Week 1 — The Rules That Never Change: A Short Algebra Report — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

Assignment 1 of the term — evaluate expressions in the right order, use the exponent rules and check them by arithmetic, evaluate an expression at a negative input, and explain a sign rule to someone new to algebra.

This week you learned to sort real numbers into families, to follow the order of operations, to name the properties of real numbers, to use the six integer-exponent rules, and to simplify and evaluate an expression. This is the first week of the course, so nothing earlier is assumed; every later week leans on the rules you practise here. In this assignment you work four short problems, interpret each one, and explain a sign rule to someone who is new to algebra. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

The task

This week's shared numbers are the powers of two, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024, which run from 2⁰ = 1 to 2¹⁰ = 1024, and the expression 3x² − 2x + 5. Every answer below can be checked by plain arithmetic, so check each one.

Work these four problems, showing your arithmetic one step per line:

1. Use the order of operations to evaluate 2 + 3 · 4². What is its value?
2. Use the product rule to write 2³ · 2⁴ as one power of 2, then check it by multiplying 2³ = 8 by 2⁴ = 16. What value do both give?
3. Use the quotient rule to write 2³ ⁄ 2⁵ as one power of 2, then check it by dividing 2³ = 8 by 2⁵ = 32. What value do both give, as a fraction or a decimal?
4. Evaluate 3x² − 2x + 5 at x = −2, putting the −2 in parentheses. What is its value?

Beside each answer, add one brief interpretation: which rule you used, what it does, and the step where a mistake is most likely. For problem 1, say what value you would get by adding first. For problem 2, say why the exponents add instead of multiply. For problem 3, say why the value is positive even though the exponent is negative. For problem 4, explain why (−2)² = 4 and why the coefficient 3 is applied after the square.

Explain it to a non-expert

A friend is helping a younger relative with homework and is puzzled: the homework sheet says −2⁴ = −16 but (−2)⁴ = 16. Write a 3–5 sentence message that explains the difference in everyday words, gives a reason tied to how the expression is written, and names one limit — for example, that the rule is an agreement about how to read the symbols, so writing parentheses around a negative number is the safest habit, or that a single check can miss an error. Explain any technical term, such as "base" or "exponent", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a coworker who is surprised that a lunch order of 5 sandwiches at 2 dollars each plus a 3-dollar delivery fee is written 3 + 5 · 2 and comes to 13 dollars, not 16. Model answer: "The multiplication happens first because it stands for the 5 sandwiches at 2 dollars each, which is 10 dollars. Then the 3-dollar fee is added, for 13 dollars. Adding the 3 and the 5 first would mean 8 sandwiches, which nobody ordered. The math only knows the order we agreed on, so if you ever mean something different, put parentheses around it."

Use that pattern for −2⁴ and (−2)⁴, with your own explanation and evidence.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

Deliverables

Submit a single algebra report containing:

• the four answers, each with the arithmetic shown one step per line and a brief interpretation of which rule you used, what it does, and the step where a mistake is most likely;
• your message to your friend, explaining −2⁴ against (−2)⁴ with a reason and naming one limit; and
• a one-line check for each answer, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled photo of your handwritten steps with a short note. Every form is graded by the same scoring criteria. 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.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

This version is designed to be completed without AI: do the arithmetic, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or Desmos to check your arithmetic. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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