--- title: "Tables from rules" subtitle: "Linear relations · Rules, tables and graphs · Lesson 2" --- # Tables from rules **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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. :::teacher-note **In one sentence:** 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). **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict $$y = 2x + 3$$ Pick the input $x = 4$. What output $y$ does the rule give? Which number goes first in the pair? :::keep-together **Which pair belongs in the table for $y = 2x + 3$?** ☐ $(4, 4)$ ☐ $(4, 27)$ ☐ $(4, 11)$ ☐ $(11, 4)$ My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** (4, 11). The input is x = 4; the output is y = 2 × 4 + 3 = 11. The pair is written (input, output). ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Making a table :::keep-together ### A table of values Here is how to make a table from a rule. 1. $y = 3x - 1$ — Make a table of values for this rule, for the inputs 0 to 3. 2. $x = 0: \; y = 3(0) - 1 = -1$ — Choose an input and put it in place of $x$. 3. $x = 1: \; y = 3(1) - 1 = 2$ — Each input gives its own output. Do not use the same number for both. 4. $x = 2: \; y = 5, \quad x = 3: \; y = 8$ — Work out the rest the same way. 5. $(0, -1), (1, 2), (2, 5), (3, 8)$ — Write each as an ordered pair: input first, output second. | $x$ (input) | 0 | 1 | 2 | 3 | |---|---|---|---|---| | $y$ (output) | −1 | 2 | 5 | 8 | **Fill in the table for $y = 4x + 2$.** {{rows 1}} | x (input) | 0 | 1 | 2 | 3 | |---|---|---|---|---| | y (output) | 2 | | | | ::: :::teacher-note **Answer:** 6, 10, 14 Put each input into the rule: 4 × 1 + 2 = 6, 4 × 2 + 2 = 10, 4 × 3 + 2 = 14. ::: :::keep-together ### 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). ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** 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). ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** 5 The first number is the input: x = 5. The rule gives the output 2 × 5 + 3 = 13. ::: :::typical ## 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.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Rules and pairs :::keep-together ### An output **For $y = 5x - 4$, what is the output when the input is 3?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 11 5 × 3 − 4 = 15 − 4 = 11. ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** 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. ::: :::keep-together ### Which pair? **For $y = 3x + 1$, which is an ordered pair from its table?** ☐ $(7, 2)$ ☐ $(2, 7)$ ☐ $(2, 2)$ ::: :::teacher-note **Answer:** $(2, 7)$ Input 2, output 3 × 2 + 1 = 7: the pair is (2, 7). ::: :::keep-together ### 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: {{field-15}} ::: :::teacher-note **Answer:** 5n + 2 It adds 5 each time, and Figure 1 is 5 + 2: s = 5n + 2. ::: ## 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. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Make a table for y = 4x − 3 with the inputs 0, 1, 2, 3. Write the pairs.** {{lines-3}} ::: :::teacher-note **Answer:** Outputs −3, 1, 5, 9. Pairs (0, −3), (1, 1), (2, 5), (3, 9). :::