Tables from rules

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.

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.

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.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Which pair?

y = 2x + 3

Pick the input $x = 4$ . What output $y$ does the rule give? Which number goes first in the pair?

Which pair belongs in the table for

y = 2x + 3

(4, 4)

(4, 27)

(4, 11)

(11, 4)

TryTry it on the board and compare the result with your prediction.

The rule machine

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?

Try it yourselfYour edits are saved in this browser.

Make a table of at least 4 ordered pairs for $y = 2x + 3$ .

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Making a table

Here is how to make a table from a rule.

Worked exampleBefore each step, say what you think comes next.

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

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

Read Jo's working one line at a time. Which number should come first?

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

x (input)	0	1	2	3
y (output)	2

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

The pair

(5, 13)

is in the table for

y = 2x + 3

. Which number is the input?13Both are inputs5

ModifyChange the problem on purpose and see what happens.

New rules and inputs

Negative inputs give sums such as $-4 + 3$ . Work this one out first: start at $-4$ on a number line, then move.

The machine now takes inputs from −5 to 5.

Need a hint?

Put −1 in place of x with brackets: 2(−1) + 3.

Try it yourselfYour edits are saved in this browser.

Fill in the table for $y = 3x - 2$. Use the machine to check.

x	1	2	3	4
y

Quick check before negative inputs: what is $-4 + 3$ ?

For $y = 2x + 3$ , put in at least 3 inputs, including negative ones.

Use the board. This ticks when you get there.

Fill in the table for $y = 10 - 2x$. What happens to the output as the input goes up?

x	0	1	2	3
y

For $y = 2x + 3$ , which input gives the output 15?

PractiseA few on your own, to check you have it.

Rules and pairs

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

Which table goes with the rule

y = x + 4

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

For

y = 3x + 1

, which is an ordered pair from its table?

(7, 2)

(2, 7)

(2, 2)

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

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

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.

Need a hint?

From x = 2 to x = 3, how much does y go up?

Another hint

That is the number in front of x. Then what do you add to make the first pair work?

Write a rule whose table includes the pairs $(2, 7)$ and $(3, 9)$ . Start with $y =$ .

For teachers

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.

Open the teacher guide: answers, solutions, slides and a worksheet.

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.

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

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

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.

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.

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).

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.

Now you can

Make a table of values from a rule by putting in inputs and working out outputs.
Write ordered pairs with the input first, (x, y).
Find the input that gives a chosen output.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson3. Plotting a rulePlot each ordered pair from a rule's table as a point, across first and then up. For a rule like y = 2x + 1, the points lie on a straight line.

Source: content/maths/linear-relations/tables-from-rules/lesson.md