Week 1: Expressions, Equations, and Literal Equations

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 Correction rule for every miss: every row here is a choice with a cost. Fill the What the student writes column, then say which row you would actually pick and what you give up for it.

Discussion prompt

Sorting first slows the student down just enough to prevent unlike-term mistakes.

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.

Put in order

Put the moves of Model: label the expression into the order they have to happen.

1. Circle each signed term
2. Group terms with matching variable parts
3. Combine each group
4. Check by substituting one simple value

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. The sign belongs to the term that follows it.

Animation

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

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

Takeaway: At one, both expressions give fifteen, so the simplified form is equivalent.

Discussion prompt

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

Hint: It starts with: Hint: make three piles

Animation

Shows: Each line of the worked example "Guided try: sort before simplifying", appearing one at a time.

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

Takeaway: At two, the original and simplified expressions both give twenty-seven.

Sort into buckets

Sort these by Reason, from memory, without looking back at Like terms must match exactly. Telling them apart on the spot is the skill; the table is only where the answer happens to be written down.

same variable part

3x and 2x; a and 4a

different variable parts

3x and 2x^2

both constants

5 and -9

g1

Reason is "same variable part" for 3x and 2x, a and 4a — that is what the table on "Like terms must match exactly" records, and it is the single property separating this group from the rest.

g2

Reason is "different variable parts" for 3x and 2x^2 — that is what the table on "Like terms must match exactly" records, and it is the single property separating this group from the rest.

g3

Reason is "both constants" for 5 and -9 — that is what the table on "Like terms must match exactly" records, and it is the single property separating this group from the rest.

Animation

Shows: Each line of the worked example "Model: combine in two passes", 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 seven; this quick check catches constant mistakes.

Predict first

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

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

Correct: Combine each group

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. Add coefficients and keep the matching variable part.

Animation

Shows: Each line of the worked example "Independent try: combine like terms", appearing one at a time.

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

Takeaway: Add coefficients and keep the matching variable part.

Predict first

A student writes this, and it looks reasonable:

Add the coefficients and keep one variable

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

Correct: This ignores the different variable parts.

This ignores the different variable parts.

Why: This ignores the different variable parts.

Annotate

From Trap: unlike terms do not merge — read this one piece at a time. What is each part doing?

On: \( 3(2)+2(2)^2=14\quad\text{but}\quad5(2)=10 \)

• This ignores the different variable parts.
• At two, the original gives fourteen, but the wrong simplification gives ten.
• The variable parts do not match, so there is no combining move.

Discussion prompt

Explain Distribution opens a package 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: distribute a positive factor", appearing one at a time.

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

Takeaway: The variable terms match; the constant has no partner.

Animation

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

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

Takeaway: Both pieces inside the package get multiplied.

Discussion prompt

Explain A negative distributor changes every sign it touches 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: simplify with a negative distributor — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Name the outside multiplier

Animation

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

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

Takeaway: Four variable terms minus six variable terms leaves negative two variable terms.

Animation

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

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

Takeaway: At four, both expressions give negative five.

Predict first

A student writes this, and it looks reasonable:

Multiply the first term, then keep the second sign unchanged

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

Correct: This misses the second multiplication by the outside negative.

This misses the second multiplication by the outside negative.

Why: This misses the second multiplication by the outside negative.

Annotate

From Trap: the negative does not stop halfway — read this one piece at a time. What is each part doing?

On: \( -3(1-4)=9\quad\text{and}\quad-3(1)+12=9\checkmark \)

• This misses the second multiplication by the outside negative.
• At one, the original gives nine, but the wrong expression gives negative fifteen.
• A negative times a negative constant becomes positive.

Discussion prompt

Explain Equations stay balanced 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.

Fill in the blanks

From Model: one-step equation — finish the line. Write what belongs on the right of the equals sign before you look.

x+7-7 = 12-7

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Addition is undone by subtraction, and both sides must change together.

Animation

Shows: Each line of the worked example "Model: one-step equation", appearing one at a time.

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

Takeaway: Addition is undone by subtraction, and both sides must change together.

Predict first

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

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

Correct: Add nine to both sides

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. Adding nine undoes subtracting nine while keeping the equation balanced.

Animation

Shows: Each line of the worked example "Independent try: one-step equation", appearing one at a time.

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

Takeaway: Adding nine undoes subtracting nine while keeping the equation balanced.

Put in order

Put the moves of Model: two-step equation into the order they have to happen.

1. Undo the addition or subtraction first
2. Undo the coefficient second
3. State the answer in test-answer form
4. Verify in the original equation

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. The minus four was applied after the multiplication by three.

Animation

Shows: Each line of the worked example "Model: two-step equation", appearing one at a time.

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

Takeaway: Three times five minus four equals eleven.

Predict first

Guided try: two-step equation 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: clear the constant first

Why: Undo plus six before dividing by the coefficient.

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 "Guided try: two-step equation", appearing one at a time.

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

Takeaway: Four times negative four plus six equals negative ten.

Discussion prompt

Explain Multi-step equations: clean sides, then undo 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: clean sides, then solve", appearing one at a time.

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

Takeaway: Undo the subtraction around the variable.

Discussion prompt

Guided try: equation with a negative distributor — 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 before solving

Animation

Shows: Each line of the worked example "Guided try: equation with a negative distributor", appearing one at a time.

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

Takeaway: Substituting negative one gives fourteen.

Predict first

A student writes this, and it looks reasonable:

Subtract five variable terms before distributing

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

Correct: The terms are not ready to combine because one side still has parentheses.

The terms are not ready to combine because one side still has parentheses.

Why: The terms are not ready to combine because one side still has parentheses.

Sort into buckets

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

Holds up

The hour is built in short loops: teacher model, guided attempt, independent attempt, then a correction note.; A missed problem is not finished when the answer is copied. It is finished when the student can name the mistake and repair it.; Sorting first slows the student down just enough to prevent unlike-term mistakes.

Breaks

Add the coefficients and keep one variable; Multiply the first term, then keep the second sign unchanged

sound

These are stated as this lesson states them — each one survives the edge cases Week 1: Expressions, Equations, and Literal Equations 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.

Discussion prompt

Variable terms are quantities too. You can add or subtract them from both sides just like constants.

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: variable on both sides — finish the line. Write what belongs on the right of the equals sign before you look.

5x - 2 = 3x + 8

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Collect variables on the side with more variable terms.

Animation

Shows: Each line of the worked example "Model: variable on both sides", appearing one at a time.

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

Takeaway: Clear the constant from the variable side.

Animation

Shows: Each line of the worked example "Independent try: variable on both sides", appearing one at a time.

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

Takeaway: Clear the constant from the variable side.

Match the pairs

Match each term to the definition this lesson gave it — not the one you would guess from the word.

• t1. term
• t2. coefficient
• t3. target variable

• d1. One signed piece of an expression. Terms are separated by plus or minus signs that are not inside parentheses.
• d2. The numerical factor attached to a variable part.
• d3. The variable that must be alone at the end.

Why: These are the working definitions of term, coefficient, target variable as Week 1: Expressions, Equations, and Literal Equations uses them. Pairing them correctly is the test of whether you could state each one with the slide switched off.

Fill in the blanks

From Model: solve a formula for height — finish the line. Write what belongs on the right of the equals sign before you look.

\fracA\checkmark___b\left(\frac______\right) = ___

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. The target is multiplied by b and by one half.

Animation

Shows: Each line of the worked example "Model: solve a formula for height", appearing one at a time.

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

Takeaway: The original formula simplifies back to A.

Predict first

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

Commit before you compute: what does Guided try: solve slope-intercept form for the input come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Divide both sides by m

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. This removes the coefficient attached to the target.

Animation

Shows: Each line of the worked example "Guided try: solve slope-intercept form for the input", appearing one at a time.

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

Takeaway: This removes the coefficient attached to the target.

Discussion prompt

Independent try: solve a perimeter formula — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Hint: clear the term that does not contain the target

Animation

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

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

Takeaway: This leaves the target term on the right.

Comparison matrix

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

Discussion prompt

Outside this lesson: where does Week 1: Expressions, Equations, and Literal Equations 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 1 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: 4x - 3(2x - 5) + 7

• A. -2x + 22 (correct)
• B. -2x - 8
• C. 10x - 8
• D. -2x + 8

Answer: A

Why: Distribute the negative three to both inside terms: 4x - 6x + 15 + 7. Then combine like terms to get -2x + 22.

Why B tempts people

This distributes -3 as -6x - 15. The second product should be positive because a negative times a negative is positive.

Why C tempts people

This treats the outside -3 as if it adds 6x and subtracts 15. The multiplier is negative three.

Why D tempts people

This gets the variable term right but loses part of the constants. The constants are 15 and 7, which make 22.

Check your understanding

Solve: 2(3x - 4) = 5x + 9

• A. x = 17 (correct)
• B. x = -17
• C. x = 1
• D. x = 9

Answer: A

Why: Distribute first: 6x - 8 = 5x + 9. Subtract 5x from both sides to get x - 8 = 9, then add 8. The solution is x = 17.

Why B tempts people

This usually comes from subtracting 9 instead of adding 8 after x - 8 = 9.

Why C tempts people

This stops after comparing coefficients. The constants still determine the value of x.

Why D tempts people

This reports the right-side constant after collecting variable terms, but x - 8 = 9 still needs one more step.

Check your understanding

Solve A = (1/2)bh for h.

• A. h = 2A / b (correct)
• B. h = A / (2b)
• C. h = 2Ab
• D. h = A - (1/2)b

Answer: A

Why: Multiply both sides by 2 to get 2A = bh. Then divide both sides by b to get h = 2A / b.

Why B tempts people

This divides by 2 instead of multiplying by 2. The factor one half must be undone by multiplying by 2.

Why C tempts people

This multiplies by b instead of dividing by b. The target is already multiplied by b.

Why D tempts people

This treats multiplication as subtraction. Literal equations still use inverse operations.

Put in order

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

1. Hint: use the Week 1 procedure
2. Solve the three problems
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. Sort, distribute, clean, undo, and verify.

Comparison matrix

From Independent mini exit ticket: refill the Task column from what you know. The rest of the table is as it appeared.

Draw it

One page, no notation unless you need it: draw how these connect — Week 1 master procedure · How this hour will run · Correction rule for every miss · First, sort the pieces · Like terms must match exactly. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.