019. Rules and Input-Output Tables

Learning Intentions

• To understand that a rule connects two varying quantities
• complete an input-output table given a rule
• find a rule in the form output $=?\times$ input $+?$ for an input-output table

Pre-requisite Summary

• Understand that a quantity can change and that two quantities can be related
• Be able to substitute a number into a simple numerical rule
• Recall multiplication facts and basic addition facts
• Understand the meaning of input and output in a table
• Be able to follow a two-step process such as multiply, then add
• Recognise that the same rule must work for every row in a table

Worked Examples

Worked Example 1

A rule connects two varying quantities.
a) Complete the input-output table for the rule output $=3\times$ input $+2$ when the inputs are $1,2,3$.
b) Describe in words how the rule connects input and output.

Worked Example 2

Complete the input-output table for the rule output $=4\times$ input $+1$.
a) Find the outputs for inputs $2,5,7$.
b) State the rule in words.

Worked Example 3

Find the rule in the form output $=?\times$ input $+?$ for the table:
a) input: $1,2,3,4$
b) output: $5,8,11,14$

Worked Example 4

Find the rule in the form output $=?\times$ input $+?$ for the table:
a) input: $2,4,6,8$
b) output: $7,13,19,25$

Worked Example 5

Use the rule output $=2\times$ input $+5$.
a) Find the output when the input is $6$.
b) Find the output when the input is $10$.
c) Complete the table for inputs $0,1,2,3$.

Worked Example 6

A table has:
a) input: $0,1,2,3,4$
b) output: $4,7,10,13,16$
Find the rule in the form output $=?\times$ input $+?$ and use it to find the output when the input is $10$.

Problems

Problem 1

A rule connects two varying quantities.
a) Complete the input-output table for the rule output $=2\times$ input $+3$ when the inputs are $1,2,3$.
b) Describe in words how the rule connects input and output.

Problem 2

Complete the input-output table for the rule output $=5\times$ input $+2$.
a) Find the outputs for inputs $2,4,6$.
b) State the rule in words.

Problem 3

Find the rule in the form output $=?\times$ input $+?$ for the table:
a) input: $1,2,3,4$
b) output: $6,10,14,18$

Problem 4

Find the rule in the form output $=?\times$ input $+?$ for the table:
a) input: $2,4,6,8$
b) output: $9,15,21,27$

Problem 5

Use the rule output $=3\times$ input $+4$.
a) Find the output when the input is $5$.
b) Find the output when the input is $9$.
c) Complete the table for inputs $0,1,2,3$.

Problem 6

A table has:
a) input: $0,1,2,3,4$
b) output: $6,9,12,15,18$
Find the rule in the form output $=?\times$ input $+?$ and use it to find the output when the input is $10$.

Exercises

Understanding and Fluency

1. Complete each table using the rule output $=2\times$ input $+1$:
   a) input: $1,2,3$
   b) input: $4,5,6$

2. Complete each table using the rule output $=3\times$ input $+4$:
   a) input: $0,1,2$
   b) input: $5,6,7$

3. Complete each table using the rule output $=5\times$ input $+2$:
   a) input: $1,3,5$
   b) input: $2,4,6$

4. Find the rule in the form output $=?\times$ input $+?$:
   a) input: $1,2,3,4$; output: $4,7,10,13$
   b) input: $1,2,3,4$; output: $6,11,16,21$

5. Find the rule in the form output $=?\times$ input $+?$:
   a) input: $0,1,2,3$; output: $2,5,8,11$
   b) input: $0,1,2,3$; output: $7,9,11,13$

6. Use the rule output $=4\times$ input $+3$:
   a) find the output for input $2$
   b) find the output for input $8$
   c) find the output for input $10$

7. Use the rule output $=6\times$ input $+1$:
   a) find the output for input $0$
   b) find the output for input $3$
   c) find the output for input $5$

8. Complete the table and then state the rule:
   a) input: $0,1,2,3$; output: $5,8,11,14$
   b) input: $0,1,2,3$; output: $1,4,7,10$

Reasoning

9. Explain why a rule must work for every row in an input-output table.

10. A student says the rule for input-output pairs $(1,5),(2,8),(3,11)$ is output $=3\times$ input. Explain the mistake.

11. Explain how the number added in a rule output $=m\times$ input $+c$ can be found from the row where the input is $0$.

12. A student finds the rule output $=4\times$ input $+2$ for the table input: $1,2,3$ and output: $6,10,14$. Explain whether the rule is correct.

Problem-solving

13. A machine uses the rule output $=3\times$ input $+2$. How many items does it produce when the input is $12$?

14. A taxi fare table follows a rule of the form output $=?\times$ input $+?$. If the inputs are $1,2,3,4$ and the outputs are $9,13,17,21$, find the rule.

15. A pattern table has inputs $0,1,2,3,4$ and outputs $8,11,14,17,20$. Find the rule and then find the output for input $15$.

16. A factory packs items using the rule output $=5\times$ input $+1$. How many items are packed for input $20$?

17. A student finds that input $2$ gives output $9$ and input $5$ gives output $18$. The rule is of the form output $=?\times$ input $+?$. Find the rule.

18. A table follows the rule output $=2\times$ input $+7$. Find the outputs for inputs $10,11,12$.

Potential Misunderstandings

• Students may think a rule only needs to work for one row instead of every row
• Students may reverse the order of operations and add before multiplying
• Students may confuse the input column with the output column
• Students may use a different rule for different rows in the same table
• Students may think the number added is the same as the common increase in the outputs
• Students may forget that the value of the output when the input is $0$ gives the constant term in rules of the form output $=m\times$ input $+c$
• Students may identify the multiplier incorrectly by looking only at one pair of numbers without checking the whole table
• Students may substitute the input incorrectly when completing the table

Next: 020. Coordinates on the Number Plane