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{{T}}
[[Category:English]]
[[Category:Slope and Rate of Change]]

{{BR}}
= Introduction =

When you walk up a ramp, watch a car climb a hill, compare distance with time, or read a graph, you are meeting the idea of '''rate of change'''. A rate of change tells you how much one quantity changes when another quantity changes. On a straight-line graph, this rate is called the '''slope'''.

This aiMOOC is designed for '''Grades 7–8'''. You will learn how to calculate slope from graphs, tables, points, equations, and real situations. You will also learn how to explain what the slope means, including its units and its sign.

A road-grade sign is a real-world example of slope. A 10% uphill grade means the road rises by 10 units for every 100 horizontal units. It does '''not''' mean that the road makes a 10-degree angle.

[[File:Latvia road sign 111.svg|350px|frameless|center]]

By the end of this course, you should be able to:
# [[English:Rate of change|Explain rate of change]]: Describe change in one quantity compared with change in another.
# [[English:Slope|Calculate slope]]: Use rise over run and the slope formula correctly.
# [[English:Linear function|Recognize constant rate of change]]: Decide when a relationship is linear.
# [[English:Graph of a function|Interpret graphs]]: Connect positive, negative, zero, and undefined slope to a graph.
# [[English:Linear equation|Connect equations and graphs]]: Explain the role of slope in equations such as y = mx + b.
# [[English:Mathematical modelling|Model real situations]]: Use slope to describe speed, cost, temperature change, elevation, and other rates.

{{#ev:youtube|https://www.youtube.com/watch?v=Iqws-qzyZwc|500|center}}

{{BR}}
= What Is Slope? =

'''Slope''' measures the steepness and direction of a line. You can think of it as a comparison between a vertical change and a horizontal change.

The most common phrase is:

'''slope = rise ÷ run'''

The '''rise''' is the change in the vertical direction, or the change in y. The '''run''' is the change in the horizontal direction, or the change in x.

In symbols:

<math>m=\frac{\Delta y}{\Delta x}</math>

The letter <math>m</math> is often used for slope. The symbol <math>\Delta</math> means "change in."

[[File:Graph of slope.png|500px|frameless|center]]

{{BR}}
== Rise and Run ==

Suppose you move from the point (2, 3) to the point (6, 11).

The horizontal change is:

<math>\Delta x=6-2=4</math>

The vertical change is:

<math>\Delta y=11-3=8</math>

So the slope is:

<math>m=\frac{8}{4}=2</math>

This means that for every 1 unit you move to the right, the line rises 2 units.

[[File:Diagram of gradient of linear function graph.svg|450px|frameless|center]]

A useful habit is to read slope as a sentence: '''"For every 1 unit of x, y changes by m units."''' This helps you connect the number to its meaning.

{{#ev:youtube|https://www.youtube.com/watch?v=MeU-KzdCBps|500|center}}

{{BR}}
== The Slope Formula from Two Points ==

If you know two points, <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math>, use:

<math>m=\frac{y_2-y_1}{x_2-x_1}</math>

The order must be consistent. If you subtract the second y-value from the first y-value, you must also subtract the second x-value from the first x-value.

For the points (1, 4) and (5, 12):

<math>m=\frac{12-4}{5-1}=\frac{8}{4}=2</math>

If you reverse both subtraction orders, you still get the same slope:

<math>m=\frac{4-12}{1-5}=\frac{-8}{-4}=2</math>

The mistake to avoid is reversing only one subtraction order.

{{BR}}
= Slope as Rate of Change =

A '''rate of change''' compares how one quantity changes in relation to another. When a graph shows a linear relationship, the slope is the constant rate of change.

For example, suppose a cyclist travels 12 kilometers in 3 hours at a constant speed. The rate of change of distance with respect to time is:

<math>\frac{12\text{ km}}{3\text{ h}}=4\text{ km/h}</math>

Here, the slope would be 4 if distance is on the y-axis and time is on the x-axis.

The units matter. A slope is not only a number. It can mean kilometers per hour, dollars per ticket, degrees Celsius per hour, centimeters per second, liters per minute, or another unit rate.

{{BR}}
== Worked Example: Cost per Item ==

A shop charges a constant amount per notebook. The table shows:

{| class="wikitable"
! Number of notebooks
! Total cost in dollars
|-
| 2
| 6
|-
| 4
| 12
|-
| 6
| 18
|}

Choose two rows. From 2 notebooks to 4 notebooks, the cost changes by 6 dollars while the number of notebooks changes by 2.

<math>m=\frac{12-6}{4-2}=\frac{6}{2}=3</math>

The slope is '''3 dollars per notebook'''. This is the rate of change.

{{BR}}
== Worked Example: Temperature Falling ==

At 1:00 p.m., the temperature is 18°C. At 5:00 p.m., it is 10°C.

<math>m=\frac{10-18}{5-1}=\frac{-8}{4}=-2</math>

The slope is '''-2°C per hour'''. The negative sign tells you that the temperature is decreasing.

{{BR}}
= Positive, Negative, Zero, and Undefined Slope =

The sign and value of a slope tell you how a line behaves from left to right.

{| class="wikitable"
! Type of slope
! What the graph does from left to right
! Example meaning
|-
| Positive
| Rises
| Distance increases as time passes
|-
| Negative
| Falls
| Temperature decreases as time passes
|-
| Zero
| Stays horizontal
| Water level remains constant
|-
| Undefined
| Is vertical
| x stays fixed while y changes
|}

[[File:Slopes of horizontal and vertical lines.png|500px|frameless|center]]

{{BR}}
== Positive Slope ==

A line with positive slope rises from left to right. If <math>m=3</math>, then y increases by 3 units for every increase of 1 unit in x.

For example, if a savings account grows by 5 dollars each week, a graph of total savings against weeks has a positive rate of change of 5 dollars per week, assuming no other deposits or withdrawals.

{{BR}}
== Negative Slope ==

A line with negative slope falls from left to right. If <math>m=-2</math>, then y decreases by 2 units for every increase of 1 unit in x.

You may write a negative slope in several equivalent ways:

<math>-\frac{2}{3}=\frac{-2}{3}=\frac{2}{-3}</math>

Keep the meaning clear: one variable increases while the other decreases.

{{BR}}
== Zero Slope ==

A horizontal line has slope 0 because its y-value does not change.

If the points are (2, 5) and (8, 5):

<math>m=\frac{5-5}{8-2}=\frac{0}{6}=0</math>

The denominator is not zero, so the calculation is defined.

{{BR}}
== Undefined Slope ==

A vertical line has undefined slope because its x-value does not change.

If the points are (4, 2) and (4, 9):

<math>m=\frac{9-2}{4-4}=\frac{7}{0}</math>

Division by zero is undefined. Therefore, a vertical line does not have a numerical slope.

{{BR}}
= Finding Slope from a Graph =

To find slope from a graph:

# Choose two clear points that lie exactly on the line.
# Find the vertical change between the points.
# Find the horizontal change between the points.
# Divide vertical change by horizontal change.
# Simplify the fraction and interpret the sign.

A helpful strategy is to draw or imagine a right triangle between the two points. The vertical leg represents rise, and the horizontal leg represents run.

If a graph rises 6 units while moving 3 units to the right:

<math>m=\frac{6}{3}=2</math>

If it falls 4 units while moving 5 units to the right:

<math>m=\frac{-4}{5}</math>

You do not have to begin at the y-intercept. Any two exact points on the same straight line give the same slope.

{{BR}}
= Finding Rate of Change from a Table =

A table can show whether a rate of change is constant.

Consider this table:

{| class="wikitable"
! x
! y
|-
| 0
| 2
|-
| 1
| 5
|-
| 2
| 8
|-
| 3
| 11
|}

Every time x increases by 1, y increases by 3. The rate of change is constant:

<math>m=\frac{3}{1}=3</math>

Because the rate of change is constant, these points lie on a straight line.

Now consider:

{| class="wikitable"
! x
! y
|-
| 0
| 1
|-
| 1
| 3
|-
| 2
| 7
|-
| 3
| 13
|}

The changes in y are 2, 4, and 6 while x increases by 1 each time. The rate is not constant, so this relationship is not linear.

{{BR}}
== When x Does Not Increase by 1 ==

Do not assume that the change in x is always 1.

Suppose the table contains the points (2, 7), (5, 16), and (8, 25). From the first row to the second:

<math>\Delta x=5-2=3</math>

<math>\Delta y=16-7=9</math>

So:

<math>m=\frac{9}{3}=3</math>

Check the next pair:

<math>m=\frac{25-16}{8-5}=\frac{9}{3}=3</math>

The constant rate of change is 3.

{{BR}}
= Slope in an Equation =

A common form for a linear equation is:

<math>y=mx+b</math>

This is called '''slope-intercept form'''.

In this form:
# <math>m</math> is the slope.
# <math>b</math> is the y-intercept, the y-value when x = 0.

[[File:Graph describing a linear function.svg|500px|frameless|center]]

For the equation:

<math>y=4x-3</math>

the slope is 4 and the y-intercept is -3.

For:

<math>y=-\frac{1}{2}x+6</math>

the slope is <math>-\frac{1}{2}</math> and the y-intercept is 6.

{{#ev:youtube|https://www.youtube.com/watch?v=IL3UCuXrUzE|500|center}}

{{BR}}
== Graphing from Slope-Intercept Form ==

Suppose:

<math>y=\frac{2}{3}x+1</math>

Start at the y-intercept, (0, 1). Then use the slope <math>\frac{2}{3}</math>: move 3 units to the right and 2 units up. Mark the new point and draw the line through the points.

A negative slope works similarly. For <math>m=-\frac{3}{4}</math>, you can move 4 units right and 3 units down.

{{BR}}
= Proportional Relationships and Slope =

A proportional relationship has the form:

<math>y=kx</math>

The constant <math>k</math> is the constant of proportionality. It is also the slope.

Because there is no added constant, the graph passes through the origin (0, 0).

For example:

<math>y=2.5x</math>

has slope 2.5 and y-intercept 0.

A linear relationship does not have to be proportional. For example:

<math>y=2.5x+4</math>

is linear with the same slope 2.5, but it is not proportional because the graph does not pass through the origin.

{{BR}}
= Real-World Meaning of Slope =

Slope becomes powerful when you connect it to context. Always ask two questions:

# What quantity is on the y-axis?
# What quantity is on the x-axis?

Then read the slope as "change in y per change in x."

Examples include:
# [[English:Speed|Distance and time]]: kilometers per hour.
# [[English:Unit price|Cost and quantity]]: dollars per item.
# [[English:Temperature|Temperature and time]]: degrees per hour.
# [[English:Elevation|Height and horizontal distance]]: meters per meter.
# [[English:Flow rate|Volume and time]]: liters per minute.

{{BR}}
== Ramps and Accessibility ==

A ramp has a physical slope. You can estimate it by measuring its vertical rise and horizontal run.

[[File:Wheelchair ramp.jpg|400px|frameless|center]]

For example, if a ramp rises 0.5 meters over a horizontal run of 6 meters:

<math>m=\frac{0.5}{6}\approx0.083</math>

As a percentage grade:

<math>0.083\times100\%\approx8.3\%</math>

This mathematical calculation describes the ramp's grade. Real accessibility requirements depend on the laws and standards that apply in a particular location, so a classroom calculation should not be treated as an official compliance check.

{{BR}}
== Percent Grade and Angle Are Different ==

Percent grade compares rise with horizontal run:

<math>\text{percent grade}=\frac{\text{rise}}{\text{run}}\times100\%</math>

A 10% grade means:

<math>\frac{10}{100}=0.10</math>

It does not mean 10 degrees.

[[File:GeoGradient.svg|400px|frameless|center]]

At this level, you do not need trigonometry to work with percent grade. The important idea is that grade is another way to express a rise-to-run ratio.

{{BR}}
= Constant and Nonconstant Rates of Change =

A nonvertical straight line has a constant slope. That means the same change in x always produces the same proportional change in y.

A curve usually has a changing rate of change. If you choose different pairs of points on a curved graph, you may get different rates.

For Grades 7–8, a useful test for a table is:
# Compare the changes in y.
# Compare the matching changes in x.
# Divide each change in y by its matching change in x.
# If the ratios are equal, the data have a constant rate of change.

A relationship can be increasing without having a constant rate. "Always going up" does not automatically mean "linear."

{{BR}}
= Comparing Two Linear Relationships =

You may be asked to compare two lines shown in different forms: one as a graph, one as a table, or one as an equation.

Suppose Line A is given by:

<math>y=3x+2</math>

Its slope is 3.

Line B is shown in a table:

{| class="wikitable"
! x
! y
|-
| 1
| 4
|-
| 3
| 8
|}

Its slope is:

<math>m=\frac{8-4}{3-1}=\frac{4}{2}=2</math>

Line A has the greater slope, so its y-value increases faster for each additional unit of x.

Be careful: a line can start higher but grow more slowly. The y-intercept tells you where it starts; the slope tells you how fast it changes.

{{BR}}
= Common Mistakes and How to Avoid Them =

'''Mistake 1: Mixing subtraction orders.''' If you calculate <math>y_2-y_1</math>, also calculate <math>x_2-x_1</math> in the same point order.

'''Mistake 2: Forgetting units.''' A slope of 5 may mean 5 kilometers per hour, 5 dollars per item, or something else. State the units when the context has units.

'''Mistake 3: Confusing slope with y-intercept.''' In <math>y=mx+b</math>, <math>m</math> is slope and <math>b</math> is the y-intercept.

'''Mistake 4: Calling every increasing graph linear.''' A curved increasing graph can have a changing rate.

'''Mistake 5: Saying a vertical line has slope 0.''' A horizontal line has slope 0. A vertical line has undefined slope.

'''Mistake 6: Thinking a negative slope must be written with a negative rise.''' You can instead use a positive rise and a negative run, as long as the ratio is negative.

{{BR}}
= Problem-Solving Strategy =

When you meet a slope or rate-of-change problem, use this routine:

# Identify the two quantities and their units.
# Decide which quantity is x and which is y.
# Choose two points or two rows.
# Compute the change in y and the change in x.
# Divide to find the rate of change.
# Simplify the result.
# Interpret the sign.
# Write the meaning in a complete sentence with units.

For example, if the slope is -4 dollars per day, do not stop at "-4." Explain that the amount decreases by 4 dollars for each additional day.

{{BR}}
= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

'''What does slope compare?'''
(Change in y with change in x)
(!The x-intercept with the y-intercept)
(!The largest x-value with the smallest y-value)
(!The total number of plotted points)

{{E}}
<br>

{{MC}}

'''A line rises 8 units while running 4 units to the right. What is its slope?'''
(2)
(!4)
(!8)
(!12)

{{E}}
<br>

{{MC}}

'''What kind of slope does a horizontal line have?'''
(Zero slope)
(!Positive slope)
(!Negative slope)
(!Undefined slope)

{{E}}
<br>

{{MC}}

'''What kind of slope does a vertical line have?'''
(Undefined slope)
(!Zero slope)
(!Positive slope)
(!Negative slope)

{{E}}
<br>

{{MC}}

'''The points are 1,3 and 5,11. What is the slope?'''
(2)
(!3)
(!4)
(!8)

{{E}}
<br>

{{MC}}

'''If a temperature drops 6 degrees in 3 hours at a constant rate, what is the rate of change?'''
(-2 degrees per hour)
(!2 degrees per hour)
(!-3 degrees per hour)
(!6 degrees per hour)

{{E}}
<br>

{{MC}}

'''In y = 4x + 7, what is the slope?'''
(4)
(!7)
(!11)
(!-4)

{{E}}
<br>

{{MC}}

'''Which statement describes a positive slope?'''
(y increases as x increases)
(!y stays constant as x increases)
(!x stays constant while y changes)
(!y decreases as x increases)

{{E}}
<br>

{{MC}}

'''A table has equal changes in x and matching equal ratios of change in y to change in x. What does this show?'''
(A constant rate of change)
(!An undefined rate of change)
(!A vertical graph)
(!A changing rate of change)

{{E}}
<br>

{{MC}}

'''A road has a 10 percent grade. What does the 10 percent describe?'''
(Rise compared with horizontal run)
(!The angle measured in degrees)
(!The road length compared with time)
(!The speed limit on the road)

{{E}}
<br>

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| Slope || Vertical change divided by horizontal change
|-
| Rise || Change in the y-direction
|-
| Run || Change in the x-direction
|-
| Positive || A line rises from left to right
|-
| Negative || A line falls from left to right
|-
| Horizontal || A line with zero steepness
|-
| Vertical || A line whose steepness is undefined
|}
{{E}}
<br>

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! Match the correct terms.
! Topic
|-
| '''Positive slope'''
| A line rises from left to right
|-
| '''Negative slope'''
| A line falls from left to right
|-
| '''Zero slope'''
| A horizontal line
|-
| '''Undefined slope'''
| A vertical line
|-
| '''Constant rate'''
| The same change ratio throughout a linear relationship
|}
{{E}}

<br>
Match each type of rate or slope to the description that belongs with it.
<br />

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| Gradient || What is another word often used for slope?
|-
| Increase || What happens to y on a line with positive slope as x grows?
|-
| Decrease || What happens to y on a line with negative slope as x grows?
|-
| Horizontal || What kind of line has zero slope?
|-
| Vertical || What kind of line has undefined slope?
|-
| Intercept || What word names the point where a graph crosses an axis?
|}
{{E}}
<br>

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Slope+and+Rate+of+Change </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.'''<br>
|type="{}"}
Slope compares the change in y with the { change in x }. The vertical change is often called the { rise }. The horizontal change is often called the { run }. A straight line has a { constant } rate of change. A line that rises from left to right has a { positive } slope. A horizontal line has slope { zero }. A vertical line has an { undefined } slope. In the equation y = mx + b, the letter m represents the { slope }. In a real situation, a slope should be explained with its { units }.
</quiz>

<br>

{{BR}}
= Open-Ended Tasks =

{{BR}}
=== Easy ===
# [[English:Slope Photo Hunt|Slope Photo Hunt]]: Find or photograph four examples of slopes in everyday life, such as ramps, roofs, stairs, or roads, and explain which seems steepest and why.
# [[English:Graph Poster|Graph Poster]]: Create a one-page poster that shows positive, negative, zero, and undefined slope with one clear graph for each case.
# [[English:Rate Story|Rate Story]]: Write a short real-world story that has a constant rate of change, create a matching table, and explain the units of the rate.
# [[English:Slope Mini-Video|Slope Mini-Video]]: Produce a one-minute video that teaches rise over run using a drawing, grid, or simple objects.

{{BR}}
=== Standard ===
# [[English:Ramp Measurement|Ramp Measurement]]: Measure the rise and horizontal run of a safe ramp or sloped surface, calculate its slope and percent grade, and describe your measurement method.
# [[English:Interview About Rates|Interview About Rates]]: Interview someone who uses rates in work or daily life, such as a driver, builder, coach, shop worker, or scientist, and connect one example to slope.
# [[English:Motion Data Experiment|Motion Data Experiment]]: Move a toy vehicle or walk a measured distance at a steady pace, record distance at equal time intervals, graph the data, and estimate the rate of change.
# [[English:Compare Two Plans|Compare Two Plans]]: Invent two linear pricing plans with different starting fees and rates, graph both plans, and explain when one plan becomes better than the other.

{{BR}}
=== Advanced ===
# [[English:Accessibility Ramp Investigation|Accessibility Ramp Investigation]]: Examine a ramp in your community, calculate its approximate grade from your own measurements, and discuss why mathematical slope matters for design without claiming legal compliance.
# [[English:Local Data Investigation|Local Data Investigation]]: Collect a small real dataset that changes over time, decide whether a linear model is reasonable, calculate a rate of change, and explain the limits of your model.
# [[English:Model Comparison Report|Model Comparison Report]]: Compare one linear and one non-linear relationship using tables and graphs, then write a report explaining how constant rate of change distinguishes them.
# [[English:Design Challenge|Design Challenge]]: Design a scale drawing of a path, track, or ramp that must meet a chosen rise and run constraint, calculate its slope in at least two equivalent forms, and justify your design choices.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Interpret a Mixed Representation|Interpret a Mixed Representation]]: Given one linear relationship as a graph and another as a table, calculate both rates of change and argue which quantity changes faster.
# [[English:Explain an Error|Explain an Error]]: Analyze a worked solution in which subtraction order is mixed in the slope formula, identify the exact error, and correct it with reasoning.
# [[English:Build a Linear Model|Build a Linear Model]]: From a real-world starting value and constant rate, write an equation in slope-intercept form, create a table, and explain what both parameters mean.
# [[English:Evaluate a Claim|Evaluate a Claim]]: Decide whether the statement "every increasing graph has a constant positive slope" is true or false and support your answer with examples.
# [[English:Transfer to Percent Grade|Transfer to Percent Grade]]: Use a measured rise and run to calculate decimal slope and percent grade, then explain why percent grade is not the same as an angle in degrees.
# [[English:Reason About Units|Reason About Units]]: Compare two numerical slopes that use different units and explain why the numbers alone are not enough to decide which situation is changing faster.

<br>
<br>

{{BR}}
= Evidence of Learning =

'''Knowledge:''' You can explain slope as a ratio of vertical change to horizontal change, distinguish positive, negative, zero, and undefined slope, and connect constant slope with linear relationships.

'''Skills:''' You can calculate slope from two points, graphs, tables, and equations; identify units; compare rates; graph a line from slope-intercept form; and check whether a table shows a constant rate of change.

'''Products:''' Strong evidence may include accurate graphs, measurement records, a mathematical model, a short explanatory video, a data investigation, or a clearly labeled poster.

'''Reasoning:''' You can explain why two points on the same straight line give the same slope, why a vertical line has undefined slope, and why a curved graph may not have a constant rate.

'''Transfer:''' You can use slope to interpret unfamiliar situations involving speed, price, temperature, elevation, ramps, flow, or other paired quantities and communicate what the rate means in context.

<br>
<br>

{{BR}}
= OERs on the Topic =

<iframe> https://en.m.wikipedia.org/wiki/Slope_(mathematics) </iframe>

<br>

{{BR}}
= Linked Learning Areas =

Slope and rate of change connect arithmetic, ratios, coordinate geometry, algebra, functions, data interpretation, and mathematical modelling. These links can help you continue learning from the same central idea.

{| align=center
{{:D-Tab}}
'''[[English:Slope and Rate of Change|Slope and Rate of Change]]'''
# [[English:Slope|Slope]]
# [[English:Rate of change|Rate of change]]
# [[English:Coordinate system|Coordinate system]]
# [[English:Ratio|Ratio]]
# [[English:Unit rate|Unit rate]]
# [[English:Linear function|Linear function]]
# [[English:Proportionality|Proportionality]]
# [[English:Slope-intercept form|Slope-intercept form]]
# [[English:Graph of a function|Graph of a function]]
# [[English:Mathematical modelling|Mathematical modelling]]
|}

{{BR}}
= aiMOOC Projects =
[[Category:English]]
[[Category:Slope and Rate of Change]]
[[Category:Mathematics]]
[[Category:Algebra]]
[[Category:Linear functions]]
[[Category:Grades 7-8]]
[[Category:STEM]]
[[Category:AI_MOOC]]
[[Category:GPT aiMOOC]]
{{MT}}

English:Slope and Rate of Change - Versionsgeschichte

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