Reading Football Rates and Starting Values

• Project a simple graph or display the table in the opening; students need notebooks or whiteboards.
• Prepare the three short football-context problems for independent work on the board or as a single handout.

Call on contrasting answers without confirming immediately. Ask, “Could the team really score 7 points every minute in a real game? Does that change what the number means in this model?” Establish that a model can simplify reality, but units and the story still determine meaning. Recall: slope is change in y divided by change in x. Today’s key move is attaching both quantities and units to the story.

Make a two-column chart: Number from representation / Meaning in the situation. Model from a table: Minutes since game started (x): 0, 5, 10, 15 Concession stand revenue (y): $0, $120, $240, $360

Then use the opening equation. “The coefficient 7 is points per minute: the predicted scoring rate. The y-intercept 14 is points when x = 0, the start of the fourth quarter.” Name the common error: students reverse the meanings because they see the intercept as the ‘first number’ or call slope simply “points.” Correct it with the question, “What happens for each 1 unit of x?” for rate, versus “What is y when x is 0?” for intercept. Compare the intercept to a game’s opening scoreboard: it is the score already on the board when the clock for this situation starts.

**Pre-teach and organize.** Before students begin, review these three terms:

• **Rate of change:** how much *y* changes for each 1 unit of *x*.
• **Y-intercept:** the value of *y* when *x = 0*; it is the starting value in this situation.
• **Context:** the story, labels, and units connected to the numbers.

Give students this four-box organizer

| What does *x* measure? | What does *y* measure? | Rate of change | Y-intercept | |---|---|---|---| | | | per | when *x = 0* |

Display a graph labeled **“Water bottles remaining at the team bench.”** The x-axis shows minutes after halftime, and the y-axis shows bottles remaining. The line passes through (0, 30), (10, 20), and (20, 10).

**Step 1: Name the quantities.** Complete the first two boxes together.

• *x* measures **minutes after halftime**.
• *y* measures **bottles remaining**.

Ask students to point to the x-axis and y-axis, then complete: “The x-axis measures , and the y-axis measures .” Check responses before continuing.

**Step 2: Find the rate of change.** Work through two points:

• From 0 to 10 minutes, the number of bottles changes from 30 to 20.
• Change in bottles: 20 − 30 = **−10 bottles**.
• Change in time: 10 − 0 = **10 minutes**.
• Rate: −10 ÷ 10 = **−1 bottle per minute**.

Add to the organizer: **−1 bottle per minute**. Say: “The negative sign means the number of bottles is going down. It does not mean there are negative bottles.” Students complete: “The bottle count changes by bottle per minute.” Check with a quick thumbs-up when they have written **−1**.

**Step 3: Find the y-intercept.** Point to (0, 30). Model: “When *x = 0* minutes, *y = 30* bottles. Therefore, the y-intercept is **30 bottles**.” Add it to the organizer. Students complete: “At the start of halftime, there are bottles at the bench.” Check that students use **30**, not 0.

**Step 4: Explain both values in context.** Model the complete response, then have students read it with a partner:

• “The rate of change is **−1 bottle per minute**. This means the team uses **1 bottle each minute** after halftime.”
• “The y-intercept is **30 bottles**. This means there are **30 bottles at the bench when 0 minutes have passed**.”

Students now write their own two sentences using this frame

> “The rate of change is bottles per minute. This means .” > > “The y-intercept is bottles. This means when .”

Pause for a quick check: ask students to underline the **units** in each sentence and circle the words that explain **when x = 0**. Invite two students to share. Correct answers by asking, “What changes for each 1 minute?” and “What is there at 0 minutes?”

**Chunked check for understanding.** Display a second graph passing through (0, 8) and (4, 20), labeled **“Fan-club donation total, dollars, after *t* weeks.”** Complete the first calculation together:

• Change in dollars: 20 − 8 = 12.
• Change in weeks: 4 − 0 = 4.
• Rate: 12 ÷ 4 = **$3 per week**.

Students independently write the y-intercept (**$8**) and then show both answers on their boards. Check before asking students to explain one value to a partner: > “The rate is per . This means .” > > “The y-intercept is . This means when .”

Restate any answer that leaves out units or the meaning of *x = 0*, and have students revise it.

Students solve the three focused problems. Reduce the written volume by requiring the calculation and one complete sentence for each value. Students may use this checklist:

1. Identify what *x* and *y* measure.
2. Find the rate using change in *y* ÷ change in *x*.
3. Include units.
4. Find the y-intercept by looking at *x = 0*.
5. Explain what each value means in the situation.

Provide the partially completed organizer below

| Problem | Rate of change and units | What the rate means | Y-intercept | What the y-intercept means | |---|---|---|---|---| | 1 | | | | | | 2 | | | | | | 3 | | | | |

1. **Table:** Hours after the stadium opens: 0, 1, 2, 3. Parking fees collected: $450, $600, $750, $900.

• Worked starting prompt: “From 0 to 1 hour, the fees change by $, so the rate is $ per hour.”
• Students identify the y-intercept by using the row where hours = 0.

1. **Graph:** “Yards remaining until a running back reaches 100 yards,” with a line through (0, 40), (2, 28), and (4, 16), where *x* is the number of carries.

• Worked starting prompt: “From 0 to 2 carries, yards remaining changes by yards. The rate is yards per carry.”
• Remind students that a negative rate means the yards remaining are decreasing.

1. **Equation situation:** A booster club begins with $275 and earns $35 for each ticket bundle sold. Let *x* be bundles sold and *y* be total dollars.

• Complete together: “The coefficient of *x*, $35, is the rate because it tells how much money is added for each 1 bundle.”
• Students identify $275 as the y-intercept and explain why it is the starting amount.
• Students complete: “$35 is not the total after 35 bundles because $35 tells the amount earned for bundle, not the total amount.”

Circulate using prompts instead of supplying answers: “What are the units?” “What changes when *x* increases by 1?” and “What is *y* when *x = 0*?” Students who finish early create a realistic football situation with a negative rate and a nonzero intercept, write its equation, and explain when the model would stop making sense, such as when the number of bottles would fall below zero.