Week 4: Exponents, Polynomials, Factoring, and Quadratics

Discussion prompt

Before we open Week 4: Exponents, Polynomials, Factoring, and Quadratics: without looking back, what was the main idea of Week 3: Slope, Linear Equations, and Systems, and what could you do by the end of it that you could not do before?

Hint: One sentence for the idea, one for the skill. If the second one is blank, that is the part to revisit.

Comparison matrix

From How this hour will run: refill the Minutes column from what you know. The rest of the table is as it appeared.

Comparison matrix

From Exponent rules count factors: every row here is a choice with a cost. Fill the Rule column, then say which row you would actually pick and what you give up for it.

Put in order

Put the moves of Model: product and quotient powers into the order they have to happen.

1. For multiplication with the same base, add exponents
2. For division with the same base, subtract exponents
3. State both simplified forms
4. Verify by expanding the factor counts

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Three copies and five copies make eight copies.

Animation

Shows: Each line of the worked example "Model: product and quotient powers", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The factor-count story matches the exponent arithmetic.

Discussion prompt

Guided try: product and quotient powers — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Hint: check that bases match

Animation

Shows: Each line of the worked example "Guided try: product and quotient powers", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Four plus two gives six, and seven minus three gives four.

Discussion prompt

When a whole group is raised to a power, every factor in that group is repeated.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Fill in the blanks

From Model: power of a product — finish the line. Write what belongs on the right of the equals sign before you look.

(x^2)^3 = x^6

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. The outside power means the entire group repeats.

Animation

Shows: Each line of the worked example "Model: power of a product", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: There are three twos and six copies of the variable.

Predict first

Your turn: work this on paper before you reveal the hint.

Commit before you compute: what does Independent try: power of a product come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Check by writing two groups

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Two copies of the group have two threes and eight variable factors.

Animation

Shows: Each line of the worked example "Independent try: power of a product", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Two copies of the group have two threes and eight variable factors.

Discussion prompt

Explain Negative exponents mean reciprocal by analogy to something with no Algebra 2 Readiness in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Animation

Shows: Each line of the worked example "Model: negative exponent", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The movement, not a negative value, is the meaning.

Predict first

A student writes this, and it looks reasonable:

Treat the negative exponent as a negative sign on the value

It is wrong. Say what breaks — and say it before you turn the page.

Correct: This changes the value instead of taking a reciprocal.

This changes the value instead of taking a reciprocal.

Why: This changes the value instead of taking a reciprocal.

Annotate

From Trap: negative exponent is not a negative coefficient — read this one piece at a time. What is each part doing?

On: \( -2^4=-16\ne\frac{1}{16} \)

• This changes the value instead of taking a reciprocal.
• At two, the wrong expression gives negative sixteen instead of positive one sixteenth.
• The exponent becomes positive after the reciprocal move.

Comparison matrix

From Polynomial vocabulary: refill the Meaning column from what you know. The rest of the table is as it appeared.

Animation

Shows: Each line of the worked example "Model: add and subtract polynomials", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Negative five plus one gives negative four; four minus nine gives negative five.

Discussion prompt

Guided try: subtract polynomials — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Hint: distribute the subtraction first

Animation

Shows: Each line of the worked example "Guided try: subtract polynomials", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Every term in the second group changes sign.

Discussion prompt

Explain Polynomial multiplication: every term meets every term to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Animation

Shows: Each line of the worked example "Model: multiply two binomials", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: At two, both forms give negative fifteen.

Animation

Shows: Each line of the worked example "Independent try: multiply binomials", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: At zero, both forms give negative twenty-four.

Predict first

A student writes this, and it looks reasonable:

Square only the first and last terms

It is wrong. Say what breaks — and say it before you turn the page.

Correct: The square means the binomial multiplied by itself.

The square means the binomial multiplied by itself.

Why: The square means the binomial multiplied by itself.

Sort into buckets

Some of these hold up and some are the exact mistakes this lesson is built to prevent. Sort them.

Holds up

An exponent tells how many copies of a base are multiplied. Rules work when those copies are being combined, grouped, or canceled.; When a whole group is raised to a power, every factor in that group is repeated.; A negative exponent does not make the value negative. It tells the factor to move across the fraction bar.

Breaks

Treat the negative exponent as a negative sign on the value; Square only the first and last terms

sound

These are stated as this lesson states them — each one survives the edge cases Week 4: Exponents, Polynomials, Factoring, and Quadratics puts it through.

flawed

Each of these is lifted from a trap in this deck: reasonable-sounding, and wrong in a way that only shows up once you rely on it.

Animation

Shows: Each line of the worked example "Model: factor out the GCF", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Distribution returns the original polynomial.

Put in order

Put the moves of Guided try: factor a GCF into the order they have to happen.

1. Hint: find the shared number and shared variable power
2. Identify the GCF
3. Divide each term by the GCF
4. Verify by distributing

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Both terms contain four and two copies of the variable.

Animation

Shows: Each line of the worked example "Guided try: factor a GCF", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The factored form multiplies back to the original.

Comparison matrix

From Trinomial factoring matches product and sum: refill the For this expression column from what you know. The rest of the table is as it appeared.

Fill in the blanks

From Model: factor a trinomial — finish the line. Write what belongs on the right of the equals sign before you look.

(x-3)(x-4) = x^2-7x+12\checkmark

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. The constants in the factors must multiply to twelve.

Animation

Shows: Each line of the worked example "Model: factor a trinomial", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Distribution returns the original trinomial.

Predict first

Independent try: factor a trinomial is about to be worked. State your hypothesis first: which rule or definition decides this one, and what is the first move it forces? Then watch whether the example agrees with you.

Correct: Hint: product negative, sum positive

Why: One number must be positive and one must be negative.

A hypothesis you wrote down is falsifiable; a vague sense of how it will go is not. If the example opens somewhere else, that gap is the thing worth chasing.

Animation

Shows: Each line of the worked example "Independent try: factor a trinomial", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The product returns the original trinomial.

Discussion prompt

Explain Difference of squares is a fast pattern to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Discussion prompt

Model: difference of squares — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Recognize both squares

Animation

Shows: Each line of the worked example "Model: difference of squares", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The factors return the original expression.

Predict first

Your turn: work this on paper before you reveal the hint.

Commit before you compute: what does Guided try: difference of squares come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Hint: rewrite each term as a square

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Four variable-squared is the square of two variable terms, and nine is the square of three.

Animation

Shows: Each line of the worked example "Guided try: difference of squares", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Four variable-squared is the square of two variable terms, and nine is the square of three.

Discussion prompt

Explain Zero product property solves factored quadratics by analogy to something with no Algebra 2 Readiness in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Animation

Shows: Each line of the worked example "Model: factor and solve", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Each input makes the quadratic equal zero.

Predict first

Your turn: work this on paper before you reveal the hint.

Commit before you compute: what does Independent try: solve by factoring come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify both solutions

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Both inputs make the original expression equal zero.

Animation

Shows: Each line of the worked example "Independent try: solve by factoring", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Both inputs make the original expression equal zero.

Discussion prompt

A basic radical simplification pulls out the largest perfect-square factor.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Fill in the blanks

From Model: simplify a radical — finish the line. Write what belongs on the right of the equals sign before you look.

(5\sqrt50\checkmark)^2 = ___

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Twenty-five is a perfect-square factor of fifty.

Animation

Shows: Each line of the worked example "Model: simplify a radical", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The square returns the original radicand.

Discussion prompt

Guided try: simplify a radical — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Hint: use the largest perfect-square factor

Animation

Shows: Each line of the worked example "Guided try: simplify a radical", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The simplified radical squares back to seventy-two.

Comparison matrix

From Daily practice map: every row here is a choice with a cost. Fill the Focus column, then say which row you would actually pick and what you give up for it.

Sort into buckets

Sort these by Questions, from memory, without looking back at Readiness assessment map. Telling them apart on the spot is the skill; the table is only where the answer happens to be written down.

6

expressions/equations; literal equations and inequalities; functions/domain; slope/linear equations; systems and graph interpretation; exponents/polynomials/factoring

4

quadratics/radicals

g1

Questions is "6" for expressions/equations, literal equations and inequalities, functions/domain, slope/linear equations, systems and graph interpretation, exponents/polynomials/factoring — that is what the table on "Readiness assessment map" records, and it is the single property separating this group from the rest.

g2

Questions is "4" for quadratics/radicals — that is what the table on "Readiness assessment map" records, and it is the single property separating this group from the rest.

Discussion prompt

Outside this lesson: where does Week 4: Exponents, Polynomials, Factoring, and Quadratics actually turn up? Name one concrete situation — a job, a piece of software someone ships, a decision somebody has to make — and say which part of Week 4 master procedure is doing the work in it.

Hint: Vague is the failure mode here. "Engineering" is not a situation; "deciding whether this build is fast enough to ship" is.

Check your understanding

Simplify: (2x^2)^3

• A. 8x^6 (correct)
• B. 2x^6
• C. 8x^5
• D. 6x^6

Answer: A

Why: The outside power applies to the coefficient and the variable power: (2x^2)^3 = 2^3(x^2)^3 = 8x^6.

Why B tempts people

This applies the power to the variable part but forgets to cube the coefficient.

Why C tempts people

This cubes the coefficient but adds exponents instead of multiplying for a power of a power.

Why D tempts people

This multiplies the coefficient by the outside exponent instead of cubing it.

Check your understanding

Expand: (x + 3)^2

• A. x^2 + 6x + 9 (correct)
• B. x^2 + 9
• C. x^2 + 3x + 9
• D. x^2 + 6x + 6

Answer: A

Why: (x + 3)^2 means (x + 3)(x + 3). The four products are x^2, 3x, 3x, and 9, which combine to x^2 + 6x + 9.

Why B tempts people

This squares only the first and last terms and skips both middle products.

Why C tempts people

This includes only one middle product. There are two 3x products.

Why D tempts people

This gets the middle term but squares the constant incorrectly.

Check your understanding

Factor: x^2 + x - 12

• A. (x + 4)(x - 3) (correct)
• B. (x - 4)(x + 3)
• C. (x + 12)(x - 1)
• D. (x + 6)(x - 2)

Answer: A

Why: The numbers 4 and -3 multiply to -12 and add to 1, so the factorization is (x + 4)(x - 3).

Why B tempts people

These numbers multiply to -12 but add to -1, not positive 1.

Why C tempts people

These numbers add to 11, not 1.

Why D tempts people

These numbers multiply to -12 but add to 4, not 1.

Check your understanding

Solve: x^2 - 7x + 12 = 0

• A. x = 3 or x = 4 (correct)
• B. x = -3 or x = -4
• C. x = 1 or x = 12
• D. x = 7 or x = 12

Answer: A

Why: The quadratic factors as (x - 3)(x - 4) = 0. Set each factor equal to zero to get x = 3 or x = 4.

Why B tempts people

The factor constants are negative, but the solutions are positive because x - 3 = 0 gives x = 3.

Why C tempts people

These numbers multiply to 12, but they do not add to -7.

Why D tempts people

These are the middle coefficient and constant, not the solutions.

Put in order

Put the moves of Independent mini exit ticket into the order they have to happen.

1. Hint: identify the pattern first
2. Solve the four prompts
3. Check each answer

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. These are quotient powers, squared binomial, difference of squares, and radical simplification.

Step through it

Step through Independent mini exit ticket one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: Prompt is x^8 / x^3
2. Step 2: Prompt is (x - 2)^2
3. Step 3: Prompt is x^2 - 9
4. Step 4: Prompt is sqrt(72)

Draw it

One page, no notation unless you need it: draw how these connect — Week 4 master procedure · How this hour will run · Exponent rules count factors · Power of a power repeats a group · Negative exponents mean reciprocal. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.