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Unit 4: Understanding Slope & Y-Intercept 8th Grade

5E Lesson Plan Math

Grade Level: 8 Subject Area: MathematicsLesson Title: Unit 4: Developing an Understanding of Slope and Y-Intercept

Lesson Length: 8 Days

THE TEACHING PROCESS

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, anduse data from a table or graph to determine the rate of change or slope and the y-intercept.

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 Grade8.1E 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 EnglishELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resourcesELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabularyELPS.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 attainmentELPS.c.1F use accessible language and learn new and essential language in the processELPS.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 expectationsELPS.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 GradeELPS.c.2A distinguish sounds and intonation patterns of English with increasing easeELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clustersELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactionsELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as neededELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken languageELPS.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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Unit 4: Understanding Slope & Y-Intercept 8th Grade• 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 SEQUENCEPhase 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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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?

Guiding students to actively relate slope with their own experiences.

Can students calculate the slope of hills, mountains, crevices.

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.

What are the students doing?

Making connections involving slope.

Observing videos.

Answering questions on prior knowledge.

Phase Two: Explore the Concept

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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.

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 y-intercept (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?

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Help students correlate b with the y-intercept. As b increases, the line crosses the y-intercept at a higher position.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

Guided practice manipulating m and b and recognizing what occurs as m and b increase, decrease, or change signs from positive to negative.

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

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Provide visuals and manipulatives to the students to reinforce slope. Any non-vertical 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.

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.


Guiding classroom discussion to examine y-intercept (b) and slope (m)

What are the various ways to depict slope?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)


Leading classroom discussions through testing hypothesis and guided arguments provided by instructor.

Phase Four: Elaborate on the Concept

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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 or 3-6 = -34-2= 2 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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Unit 4: Understanding Slope & Y-Intercept 8th GradeActivity 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.

Equation answers provided below:

A. y = x - 1B. y = -2x - 8C. y = ¾x – 6½D. y = -2/3x + 5E. y = -3F. y = 4/3x + 4

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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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Encouraging critical thinking.

What happens to a bridge when its slope changes from 12 to -12? Bridge flips like a seesaw.

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.


Answering critical thinking word problems involving multiple steps.

Finding solutions to real-world problems involving slope.

Phase Five: Evaluate students' Understanding of the Concepts

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Day 8

Activity: Assess students based on IFD (PI, performance indicator now PA)

Sample problem listed below to assess students understanding of slope and y-intercept.

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 y-intercept 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.


Assess students understanding of slope and y-intercept through PI (performance indicator) and a formal assessment.


Demonstrating mastery and understanding of slope and y-intercept through formal assessment involving real-world critical thinking questions.

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