Equivalent Linear Representations

Grade level: Grade 8 (Canadian curriculum: BC/Ontario-aligned)

Place in the unit

This lesson develops the central idea from rea.m08.algebra.patterns-and-relations.representation-changes.overview: the goal is to express the same linear relation in different forms, not to invent a new one. It also builds on the broader unit ideas in rea.m08.algebra.patterns-and-relations and connects naturally to contextual modelling in rea.m08.algebra.patterns-and-relations.context-models.overview.

Learning goals

By the end of this lesson, you should be able to:

• recognize whether a word rule, table, equation, and graph represent the same linear relation;
• translate between these forms while keeping the rate of change and starting value unchanged;
• explain what the slope and y-intercept mean in context;
• check whether two representations are equivalent.

Big idea

A linear relation is like one straight story told in four languages:

• words;
• table of values;
• equation;
• graph.

If the relation is equivalent across these forms, then all of the following must match:

• the same independent and dependent variables;
• the same starting value;
• the same constant rate of change;
• the same ordered pairs.

Changing the form is allowed. Changing the pattern is not.

Intuition first

Imagine a taxi fare:

• there is an amount you pay before moving;
• then the cost increases by the same amount for each kilometre.

You can describe that situation in words, list values in a table, write an equation, or draw a straight line. Each form shows the same pattern from a different angle.

A helpful visual picture:

• the starting value is where the line crosses the vertical axis;
• the rate of change is the "rise for each run" as you move across the graph;
• in a table, the rate of change is the amount added to the output whenever the input increases by 1;
• in words, it is often signalled by phrases like "starts at" and "for each".

Core vocabulary and notation

Let:

• x = input, independent variable;
• y = output, dependent variable.

A linear relation is often written as:

• y = mx + b

where:

• m is the constant rate of change (slope);
• b is the starting value (y-intercept), meaning the value of y when x = 0.

Representation checklist

Two linear representations are equivalent only if they produce the same points.

Check these three features:

1. Does the relation have the same variables and meaning?
2. Is the rate of change the same?
3. Is the starting value the same?

If any one of these changes, the representation is not equivalent.

How each representation shows the same relation

1. Words

Look for:

• starting value: "begins at," "starts with," "initially," "flat fee of";
• rate of change: "each time," "for every 1," "increases by," "decreases by".

Example wording:

• "The cost starts at $4 and increases by $2 for each ride."

2. Table

Look for:

• equal changes in y for equal changes in x;
• the value when x = 0, if shown.

Example:

x	0	1	2	3
y	4	6	8	10

Here:

• starting value is 4;
• rate of change is +2 each time x increases by 1.

3. Equation

In y = mx + b:

• m tells how steep the line is and how fast the output changes;
• b tells where the graph crosses the y-axis.

Example:

• y = 2x + 4

4. Graph

On the graph of a linear relation:

• the graph is a straight line;
• the y-intercept is (0, b);
• the slope m tells how many units up/down the line moves when x increases.

For y = 2x + 4:

• start at (0, 4);
• move right 1 and up 2 to get more points.

Procedures

Procedure A: From words to equation, table, and graph

1. Identify the input and output.
2. Find the starting value.
3. Find the constant rate of change.
4. Write the equation y = mx + b.
5. Substitute several x-values to make a table.
6. Plot the points and draw a straight line.
7. Check that the graph and table match the wording.

Procedure B: From a table to an equation

1. Check that the change in y is constant for equal changes in x.
2. Find the rate of change m.
3. Find the value of y when x = 0; that is b.
4. Write y = mx + b.
5. Test the equation with one or two points from the table.

Procedure C: From a graph to an equation

1. Find the y-intercept (0, b).
2. Choose another clear point on the line.
3. Compute slope as rise/run.
4. Write y = mx + b.
5. Check using one plotted point.

Procedure D: Decide whether two representations are equivalent

1. Compare starting values.
2. Compare rates of change.
3. Test one or two ordered pairs.
4. If all match, the representations are equivalent.

Worked examples

Example 1: Words to all other forms

A music app charges a monthly fee of $5, plus $3 for each song downloaded.

Step 1: Define variables.

• x = number of songs downloaded
• y = total cost in dollars

Step 2: Find the starting value. Even with 0 songs, the monthly fee is $5. So b = 5.

Step 3: Find the rate of change. The cost increases by $3 for each song. So m = 3.

Step 4: Write the equation. y = 3x + 5

Step 5: Make a table.

x	0	1	2	4
y	5	8	11	17

Calculations:

• if x = 0, y = 3(0) + 5 = 5
• if x = 1, y = 3(1) + 5 = 8
• if x = 2, y = 3(2) + 5 = 11
• if x = 4, y = 3(4) + 5 = 17

Step 6: Describe the graph.

• The line crosses the y-axis at (0, 5).
• From there, every step right 1 goes up 3.
• Points such as (1, 8) and (2, 11) lie on the line.

Conclusion: The words, table, equation y = 3x + 5, and graph all represent the same linear relation.

Example 2: Table to equation and words

A table is given:

x	0	2	4	6
y	7	11	15	19

Step 1: Check for constant change. As x increases by 2, y increases by 4.

So the rate per 1 unit of x is: m = 4/2 = 2

Step 2: Find the starting value. When x = 0, y = 7. So b = 7.

Step 3: Write the equation. y = 2x + 7

Step 4: Verify with another table value. For x = 4: y = 2(4) + 7 = 8 + 7 = 15 This matches the table.

Step 5: Write a word description. A correct word rule is:

• "The output starts at 7 and increases by 2 for each increase of 1 in the input."

Graph description:

• y-intercept (0, 7)
• slope 2
• straight line through the table points

Example 3: Graph to equation and table

A line crosses the y-axis at -2 and also passes through the point (3, 4).

Step 1: Identify the y-intercept. Since the line crosses the y-axis at -2, we have b = -2.

Step 2: Find the slope. Use the points (0, -2) and (3, 4).

rise = 4 - (-2) = 6

run = 3 - 0 = 3

m = rise/run = 6/3 = 2

Step 3: Write the equation. y = 2x - 2

Step 4: Build a table.

x	0	1	2	3
y	-2	0	2	4

Calculations:

• x = 0: y = 2(0) - 2 = -2
• x = 1: y = 2(1) - 2 = 0
• x = 2: y = 2(2) - 2 = 2
• x = 3: y = 2(3) - 2 = 4

Step 5: Write the word rule.

• "The output starts at -2 and increases by 2 for each increase of 1 in the input."

Conclusion: A graph, equation, table, and word statement can still be equivalent even when the starting value is negative.

Example 4: Decide whether two representations are equivalent

Representation A:

• Equation: y = 4x + 1

Representation B:

x	0	1	2	3
y	1	5	9	13

Representation C:

• Word rule: "Start at 4 and add 1 each time."

Step 1: Compare A and B. Equation A has:

• starting value 1
• rate of change 4

Table B has:

• when x = 0, y = 1, so starting value 1
• outputs increase 1 -> 5 -> 9 -> 13, so change is +4

A and B match. So A and B are equivalent.

Step 2: Compare A and C. Equation A means:

• start at 1
• add 4 each time

Word rule C says:

• start at 4
• add 1 each time

These are reversed. So C is not equivalent to A.

Step 3: Confirm with a test value. For x = 2 in A: y = 4(2) + 1 = 9

Using C:

• if you start at 4 and add 1 each time, then at x = 2 the output would be 6

Since 9 != 6, they are not equivalent.

Conclusion: Equivalent representations must keep both the starting value and the rate of change the same.

Example 5: Context with decreasing relation

A container begins with 20 L of water and loses 3 L each minute.

Step 1: Define variables.

• x = time in minutes
• y = amount of water in litres

Step 2: Starting value. At time 0, there are 20 L. So b = 20.

Step 3: Rate of change. The amount decreases by 3 L each minute. So m = -3.

Step 4: Equation. y = -3x + 20

Step 5: Table.

x	0	1	2	5
y	20	17	14	5

Step 6: Graph description.

• Start at (0, 20).
• Move right 1 and down 3 each time.
• The line slopes downward because the quantity is decreasing.

Conclusion: Equivalent linear representations also work for decreasing patterns, as long as the rate of change remains constant.

Common misconceptions and repairs

Misconception 1

"If two representations use the same numbers, they must be equivalent."

Repair: The numbers must play the same roles. In y = 4x + 1, 4 is the rate and 1 is the starting value. Swapping them changes the relation.

Misconception 2

"The first number in a table is always the slope."

Repair: Slope is not a table entry. It is the change in y divided by the change in x.

Misconception 3

"A straight-line graph always means the relation starts at 0."

Repair: A line may cross the y-axis above, below, or at 0. The crossing point gives the starting value.

Misconception 4

"Decreasing relations are not linear."

Repair: They are linear if the change is constant. A negative slope still makes a straight line.

Misconception 5

"Equivalent means the equations must look identical."

Repair: Equivalent means they describe the same ordered pairs. A table, a graph, and an equation can all be equivalent even though they look very different.

Practice

Solve each question, then check the answers.

Questions

1. A babysitter charges $12 to show up and $8 per hour. Write the equation and a table for 0, 1, 2, 3 hours.
2. A table shows:

x	0	1	2	3
y	-1	2	5	8

Write the equation and a word rule. 3. A graph crosses the y-axis at 6 and rises 1 for every run of 2. Write the equation. 4. Are these equivalent?

• Equation: y = -2x + 10
• Word rule: "Start at 10 and decrease by 2 for each increase of 1 in x."

5. Are these equivalent?

• Table: (0, 3), (1, 7), (2, 11)
• Equation: y = 3x + 4

Practice answers

1. Equation: y = 8x + 12

x	0	1	2	3
y	12	20	28	36

2. The change is +3, and when x = 0, y = -1. Equation: y = 3x - 1 Word rule: "Start at -1 and increase by 3 each time."

3. Slope m = 1/2, y-intercept b = 6. Equation: y = (1/2)x + 6

4. Yes. Both have starting value 10 and rate of change -2.

5. No. The table has slope 4 and starting value 3, so its equation is y = 4x + 3, not y = 3x + 4.

Self-check summary

When translating a linear relation, always ask:

• What is the starting value?
• What is the constant rate of change?
• Do all forms give the same ordered pairs?

If the answer to all three is yes, the representations are equivalent.

Connections forward

This lesson prepares you to:

• model real situations with linear equations and graphs;
• interpret slope and intercept in context;
• compare linear relations efficiently.

For broader unit structure, see rea.m08.algebra.patterns-and-relations. For contextual applications, continue with rea.m08.algebra.patterns-and-relations.context-models.overview.