Substitute Guide: The Pythagorean Theorem & Distance
Learning goal (plain language)
Students practice the Pythagorean Theorem — a² + b² = c² for right triangles — and use it to find a missing leg or hypotenuse, to measure the distance between two points on a grid (the distance formula), to solve a real-world ladder problem, and to test whether a triangle is a right triangle using the converse. You do not need a math background to run this. Every formula, a worked example, and a labeled graph are printed in the student packet. Students show their work; you do not need to teach the method — the packet does the teaching. Your job is to hand it out, read the start script, keep the room on task, and collect the packets.
Standards alignment
Framework: 19 TAC §111.41 (Geometry). Knowledge & skills for right-triangle relationships and coordinate geometry — the Pythagorean Theorem and its converse, and finding distance on the coordinate plane.
- Provisional — G.6(D): apply the Pythagorean Theorem and its converse to solve problems and to determine whether a triangle is a right triangle.
- Provisional — G.2(B): use the distance formula on the coordinate plane to find the length of a segment between two points.
- Provisional — G.5(A): use right-triangle relationships and length calculations to solve mathematical and real-world problems.
Provisional — pending educator verification against the current official TAC source.
(Standards are paraphrased here, not quoted. Please confirm codes and wording against your official source. This packet is not claimed to be "aligned to the TEKS" until reviewed.)
Setup Before class (5 min)
- Count and hand out one student packet per student.
- Confirm each student has a pencil. A basic or scientific calculator is allowed but not required; students who have one may use it (helpful for square roots).
- Write the date on the board and remind students to fill in the name line.
- Keep the answer key with you — it is a separate file and is not in the student packet.
- Standard vs. block: on a ~50-minute period, students do items 1–8, 10, and ACE (item 9 is optional). On a ~90-minute block, students also complete item 9, the multi-step composite-distance task.
Materials What's needed
- Printed student packet (5–7 pages) per student.
- Pencil per student; a basic four-function or scientific calculator is optional.
- This guide and the separate answer key (teacher only).
- No technology, no internet, no special supplies.
Timing Suggested pacing (standard ~50 min · block ~90 min)
| Segment | Standard (~50 min) | Block (~90 min) | What students do |
|---|---|---|---|
| Start — Retrieval warm-up | 6 min | 8 min | Q1–Q2 (squaring & roots; horizontal/vertical distance) |
| Build — Study the math | 8–12 min | 12–15 min | Read reference box, worked example, Figure 1; Q3 |
| Apply — Use what you know | 20–26 min | 28–34 min | Q4–Q8 (missing side, distance, ladder, error analysis, converse) |
| Block — Extra applied task | optional | 12–15 min | Q9 (composite path vs. straight-line distance) |
| Explain — Justify (CER) | 6–10 min | 8–12 min | Q10 claim/evidence/reasoning (converse) |
| Close — ACE | 5 min | 5–7 min | Articulate / Connect / Extend |
| Continue — early finishers | optional | optional | Build Pythagorean triples; test 5-6-8 |
Script Read aloud to start
"Today's assignment is a paper math packet called The Pythagorean Theorem and Distance. You will work by yourself with a pencil. A basic or scientific calculator is allowed if you have one, but you do not need one. The most important rule: show your work — write down the squares you compute and the square root you take, not just a final number. When an answer is not a perfect square, give both the exact square root and a rounded decimal. Everything you need is printed in the packet, including a worked example, a labeled graph, and all the formulas. Start with the warm-up, then read the Build section before the Apply questions. If a question is tricky, skip it and come back. Put your name, class period, and date at the top now. You have about [50 / 90] minutes; raise your hand if you need help reading a question."
Support When students ask for help
- Students show their work; you do not need to teach the method. Point them back to the printed reference box and the worked example — the method is right there.
- You may read any part aloud to a student or the class.
- If a student is stuck, prompt with "What does the worked example do?" or "Which side is across from the right angle — that is the hypotenuse" rather than giving the answer.
- For a missing side, remind them: to find the hypotenuse you add the squares then square-root; to find a leg you subtract (a² = c² − b²) then square-root.
- Remind students that the distance formula is just a² + b² = c² — the horizontal change is one leg and the vertical change is the other.
- It is fine if a student cannot finish the optional early-finisher challenge.
Access Accommodations & language support
- Read-aloud of any question is allowed for all students.
- Point students to the worked example, the formulas in the reference box, and the sentence stems for Q10 and ACE.
- A calculator is allowed for any student who wants one — this removes the square-root arithmetic barrier so students can focus on the reasoning.
- Extended time is fine; the core can stop after Q10 if time runs short.
- Students may show work with numbers and short phrases if writing full sentences is a barrier, as long as the reasoning is clear.
- The packet is grayscale-safe — Figure 1 uses black lines, labeled vertices, a right-angle mark, and text, so it prints and reads clearly in black and white.
Tools Allowed & not allowed
Allowed: pencil, the printed packet, a basic or scientific calculator, quiet self-read-aloud. Not needed / not allowed: phones for messaging, internet, AI tools, or getting answers from another student.
Teacher follow-up (for the returning teacher)
The highest-value items to review next class are the error analysis (Q7) and the justification (Q10), which reveal the "add the legs" error and whether students can use the converse instead of assuming a scalene triangle is right. Watch for mislabeled hypotenuses on the ladder item (Q6), forgetting to square-root (Q3, Q5), and dropping the exact radical when the answer is not a perfect square (Q3 → √180, Q9c → √116). The distance items (Q5, Q9) show who connects the distance formula back to a² + b² = c². The separate answer key lists worked steps, alternate strategies, misconceptions, and a 3-point rubric for Q10.