Assisting Teacher Handoff Sheet

Session 3.1 - Linear Functions: Slope, Intercepts, Tables | Week 3, Tuesday

FOR THE ASSISTING TEACHER: Read the information below before the primary instructor leaves at 2:00. You are supervising worksheet time from 2:00-3:00. Students work independently from 3:00-3:30.

What Students Just Learned (30-second summary)

In the 30-minute Socratic discussion, students learned that slope (m) is the rate of change (how much output increases per input, like $15 per hour or 50 gallons per minute) and y-intercept (b) is the starting value (what y equals when x = 0, like a $100 bonus or 500 gallons already in the pool). Together, they write equations as y = mx + b. They also learned that in a table, constant first differences tell you the slope.

The Worksheet

Students are working on Worksheet 3.1 (35 points, 11 problems across 4 parts). There is a Quick Reference box at the top with formulas, definitions, and key vocabulary. Point students there first if they're stuck.

Part A (Story): Real-world applications (savings, phone plan, pool draining). Students write equations, identify slope and intercept, and explain their meaning. Most accessible.

Part B (Visual): Tables, first differences, and the slope formula. Students calculate Δy to find patterns. Check that they understand "constant differences = linear."

Part C (Traditional): Identifying m and b from equations, building tables, and using the slope formula with two points. Most technical. Students often struggle with negative slope here.

Part D (Synthesis): Real-world interpretation (teacher salary) and explaining why slope is called "rate of change." Conceptual, reflective.

Top 3 Mistakes to Watch For

1. "Slope is just rise over run."
Actually, slope is the rate of change. The formula m = Δy / Δx is how you calculate it, but the meaning is "how much does y change for every 1-unit increase in x?" For a job paying $18/hour, slope is 18 (dollars per hour). For a pool draining 15 gallons/hour, slope is -15 (gallons per hour). Help students connect the number to the context.
2. Negative slope confusion.
Students say "negative slope means the line goes down" or "it is bad." Wrong. Negative slope means as x increases, y decreases. A pool draining has negative slope (time increases, water decreases). A savings account with withdrawals has negative slope. The line does go down-right on a graph, but emphasize: "higher input → lower output."
3. Forgetting y-intercept is at x = 0.
Students look at the table and pick any y-value as the intercept. Correct: "The y-intercept is what y equals when x = 0. Look for where x = 0 in the table." If x = 0 is not shown, students extend the pattern backward using slope.

If a Student Is Stuck

First: Point them to the Quick Reference box at the top of the worksheet. It has the formula and definitions.

Second: Ask them "What does slope mean?" or "What is the y-intercept?" If they can explain in plain English, they understand. If not, re-teach using a real-world story (phone plan, hourly pay, pool).

Third: The Session 3.1 Reading has worked examples. It's available in Canvas. Have them read the lane that matches their question (Story lane for "what does this mean in real life?" Visual lane for tables and patterns, Traditional lane for equations).

Video: Math Antics "What is Slope?" on YouTube (~10 min). Great visual explanation if a student needs reinforcement.

Priority Problems

If students are running low on time, have them prioritize: Problems 1, 4, 7, 10 (one representative from each part). These cover the core ideas:

Problems 2, 3, 5, 6, 8, 9, 11 can be finished at home if time is tight.

Common Workflow Issues & Solutions

Issue 1: "I don't know what equation to write." Reframe: "In this problem, what number changes? That is your slope. What is the starting value? That is your y-intercept. Then write y = (slope)x + (intercept)."

Issue 2: "How do I know if it is linear?" Answer: "Check the first differences in the table. If they are all the same number, it is linear. If they change, it is not."

Issue 3: "What if x = 0 is not in the table?" Answer: "Use slope and work backward. If x = 1 gives y = 10 and slope is 3, then x = 0 gives y = 10 - 3 = 7. That is your y-intercept."

Issue 4: "Why is the slope negative?" Answer: "Because the quantity is getting smaller. Money leaving a bank account, water draining from a pool, temperature dropping. As time goes forward, the amount goes down. That is negative slope."

Transition to Independent Work (3:00 PM)

At 3:00, announce: "You now have 30 minutes of independent work. You can ask questions, but try to work on your own. Focus on problems you haven't finished yet. If you get stuck, re-read the Quick Reference box or look at the Session 3.1 Reading. You'll have time before next class to ask questions."

During this time, circulate. Do not give answers, but ask guiding questions: "What is the slope?" "What is the y-intercept?" "How do you know if it is linear?" Let them think.

Materials You'll Need