Week 3: Slope, Linear Equations, and Systems

Discussion prompt

Before we open Week 3: Slope, Linear Equations, and Systems: without looking back, what was the main idea of Week 2: Inequalities, Graphing, and Functions, 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 Ordered pairs have roles: every row here is a choice with a cost. Fill the Role column, then say which row you would actually pick and what you give up for it.

Discussion prompt

Slope compares output change to input change.

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: slope from two points into the order they have to happen.

1. Subtract the output values in one order
2. Subtract the input values in the same order
3. Divide vertical change by horizontal change
4. Verify by reading the rate between the points

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 denominator must use the same point order.

Animation

Shows: Each line of the worked example "Model: slope from two points", appearing one at a time.

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

Takeaway: Four input steps produce eight output steps, which is two per step.

Discussion prompt

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

Hint: It starts with: Hint: keep the subtraction order matched

Animation

Shows: Each line of the worked example "Guided try: slope from points", appearing one at a time.

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

Takeaway: Four input steps at negative three per step creates negative twelve output change.

Predict first

A student writes this, and it looks reasonable:

Put horizontal change on top

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

Correct: The subtraction is real, but the fraction is upside down.

The subtraction is real, but the fraction is upside down.

Why: The subtraction is real, but the fraction is upside down.

Annotate

From Trap: flipping the slope fraction — read this one piece at a time. What is each part doing?

On: \( (2,5)\quad\text{and}\quad(6,13) \)

• The subtraction is real, but the fraction is upside down.
• A rate of one half would make four input steps produce only two output steps.
• Slope is output change divided by input change.

Step through it

Step through Slope signs tell direction one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: As input increases is left to right
2. Step 2: As input increases is left to right
3. Step 3: As input increases is left to right

Predict first

The table runs: 1 | 7 · 2 | 10 · 3 | 13

In Model: slope from a table, given the rows so far: what is the next one — the row where x is 4?

Correct: 4 | 16

Why: The relationship between the columns, not the individual numbers, is what generates the next row. From input one to four, output changes by nine over three input steps.

Animation

Shows: Each line of the worked example "Model: slope from a table", appearing one at a time.

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

Takeaway: From input one to four, output changes by nine over three input steps.

Predict first

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

Commit before you compute: what does Independent try: slope from a table 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 across the full table

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. From zero to four, output changes from five to negative three, a change of negative eight over four.

Animation

Shows: Each line of the worked example "Independent try: slope from a table", appearing one at a time.

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

Takeaway: From zero to four, output changes from five to negative three, a change of negative eight over four.

Discussion prompt

Explain Slope-intercept form separates two jobs 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: identify slope and intercept", appearing one at a time.

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

Takeaway: The line drops four output units per one input unit and starts at nine.

Fill in the blanks

From Guided try: identify from an equation — finish the line. Write what belongs on the right of the equals sign before you look.

\frac-6\checkmark___(0)-6 = ___

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. The coefficient of the input is one half.

Animation

Shows: Each line of the worked example "Guided try: identify from an equation", appearing one at a time.

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

Takeaway: At input zero, the output is negative six.

Discussion prompt

Explain A line needs a rate and an anchor 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: write from slope and point — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Start in point-slope form

Animation

Shows: Each line of the worked example "Model: write from slope and point", appearing one at a time.

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

Takeaway: This starts the conversion to slope-intercept form.

Predict first

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

Commit before you compute: what does Guided try: write from slope and point 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: point-slope form prevents guessing the intercept

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. Substitute the point carefully; subtracting a negative becomes addition.

Animation

Shows: Each line of the worked example "Guided try: write from slope and point", appearing one at a time.

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

Takeaway: Put the equation into slope-intercept form.

Discussion prompt

Explain Two points can create the line 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: write from two points — finish the line. Write what belongs on the right of the equals sign before you look.

y-2 = 3(x-1)

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Output change is six and input change is two.

Animation

Shows: Each line of the worked example "Model: write from two points", appearing one at a time.

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

Takeaway: Both original points satisfy the final equation.

Put in order

Put the moves of Independent try: write from two points into the order they have to happen.

1. Hint: find slope before writing the equation
2. Find the slope
3. Use the intercept from the point with input zero
4. Write the equation
5. Verify with the second point

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 first point already reveals the intercept, but still verify with the second.

Animation

Shows: Each line of the worked example "Independent try: write from two points", appearing one at a time.

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

Takeaway: When input is zero, output is the intercept.

Predict first

A student writes this, and it looks reasonable:

Put the point output directly into the intercept spot

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

Correct: This assumes the point is on the vertical axis, but its input is four.

This assumes the point is on the vertical axis, but its input is four.

Why: This assumes the point is on the vertical axis, but its input is four.

Annotate

From Trap: using the point output as the intercept — read this one piece at a time. What is each part doing?

On: \( y=-3x+13\quad\Rightarrow\quad-3(4)+13=1\checkmark \)

• This assumes the point is on the vertical axis, but its input is four.
• Input four gives negative eleven, not one.
• The point anchors the line away from the vertical axis.

Discussion prompt

Explain A system asks for a shared solution 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.

Fill in the blanks

From Model: solve by substitution — finish the line. Write what belongs on the right of the equals sign before you look.

2x+1 = -x+7

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. At the intersection, both equations have the same output.

Animation

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

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

Takeaway: At the intersection, both equations have the same output.

Predict first

Guided try: substitution 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: both equations already equal the output

Why: Set the right sides equal to find the input.

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: substitution", appearing one at a time.

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

Takeaway: At the shared solution the outputs match.

Discussion prompt

Elimination adds or subtracts equations so one variable disappears.

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.

Discussion prompt

Model: solve by elimination — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Add the equations

Animation

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

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

Takeaway: The ordered pair satisfies both equations.

Animation

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

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

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

Comparison matrix

From Special systems have clues: refill the What happens column from what you know. The rest of the table is as it appeared.

Animation

Shows: Each line of the worked example "Model: recognize no solution", appearing one at a time.

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

Takeaway: The variable terms cancel and leave a false statement.

Animation

Shows: Each line of the worked example "Guided try: recognize infinitely many solutions", appearing one at a time.

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

Takeaway: Input zero gives output six in both equations.

Predict first

Rewrite and identify slope and intercept.

Commit before you compute: what does Model: rewrite to slope-intercept form 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 by substituting an easy point

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. Input zero gives output six, and the original equation stays true.

Animation

Shows: Each line of the worked example "Model: rewrite to slope-intercept form", appearing one at a time.

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

Takeaway: Input zero gives output six, and the original equation stays true.

Discussion prompt

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

Hint: It starts with: Hint: isolate the output variable

Animation

Shows: Each line of the worked example "Guided try: rewrite and read a line", appearing one at a time.

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

Takeaway: Input zero gives output negative five, and the original equation is true.

Predict first

A student writes this, and it looks reasonable:

Use one order for outputs and the opposite order for inputs

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

Correct: The mismatched order flips the slope sign.

The mismatched order flips the slope sign.

Why: The mismatched order flips the slope sign.

Sort into buckets

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

Holds up

Week 3 is about turning linear work into a dependable routine before Algebra 2 graphing gets more demanding.; The first coordinate is the input. The second coordinate is the output.; Slope compares output change to input change.

Breaks

Put horizontal change on top; Put the point output directly into the intercept spot

sound

These are stated as this lesson states them — each one survives the edge cases Week 3: Slope, Linear Equations, and Systems 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.

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.

Discussion prompt

Outside this lesson: where does Week 3: Slope, Linear Equations, and Systems 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 3 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.

Eliminate the wrong options

What is the slope through (2, 5) and (6, 13)?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. 2
• B. 1/2
• C. -2
• D. 4

Survives elimination: A

Why: The vertical change is 13 - 5 = 8 and the horizontal change is 6 - 2 = 4. Slope is 8 / 4 = 2.

Check your understanding

What is the slope through (2, 5) and (6, 13)?

• A. 2 (correct)
• B. 1/2
• C. -2
• D. 4

Answer: A

Why: The vertical change is 13 - 5 = 8 and the horizontal change is 6 - 2 = 4. Slope is 8 / 4 = 2.

Why B tempts people

This flips the slope fraction. Slope is vertical change over horizontal change.

Why C tempts people

This has the right size but wrong sign. Both coordinates increased, so the slope is positive.

Why D tempts people

This reports the horizontal change, not the ratio.

Predict first

Which line has slope -3 and passes through (4, 1)?

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: y = -3x + 13

Why: Use y - 1 = -3(x - 4). Distribute to get y - 1 = -3x + 12, then add 1 to get y = -3x + 13.

Check your understanding

Which line has slope -3 and passes through (4, 1)?

• A. y = -3x + 13 (correct)
• B. y = -3x + 1
• C. y = 3x - 11
• D. y = -3x - 13

Answer: A

Why: Use y - 1 = -3(x - 4). Distribute to get y - 1 = -3x + 12, then add 1 to get y = -3x + 13.

Why B tempts people

This uses the point output as the intercept. The point is not the intercept unless the input is zero.

Why C tempts people

This changes the slope sign. The requested slope is negative three.

Why D tempts people

This keeps the slope but moves the intercept in the wrong direction.

Check your understanding

Solve the system: y = 2x + 1 and y = -x + 7

• A. (2, 5) (correct)
• B. (2, -5)
• C. (5, 2)
• D. x = 2

Answer: A

Why: Set 2x + 1 = -x + 7. Then 3x = 6, so x = 2. Substitute back to get y = 5, so the solution is (2, 5).

Why B tempts people

This finds the correct input but substitutes with a sign mistake.

Why C tempts people

This swaps the coordinates. Ordered pairs are input first, output second.

Why D tempts people

This stops halfway. A system solution needs both coordinates.

Eliminate the wrong options

What is true about y = 2x + 1 and y = 2x - 3?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. no solution
• B. one solution
• C. infinitely many solutions
• D. solution is (2, 1)

Survives elimination: A

Why: The lines have the same slope but different intercepts, so they are parallel and never intersect. The system has no solution.

Check your understanding

What is true about y = 2x + 1 and y = 2x - 3?

• A. no solution (correct)
• B. one solution
• C. infinitely many solutions
• D. solution is (2, 1)

Answer: A

Why: The lines have the same slope but different intercepts, so they are parallel and never intersect. The system has no solution.

Why B tempts people

A one-solution system needs the lines to intersect. Parallel lines do not.

Why C tempts people

Infinitely many solutions would mean the two equations are the same line.

Why D tempts people

The ordered pair is not on both lines; this choice guesses instead of comparing the equations.

Put in order

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

1. Hint: each problem uses a different Week 3 routine
2. Solve the three 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. One is slope, one is line writing, and one is a system.

Comparison matrix

From Independent mini exit ticket: refill the Check 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 3 master procedure · How this hour will run · Ordered pairs have roles · Slope is rate of change · Slope signs tell direction. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.