Tables of values and the Cartesian plane: a week of ready-to-teach maths

Five lessons for Year 7 Algebra. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus growing pattern cards, rule and table cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: pattern cards (Day 3) and rule and table cards (Day 4).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A rule such as y = 3x + 2 is a recipe for turning any x into a y. Working it out for several x values fills a table of values, and each x with its y is a coordinate pair (x, y) that can be plotted on the Cartesian plane: across, then up. For every rule this week the points line up, so the graph is a straight line. A growing pattern gives a table too: count each figure. The journey also runs backwards: the steady step in a table is the number that multiplies x, and one pair then gives the constant, which is where the line crosses the y-axis.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write a rule

The number in front of the letter multiplies it: 3x means 3 × x. Multiply before adding, so when x is 4, y = 3 × 4 + 2 = 14. Write a pair as (x, y), across first. In a growing pattern, n stands for the figure number. Calculators are not needed this week.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 turns a rule into a table; Day 2 turns the table into points on the plane; Day 3 starts from a picture instead of a rule; Day 4 runs the journey backwards, from a table to its rule; and Day 5 puts two rules on one plane to see what each number in a rule does. It builds on formulas (AC9M7A01) and on reading graphs of data (AC9M7A04), and it leads into graphing linear relations in Year 8 (AC9M8A02).

Materials for the week (one trip)

• From the classroom: pencils, rulers, two coloured pencils each, scissors, small envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 6 pattern cards (Day 3).
• Cut-out sheet 2, one per pair: 8 rule cards and 8 table cards (Day 4).
• Toothpicks or counters for building patterns, if you have them (Day 3, not essential).
• No calculators and no other equipment to buy.

Words for the word wall

rule, substitute, table of values, input, output, coordinate pair, Cartesian plane, x-axis, y-axis, origin, plot, straight line, growing pattern, figure number, step, coefficient, multiplier, constant, check, steeper, crosses the y-axis

Dear families

This week in maths, Year 7 is turning rules into tables of values, plotting the values as points on a grid, and finding the rule behind a growing pattern of shapes.

Try this at home

• Build squares in a row with toothpicks or dry pasta. Count the sticks for 1, 2 and 3 squares, then predict 10 squares.
• A ride costs $4 to start plus $2 for each kilometre. Make a table of the cost for 1 to 5 kilometres.
• Play “guess my rule”: one person thinks of a rule such as “times 3, add 1”; the other calls out numbers, hears the answers and guesses the rule.
• On grid paper, plot (1, 3), (2, 6) and (3, 9). What rule do the points follow? (y = 3x.)

Rules this week

Fill one row a day with a rule from class, one pair from its table, and a check that the pair fits.

Printed from the free seegongsik Tables of values and the Cartesian plane teaching pack · seegongsik.com/au/y7/algebra/AC9M7A05/pack

From a rule to a table

A rule turns each x into a y. Students substitute x values into rules, multiplying first, fill tables of values, notice the steady step along the y row, and write rules from words.

We are learning to

• substitute a number into a rule, multiplying before adding or subtracting,
• fill a table of values from a rule,
• see how the y row grows and link the step to the rule.

Success criteria

• I can work out y for any x, showing one line of working.
• I can say why the y row of a table goes up by the same amount each time.

You need

The worksheet and the unit open on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the table together; start the second with Part A.

Watch for these ideas

• Reading 3x as a two-digit number. 3x means 3 × x.
• Adding before multiplying: 3 × 4 + 2 is 14, not 18.
• Expecting the y row to grow by the multiplier when the x row jumps by more than 1.

Answers

• Part A: y = 3x + 2: 2, 5, 8, 11, 14; y = x + 5: 5, 6, 7, 8, 9; y = 6x − 5: 1, 7, 13, 19, 25; y = 5x: 0, 5, 10, 15, 20.
• Part B: x = 1: 6 × 1 − 2 = 4; x = 5: 6 × 5 − 2 = 28; x = 10: 6 × 10 − 2 = 58; the last x is 12 (70 + 2 = 72, and 72 ÷ 6 = 12).
• Part C: y = 3x − 1, 3 × 6 − 1 = 17; y = x + 7, 6 + 7 = 13; y = 10x + 5, 10 × 6 + 5 = 65.
• Part D: 6, 16, 26, 36; x goes up 2 each time, and 5 × 2 = 10. The rule is y = 7x + 3.

From a rule to a table

A. Fill each table

Put each x into the rule. Multiply first.

B. Gaps in a table

The rule is y = 6x − 2. Fill the gaps. For the last column, work backwards: which x gives y = 70?

C. From words to a rule

Write each rule with x and y, then work out y when x = 6.

1. Multiply x by 3, then subtract 1.   Rule: ____________   When x = 6, y = ________
2. Add 7 to x.   Rule: ____________   When x = 6, y = ________
3. Multiply x by 10, then add 5.   Rule: ____________   When x = 6, y = ________

D. Stretch

1. Fill the table for y = 5x − 4. The rule multiplies by 5, so why does the y row go up by 10?

   x	2	4	6	8
   y
2. Write a rule whose table, for x = 0, 1, 2, 3, gives y = 3, 10, 17, 24.   ____________________

Plotting the pairs

Each x with its y is a coordinate pair (x, y): across, then up. Students read and plot points, plot a table and see the points line up, and test whether a point fits a rule by substituting.

We are learning to

• read and write points as (x, y), across first, then up,
• plot the pairs from a table on the Cartesian plane,
• test whether a point fits a rule.

Success criteria

• I can plot a pair and its reverse, such as (2, 6) and (6, 2), in the right places.
• I can show by substituting whether a point is on a rule’s line.

You need

The worksheet, a ruler, a sharp pencil, and Monday’s table for y = 4x + 1.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after plotting the table; start the second with Part A.

Watch for these ideas

• Plotting (x, y) as (y, x). Say “across, then up” every time.
• Counting from 1 at the origin. The origin is 0 on both axes.
• Joining the points freehand. Use a ruler: a point off the line usually means a slip in the table.

Answers

• Part A: A (1, 6), B (6, 1), C (0, 5), D (7, 0), E (4, 4), F (2, 7).
• Part B: y = 2, 3, 4, 5, 6, 7; the points line up; at x = 3½, y = 5½.
• Part C: (2, 7) on (3 × 2 + 1 = 7); (5, 16) on (3 × 5 + 1 = 16); (4, 12) not on (3 × 4 + 1 = 13); (7, 21) not on (3 × 7 + 1 = 22); (0, 1) on (3 × 0 + 1 = 1).
• Part D: y = −3, −2, −1, 0, 1, 2; it crosses the x-axis at (3, 0); x = 0, 1, 2 give points below it.

Plotting the pairs

A. Read the points

B. From a table to points

C. On the line or not?

Substitute each x into y = 3x + 1. Is the point on the rule’s line?

1. (2, 7)   Working: ____________________   On the line? ______
2. (5, 16)   Working: ____________________   On the line? ______
3. (4, 12)   Working: ____________________   On the line? ______
4. (7, 21)   Working: ____________________   On the line? ______
5. (0, 1)   Working: ____________________   On the line? ______

D. Stretch

Growing patterns

A growing pattern can be counted into a table, and the picture shows why its rule works. Students count figures, fill tables, find each rule, point to its parts in the picture, and plot the counts.

We are learning to

• draw the next figure of a growing pattern and count it,
• fill a table from a pattern and write its rule,
• show each number in the rule in the picture.

Success criteria

• I can find the rule for a pattern and use it for figure 10.
• I can say why we plot a pattern’s points but do not join them.

You need

Pattern cards from cut-out sheet 1 (one set per pair), toothpicks or counters if you have them, the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the pattern cards; start the second with Part A.

Watch for these ideas

• Taking figure 1’s count as the constant. Squares in a row start at 4, but the rule is 3n + 1.
• Counting every figure from scratch instead of seeing what each new figure adds.
• Joining the plotted points. There is no figure 2½, so the points stay apart.

Answers

• Cards: 1 3n + 1, figure 10 has 31; 2 2n + 2, figure 10 has 22; 3 4n + 1, figure 10 has 41; 4 5n + 1, figure 10 has 51; 5 2n + 6, figure 10 has 26; 6 4n, figure 10 has 40.
• Part A: 7, 12, 17, 22, 27; rule 5n + 2; figure 10 has 52; 77 matchsticks make figure 15.
• Part B: (1, 7), (2, 12), (3, 17), (4, 22). They line up, but are not joined: only whole figures exist.
• Part C: Ana and Ben (figure 3 has 17); Kiri’s 7n gives 21, which fits figure 1 only.
• Part D: 1, 4, 9, 16; the steps grow (3, 5, 7), so the points curve upwards. Any figures with 5, 8, 11 parts.

Growing patterns

A. Squares two rows high

B. Plot the pattern

C. Three rules

For the pattern in Part A, Ana wrote 5n + 2, Ben wrote 2 + 5n, Kiri wrote 7n. Test each rule on figure 3. Who is right?

D. Stretch

1. A square of dots grows: figure n has n rows of n dots. Fill the table. Is the step steady? Would the points line up?

   figure n	1	2	3	4
   dots

2. Draw figures 1 to 3 of your own pattern with the rule 3n + 2.

From a table to a rule

A table can hand back its rule: the steady step in y for each step of 1 in x is the multiplier, and one pair then gives the constant. Students find rules, check them on a second column, and match rule cards to table cards.

We are learning to

• find the multiplier from the step in the y row,
• find the constant at x = 0, or by working back from one pair,
• check a rule on a second column before trusting it.

Success criteria

• I can find the rule for a table and check it on two columns.
• I can explain why the step is not the multiplier when x jumps by 2.

You need

Rule and table cards from cut-out sheet 2 (one set per pair) and the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the card match; start the second with Part A.

Watch for these ideas

• Taking the first y as the constant when the table starts at x = 1. Work back one step to x = 0.
• Taking the step as the multiplier when x jumps by 2. Halve the step first.
• Trusting a rule after one column. A second column catches a wrong rule.

Answers

• Part A: y = 3x + 6; y = 6x; y = 5x + 3; y = 6x − 4.
• Part B: y = x + 3 gives 5 at x = 2, but the table has 7; the rule is y = 3x + 1, and at x = 20, y = 3 × 20 + 1 = 61.
• Part C: up 8 for each 2, so 4 for each 1: y = 4x + 3; up 6 for each 2, so 3 for each 1; 10 − 6 = 4: y = 3x + 4.
• Part D: y = 20 − 3x; y = 2 when x = 6. At x = 0 the multiplied part is 0, so y is the constant.
• Cards: 1 D, 2 G, 3 B, 4 F, 5 H, 6 A, 7 E, 8 C.

From a table to a rule

A. Find the rule

Find the step in y for each step of 1 in x, then the constant. Check on a second column.

B. Check before you trust

Kai looked at this table and wrote y = x + 3, because 1 + 3 = 4. Check Kai’s rule on the second column. What is the right rule? Use it to find y when x = 20.

C. Steps of two

The x row jumps by 2 each time. Find each rule.

D. Stretch

1. Find the rule for this falling table. For which x is y = 2?

   x	0	1	2	3
   y	20	17	14	11
2. Explain why a table that starts at x = 0 shows the constant straight away.

Two rules on one plane

Plotting two rules on one set of axes shows what each number in a rule does: the multiplier sets how steep the line is, and the constant sets where it crosses the y-axis. Students plot, compare and predict lines.

We are learning to

• plot two rules on one Cartesian plane from their tables,
• explain what the multiplier and the constant do to a line,
• predict a line’s crossing point and steepness from its rule.

Success criteria

• I can say what is the same and what is different about two lines, using their rules.
• I can say where a line crosses the y-axis without plotting it.

You need

The worksheet, a ruler, two coloured pencils, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 4; start the second with problem 5.

Watch for these ideas

• Thinking a bigger constant makes a steeper line. The constant only lifts the line.
• Starting the line at the multiplier: y = 3x + 2 crosses the y-axis at 2, not 3.
• Bending a line to reach a point. A point off the line means a slip in the table.

Answers with working

1. y = 3x: 0, 3, 6, 9; y = 3x + 2: 2, 5, 8, 11.
2. two straight lines, one 2 above the other.
3. same steepness, up 3 for each 1 across; different heights: y = 3x + 2 is 2 higher at every x.
4. y = 3x at (0, 0), y = 3x + 2 at (0, 2); the constant.
5. C is y = x + 2, A is y = 2x + 2, B is y = 4x + 2: all cross at (0, 2), and the bigger multiplier is steeper.
6. at (0, 4); steeper, because 5 is more than 2.
7. y = x + 6: 6, 7, 8, 9, 10; y = 3x: 0, 3, 6, 9, 12; they cross at (3, 9).
8. up 18 for 3 across, so 6 for each 1: y = 6x + 1; for example (1, 7), as 6 × 1 + 1 = 7.

Two rules on one plane

Problems (7 and 8 are the stretch)

1. Fill both tables above.
2. Plot both rules on the grid, one colour each, and join each set of points with a ruler.
3. What is the same about the two lines, and what is different? Use numbers.
4. Where does each line cross the y-axis? Which number in each rule tells you?
5. The graph shows three lines. Which line is y = x + 2, which is y = 2x + 2, and which is y = 4x + 2? Explain how you know.
6. Without plotting, where does y = 5x + 4 cross the y-axis? Is it steeper or less steep than y = 2x + 4?
7. Stretch. Make tables for y = x + 6 and y = 3x for x = 0 to 4. Where would their lines cross?
8. Stretch. A line passes through (0, 1) and (3, 19). Find its rule and one more point on it.

Pattern cards

Cut out the 6 cards. Day 3: draw figure 4, fill a table for figures 1 to 4 and figure 10, and write the rule.

Rule and table cards

Cut out the 16 cards. Day 4: match each rule card to its table card, and show one row of working as proof.

Rules

Tables

What we know: Tables of values and the plane

Work on your own. Show one line of working for each answer.

5. Fill the table, then write the rule. Rule: ______________

   figure n	1	2	3	4
   matchsticks

6. How many matchsticks are in figure 10? ________ Which figure uses 87? ________
7. Find the rule for this table. ______________

   x	0	2	4	6
   y	5	11	17	23

8. Which line is steeper, y = 2x + 6 or y = 5x + 1? Which crosses the y-axis higher? Explain with the numbers in the rules. ________________________________

Answers and marking guide

Answers with working

1. 7, 11, 15, 19.
2. No: 4 × 5 + 7 = 27, not 26.
3. The points (0, 7), (1, 11), (2, 15), (3, 19) lie on one straight line.
4. Jo swapped the numbers: the pair is (3, 8), 3 across and 8 up.
5. 10, 17, 24, 31; the rule is 7n + 3 matchsticks.
6. 7 × 10 + 3 = 73; figure 12, as 87 − 3 = 84 and 84 ÷ 7 = 12.
7. y goes up 6 for each 2, so 3 for each 1, and y = 5 at x = 0: y = 3x + 5.
8. y = 5x + 1 is steeper (5 for each 1 across); y = 2x + 6 crosses higher, at (0, 6).

A quick three-level guide

Eight questions, four ideas. A student at standard fills a table from a rule or a pattern, plots the pairs across then up, and reads a rule back out of a table.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: rule to table, plotting, growing patterns, table to rule, and two rules on one plane.