so try twice the distance: 2, 4, 6. Each fare is 5 more: the fare is five more than twice the distance.
  2. Step 212 km: 2 × 12 + 5 = 24 + 5 = GH₵29.
  3. Step 3GH₵45: undo in reverse order: 45 − 5 = 40, then 40 ÷ 2 = 20 km.
  4. Step 4Check: 2 × 20 + 5 = 45 ✓

Answer: Five more than twice the distance; GH₵29; 20 km

Summary

  • Twice means ×2, thrice means ×3; 'one more than' means +1 and 'one less than' means −1.
  • To find a rule from a table, see how much the output goes up when the input goes up by 1 — that is the multiplier.
  • Then compare the output with 'multiplier × input' to see what is added or subtracted.
  • Word order matters: 'one more than twice x' is 2x + 1, but 'twice one more than x' is 2(x + 1).
  • To find an input from an output, undo the operations in reverse order.

Practice: 15 questions

5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.

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1 easy
Which phrase means ?
2 easy
Inputs 1, 2, 3, 4 give outputs 3, 6, 9, 12. Which rule fits?
3 easy
The rule is 'one less than x'. What is the output when x = 10?
4 easy
The rule is 'one more than twice x'. What is the output when x = 6?
5 easy word problem
A packed breakfast costs GH₵3 per worker, so the cost is three times the number of workers. What is the cost for 9 workers?
GH₵
6 medium
Inputs 1, 2, 3, 4 give outputs 7, 10, 13, 16. Which rule fits?
7 medium
The rule is 'twice one more than x'. What is the output when x = 7?
8 medium word problem
On a school trip to Cape Coast Castle, lunch costs GH₵4 per pupil, so the total cost is four times the number of pupils. The total cost was GH₵96. How many pupils went?
pupils
9 medium
Inputs 1, 2, 3, 4 give outputs 0, 4, 8, 12.

  1. (a) Which rule fits the table?

  2. (b) Use the rule to find the output when x = 10.

10 medium
The rule is 'one less than thrice x'. The output is 20. Find x.
11 hard word problem
A taxi fare is GH₵7 for 1 km, GH₵9 for 2 km and GH₵11 for 3 km, and continues in the same way.

  1. (a) Which rule describes the fare?

  2. (b) What is the fare for 15 km (in GH₵)?

  3. (c) How many kilometres can you travel for GH₵45?

12 hard
Which of these is NOT the same rule as 'four times one less than x'?
13 hard
Inputs 1, 2, 3, 4, 5 give outputs 5, 11, 17, 23, 29. Write the rule for the output using x.
14 hard word problem
At a church harvest in Ho, tables are joined in a long row. 1 table seats 6 people, 2 tables seat 10 people and 3 tables seat 14 people. How many people can sit at 12 tables joined in a row?
people
15 hard word problem
Fatima has GH₵20 in her MoMo wallet and saves GH₵15 every week. Her savings are twenty more than fifteen times the number of weeks. After how many weeks will she have GH₵170?
weeks