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Math · Patterns & Functions

Patterns & Functions: Input–Output Tables, Rules, Coordinates, Predict

Move from equations and inequalities into function thinking by reading input–output tables, applying and reversing rules, writing ordered pairs, plotting points and predicting values.

Grades 5–6 · 40 practice questions · Original material

Learning objectiveUse input–output tables to identify and apply rules, connect each input and output as an ordered pair, reverse simple rules to find missing values, and distinguish a rule that fits every row from one that fits only some examples.

The idea

A function rule connects an input to exactly one output. A table is one way to show the connection: choose an input, apply the rule, and record the output. For example, the rule y = x + 3 sends 2 to 5 and 7 to 10.

Each table row can also be written as an ordered pair (x, y). The first coordinate is the input and the second is the output. Plotting several ordered pairs can reveal how a rule behaves as the input changes.

A pattern is not proved by one row. A proposed rule must work for every given input–output pair. When a value is missing, work forward with the rule or undo the rule with an inverse operation.

Worked examples

Example 1 · Apply a rule

Rule: y = x + 4. What is the output when x = 6?

Start with the input 6.
Add 4 because the rule is x + 4.
6 + 4 = 10.

y = 10.

Example 2 · Complete a table

Rule: y = 3x. Inputs are 1, 2 and 5.

Multiply each input by 3.
1 → 3, 2 → 6, 5 → 15.
Check that the same rule was used for every row.

The outputs are 3, 6 and 15.

Example 3 · Find a missing input

Rule: y = x + 7. The output is 12. Find x.

The rule added 7.
Undo +7 by subtracting 7 from 12.
12 − 7 = 5.

x = 5.

Example 4 · Write an ordered pair

A table row has input 4 and output 9.

Input is the first coordinate.
Output is the second coordinate.
Write the pair in (x, y) order.

(4, 9).

Example 5 · Test a proposed rule

Pairs are (1, 4), (2, 7), (3, 10). Does y = 3x + 1 fit?

Test x = 1: 3(1)+1 = 4.
Test x = 2: 3(2)+1 = 7.
Test x = 3: 3(3)+1 = 10.

Yes. The rule fits every given pair.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Switching the order and writing (output, input).

Why this fails: Ordered pairs use (x, y): input first, output second.

Finding a rule that works for only one row and stopping.

Why this fails: A valid rule must match every given pair in the table.

Using the output as the next input unless the problem says to.

Why this fails: A function table usually treats each row independently.

Trying to reverse y = 2x + 3 by subtracting 2 and dividing by 3.

Why this fails: Undo operations in reverse order: subtract 3, then divide by 2.

Assuming every pattern has to add the same amount.

Why this fails: Some rules multiply, combine operations, or follow other consistent relationships.

Foundation

Foundation practice

Apply one-step rules and read simple input–output tables.

Apply rule

1. Rule y = x + 5. Find y when x = 3.

Apply rule

2. Rule y = 2x. Find y when x = 7.

Apply rule

3. Rule y = x − 4. Find y when x = 11.

Table

4. For y = 3x, what output matches input 4?

Table

5. For y = x + 2, what output matches input 9?

Ordered pair

6. Input 5 gives output 8. Write the ordered pair.

Ordered pair

7. Which is the input in (6, 13)?

Reverse rule

8. Rule y = x + 6. If y = 15, find x.

Reverse rule

9. Rule y = 4x. If y = 20, find x.

Pattern check

10. Do pairs (1,3), (2,4), (3,5) fit y = x + 2?

Core

Core practice

Use two-step rules, complete missing values and connect tables to coordinates.

Apply rule

11. Rule y = 2x + 1. Find y when x = 6.

Apply rule

12. Rule y = 3x − 2. Find y when x = 5.

Reverse rule

13. Rule y = 2x + 3. If y = 17, find x.

Reverse rule

14. Rule y = 5x − 4. If y = 21, find x.

Table

15. For y = x + 7, complete outputs for x = 0, 3, 8.

Table

16. For y = 2x − 1, complete outputs for x = 1, 4, 6.

Coordinates

17. Write the ordered pairs for inputs 0,1,2 under y = 2x.

Coordinate meaning

18. What does the point (4, 9) mean in an input–output graph?

Rule test

19. Do (1,5), (2,8), (3,11) fit y = 3x + 2?

Rule test

20. Do (1,4), (2,6), (3,9) fit y = 2x + 2?

Reasoning

Reasoning practice

Infer rules from several pairs and explain why a rule does or does not fit.

Infer rule

21. Pairs are (1,4), (2,5), (3,6). Give a rule.

Infer rule

22. Pairs are (1,5), (2,10), (4,20). Give a rule.

Infer rule

23. Pairs are (0,2), (1,4), (2,6). Give a rule.

Missing value

24. For y = 3x + 1, which input gives output 25?

Compare rules

25. At x = 4, which is larger: y = 2x + 1 or y = x + 6?

Compare rules

26. At x = 7, compare y = 2x and y = x + 5.

Explain

27. Why is one matching row not enough to prove a rule?

Error analysis

28. A student says (3,8) means input 8 and output 3. Correct the error.

Prediction

29. A rule gives (2,7), (3,10), (4,13). Predict the output for x = 5.

Graph reasoning

30. Points (0,1), (1,3), (2,5) rise by how much in y when x increases by 1?

Stretch

Stretch practice

Connect rules, tables and graphs in multi-step situations.

Real-world rule

31. A taxi-style classroom model starts at $4 and adds $2 per kilometre. Write a rule for cost C after k kilometres.

Real-world rule

32. Using C = 2k + 4, find the cost at k = 6.

Reverse situation

33. Using C = 2k + 4, what k gives C = 20?

Table to rule

34. Outputs are 4, 7, 10, 13 for inputs 0,1,2,3. Give a rule.

Rule to point

35. For y = 4x − 3, give the point when x = 5.

Point test

36. Does (6, 20) lie on y = 3x + 2?

Point test

37. Does (4, 15) lie on y = 3x + 1?

Compare growth

38. Which grows faster as x increases: y = 4x + 1 or y = 2x + 10?

Boundary of evidence

39. Pairs (1,3), (2,5), (3,7) fit y = 2x + 1. Does that prove no other rule could ever match those three points?

Create

40. Give three ordered pairs that fit y = x − 2.

Answer key

1. 8
3 + 5 = 8.
2. 14
2 × 7 = 14.
3. 7
11 − 4 = 7.
4. 12
Multiply 4 by 3.
5. 11
Add 2 to 9.
6. (5, 8)
Input comes first and output comes second.
7. 6
The first coordinate is the input x.
8. 9
Undo +6 by subtracting 6.
9. 5
Undo ×4 by dividing by 4.
10. Yes.
Each output is 2 more than its input.
11. 13
2 × 6 + 1 = 13.
12. 13
3 × 5 − 2 = 13.
13. 7
Subtract 3 to get 14, then divide by 2.
14. 5
Add 4 to get 25, then divide by 5.
15. 7, 10, 15
Add 7 to each input.
16. 1, 7, 11
Double each input, then subtract 1.
17. (0,0), (1,2), (2,4)
Evaluate the same rule for each input.
18. Input 4 produces output 9.
The coordinates record x first and y second.
19. Yes.
The rule gives 5, 8 and 11.
20. No.
The rule predicts 4, 6 and 8; the third pair fails.
21. y = x + 3
Each output is 3 more than its input.
22. y = 5x
Each output is five times the input.
23. y = 2x + 2
The output starts at 2 and rises by 2 for each +1 in x.
24. 8
25 − 1 = 24 and 24 ÷ 3 = 8.
25. y = x + 6
At x = 4, the outputs are 9 and 10, so y = x + 6 is larger by 1.
26. y = 2x is larger by 2.
The outputs are 14 and 12.
27. Another row may fail the proposed rule.
A rule must account for all given input–output pairs.
28. Input is 3 and output is 8.
Ordered pairs are written (x, y).
29. 16
The consistent rule is y = 3x + 1.
30. 2
The y-values increase 1→3→5.
31. C = 2k + 4
The fixed start is 4 and each kilometre adds 2.
32. $16
2×6 + 4 = 16.
33. 8 km
20 − 4 = 16; 16 ÷ 2 = 8.
34. y = 3x + 4
The starting output is 4 and each +1 input adds 3.
35. (5, 17)
4×5 − 3 = 17.
36. Yes.
3×6 + 2 = 20.
37. No.
The rule gives 13, not 15.
38. y = 4x + 1
Its output rises by 4 per +1 in x, compared with 2.
39. No.
Those points support the rule, but a finite set of points can be matched by more than one more complicated rule.
40. For example (2,0), (5,3), (9,7).
Each output must be exactly 2 less than the input.

Related practice

Inequalities Two-step equations Math library