Authentic Lessons for 21st Century Learning

Slope Matters

Slope-Intercept Form

Michell Eike, Lindsey Link, Cassie Nash | Published: May 31st, 2022 by K20 Center


In this lesson, students will use the Desmos Studio graphing calculator to explore how changing the y-intercept or slope of a line affects the graph. They will find real items to serve as examples of lines with defined slopes and use Desmos Studio to find the linear functions that represent those items. The lesson is excellent for the introduction of slope-intercept form of a line or as a review before working with systems of equations.

Essential Question(s)

What components are necessary for a function to create a line?



Students make notes and observations of photos that have strong lines with positive or negative slopes during an I Notice, I Wonder learning activity.


Students use the online Desmos Studio graphing calculator to explore how changing the number in front of x (slope) or the number after the plus sign (the y-intercept) influences the graph of a line.


Using the results from the Explore portion of the lesson, students generate a "rule" for the relationship between the linear equation and the graph and justify their decision using the Commit and Toss strategy. Then students complete a foldable, identifying the components of slope-intercept form.


Students find real-world examples of lines with defined slopes and then use digital photos and the Desmos Studio graphing calculator to generate a linear function that represents their pictures.


Students demonstrate their ability to predict the effect on a linear function when the slope changes and use a Muddiest Point prompt to reflect on their learning through an Exit Ticket.


  • Lesson Slides (attached)

  • Slope Pictures (attached; printed front only)

  • Predicting Changes (attached; one per student; printed front/back)

  • Commit and Toss (attached; one half per student; printed front only)

  • Foldable handout (attached; one per student; printed front only)

  • Exit Ticket (attached; one per student; printed front only)

  • Paper

  • Pencils

  • Colored pencils

  • Playing cards (one deck per student pair)

  • Digital cameras or phones with cameras

  • Student devices with Internet access

  • Foldable Definitions handout (optional; attached; one half per student; printed font only)

  • Scissors (optional)

  • Glue sticks (optional)


15 Minute(s)

Introduce the lesson using the attached Lesson Slides. Display slide 3 and share the lesson's essential question: How can we predict what the graph of a linear function will look like? Display slide 4 and share the lesson's learning objectives. Review each of these with students to the extent you feel necessary.

Instruct students to find a partner or assign students partners. Display slide 5 and share the I Notice, I Wonder instructional strategy.

On a piece of paper, have students make a table like the one on slide 5. Tell students that they will be walking around the classroom and reflecting on different pictures. Explain that they will write down everything they notice from a picture in the left column and anything they wonder or have a question about in the right column. If necessary, give students an example of what they might write, "I noticed ____ about picture #1." Do not mention lines or slopes at this time. Have students walk around the room to the pictures and, using the strategy, reflect on the pictures. Give students some time to reflect and talk with their partner about their discoveries.

Have the class come together to share their notices and wonders. Ask if anyone noticed anything that all of the pictures had in common.


20 Minute(s)

Pass out the attached Predicting Changes handout to each student and give each pair of students a deck of cards.

Display slide 6 and review how to use the Desmos Studio graphing calculator with your students.

Instruct each students pair to type the equation y = 2x + 3 into the entry box on the left column of the page. Then model to students how to graph the line shown on the computer onto the first coordinate plane on the Predicting Changes handout.

Display slide 7 and explain the activity to your students. They are to draw a card. The card determines what the student needs to change about the previous equation, following the card rules on slide 8. Using the deck of cards and the instructions, have student pairs complete the front side of the handout (or three manipulations). Here students are focusing on the procedure of the activity. Show slide 8 and review the card rules with the class. A summary of the card rules (the image on slide 8) is also on the Predicting Changes handout for students to quickly reference. Display slide 9 and ensure students understand what they are expected to do by reviewing this example with the class.

As students are completing the front side of the handout, direct students to get a colored pencil or pass out colored pencils. Each student needs one colored pencil.

Instruct students to now look at the back side of the Predicting Changes handout. Display slide 10 and review the new process with the class. The card drawing and equation changing process is the same as before, but now students are to predict what will happen to the graph after the new equation is written, before they use the Desmos Studio graphing calculator. Students should draw their predicted line using a colored pencil and the actual line with a regular pencil. Display slide 11 to use as an example as needed.


25 Minute(s)

Display slide 12 and have your students reflect on the previous activity by answering the following questions: Were your predictions correct? If not, do you know why?

If time allows, ask for volunteers to share their thoughts with the class.

Pass out the Commit and Toss handout to each student and show slide 13. Direct students to independently answer 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?

Using the Commit and Toss strategy, have students crumple their papers and toss them into a pile. After each student picks up a classmate's paper, 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, have students share out their statements and any agreements or disagreements.

Display slide 14 and have students talk about what they think the words slope and y-intercept mean. While students are discussing, pass out the attached Foldable handout and a pair of scissors to each student.

Show slide 15 and demonstrate to students how to fold and cut the foldable.

Once students complete the folding and cutting of the foldable, ask for volunteers to share their definition of slope. Ask which letter on the foldable they think they should label as the slope, then show slide 16.

Repeat this again with the y-intercept and use slide 17.

Show slide 18 and ensure that students wrote the definitions of slope and y-intercept under the correct flaps.

Display slide 19 and explain to students that x is the independent variable and y is the dependent variable. Tell students that these two variables will remain variables, while the variables m and b will be specific numbers.

Show slide 20 and tell students that they are adding reminders instead of definitions under the equals sign and plus sign.


20 Minute(s)

Display slide 21 and instruct your student pairs to use a digital camera or a cell phone to photograph two examples of slope in the school building. Allow approximately ten minutes for students to take the photographs.

Instruct students to add their picture to their Desmos Studio graph by clicking the "Add Item" button (plus sign) in the top-left corner of the screen and selecting "image." This can be accomplished several ways. Students could access the Desmos Studio graphing calculator from their device and allow Desmos Studio to access their photo folder, or they could get the image to their school device via email, etc. and access the Desmos Studio graphing calculator from there.

Now direct students to determine the slope-intercept equations for their images.

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.


10 Minute(s)

Display slide 22 and pass out the attached Exit Ticket handout to each student. Using the Exit Ticket strategy, students demonstrate understanding of "How does changing the equation change the graph?" by answering questions about a specific equation and graph on the handout.

Introduce the Muddiest Point strategy to help students reflect on the lesson. Students will write their responses on their Exit Ticket handout.

Collect completed handouts to assess student learning.