Grade 8 Equivalent Linear Representations: Unit Mastery Quiz with Full Solutions

Purpose

Use this quiz after studying Grade 8 linear patterns and relations. The focus is not on learning the topic for the first time, but on showing that you can recognize and create equivalent linear representations: table, graph, word rule, and equation.

This assessment assumes you already have basic fluency with expressions, ratios, and simple linear modeling. For adjacent review, see:

• rea.m08.algebra.expressions-and-substitution.expression-parts.unit-mastery-quiz
• rea.m08.number.ratios-rates-proportions.equivalent-ratios.unit-mastery-quiz
• rea.m08.algebra.solving-linear-equations.linear-modeling.unit-mastery-quiz

Coverage

The 10 questions sample the full unit:

• identifying constant rate of change
• finding initial value
• deciding whether two representations are equivalent
• writing equations from tables, graphs, and words
• making tables from equations
• interpreting slope and intercept in context
• spotting and repairing common representation errors

Recommended Scoring

Total: 20 marks

• Most questions are worth 2 marks.
• Award 1 mark when the method is correct but arithmetic or reading is partly off.

Mastery Threshold Recommendation

Recommended mastery threshold: 16/20 (80%), with these conditions:

• the learner must correctly answer at least 4 of Questions 5-10, because those require converting between representations rather than only reading one form;
• no more than one error may involve confusing slope/rate of change with initial value.

If a learner scores below mastery, reteach by representation pair:

• table <-> equation
• graph <-> equation
• words <-> equation
• context <-> interpretation of slope and intercept

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Quiz

Question 1 (2 marks)

A table shows a linear relation.

x	0	1	2	3
y	5	8	11	14

1. Find the rate of change.
2. Write an equation for the relation.

Question 2 (2 marks)

A relation is given by the equation y = 4x - 3.

1. Complete a table for x = 0, 1, 2, 3.
2. State the initial value.

Question 3 (2 marks)

A graph of a line passes through (0, 2) and (4, 10).

1. Find the slope.
2. Write the equation.

Question 4 (2 marks)

A pattern starts at 7 and increases by 6 each step.

1. Write the verbal rule in words.
2. Write the equation relating step number x to value y.

Question 5 (2 marks)

Do these two representations describe the same linear relation? Explain.

• Equation: y = 2x + 1
• Table:

x	0	1	2	3
y	1	3	5	8

Question 6 (2 marks)

A taxi fare is $3 to start plus $2 for each kilometre.

1. Write an equation for total cost C in terms of kilometres k.
2. What does the 3 mean? What does the 2 mean?

Question 7 (2 marks)

A student says that the table below matches y = 3x + 2.

x	1	2	3	4
y	5	8	11	14

Is the student correct? Justify your answer using substitution or rate-and-start reasoning.

Question 8 (2 marks)

Write a table of values for the relation y = -2x + 6 using x = 0, 1, 2, 3. Then state whether the relation is increasing or decreasing.

Question 9 (2 marks)

Two students describe the same line in different ways:

• Student A: "It crosses the y-axis at 4 and goes up 5 for every 2 across."
• Student B: y = 2.5x + 4

Are these equivalent representations? Explain why or why not.

Question 10 (2 marks)

A plant is 12 cm tall now and grows 1.5 cm each week.

1. Write an equation for height h after w weeks.
2. Find the height after 8 weeks.
3. Explain how the equation, a table, and a graph would all show the same relation.

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Full Marking Solutions

Solution 1

The y-values increase by +3 each time x increases by 1, so the rate of change is 3.

When x = 0, y = 5, so the initial value is 5.

Equation: y = 3x + 5

Marking:

• 1 mark for rate 3
• 1 mark for equation y = 3x + 5

Solution 2

Substitute each x-value into y = 4x - 3:

• x = 0: y = 4(0) - 3 = -3
• x = 1: y = 4(1) - 3 = 1
• x = 2: y = 8 - 3 = 5
• x = 3: y = 12 - 3 = 9

Table:

x	0	1	2	3
y	-3	1	5	9

Initial value: -3

Marking:

• 1 mark for a correct table
• 1 mark for initial value -3

Solution 3

Slope: m = (10 - 2) / (4 - 0) = 8 / 4 = 2

The point (0, 2) shows the y-intercept is 2.

Equation: y = 2x + 2

Marking:

• 1 mark for slope 2
• 1 mark for equation y = 2x + 2

Solution 4

Verbal rule: start at 7 and add 6 for each step.

Since the value at step 0 is 7 and the rate is 6, the equation is: y = 6x + 7

Marking:

• 1 mark for a correct verbal rule
• 1 mark for equation y = 6x + 7

Solution 5

Check the equation y = 2x + 1 against the table.

• If x = 0, y = 1 which matches.
• If x = 1, y = 3 which matches.
• If x = 2, y = 5 which matches.
• If x = 3, the equation gives y = 7, but the table shows 8.

So the two representations are not equivalent.

Explanation: equivalent representations must match for all corresponding values, not just some of them.

Marking:

• 1 mark for saying they are not equivalent
• 1 mark for a valid check and explanation

Solution 6

The starting fee is $3, and the cost increases $2 for each kilometre.

Equation: C = 2k + 3

Interpretation:

• 3 is the starting cost before any kilometres are travelled
• 2 is the cost per kilometre

Marking:

• 1 mark for equation C = 2k + 3
• 1 mark for interpreting both numbers correctly

Solution 7

Test y = 3x + 2 with the table.

• If x = 1, y = 3(1) + 2 = 5
• If x = 2, y = 3(2) + 2 = 8
• If x = 3, y = 3(3) + 2 = 11
• If x = 4, y = 3(4) + 2 = 14

Every pair matches, so the student is correct.

Another valid explanation: the y-values go up by 3 each time, so slope is 3, and working backward from (1, 5) gives start value 2.

Marking:

• 1 mark for saying the student is correct
• 1 mark for valid justification

Solution 8

Substitute into y = -2x + 6:

• x = 0: y = 6
• x = 1: y = 4
• x = 2: y = 2
• x = 3: y = 0

Table:

x	0	1	2	3
y	6	4	2	0

Because the y-values go down as x increases, the relation is decreasing.

Marking:

• 1 mark for a correct table
• 1 mark for identifying it as decreasing

Solution 9

Student A says the line crosses the y-axis at 4 and rises 5 for every 2 across. That means slope m = 5/2 = 2.5 and intercept b = 4.

So Student A's description becomes: y = 2.5x + 4

This is exactly Student B's equation, so the representations are equivalent.

Marking:

• 1 mark for saying they are equivalent
• 1 mark for connecting 5/2 to 2.5 and identifying intercept 4

Solution 10

The plant starts at 12 cm and grows 1.5 cm each week.

Equation: h = 1.5w + 12

After 8 weeks: h = 1.5(8) + 12 = 12 + 12 = 24

So the height is 24 cm.

How the representations agree:

• Equation: shows growth rule directly, h = 1.5w + 12
• Table: each week, height increases by 1.5, starting from 12
• Graph: a straight line crossing the vertical axis at 12 with slope 1.5

These are equivalent because they all show the same starting height and the same constant rate of growth.

Marking:

• 1 mark for equation h = 1.5w + 12
• 1 mark for 24 cm and explanation of equivalence across forms

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Common Errors to Watch For

• Confusing slope with initial value. Repair: ask, "What happens when x = 0?" That gives the initial value.
• Checking only one row when deciding equivalence. Repair: equivalent representations must agree for all corresponding values.
• Using the wrong sign for decreasing relations. Repair: if values go down as x goes up, the slope must be negative.
• Reading a graph's rise correctly but forgetting run. Repair: slope is rise/run, not just rise.
• Starting the pattern at step 1 when the equation uses step 0 as the intercept. Repair: decide clearly what x means before writing the equation.

Interpreting Results

• 18-20 marks: strong unit mastery; ready for more formal linear-relation work.
• 16-17 marks: meets mastery; minor repair only.
• 12-15 marks: partial understanding; reteach conversions between representations.
• 0-11 marks: substantial reteaching needed, especially slope, intercept, and equivalence checks.

Suggested Next Links

After this quiz, move naturally to linear modeling and later Grade 9 work connecting equivalent forms of linear relations. For adjacent Grade 8 support, use the linked records instead of repeating prerequisite drills here.