8th Grade
Unit 4: Understanding Slope & Y-Intercept
5E Lesson Plan Math
Grade Level: 8
Lesson Title: Unit 4: Developing an
Understanding of Slope and Y-Intercept
THE TEACHING PROCESS
Subject Area: Mathematics
Lesson Length: 8 Days
Lesson Overview
This unit bundles student expectations that address using tables and graphs to
develop the understanding of slope and y-intercept. According to the Texas
Education Agency, mathematical process standards including application, a
problem-solving model, tools and techniques, communication, representations,
relationships, and justifications should be integrated (when applicable) with
content knowledge and skills so that students are prepared to use mathematics in
everyday life, society, and the workplace.
During this unit, students use similar right triangles to develop an understanding of
slope. This approach lends itself to the development of the formula for slope by
determining the ratio of the change in y- values and x-values is the same for any
two points on the same line. Students use data from a table or graph to determine
the rate of change or slope and the y-intercept.
Unit Objectives:
Students will…
use similar right triangles to develop an understanding of slope,
develop the formula for slope by determining the ratio of the change in y- values
and x-values on the same line, and
use data from a table or graph to determine the rate of change or slope and the yintercept.
Standards addressed:
TEKS:
8.1A Apply mathematics to problems arising in everyday life, society, and the
workplace.
8.1B Use a problem-solving model that incorporates analyzing given information,
formulating a plan or strategy, determining a solution, justifying the solution, and
evaluating the problem-solving process and the reasonableness of the solution.
8.1C Select tools, including real objects, manipulatives, paper and pencil, and
technology as appropriate, and techniques, including mental math, estimation, and
number sense as appropriate, to solve problems.
8.1D Communicate mathematical ideas, reasoning, and their implications using
multiple representations, including symbols, diagrams,graphs, and language as
appropriate.
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Unit 4: Understanding Slope & Y-Intercept
8th Grade
8.1E Create and use representations to organize, record, and communicate
mathematical ideas.
8.1F Analyze mathematical relationships to connect and communicate
mathematical ideas.
8.1G Display, explain, and justify mathematical ideas and arguments using precise
mathematical language in written or oral communication.
8.4A Use similar right triangles to develop an understanding that slope, m, given
as the rate comparing the change in y-values to the change in x-values, (y2 - y1)/
(x2 - x1), is the same for any two points (x1, y1) and (x2, y2) on the same line.
8.4C Use data from a table or graph to determine the rate of change or slope
and y-intercept in mathematical and real-world problems.
ELPS:
ELPS.c.1 The ELL uses language learning strategies to develop an awareness of
his or her own learning processes in all content areas. In order for the ELL to meet
grade-level learning expectations across the foundation and enrichment
curriculum, all instruction delivered in English must be linguistically accommodated
(communicated, sequenced, and scaffolded) commensurate with the student's
level of English language proficiency. The student is expected to:
ELPS.c.1A use prior knowledge and experiences to understand meanings in
English
ELPS.c.1B monitor oral and written language production and employ selfcorrective techniques or other resources
ELPS.c.1C use strategic learning techniques such as concept mapping, drawing,
memorizing, comparing, contrasting, and reviewing to acquire basic and gradelevel vocabulary
ELPS.c.1D speak using learning strategies such as requesting assistance,
employing non-verbal cues, and using synonyms and circumlocution (conveying
ideas by defining or describing when exact English words are not known)
ELPS.c.1E internalize new basic and academic language by using and reusing it
in meaningful ways in speaking and writing activities that build concept and
language attainment
ELPS.c.1F use accessible language and learn new and essential language in the
process
ELPS.c.1G demonstrate an increasing ability to distinguish between formal and
informal English and an increasing knowledge of when to use each one
commensurate with grade-level learning expectations
ELPS.c.1H develop and expand repertoire of learning strategies such as
reasoning inductively or deductively, looking for patterns in language, and
analyzing sayings and expressions commensurate with grade-level learning
expectations.
ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and
electronic media to gain an increasing level of comprehension of newly acquired
language in all content areas. ELLs may be at the beginning, intermediate,
advanced, or advanced high stage of English language acquisition in listening. In
order for the ELL to meet grade-level learning expectations across the foundation
and enrichment curriculum, all instruction delivered in English must be linguistically
accommodated (communicated, sequenced, and scaffolded) commensurate with
the student's level of English language proficiency. The student is expected to:
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Unit 4: Understanding Slope & Y-Intercept
8th Grade
ELPS.c.2A distinguish sounds and intonation patterns of English with increasing
ease
ELPS.c.2B recognize elements of the English sound system in newly acquired
vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C learn new language structures, expressions, and basic and academic
vocabulary heard during classroom instruction and interactions
ELPS.c.2D monitor understanding of spoken language during classroom
instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm
understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape,
video, DVD, and CD ROM to build and reinforce concept and language attainment
Misconceptions:
•
Some students may think that the slope in a linear relationship
is m =
, since the x-coordinate (horizontal) always comes
before the y-coordinate (vertical) in an ordered pair, instead of the correct
representation that slope in a linear relationship is m =
.
•
Some students may think that the intercept coordinate is the zero
term instead of the non-zero term, since intercepts are associated with
zeros. In other words, students may think (0, 4) would be the x-intercept
because the 0 is in the x coordinate.
Underdeveloped Concepts:
•
Some students may confuse corresponding sides of similar triangles.
Vocabulary:
•
Rate – a multiplicative comparison of two different quantities where
the measuring unit is different for each quantity
•
Similar shapes – shapes whose angles are congruent and side
lengths are proportional (equal scale factor)
•
Slope – rate of change in y (vertical) compared to the rate of change
in x (horizontal),
or
or
, denoted
as m in y = mx + b
•
y-intercept – y-coordinate of a point at which the relationship
crosses the y-axis meaning the x-coordinate is equal to zero, denoted
as b in y = mx + b
Related Vocabulary:
•
•
•
•
•
Congruent
Constant of proportionality
Corresponding angles
Corresponding sides
Horizontal
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
• Input
• Linear
• Negative
• Non-proportional
• Origin
• Output
• Positive
• Proportional
• Rate of change
• Ratio
• Right angle
• Right triangle
• Rise
• Run
• Scale factor
List of Materials:
White Boards
Expo Markers
Graph Paper
Handouts
Day 2-3 Homeschoolmath.net/worksheets/graphing.php
INSTRUCTIONAL SEQUENCE
Phase One: Engage the students
Day 1 Activity 1: Introduce the concept of slope and y-intercept.
Ask the students what they know about slopes. Relate their prior knowledge in
order to introduce a new concept or to review prior material then have the students
give real-world examples of how slope is shown such as mountains and hills,
spider-webs, ascending and descending planes, throwing balls, footballs, etc.
Day 1 Activity 2: Review
Students will need to have prior knowledge of the coordinate plane including:
plotting points, positive & negative directions, the x & y axis, and the origin.
Review as needed which may take several days.
Introduce y = mx+b and what each variable relates to on the coordinate plane.
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
Day 1 Activity A: Video Clips
Entertaining video clips are available on youtube, teachertube, united streaming,
and several other locations to introduce slope and y-intercept. Videos need to be
adapted to your specific class and range from 1 to 10 minutes, but can quickly
capture the students’ engagement into the lesson.
What’s the teacher doing?
What are the students doing?
Guiding students to actively relate slope
with their own experiences.
Making connections involving slope.
Observing videos.
Can students calculate the slope of hills,
mountains, crevices.
Answering questions on prior
knowledge.
Ask students vocabulary terms they should
be familiar with such as horizontal, vertical,
ascending, & decending.
Determine if students correlate going up
with being positive and going down with
negative.
Phase Two: Explore the Concept
Day 2-3
Activity: Introducing the formula: y = mx + b and y = kx through guided
worksheets
Have students draw on graph paper, or white boards sample linear equations and
relate guide them to noticing patterns with b. b represents the y-intercept or the
location the line passes the y-axis on the coordinate plane.
Slope(m) needs to be taught in multiple ways including rate of change in y(vertical
change) compared to the rate of change in x(horizontal change). This is
commonly expressed as rise over run. Below are same illustrations of how slope
can be depicted.
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
Misconception: some students may reverse the order, placing x on top and y on
bottom.
Sample problems should be worked in front of the class explaining slope and rise
over run. Homeschoolmath.net/worksheets/graphing.php allows you to
create customizable worksheets that have generated problems for the students.
Samples are provided below.
Have students work on worksheets from
Homeschoolmath.net/worksheets/graphing.php (permission given from Maria
Miller to include link and use of website)
Depending on students, teachers may wish to teach slope (m) one day and yintercept (b) the following day. Some students will require multiple days of
repetition and providing warm-ups on a daily basis will help with retention.
What’s the teacher doing?
What are the student’s doing?
Help students correlate b with the yintercept. As b increases, the line crosses
the y-intercept at a higher position.
Guided practice manipulating m and
b and recognizing what occurs as m
and b increase, decrease, or change
signs from positive to negative.
Help students correlate m with slope
(rise/run).
Ask the students whether negative slopes
ascend or descend.
Ask students what happens as slope
increases from 1 to 10. What happens
when slope decreases from 1 to 1/10
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
Phase Three: Explain the Concept
Day 4-5
Activity 1: Interactive classroom discussions.
Group students in a semicircle and ask them questions to allow them to think on
their own and with their peers why something works the way it does. Some
students will need more guidance than others.
Explain how the slope is determined by using the origin as one of the points the
linear equation passes through. This removes b from the equation and lets the
students deal solely with m. Have the students work various chart problems in
order to correlate why m = rise over run and not run over rise.
Ask students to determine what happens when slope is positive or negative. Ask
students to calculate the slope of vertical and horizontal lines. Below are four
samples students should be familiar with.
Explain how the y-intercept is determined by having students examine sample
questions where x = 0. Most students relate the x with the x-axis and by saying
x=0, they quickly want to draw a horizontal line. Stress how x = 0 is the y-axis.
Visually demonstrate this.
Activity 2: Provide students with similar right triangles on a coordinate
graph
Provide visuals and manipulatives to the students to reinforce slope. Any nonvertical or non-horizontal linear equation can be made into a triangle. Students
can determine the slope of a linear equation by drawing a vertical line from one
point and a horizontal line from the other point. They can then visually see the rise
over run used to form the slope of the third line.
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
This is greatly simplified using graph paper and whole numbers. As students gain
confidence, replace whole numbers with rational numbers.
The websites below provide worksheets, interactive programs and descriptive
examples on teaching slope and y-intercept.
Ixl.com (membership required) – v9 interactive program allowing students to
calculate slope with a built in difficulty modifier.
KhanAcademy.org provides multiple videos on determining slope, y-intercept and
related concepts.
Softschools.com provides a free interactive program illustrating various slopes
created between two points under geometry section.
Algebra-class.com/slope-intercept-form.html provides additional descriptive
examples of graphing and using slope.
What’s the teacher doing?
What are the students doing?
Guiding classroom discussion to examine
y-intercept (b) and slope (m)
Leading classroom discussions
through testing hypothesis and
guided arguments provided by
What are the various ways to depict slope? instructor.
M, rise over run, y/x, unit rate, etc.
What does the y-intercept represent?
Starting point, answers will vary depending
on the problem (cell phones- starting cost,
plane- starting altitude, etc)
Phase Four: Elaborate on the Concept
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
Not all linear equations will take the form of y = mx + b. Students will need to
isolate y on some occasions. Manipulating equations will be required in this case.
Example: y - 2 = 1/2x + 12
In order to benefit from the formula y = mx + b, students will have to isolate the y
variable by moving the -2. If students have no prior experience solving for a
variable, they will require multiple days to teach this activity.
Students will also be required to know m is the change in y-values over the
change in x-values or (y2-y1/x2-x1).
This can be taught giving the students two points on a linear equation and having
them determine the slope.
Example: points A(4,6) and B(2,3) are on the same line. What is the slope. This
is determined by subtracting the y values as the numerator and the x values as the
denominator. Students must be consistent on placing A – B or B – A.
6-3= 3
4-2= 2
or
3-6 = -3
2-4 = -2
Students will need to be able to analyze data from charts and tables in order to
find slope and y-intercept. This requires plotting data, and utilizing the formulas
learned.
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
Activity 1: Matching equations with graphs.
Separate students into small groups or pairs and provide them with the equations
and graphs below. Have the students determine which equations belong with
which graphs.
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
Equation answers provided below:
A.
B.
C.
D.
E.
F.
y=x-1
y = -2x - 8
y = ¾x – 6½
y = -2/3x + 5
y = -3
y = 4/3x + 4
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
Activity 2: Provide students with similar right triangles and a coordinate
graph.
Have students determine whether all similar right triangles have the same slope.
(Slopes will vary depending on placement)
Have students group up and provide the with the cut outs shown below. Have
them determine when the slopes would be equal.
The first solution below would represent two triangles with the same slope. The
second solution shows two similar triangles that have different slopes.
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
What’s the teacher doing?
What are the students doing?
Encouraging critical thinking.
Answering critical thinking word
problems involving multiple steps.
What happens to a bridge when its slope
changes from 12 to -12? Bridge flips like a
seesaw.
Finding solutions to real-world
problems involving slope.
If an airplane descends at 50 miles a
minute, at an altitude of 2000, how long
before it lands? What does the y-intercept
represent? Can the y-intercept be
negative? 40 minutes, starting altitude,
only if the plane is below sea level.
Phase Five: Evaluate students'
Understanding of the Concepts
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8th Grade
Unit 4: Understanding Slope & Y-Intercept
Day 8
Activity: Assess students based on IFD (PI, performance indicator now PA)
Sample problem listed below to assess students understanding of slope and yintercept.
Bennett noticed a beetle crawling towards him. The table below shows the
distance the beetle was from him after each second.
a) Use the data in the table to graph the relationship between the number of
seconds and distance traveled.
b) Use the table of data or graph to determine the rate of change, or slope, and yintercept and explain what each of them represents in the context of the problem
situation.
c) Write an equation to represent the problem situation where x represents the
time in seconds and y represents the distance the beetle is from Bennett.
d) Use the graph to describe how similar right triangles can be used to justify how
the slope of the line representing the problem situation is the same for any two
points on the line.
What’s the teacher doing?
What are the students doing?
Assess students understanding of slope
and y-intercept through PI (performance
indicator) and a formal assessment.
Demonstrating mastery and
understanding of slope and yintercept through formal assessment
involving real-world critical thinking
questions.
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