Week 1 Model/Build Studio: The Phone-Tree Model

Part 1 — The Big Picture

This week you learned that an exponent counts repeated multiplication, that six rules tell you how powers combine, and that every result can be checked. In this Studio you build a real model from numbers you generate yourself: a phone tree in which each person calls 5 new people. You write the model with exponents, test each exponent rule in Desmos or GeoGebra, check two expressions by substitution, and then catch a chatbot's algebra slips on the same problems.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: How many new volunteers does a 5-call phone tree reach at each level, and do the exponent rules give the same numbers that plain arithmetic and Desmos give?

Workshop data: Community garden phone-tree model

Source: A workshop-only model you build yourself: one organizer starts a phone tree, each person calls 5 people who have not been called yet, and the model gives the number of new volunteers called at each level.

• Rows: 7 levels of the tree, level 0 through level 6, one row per level in the first scaffold table
• Model: Each level has 5 times as many new volunteers as the level before it, starting from 1 organizer at level 0
• Substitution values: a = −2 and a = −5, used in the third scaffold table

Part 3 — Materials & Procedure

Tools and materials

• Desmos (free) or GeoGebra
• A document or notebook for your written answers

Ways of working

• build-a-model-from-real-numbers
• apply-the-exponent-rules-step-by-step
• verify-by-substitution

Procedure

1. Open the free Desmos graphing calculator at desmos.com/calculator, or GeoGebra's graphing calculator, and start a blank graph. Tool: Desmos (free) or GeoGebra. Check: An empty expression list appears beside an empty graph.
2. On seven separate lines, type 5^0, 5^1, 5^2, 5^3, 5^4, 5^5, and 5^6. Copy each value into the first scaffold table, then check each row by multiplying the row above it by 5. Tool: Desmos (free) or GeoGebra. Check: Each line shows a whole number, and each value is 5 times the value on the line above it.
3. On a new line, type y=5^x to graph the model, then click the point where the curve crosses the vertical axis. Tool: Desmos (free) or GeoGebra. Check: The curve rises from left to right, more steeply as it goes, and the labelled crossing point matches level 0 of your table.
4. Test the product rule: type 5^2*5^3 on one line and 5^(2+3) on the next. Then test the quotient rule with 5^6/5^4 and 5^(6-4), the power rule with (5^2)^3 and 5^(2*3), the zero exponent with 5^3/5^3 and 5^0, and the negative exponent with 5^2/5^3 and 5^(-1). Record each pair in the second scaffold table, and check each one with your own arithmetic from the first table. Tool: Desmos (free) or GeoGebra. Check: The two lines in each pair show the same value.
5. Compare a negated power with a negative base: type -5^4 on one line and (-5)^4 on the next. Record both in the third scaffold table. Tool: Desmos (free) or GeoGebra. Check: The two values have the same size and opposite signs.
6. Check a simplification by substitution. Type a=-2 on a new line, then type 7a-4(a-3) and, on the next line, your own simplified form of it. Write both values in the third scaffold table, then show the same substitution by hand, one step per line. Tool: Desmos (free) or GeoGebra. Check: When your simplified form is right, its line shows the same value as the original expression's line.
7. Change the line a=-2 to a=-5, then type 2a^2 and (2a)^2 on two new lines. Record both values in the third scaffold table and explain in one sentence which number the exponent belongs to in each. Tool: Desmos (free) or GeoGebra. Check: The two lines show different values, and the second is larger.

Fallback route

When to use it: Use this route if neither Desmos nor GeoGebra is available on your device.

Fill in every table by hand, using repeated multiplication for each power and a basic calculator for the arithmetic checks, and write the hand check for every row. Answer every analysis question and complete the critique step the same way, and submit the same items for equal credit.

This route earns the same credit as the primary route.

Part 4 — The Scaffold

Build the model level by level, because each row should be 5 times the row above it, and that pattern is what an exponent counts.

Student scaffold table 1

Level n	New volunteers at this level, written as a power of 5	Value from Desmos or GeoGebra	Check: 5 times the row above
0	5⁰		(start: the organizer)
1	5¹
2	5²
3	5³
4	5⁴
5	5⁵
6	5⁶

Test each exponent rule two ways, because a rule you can check by arithmetic never has to be taken on faith.

Student scaffold table 2

Rule	Expression	Written as one power of 5	Value from the tool	Arithmetic check from your first table
Product rule	5² · 5³
Quotient rule	5⁶ ⁄ 5⁴
Power rule	(5²)³
Zero exponent	5³ ⁄ 5³
Negative exponent	5² ⁄ 5³

Substitute and compare, because two expressions that should be equal must give the same value, and two that should differ must not.

Student scaffold table 3

Expression	Value of a	Value from the tool	Hand check, one step per line
−5⁴	(no variable)
(−5)⁴	(no variable)
7a − 4(a − 3)	−2
Your simplified form of 7a − 4(a − 3)	−2
2a²	−5
(2a)²	−5

Part 5 — Analysis Questions

1. Using your first table, how many new volunteers are called at level 6, and how many people are in the tree altogether from level 0 through level 6? Show the addition.
2. In your second and third tables, which pairs of lines gave matching values and which did not? Name the pairs that differ and say what is different about how each pair is written.
3. Explain, by writing out the factors, why 5² · 5³ is 5⁵ and not 5⁶, and why 5² ⁄ 5³ is a positive number even though its exponent is negative.
4. Write the model for the number of new volunteers at level n as one power of 5, and explain what 5⁰ = 1 means in the phone tree. Then classify √625 and √125 as rational or irrational, using your table.
5. Choose one and answer it in 3–4 sentences: (a) a real phone tree stops growing long before level 20 — name one reason the 5ⁿ model breaks down in practice and what you would change about it; or (b) you checked each rule with a tool and by hand — say which check you would trust if the two disagreed, and why.

Part 6 — AI-Critique Moment

Choose one equal-credit critique route. Route A: ask an approved AI tool to critique your completed work, then check its response against your own evidence. Route B is the no-AI equivalent printed below.

Approved tools: Claude, Gemini, ChatGPT, Grok, Microsoft Copilot.

Prompt to paste:

I've pasted my completed Week 1 Studio tables above. A community garden starts a phone tree: one organizer calls 5 volunteers, each of them calls 5 new volunteers, and so on, so level n of the tree has 5ⁿ new volunteers. Please answer these, showing every step. 1. Evaluate −5⁴ and (−5)⁴. 2. Write 5² · 5³ as a single power of 5 and give its value. 3. Write 5⁻² as a number. 4. Is √625 rational or irrational? Is √125? 5. Evaluate 2a² when a = −5. Then, as stretch questions for later in the course: 6. Simplify (x + 3) ⁄ x. 7. Solve −3x > 9. 8. In y = −2x + 5, what are the slope and the y-intercept? 9. Is x = −2 a solution of √(x + 3) = x + 1? 10. Rewrite log(x + 4) as a sum of two logarithms.

Test the response for these likely failures:

• Chatbots often square away the minus sign outside the parentheses and report −5⁴ as 625, the same as (−5)⁴.
• Chatbots sometimes multiply the exponents when multiplying powers, writing 5² · 5³ = 5⁶.
• Chatbots occasionally read a negative exponent as a negative number, giving 5⁻² = −25.
• Chatbots sometimes call every square root irrational, or call √125 rational because 125 is a power of 5.
• Chatbots sometimes square the coefficient along with the variable, reading 2a² at a = −5 as (2 · −5)² = 100.
• On the stretch question, chatbots often cancel the x in (x + 3)⁄x and report 3.
• On the stretch question, chatbots often square both sides and accept every candidate without checking it in the original equation.
• On the stretch question, chatbots sometimes split log(x + 4) into log(x) + log(4).
• On the stretch question, chatbots sometimes divide −3x > 9 by −3 and keep the sign, giving x > −3.
• On the stretch question, chatbots sometimes swap the two numbers in y = −2x + 5.

Write 3–5 sentences naming one thing the chatbot got right and at least two places it went wrong or overreached on questions 1–5, quoting its words and correcting each with evidence from your own tables or Desmos lines. For questions 6–10, mark each chatbot answer as one to check later in the course and note any that disagree with the checks listed here.

Equivalent route (no AI required)

No AI account is required for this Studio. This route assesses the same objective, Objective 1 (classify real numbers, and simplify and evaluate algebraic expressions using the properties of real numbers, the order of operations, and the rules of integer exponents), with comparable effort, about the same 90 minutes, under the same rubric for the same credit. Instead of the chatbot, critique this flawed analysis, printed here in full: "−5⁴ = 625, the same as (−5)⁴. 5² · 5³ = 5⁶, because 2 × 3 = 6. 5⁻² = −25. Both √625 and √125 are irrational, because they are written with a root sign. 2a² at a = −5 is (−10)² = 100. Also, (x + 3)⁄x = 3 after cancelling the x; −3x > 9 gives x > −3; y = −2x + 5 has slope 5 and y-intercept −2; x = −2 solves √(x + 3) = x + 1; and log(x + 4) = log(x) + log(4)." Mark it up: quote each flawed sentence, name what is wrong, and correct it with evidence from your own tables, flagging the last five claims as ones to check later in the course. Submit the marked-up analysis in place of the chatbot report and share link.

Part 7 — What to Submit

• All three completed scaffold tables
• Written answers to all five analysis questions
• The chatbot critique report with its share link, or the equivalent route's marked-up flawed analysis

Deliverable:

A short Studio report on the phone-tree model: three completed tables (the model's values level by level, five exponent rules each checked two ways, and six substitutions with hand checks), written answers to five analysis questions, and a critique of a chatbot's work on the same problems, or, on the equivalent route, a marked-up correction of the printed flawed analysis. Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text. Choose a written, recorded, or visual form for your answers; every form is graded by the same scoring criteria.

Rubric

The rubric below shows how each submitted part is evaluated; bands are full, partial, or none.

Workshop evaluation criteria

Criterion	Full	Partial	None
Building the model from the numbers	All seven levels are filled in, each written as a power of 5 and as a value, each row is checked as 5 times the row above, and the model for level n is written as one power of 5.	The table is mostly complete, but one or two values or checks are missing or wrong, or the model is stated without the level-by-level check.	No model is built, or the values are invented rather than computed.
Applying the exponent rules step by step	All five rules are written as one power of 5, each value matches the arithmetic check, and question 3 explains by writing out factors why exponents add and why a negative exponent gives a reciprocal.	Most rules are applied correctly, but a single power, a check, or the factor explanation is missing or wrong.	The rules are not applied, or the exponents are combined with the wrong operation throughout.
Verifying by substitution	Every substitution in the third table shows hand steps with negatives in parentheses, the simplified form of 7a − 4(a − 3) matches the original at a = −2, and the differing pairs are named and explained.	Substitutions are recorded, but hand steps are missing for some rows, or one differing pair is not explained.	No substitution check is shown.
Analysis, classification, and limitation	Questions 1, 2, and 4 are answered with values from the tables, √625 and √125 are classified with a reason, and the chosen option of question 5 gives a specific, reasoned answer.	Most questions are answered, but one is missing, unsupported, or misclassifies a root.	The analysis questions are not answered.
Critique of the chatbot or of the printed flawed analysis	On either route, at least two errors on questions 1–5 are quoted and corrected with evidence from the student's own tables, one correct point is acknowledged, and the stretch answers are flagged for later; both routes earn this band equally.	On either route, errors are named but corrected without evidence, or only one error is caught.	No critique appears, or pasted output is submitted without any checking.

Quality Gate

• Every value in the tables comes from your own tool lines or your own arithmetic, never from memory or a chatbot.
• Each row of the first table is 5 times the row above it.
• Every rule in the second table is checked two ways, and every pair that should match does match.
• Every substitution shows its hand steps, with each negative value in parentheses.
• No value, rule, or source is invented anywhere in the submission.
• The critique quotes the chatbot or the flawed analysis and corrects at least two errors with evidence from your tables.