Tables from rules

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 12 min · Modify 15 min · Practise and Make 10 min

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.

Running the lesson

1. Predict: take a show of hands. (4, 4) and (11, 4) are the common wrong answers. Ask what each number means.
2. Run: as students add pairs, ask: which row are your inputs? Which are outputs? Who chose the inputs?
3. Investigate: model the worked example live. Write the substitution for each input, then the pair, saying 'input first'.
4. Modify: 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.
5. Practise: for the 'which table' question, ask students to check one column of each table with the rule.
6. Make: compare methods for finding the rule: the jump in y (2) and then the leftover (3).

When students get stuck

Answers to the checks

1. which pair?: Which pair belongs in the table for $y = 2x + 3$?
   (4, 11). The input is x = 4; the output is y = 2 × 4 + 3 = 11. The pair is written (input, output).
2. the rule machine: Make a table of at least 4 ordered pairs for $y = 2x + 3$.
   
3. A table of values: Fill in the table for $y = 4x + 2$.
   6, 10, 14
   Put each input into the rule: 4 × 1 + 2 = 6, 4 × 2 + 2 = 10, 4 × 3 + 2 = 14.
4. Find the mistake: Jo makes a pair for $y = 2x + 1$. Click the line where her working goes wrong.
   line 4
   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).
5. Input or output?: The pair $(5, 13)$ is in the table for $y = 2x + 3$. Which number is the input?
   5
   The first number is the input: x = 5. The rule gives the output 2 × 5 + 3 = 13.
6. Subtraction: Fill in the table for $y = 3x - 2$. Use the machine to check.
   1, 4, 7, 10
   3 × 1 − 2 = 1, 3 × 2 − 2 = 4, 3 × 3 − 2 = 7, 3 × 4 − 2 = 10.
7. Quick check: Quick check before negative inputs: what is $-4 + 3$?
   -1
   Start at −4 and move 3 to the right: −1. You need this for x = −2 in y = 2x + 3: 2(−2) + 3 = −4 + 3 = −1.
8. Negative inputs: For $y = 2x + 3$, put in at least 3 inputs, including negative ones.
   
   A negative input works the same way: x = −2 gives y = 2(−2) + 3 = −1.
9. Going down: Fill in the table for $y = 10 - 2x$. What happens to the output as the input goes up?
   10, 8, 6, 4
   10 − 2 × 0 = 10, then 8, 6, 4: the output goes down by 2 each time.
10. Backwards: For $y = 2x + 3$, which input gives the output 15?
    6
    2x + 3 = 15, so 2x = 12 and x = 6. The pair is (6, 15).
11. An output: For $y = 5x - 4$, what is the output when the input is 3?
    11
    5 × 3 − 4 = 15 − 4 = 11.
12. Which table?: Which table goes with the rule $y = x + 4$?
    x: 0, 1, 2 and y: 4, 5, 6
    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.
13. Which pair?: For $y = 3x + 1$, which is an ordered pair from its table?
    $(2, 7)$
    Input 2, output 3 × 2 + 1 = 7: the pair is (2, 7).
14. Review: Review: a pattern has 7, 12, 17, 22 tiles in Figures 1 to 4. Write a rule for the tiles $s$ in figure $n$.
    5n + 2
    It adds 5 each time, and Figure 1 is 5 + 2: s = 5n + 2.
15. find the rule: Write a rule whose table includes the pairs $(2, 7)$ and $(3, 9)$. Start with $y =$.
    2x + 3
    y = 2x + 3: the output goes up 2 when the input goes up 1, and 2 × 2 + 3 = 7.

Solutions

One possible solution for each task. Other correct programs are fine.

new rules and inputs › Subtraction

Open task: any program that does what the step asks.

new rules and inputs › Quick check

Open task: any program that does what the step asks.

new rules and inputs › Negative inputs

Open task: any program that does what the step asks.

new rules and inputs › Going down

Open task: any program that does what the step asks.

new rules and inputs › Backwards

Open task: any program that does what the step asks.

Stretch and support

Stretch: For y = 10 − 2x, which input gives the output 0? A negative output? What does that say about the line?

Support: Give a two-row table with 'in' and 'out' already labelled, and the substitution started: 2( ) + 3.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Tables from rules".

The idea: Generating tables from function rules: I can Students generate a table of ordered pairs from a given function rule and identify the relationship between input and output values.

What understanding looks like:
- Correctly generates a table of ordered pairs from a given function rule.
- Labels the table with input (x) and output (y) columns.
- Does not substitute the same value for both x and y.

Mistakes students often make here:
- A student writes the same number for x and y.
- A student writes pairs the wrong way round (Jo's mistake).
- A student puts an output in as an input when working backwards.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Generating tables of values from patterns), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Tables from rules".

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).

What each step is for

• which pair?: Committing to a pair before using the machine shows whether students keep input and output separate, and in order.
• the rule machine: Putting inputs in one at a time and seeing the substitution makes the table's two rows mean something: what goes in, and what comes out.
• making a table: Each output comes from putting its own input into the rule; the pair keeps them together, input first.
• new rules and inputs: 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.
• rules and pairs: Working out outputs, matching a table to its rule and keeping pairs in order, mixed; the review is a pattern rule from the last lesson.
• find the rule: Writing a rule that makes given pairs reverses the lesson: from the table back to the rule.

Where it 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).