Materials & Prep
Project the opening table and graph. Prepare a half-sheet independent problem set and exit ticket, or display them for students to complete in notebooks. Usual classroom setup only.
Opening
0–7 minDisplayA football team’s total yards after each completed quarter:
Quarter elapsed (x): 0, 1, 2, 3, 4 Total yards (y): 35, 95, 155, 215, 275
Ask“The rate of change is 60. What exactly should a sports announcer say happened? And what does 35 mean? Commit to a one-sentence answer silently, then compare with a partner.”
Listen forStudents may say “they got 60 yards” without the unit or time interval, or call 35 the yards in the first quarter. Take two contrasting responses without confirming yet. Recall: “How did you calculate slope from a table last class?” Students state change in y divided by change in x.
Direct Instruction
7–16 minConnectExplain that slope is not finished when it is a number. Its meaning must name what changes, how much, and for what change in x. The y-intercept is the starting amount, when x = 0.
ModelUse the displayed table. “I choose two rows: from quarter 1 to quarter 3, yards change from 95 to 215, so the change is 120 yards. Quarters change from 1 to 3, so the change is 2 quarters. 120 ÷ 2 = 60 yards per quarter. I do not say ‘60 yards’ because that leaves out the rate. In this situation, the team gained 60 total yards for each quarter played.”
Think aloud“For the y-intercept, I look for x = 0, not the first score or first row I happen to notice. At 0 quarters, the team already has 35 yards. So 35 means the team had 35 total yards before the first quarter began.”
ConfrontReturn to the opening responses. “If 35 were yards gained during the first quarter, the total after one quarter would be 35, but the table says 95.” Students revise their sentence. Emphasize that a y-intercept can represent a genuine starting amount, even when it may feel unusual in a sports context.
Guided Practice
16–27 minDisplayA graph titled “Total funds raised for football equipment.” The line includes points (0, 120), (2, 220), (4, 320), where x is weeks and y is dollars.
Have studentsIn pairs, annotate the graph with a slope triangle, calculate the rate of change, circle the y-intercept, and complete:
- The amount raised changes by dollars per .
- This means .
- At 0 weeks, . This means .
DiscussCall on pairs to justify their wording using a point or the axes as evidence. Expected: $50 per week; the group raises $50 each week; $120 was already raised at the start.
Check for understandingDisplay a quick table: minutes remaining 12, 10, 8; home-team score 14, 20, 26. Students hold up or write: “The rate is , meaning .” Scan for “3 points per minute” and a complete contextual statement. Clarify that a negative change in x can still produce a positive slope if y also decreases, but interpreting time remaining requires careful language: the score rises 3 points for each minute that passes, not for each minute remaining.
Independent Work
27–39 minStudents complete three focused items, showing slope work and writing both interpretations in full sentences.
- Table: Game time elapsed (minutes) 0, 5, 10, 15; total rushing yards 18, 43, 68, 93. Determine and interpret the rate and y-intercept.
- Graph: A line for “Water bottles sold at the concession stand,” with points (0, 24), (3, 60), (6, 96), x in hours. Determine and interpret the rate and y-intercept.
- Compare: Two teams’ practice-fund graphs are shown. Team A: y = 40x + 100. Team B passes through (0, 160) and (4, 280). State which team starts with more money, which raises money faster, and support each claim with the relevant intercept or rate and its meaning.
ScaffoldProvide the sentence frame, “The rate of change is [y-units] per [x-unit], which means . The y-intercept is [y-units], which means when is 0, .” Students who need an entry point first label the x- and y-axis quantities and units before calculating.
CirculateAsk, “What does your denominator tell you?” and “What was true when x was zero?” Do not accept an interpretation that merely repeats a number.
ExtensionStudents write a two-point table or describe a graph for a football-related situation with a negative rate of change, then give precise interpretations of both slope and intercept. Partners check whether the starting amount and units make sense.
Closing
39–45 minExit ticketA graph of “Distance left in a 100-yard drill” includes points (0, 100), (4, 60), and (8, 20), with x in seconds and y in yards. Determine the rate of change and y-intercept. Then explain each value in this situation in two complete sentences.
Look for-10 yards per second, meaning the distance remaining decreases by 10 yards each second; 100 yards, meaning the player had 100 yards remaining at 0 seconds. Sort responses for next-day support: calculation error, missing units, or inaccurate contextual meaning.