Slope of a Line

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Slope of a Line

Original Lesson

Entry: 

  • I. Intro: Teacher Presentation (5 min)
  • II. Demonstration: Slope and Steepness (15 min)
  • III. Concept Attainment Demonstration (10 min)
  • IV. Formative Assessment with Paper Quiz Questions (10 min)

I. Intro: Teacher Presentation (10 min)

OBJECTIVE: "Today we will look at graphs of straight-line equations and calculate the slopes of lines."

PREPARATION: [Students open math journals to take notes. They have graph paper in their math journals.]

PRIOR KNOWLEDGE: "As you recall, every point on a graph is defined by an X and Y value. Two points define a line. We tend to define lines on Cartesian planes (graphs) although they can exist in three-dimensional space as well.

Equations with two variables can create shapes on a two-dimensional graph. To graph an equation, we take a set of X values (called the "domain") and plug them into the equation to generate Y values (called the "range"). This creates a set of points, which have a shape associated with the kind of equation."

NEW MATERIAL:  "Today we will explore straight-line equations. Straight-line equations, or "linear" equations, graph as straight lines, and have simple variable expressions with no exponents on them. Exponents create curved lines, like parabolas or hyperbolas, and we will use them next week to draw faces! But today, just lines.

The “angle” of the line (measured by the line as it connects with the X axis) is called the slope. On a road, we speak of “% grades” - which IS an angular measure. However, slope is not an angular measure, the way equilateral triangles have 60-degree angles. Slope is a ratio of "rise" (Y) over "run" (X), a measure of how the height changes (the "rise" of a hill) as the horizontal distance changes (as you "run" alongside)."

II. Demonstration Slope and Steepness (15 min)

Use coordinate axis whiteboard to explain the following topics.

  1. Slope tells about line direction and angle of a line.

  2. Horizontal lines have a slope of 0. The closer a line is to horizontal, the closer the slope is to 0.

  3. The higher the absolute value of the slope, the steeper the line will be.

  4. Slope Triangles can be used to calculate slope as Rise (Change in Y) over Run (Change in X).

III. Concept Attainment Demonstration  (5 min)

Demonstrate relationships between Domain and Range values, Line Attributes, and Slope.

The values for Y are calculated using an equation for each value of X. We will use graphs like this many times from now on. This equation is called the "slope-intercept" form because for every value of x, we multiply and generate graphs from it:

y = mx + b

This is called the slope-intercept form because "m" is the slope and "b" gives the y-intercept: the value where the line intersects the Y axis.

Let's explore some changes to the lines as we change the equations. For each equation change, I will calculate new Y values and draw a line.  I will then draw a "Slope Triangle" to calculate rise / run to see how the slope triangle method gives us the same value as Slope (M) in the equation.

IV. Formative Assessment with Paper Quiz Questions (10 min)

I will now hand out a quiz worksheet, with ten (10) 3-part questions. Because this is new material, the quiz is not graded. However, take the quiz in pen, and do not cross out anything. I want to see how many examples it takes before you are able to answer the questions without guessing.

You will answer each of the three parts for each new line I draw, and then we'll talk. Here are the parts you complete for each example:

  1. What is the y-intercept (where does the line cross the Y axis?)

  2. Is the slope positive or negative? Write + for positive, - for negative.

  3. What is the slope of the line?

When you have finished the example, turn your paper over. When everyone's quiz is face down, we will discuss the correct answers, and then do the next round. If you do not understand why your answer was incorrect, ask a question before we move on to the next round, since other students may have the same question and this helps all of us.

Hand in your quiz at the end of the class. Although it is not part of your grade, I want to see how each of you did. Let's start!"

Substitution

Adoption: 

NOTE:  Although this is positioned as a substitution lesson (where technology has no significant impact), the use of a video and spreadsheet modeling save the teacher time and reach students in more direct ways than a lecture and whiteboard demonstration would. 

  • I. Intro: Teacher Presentation (10 min)
  • II. Visualization Video: Slope and Steepness (15 min)
  • III. Concept Attainment Demonstration with Excel (5 min)
  • IV. Formative Assessment with Paper Quiz Questions (10 min)

I. Intro: Teacher Presentation (10 min)

OBJECTIVE: Students will be able to recognize graphs of straight-line equations and calculate the slopes of lines.

PREPARATION: [Students open math journals to take notes. They have graph paper in their math journals.]

PRIOR KNOWLEDGE: "As you recall, every point on a graph is defined by an X and Y value. Two points define a line. We tend to define lines on Cartesian planes (graphs) although they can exist in three-dimensional space as well.

Equations with two variables can create shapes on a two-dimensional graph. To graph an equation, we take a set of X values (called the "domain") and plug them into the equation to generate Y values (called the "range"). This creates a set of points, which have a shape associated with the kind of equation."

NEW MATERIAL:  "Today we will explore straight-line equations. Straight-line equations, or "linear" equations, graph as straight lines, and have simple variable expressions with no exponents on them. Exponents create curved lines, like parabolas or hyperbolas, and we will use them next week to draw faces! But today, just lines.

The “angle” of the line (measured by the line as it connects with the X axis) is called the slope. On a road, we speak of “% grades” - which IS an angular measure. However, slope is not an angular measure, the way equilateral triangles have 60-degree angles. Slope is a ratio of "rise" (Y) over "run" (X), a measure of how the height changes (the "rise" of a hill) as the horizontal distance changes (as you "run" alongside)."

II. Visualization Video: Slope and Steepness (15 min)

VALUE OF INSTRUCTIONAL VIDEO: Use real-world context, interactions, and motion to engage students and deepen understanding. 

This should be an opportunity for students to be critical viewers of this video on slope and steepness. Don't forget to remind student to stop to take notes during the video. ​

  1. Slope tells about line direction and angle of a line.

  2. Horizontal lines have a slope of 0. The closer a line is to horizontal, the closer the slope is to 0.

  3. The higher the absolute value of the slope, the steeper the line will be.

  4. Slope Triangles can be used to calculate slope as Rise (Change in Y) over Run (Change in X).

III. Concept Attainment Demonstration with Excel (5min)

  • Value of Spreadsheet Modeling: Demonstrate relationships between Domain and Range values, Line Attributes, and Slope.

[Project Slope.xls onto the screen. (Excel Spreadsheet)]

"This spreadsheet uses an equation to create a value of Y for each value of X. This equation is called the "slope-intercept" form because for every value of x, we multiply and generate graphs from it:

y = mx + b

This is called the slope-intercept form because "m" is the slope and "b" gives the y-intercept: the value where the line intersects the Y axis.

This spreadsheet generates random values for M and B, creating new values of Y for each X value, and changing the shape of the line. Let's explore a few rounds. For each round, draw a "Slope Triangle" in your mind and calculate rise / run to see how the slope triangle method gives you the same value as Slope in the graph."

IV. Formative Assessment with Paper Quiz Questions (10min)

  • Value of Spreadsheet Modeling:: Generate random problems for students to solve using visual cues. Don't forget that the formative assessment aspect of this assignment is essential. The quiz/exit tickets provide essential data. 

"Now I will hide all values. You will need to analyze the line to determine what M and B are when I generate new X and Y values.

I will now hand out a quiz worksheet, with ten (10) 3-part questions. Because this is new material, the quiz is not graded. However, take the quiz in pen, and do not cross out anything. I want to see how many examples it takes before you are able to answer the questions without guessing. You will answer each of the three parts for each question every time I change the graph, and then we'll talk.

Here are the parts you complete for each example:

  1. What is the y-intercept (where does the line cross the Y axis?)

  2. Is the slope positive or negative? Write + for positive, - for negative.

  3. What is the slope of the line?

When you have finished the example, turn your paper over. When everyone's quiz is face down, we will discuss the correct answers, and then do the next round. If you do not understand why your answer was incorrect, ask a question before we move on to the next round, since other students may have the same question and this helps all of us.

Hand in your quiz at the end of the class. Although it is not part of your grade, I want to see how each of you did. Let's start!"

Augmentation

Adaptation: 

NOTE: Because students are using computers, there is opportunity for individualized note-taking, and the teacher can electronically assign formative assessments and review data. However, this classroom is still fully teacher-centered.

  1. I. Intro: Teacher Presentation (10 min) - same as Substitution
  2. II. Visualization Video: Slope and Steepness (15 min)
  3. III. Concept Attainment Demonstration with Excel (5min)
  4. IV. Calculating Slope: Formative Assessment with Online Quiz (10min)

I. Intro: Teacher Presentation (10 min) - same as Substitution

OBJECTIVE: Today we will look at graphs of straight-line equations and calculate the slopes of lines.

PREPARATION: [Students open math journals to take notes. They have graph paper in their math journals.]

PRIOR KNOWLEDGE:“As you recall, every point on a graph is defined by an X and Y value. Two points define a line. We tend to define lines on Cartesian planes (graphs) although they can exist in three-dimensional space as well.

Equations with two variables can create shapes on a two-dimensional graph. To graph an equation, we take a set of X values (called the "domain") and plug them into the equation to generate Y values (called the "range"). This creates a set of points, which have a shape associated with the kind of equation.”

NEW MATERIAL:  “Today we will explore straight-line equations. Straight-line equations, or "linear" equations, graph as straight lines, and have simple variable expressions with no exponents on them. Exponents create curved lines, like parabolas or hyperbolas, and we will use them next week to draw faces! But today, just lines.

The “angle” of the line (measured by the line as it connects with the X axis) is called the slope. On a road, we speak of “% grades” - which IS an angular measure. However, slope is not an angular measure, the way equilateral triangles have 60-degree angles. Slope is a ratio of "rise" (Y) over "run" (X), a measure of how the height changes (the "rise" of a hill) as the horizontal distance changes (as you "run" alongside).

II. Visualization Video: Slope and Steepness (15 min)

  • Value of Video Technology: Use context, interactions, and motion to engage students and deepen understanding.

  • Value of 1:1 Access: Students can watch video at their own rate, close up, and pause and replay to take notes. Doing this in class supports the habit at home for flipped lessons.

Let's watch this 4-minute video about slope and steepness one time together. [Play video] Any questions?

Now, open your Chromebooks, take out your earbuds, and go to our Google Classroom page, where you will find this video under today's lesson.

You now have 12 minutes to watch it a few times more. Open your math notebooks to a new page, and entitle that page "Slope and Steepness". Take some notes on that page, pausing the video as long as you need to write each item. Your notes should answer these four questions:

  1. What does slope tell us about a line?

  2. What slope do horizontal lines have?

  3. What is the relationship between the absolute value of a line's slope and the steepness of the line?

  4. How do you calculate the slope of a line using slope triangles? Use the words "rise" and "run" in your answer.

When I check your notebooks at the end of the week, if you have all four questions answered correctly, you will earn a check. Less than four, check minus. For a check plus, draw an example of a slope triangle, showing rise and run values and the calculation of slope, the way the video showed this.

III. Concept Attainment Demonstration with Excel (5min)

  • Value of Spreadsheet Technology: Demonstrate relationship between Domain and Range values, Line Attributes, and Slope.

[Project Slope.xls onto the screen. (Excel Spreadsheet)]

This spreadsheet uses an equation to create a value of Y for each value of X. We will use graphs like this many times from now on. This equation is called the "slope-intercept" form because for every value of x, we multiply and generate graphs from it:

y = mx + b

This is called the slope-intercept form because "m" is the slope and "b" gives the y-intercept: the value where the line intersects the Y axis.

This spreadsheet generates random values for M and B, creating new values of Y for each X value, and changing the shape of the line. Let's explore a few rounds. For each one, draw a "Slope Triangle" in your mind and calculate rise / run to see how the slope triangle method gives you the same value as Slope in the graph.

IV. Calculating Slope: Formative Assessment with Online Quiz (10min)

  • Value of Spreadsheet Technology: Generate random problems for students to solve using visual cues.  [Calculating Slope]

  • Value of Google Forms: Can use Flubaroo to automatically assess concept attainment through student performance, with easy visual display. [Quiz Form]

  • Value of Flubaroo: Automatically grade "one right answer" quiz questions based on given correct answer set, provide instant visual feedback as well as email feedback. [Responses]

Now I will change the graph but hideall values. Each time I change the graph, generating new X and Y values, you will need to analyze the lines and answer three questions:

  1. What is the y-intercept (where does the line cross the Y axis?)

  2. Is the slope positive or negative? Enter + for positive, - for negative.  

What is the slope of the line? Express as an integer. I will change the graph five times.

  1. On Google Classroom, under the video from today's lesson,  is a link to today's quiz form.

Using the form, enter your answers for each line.We will discuss the correct answers, and then try another round. If you do not understand why your previous answer was incorrect, ask a question before we move on to the next round.Other students may have the same question and this may help all of us. On the fifth round, we will NOT discuss the correct answer, so please be silent while other students complete this. Because this is new material, the quiz is not graded.

Do not hit SUBMIT until you have responded to all five lines.  Use the question box at the bottom of the form to give me feedback about today's lesson, and well you feel you understand what "slope" means.

Hopefully, by the time we have done our fifth set, everyone will have the right answers. I will look these over tonight. If I see that anyone is having difficulty, we'll do some pair tutoring in class tomorrow.  

 

Modification

Infusion: 

NOTE: By introducing more frequent independent activities and a blended group assignment, the lesson moves towards a more student-centered classroom. Here is a screenshot of Google Classroom as used:

  • I. Intro: Bell-Ringer (5 min)
  • II. Visualization Video: Slope and Steepness (15 min)
  • III. Concept Attainment Demonstration with Excel (5 min)
  • IV. "What's My Line" Game (15 min)

Classroom Screenshot

I. Intro: Bell-Ringer (5 min)

  • Value of Google Form Bell-Ringer: Review and assess prior knowledge before introducing new material.  Form Link.

  • Value of Flubaroo: Provide immediate form results to inform instruction. Flubaroo Tutorial.

II. Visualization Video: Slope and Steepness (15 min)

  • Value of Instructional Video: Use context, interactions, and motion to engage students and deepen understanding. Video Link.

  • Value of 1:1 Access: Students can watch video at their own rate, close up, and pause and replay to take notes. Doing this in class models and supports the habit at home for flipped lessons.

  • Value of Google Template for taking notes: Use conceptual or graphic organizers to prime students, and then they keep their own versions. In this lesson, this organizer is a "digital worksheet",  an example of "Substitution". Template Link.

III. Concept Attainment Demonstration with Excel (5 min)

  • Value of Spreadsheet Technology: Demonstrate relationship between Domain and Range values, Line Attributes, and Slope. SlopeVisualizer File.

   NOTES TO TEACHER:  

    1. At some point, hide rows 1-2 and begin to reveal answers by switching from the "Line" worksheet to the "Answers" Worksheet

    2. When done, show students how to play "What's My Line" with the Google Version of the sheet (see below)

IV. "What's My Line" Game

  • Value of Google Sheets over Excel: Students can access spreadsheet via web. Link to What's My Line.

  • Value of Competitive / Cooperative Group Games: Effective use of gamified learning can increase student engagement and provides opportunities for peer learning..

GAME INSTRUCTIONS:

Teams take turn showing the graph on one laptop and and "Rolling the Dice" (by refreshing the web page, which is F5 or the refresh symbol (see right) or just hitting enter when your cursor is in the address bar).  When it is your team's turn to enter an answer, look at the graph and discuss together your answers to the four questions.

When you are ready, tell the other team your answers. The presenting team will then click the "Answers" worksheet to see what the correct answers were.  If you get it right, take a point! Explain it to each other if you got it wrong.  If neither of the people on your team understand why your answer was incorrect, ask the presenting team if they can explain it to you.  If nobody on either team can explain the answers, raise your hands for help.

Then switch roles. Once everyone on each team can get the right answer every time, you have free time!  Then you have free time, unless you are called for help with today's Bell Ringer.

 

Redefinition

Transformation: 

[Under Construction]

NOTE: Students are given choice of assignments for which products are constructed independently or in small-group collaboration, and tie the material to real-world situations where understanding line slope would be useful.  This assignment addresses all five elements of student-centered learning articulated by the Technology Integration Matrix (see http://myinstructionaldesigns.com/blog/rediscovering-tim). 

  • I.Warm-up: Do the Bell Ringer (5 mins)
  • II. Instructional Video and Concept Attainment: (20 mins)
  • III. Application Activity
  • IV. Wrap-Up
  1. Warm-up: Do the Bell Ringer (5 mins)

  2. Instructional Video and Concept Attainment: (20 mins)

    1. Download Google Doc Note-Taking template

    2. Watch “Steepness” video, take notes, submit notes as today’s assignment, complete quiz
      OR ALT Watch the Flocabulary “Linear Equations” video, take notes, submit notes, and complete exercise.
       

  3. Main Activity: (20 mins) I created three assignments, let me know preference. Activity will require a worksheet/template

    Historic Time(line)s
    Working in pairs and using Google Drawing, students will upload an image of a famous pyramid (or other geometric wonder of the world), research basic information (name, location, date, height), identify and trace at least two lines, and estimate the corresponding linear equations for each answer: Where does the line cross the Y axis?; Is the slope positive or negative?; How far between are the points along the Y axis?; What is the slope of the line? They may use a slope slider to check work. If time allows, students present work to class in chronological order of building (oldest to newest). preliminary sample

    Slope Name Art Illustration
    Working individually and using Google Drawing, students will type their names in uppercase letters (large size and using sans serif font). Next, student will identify and trace at least two lines, and estimate the corresponding linear equations for each answer: Where does the line cross the Y axis?; Is the slope positive or negative?; How far between are the points along the Y axis?; What is the slope of the line? They may use a slope slider to check work. If time allows, students present work in Gallery Walk format. (see similar assignment here)

    What’s My Line?
    Students will pick two variables that they think may have a relationship. For Example: Height Vs. Weight, Time at mall Vs. Money spent, Foot size Vs. How high you can jump, Fingernail length Vs. Finger length, etc. Students then need to collect data (the more the merrier). The students/teacher may choose the method that is used to have students collect the data. The teacher may assign the collection process as a homework activity or it could be done during class. Have students collect data in a table or list format (see attached worksheets). -Note: I pulled this copy from elsewhere so I would need to re-word if you decide to use
     

  4. Wrap-Up:

    1. Complete online exit ticket

    2. Meet with teacher for 1:1 or small group instruction

    3. Revisit/review any resource from lesson

Explore a new resource (such as Game Over Gopher)

Standards

Content Standards: 

Standard 3 - Mathematics

Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry.

Key Idea 1 - Mathematical Reasoning

Students use mathematical reasoning to analyze mathematical situations, make conjectures, gather evidence, and construct an argument.

Intermediate Performance Indicators:

  • make and evaluate conjectures and arguments using appropriate language
  • make conclusions based on inductive reasoning

Key Idea 2 – Number and Numeration

Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.

Intermediate Performance Indicators:

  • understand and apply ratios, proportions, and percents through a wide variety of hands-on explorations.

Key Idea 4 - Modeling/Multiple Representation

Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.

Intermediate Performance Indicators:

  • visualize, represent, and transform two- and three-dimensional shapes
  • use the coordinate plane to explore geometric ideas
  • represent numerical relationships in one- and two-dimensional graphs
  • use variables to represent relationships
  • use concrete materials and diagrams to describe the operation of real world processes and systems
  • investigate both two- and three-dimensional transformations
  • use appropriate tools to construct and verify geometric relationships

Key Idea 7 – Patterns and Functions

Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.

Intermediate Performance Indicators:

  • recognize, describe, and generalize a wide variety of patterns and functions
  • describe and represent patterns and functional relationships using tables, charts and graphs, algebraic expressions, rules, and verbal descriptions
  • develop methods to solve basic linear and quadratic equations
  • develop an understanding of functions and functional relationships: that a change in one quantity (variable) results in change in another
  • verify results of substituting variables
  • explore relationships involving points, lines, angles, and planes
  • use patterns and functions to represent and solve problems

Strategy

Instructional Strategy: 

A video is substituted for a lecture or a textbook to help students apply kinesthetic real-world experience to understand a mathemtical concept.  Then, an interactive modeling tool is used to perform two-dimensional transformations. 

Tags

Files: 

Tutorials: