Summary
In this lesson, students will use the Internet site Desmos.com to explore how changing the y-intercept, or slope of a line, changes a graph. They will find real items to serve as examples of slope and use Desmos to find the linear functions that represent those items. The lesson is excellent for the introduction of slope-intercept, or as a review before working with systems of equations.
Essential Question(s)
What components are necessary for a function to create a line?
Snapshots
Engage: Students view photos that demonstrate slope (skier, hill, stair rail) and then complete the I Notice, I Wonder learning activity.
Explore: Students use an online tool at Desmos.com to explore how changing the slope or y-intercept influences the graph of a line.
Explain: Using the data collected in the Explore portion of the lesson, students generate a "rule" for how slope and y-intercept changes affect a line, and justify their decision using the Commit and Toss strategy.
Extend: Students find real-world examples of slope and then use digital photos and Desmos.com to generate a linear function that represents their pictures.
Evaluate: Each student completes a foldable, identifying the components of a linear function. Using the formula y = mx + b, derived during the Extend portion of the lesson, each student group trades real-world functions with another group, and creates a graph, labeling the slope and y-intercept.
Materials
Whiteboard or chart paper
Computers or handheld devices (one per student pair)
Internet or WiFi
Colored pencils
Playing cards (one deck per student pair)
Digital cameras or phones with cameras
Slope Foldable handout (attached; one per student)
Foldable Printed Definitions (attached; optional if needed)
Desmos Graph Results (attached; one per student)
Rules for Explore (attached; one per student pair)
Exit Ticket (attached; one per student)
Slope Pictures (attached)
Lesson Slides (attached)
Engage
Introduce the lesson using the attached Lesson Slides. Display slide 3 and share the lesson’s Essential Question: What components are necessary for a function to create a line?
Display slide 4 and share the lesson’s learning objective: students will use different tools to explore how changing the y-intercept or slope of line, changes a graph.
Display slide 5 and inform your students that they will be using the I Notice, I Wonder strategy to reflect on the pictures. Have them get out a piece of paper and create a T-Chart. On the left of the T-chart write I Notice, and on the right write I Wonder. Have them write down everything they notice from the picture on the left column and anything they are wondering or questioning about on the right side of the chart. Give students some time to reflect and then allow them to talk to their elbow partner about their discoveries.
If you prefer to keep it for later, make a class list for each category on a whiteboard or on chart paper.
Explore
Display slide 6 and review how to use Desmos with your students.
Pair students and assign them a computer or a handheld device that has Internet access. Pass out the attached "Desmos Graph Results" handout sheet, and each pair of students should be provided with a deck of cards.
Have student pairs log into Desmos.com and click on the red button labeled "start graphing."
Instruct each pair of students to type the equation y = 2x + 3, into the box in the upper left hand corner of the page at Desmos.com.
As an example, you should model how to graph the line shown on the computer onto a copy of the attached "Desmos Graph Results" handout.
Instruct students to make the necessary changes using their computers or handheld devices, but not to write this example down, as it is merely to ensure that the process for changing the equations is understood. Have your students discuss what change is needed for the example, and then you should make the change on a projector.
Display slide 7 and pass out the attached "Rules for Explore" handout to your student pairs. Using the deck of cards and the instructions, have student pairs complete the first side of the page (or three manipulations). They should predict what will happen to the graph after the new equation is written, but BEFORE they make the change in the Desmos equation box. The students should draw their predicted line using a colored pencil, and the actual line with a regular pencil.
Explain
Display slide 8 and have your students think about the answers to the following questions: Were the predictions correct? If not, why?
On a sheet of notebook paper, have them write their answers to the following questions:
What does changing the number in front of the x do to the graph?
What does changing the number after the plus sign do to the graph?
What do these numbers represent, and how do they interact?
Using the directions for the Commit and Toss strategy, have your students write whether they agree or disagree with what is on other students' papers, and give the reasons why. Once they are done return the papers to their original owners to be read. Have students share out their statements and any agreements or disagreements.
Display slide 9 and lead a class discussion on defining slope, the y-intercept, the naming of x as the domain (or independent variable), and the naming of y as the range (or dependent variable).
Extend
Display slide 10 and instruct your student pairs to use a digital camera or a cell phone to photograph two examples of slope in the school building. The teacher should allow approximately ten minutes for students to take the photographs.
Using email or a direct Internet connection, the photographs should be uploaded into the Desmos program.
Have students use Desmos to overlay a coordinate graph on their photographs, and then determine the slope-intercept equations for their objects.
Instruct students to explain, either in writing or in a class discussion, why they know their equations are correct, and what helped them decide their answers are correct. Students need to include why they chose the objects they photographed.
Evaluate
Display slide 11, pass out to each student a copy of the attached “Slope Foldable” handout and instruct your students to fold on the solid line, and cut on the dotted lines. Under each flap, have the students write the definition of the symbol, and what it represents on the graph.
Once the foldables are completed, have your students trade the equation generated from the picture earlier to another group. Have each student from the group graph the formula given, then label the slopes as either positive or negative. Have each student pair find the coordinates of any points along the lines and label the y-intercept.
For homework, if the attached “Exit Ticket” handout was not used in the previous lesson, this is a good place to insert it.
Resources
Allan, B. (1988). "Observatory" [sculpture]. http://static1.squarespace.com/static/5254e175e4b0218181124c42/t/565faef7e4b069d2e2e2d907/1449111288382/?format=500w
Bored, R. (Photographer). (2015, April 10). "French stuntman Éric Barone broke the cycling speed record on a ski slope in the French Alps" [digital image]. "Wired." New York, N.Y. : Conde Nast. https://www.wired.com/wp-content/uploads/2015/04/speed-bike-16x9-1024x576.jpg
Elms. G (Photographer). (n.d.). Untitled [digital image]. Lonely Planet Images, Getty Images. https://aos.iacpublishinglabs.com/question/aq/700px-394px/slope-used-real-life_2f701c6221b54a6c.jpg?domain=cx.aos.ask.com
K20 Center. (n.d.). Commit and Toss strategy. Strategies. https://learn.k20center.ou.edu/strategy/119
K20 Center. (n.d.). I Notice, I Wonder strategy. Strategies. https://learn.k20center.ou.edu/strategy/180
Lisonbee, A. (Photographer). (2011, May 26). "Lots of tilt" [digital image]. http://www.fatcyclist.com/2011/05/26/guest-post-how-to-make-photos-of-steep-hills-looksteep
Novello, M.(2014, July 11). "A view of a typical San Francisco road, USA" [digital image]. http://www.digitaltrends.com/mobile/parking-space-auction-app-disables-service/
Yano, T. (Photographer). (2010, August 30). "House of slope by FujiwaraMuro Architects" [digital image]. http://www.spoon-tamago.com/wp-content/uploads/2010/08/house-of-slope-FujiwaraMuro-3.jpg
