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# Tables from rules

Linear relations · Rules, tables and graphs

<!-- Speaker note: Students generate tables of ordered pairs from a rule, keeping input and output separate and in order, including negative inputs and working back from an output (AC9M7A05, node 2). About 45 minutes (Predict and Run 8 min · Investigate 12 min · Modify 15 min · Practise and Make 10 min). -->

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## Today

**We are learning to** a rule turns each input x into an output y. Choose inputs, work out each output, and write them as a table of ordered pairs (x, y), input first.

- I can make a table of values from a rule by putting in inputs and working out outputs.
- I can write ordered pairs with the input first, (x, y).
- I can find the input that gives a chosen output.

<!-- Speaker note: Before class: Do the Run and Modify boards yourself first. The rule machine shows the substitution for every input (2(4) + 3 = 11): point at it, because it is the habit from the Equations unit. The progression data names a specific mistake for this step: putting the same value in for both x and y (the pair (4, 4)). The Predict plants it, with the swapped pair (11, 4). The next lesson plots these pairs as points: errors here carry straight into the graph, so take time over order. -->

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## Why it matters

> A spreadsheet column, a recipe scaled for more people, a taxi meter: each one takes a number in and gives a number out by the same rule, every time.

<!-- Speaker note: In the last lesson, a pattern gave a rule. Now we start from the rule. A **rule** such as $y = 2x + 3$ takes an **input** $x$ and gives an **output** $y$. A **table of values** lists inputs and their outputs. Each column is an **ordered pair** $(x, y)$: input first, output second. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Which pair belongs in the table for $y = 2x + 3$?**

- $(4, 4)$
- $(4, 27)$
- $(4, 11)$
- $(11, 4)$

<!-- Speaker note: What happens: (4, 11). The input is x = 4; the output is y = 2 × 4 + 3 = 11. The pair is written (input, output). Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. (4, 4) and (11, 4) are the common wrong answers. Ask what each number means. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Flinear-relations%2Ftables-from-rules%2Fstep-2&code=widget%3A+rule%0Arule%3A+y+%3D+2x+%2B+3%0Ax%3A+0%2C+10%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/tables-from-rules/?full=step-2#step-2)

<!-- Speaker note: Type an input $x$ and press **Put it in**. The machine shows the working and adds the pair to the table. Try $x = 0, 1, 2, 3$. Was your predicted pair right? -->

<!-- Speaker note: Teacher: as students add pairs, ask: which row are your inputs? Which are outputs? Who chose the inputs? -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* (4, 11). The input is x = 4; the output is y = 2 × 4 + 3 = 11. The pair is written (input, output).

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## A table of values

Here is how to make a table from a rule.

| $x$ (input) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| $y$ (output) | −1 | 2 | 5 | 8 |

$$y = 3x - 1$$

1) $x = 0: \; y = 3(0) - 1 = -1$ — Choose an input and put it in place of $x$.
2) $x = 1: \; y = 3(1) - 1 = 2$ — Each input gives its own output. Do not use the same number for both.
3) $x = 2: \; y = 5, \quad x = 3: \; y = 8$ — Work out the rest the same way.
4) $(0, -1), (1, 2), (2, 5), (3, 8)$ — Write each as an ordered pair: input first, output second.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write y = 3x - 1 and say "Make a table of values for this rule, for the inputs 0 to 3." Line 1: write x = 0: y = 3(0) - 1 = -1 and say "Choose an input and put it in place of x." Line 2: write x = 1: y = 3(1) - 1 = 2 and say "Each input gives its own output. Do not use the same number for both." Line 3: write x = 2: y = 5, x = 3: y = 8 and say "Work out the rest the same way." Line 4: write (0, -1), (1, 2), (2, 5), (3, 8) and say "Write each as an ordered pair: input first, output second." -->

<!-- Speaker note: Teacher: model the worked example live. Write the substitution for each input, then the pair, saying 'input first'. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A table of values

**Fill in the table for $y = 4x + 2$.**

| x (input) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y (output) | 2 | ? | ? | ? |

* Answer: 6, 10, 14

<!-- Speaker note: Put each input into the rule: 4 × 1 + 2 = 6, 4 × 2 + 2 = 10, 4 × 3 + 2 = 14. -->

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## Find the mistake

**Jo makes a pair for $y = 2x + 1$. Click the line where her working goes wrong.**

```
 1  Rule: y = 2x + 1. Make a pair for the input x = 3.
 2  y = 2 × 3 + 1
 3  y = 7
 4  The pair is (7, 3).
```

* Answer: line 4

<!-- Speaker note: Line 4: the input comes first in an ordered pair. The input is 3 and the output is 7, so the pair is (3, 7). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Input or output?

**The pair $(5, 13)$ is in the table for $y = 2x + 3$. Which number is the input?**

- 13
- Both are inputs
- 5

* Answer: 5

<!-- Speaker note: The first number is the input: x = 5. The rule gives the output 2 × 5 + 3 = 13. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## New rules and inputs



- **Subtraction:** 
- **Quick check:** Negative inputs give sums such as $-4 + 3$. Work this one out first: start at $-4$ on a number line, then move.
- **Negative inputs:** The machine now takes inputs from −5 to 5.
- **Going down:** 
- **Backwards:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/tables-from-rules/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a rule with subtraction, negative inputs (after a quick check of adding to a negative, from Number Foundations), a rule whose output goes down, and a table that starts from the output. Hints, in order: Put −1 in place of x with brackets: 2(−1) + 3. For students who need more support: Give a two-row table with 'in' and 'out' already labelled, and the substitution started: 2( ) + 3. For early finishers: For y = 10 − 2x, which input gives the output 0? A negative output? What does that say about the line? When students get stuck: A student writes the same number for x and y. Ask: What does the rule do to the input? Fix: Write the substitution with the input in brackets, work it out, and only then write the pair. / A student writes pairs the wrong way round (Jo's mistake). Ask: Which number did you choose, and which did the rule give you? Fix: The one you chose is the input and goes first. Label the table rows 'in' and 'out'. / A student puts an output in as an input when working backwards. Ask: Is 15 the number going in or coming out? Fix: Write the rule equal to the output: 2x + 3 = 15. Then undo the steps, as in the Equations unit. -->

<!-- Speaker note: Teacher: the quick check (−4 + 3) retrieves adding to a negative from Number Foundations. If several students miss it, do two or three on a number line before the negative inputs. Then ask: where do the brackets go? 2(−3) + 3. Multiplying a negative (2 × −3) is Year 8 work: here the machine shows the result, so read it together rather than teach the rule. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## An output

**For $y = 5x - 4$, what is the output when the input is 3?**

* Answer: 11

<!-- Speaker note: 5 × 3 − 4 = 15 − 4 = 11. -->

<!-- Speaker note: Teacher: for the 'which table' question, ask students to check one column of each table with the rule. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Which table?

**Which table goes with the rule $y = x + 4$?**

- x: 4, 5, 6 and y: 0, 1, 2
- x: 0, 1, 2 and y: 4, 5, 6
- x: 0, 1, 2 and y: 0, 4, 8

* Answer: x: 0, 1, 2 and y: 4, 5, 6

<!-- Speaker note: Each output is 4 more than its input: 0 + 4 = 4, 1 + 4 = 5, 2 + 4 = 6. The third table has inputs and outputs swapped. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Which pair?

**For $y = 3x + 1$, which is an ordered pair from its table?**

- $(7, 2)$
- $(2, 7)$
- $(2, 2)$

* Answer: $(2, 7)$

<!-- Speaker note: Input 2, output 3 × 2 + 1 = 7: the pair is (2, 7). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review

**Review: a pattern has 7, 12, 17, 22 tiles in Figures 1 to 4. Write a rule for the tiles $s$ in figure $n$.**

* Answer: 5n + 2

<!-- Speaker note: It adds 5 each time, and Figure 1 is 5 + 2: s = 5n + 2. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: find the rule

Use what you know from the pattern lesson: how much does the output go up when the input goes up by 1? Then check your rule with both pairs.

Stretch: write a rule of your own, then give a partner two of its pairs and see if they can find it.

<!-- Speaker note: What this step is for: Writing a rule that makes given pairs reverses the lesson: from the table back to the rule. Hints, in order: From x = 2 to x = 3, how much does y go up? / That is the number in front of x. Then what do you add to make the first pair work? -->

<!-- Speaker note: Teacher: compare methods for finding the rule: the jump in y (2) and then the leftover (3). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Make a table for y = 4x − 3 with the inputs 0, 1, 2, 3. Write the pairs.**

* Outputs −3, 1, 5, 9. Pairs (0, −3), (1, 1), (2, 5), (3, 9).

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: plotting the pairs as points. The pairs from a rule like y = 2x + 3 lie on a straight line, and the rule's numbers show in how steep it is and where it starts. Where it leads in class: filling a spreadsheet column with a formula, and conversion tables (centimetres to inches, dollars to another currency). -->