Week 2: Inequalities, Graphing, and Functions

Discussion prompt

Before we open Week 2: Inequalities, Graphing, and Functions: without looking back, what was the main idea of Week 1: Expressions, Equations, and Literal Equations, 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 Equation answers versus inequality answers: every row here is a choice with a cost. Fill the Answer type column, then say which row you would actually pick and what you give up for it.

Put in order

Put the moves of Model: one-step inequality into the order they have to happen.

1. Subtract three from both sides
2. Decide endpoint type
3. Shade the correct side
4. Check with one inside and one outside 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. Undo the added constant while keeping the inequality equivalent.

Animation

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

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

Takeaway: Four works, zero works, and five does not.

Discussion prompt

Guided try: one-step inequality — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Hint: undo subtraction

Animation

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

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

Takeaway: The endpoint is open because three is not included.

Sort into buckets

Sort these by Circle, from memory, without looking back at Graphing inequality endpoints. Telling them apart on the spot is the skill; the table is only where the answer happens to be written down.

open

less than; greater than

closed

less than or equal; greater than or equal

g1

Circle is "open" for less than, greater than — that is what the table on "Graphing inequality endpoints" records, and it is the single property separating this group from the rest.

g2

Circle is "closed" for less than or equal, greater than or equal — that is what the table on "Graphing inequality endpoints" records, and it is the single property separating this group from the rest.

Animation

Shows: Each line of the worked example "Model: read a graph from notation", appearing one at a time.

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

Takeaway: Negative three and zero work; negative four does not.

Predict first

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

Commit before you compute: what does Independent try: graph from notation 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 with sample values

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. One works, two does not, and three does not.

Animation

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

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

Takeaway: One works, two does not, and three does not.

Discussion prompt

Most inequality moves are equation moves. The exception happens when multiplying or dividing by a negative number.

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

Explain Why the symbol reverses 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: solve and graph with a flip", appearing one at a time.

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

Takeaway: Negative three works; negative five fails.

Animation

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

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

Takeaway: Negative four works, zero works, and negative five fails.

Predict first

A student writes this, and it looks reasonable:

Subtract five from both sides

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

Correct: The division happened, but the order reversal was missed.

The division happened, but the order reversal was missed.

Why: The division happened, but the order reversal was missed.

Annotate

From Trap: forgetting the flip — read this one piece at a time. What is each part doing?

On: \( -2(-5)+5=15\not<13 \)

• The division happened, but the order reversal was missed.
• Negative five is on the claimed side, but it fails the original inequality.
• Negative division reflects the number line.

Comparison matrix

From Compound inequalities: and versus or: refill the Graph behavior column from what you know. The rest of the table is as it appeared.

Discussion prompt

Model: graph an and statement — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Handle the left boundary

Animation

Shows: Each line of the worked example "Model: graph an and statement", appearing one at a time.

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

Takeaway: Negative three works, four does not, and zero works.

Animation

Shows: Each line of the worked example "Guided try: graph another and statement", appearing one at a time.

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

Takeaway: Zero and five work; negative one does not.

Animation

Shows: Each line of the worked example "Model: graph an or statement", appearing one at a time.

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

Takeaway: Negative three works, three works, and zero fails.

Put in order

Put the moves of Independent try: translate words to compound inequality into the order they have to happen.

1. Hint: at least means included
2. Translate the lower boundary
3. Translate the upper boundary
4. Combine with and
5. Check with examples

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. Write the lower boundary first, then the upper boundary.

Animation

Shows: Each line of the worked example "Independent try: translate words to compound inequality", appearing one at a time.

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

Takeaway: Ten and seventeen work; eighteen does not.

Discussion prompt

Explain Function notation is not multiplication 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: evaluate a function — finish the line. Write what belongs on the right of the equals sign before you look.

f(x) = 2x^2-3\qquad f(-2)

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Order of operations puts the exponent before the coefficient multiplication.

Animation

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

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

Takeaway: Input negative two produces output five.

Predict first

Guided try: evaluate with a positive input 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: substitute the input twice

Why: The same input goes into every copy of the variable.

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: evaluate with a positive input", appearing one at a time.

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

Takeaway: The input-output pair is three, negative three.

Predict first

A student writes this, and it looks reasonable:

Read the notation as a product

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

Correct: This ignores the rule that defines the function.

This ignores the rule that defines the function.

Why: This ignores the rule that defines the function.

Annotate

From Trap: treating function notation like multiplication — read this one piece at a time. What is each part doing?

On: \( f(x)=2x^2-3\qquad f(-2) \)

• This ignores the rule that defines the function.
• The answer still has the function name in it, so no numerical output was found.
• The notation asks for the output when the input is negative two.

Step through it

Step through Domain and range from a table one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: input is -2
2. Step 2: input is 0
3. Step 3: input is 3

Discussion prompt

Model: find domain and range from a table — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: List the input values

Animation

Shows: Each line of the worked example "Model: find domain and range from a table", appearing one at a time.

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

Takeaway: Every listed value appears in its correct column.

Predict first

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

Commit before you compute: what does Independent try: domain and range 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 by matching each table row

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. Every table row uses an input from the domain and output from the range.

Animation

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

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

Takeaway: Every table row uses an input from the domain and output from the range.

Discussion prompt

Some formulas forbid inputs that make a denominator zero.

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: find a formula domain — finish the line. Write what belongs on the right of the equals sign before you look.

r(x) = \frac{1}{x-4}

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. This identifies the input that would break the formula.

Animation

Shows: Each line of the worked example "Model: find a formula domain", appearing one at a time.

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

Takeaway: At four, the denominator is zero, so four is not allowed.

Fill in the blanks

From Guided try: another formula domain — finish the line. Write what belongs on the right of the equals sign before you look.

p(-2) = \frac{5}{0}\quad\text{not allowed}

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Solve the small equation that makes the denominator zero.

Animation

Shows: Each line of the worked example "Guided try: another formula domain", appearing one at a time.

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

Takeaway: All real inputs are allowed except negative two.

Predict first

A student writes this, and it looks reasonable:

Use an open circle at the boundary

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

Correct: This says the boundary value is not included, which contradicts the equality part.

This says the boundary value is not included, which contradicts the equality part.

Why: This says the boundary value is not included, which contradicts the equality part.

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 equation often has one exact answer. An inequality usually has a whole set of answers.; The symbol decides whether the boundary point is included. The direction decides which side gets shaded.; Most inequality moves are equation moves. The exception happens when multiplying or dividing by a negative number.

Breaks

Subtract five from both sides; Read the notation as a product

sound

These are stated as this lesson states them — each one survives the edge cases Week 2: Inequalities, Graphing, and Functions 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.

Predict first

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

Commit before you compute: what does Guided try: evaluate with zero 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 with an input-output pair

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. The ordered pair for this evaluation is zero, two.

Animation

Shows: Each line of the worked example "Guided try: evaluate with zero", appearing one at a time.

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

Takeaway: The ordered pair for this evaluation is zero, two.

Discussion prompt

Independent try: mixed Week 2 practice — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Hint: these are three different jobs

Animation

Shows: Each line of the worked example "Independent try: mixed Week 2 practice", appearing one at a time.

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

Takeaway: Four satisfies the inequality boundary, five breaks the function, and zero gives the listed output.

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 2: Inequalities, Graphing, and Functions 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 2 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

Solve: -2x + 5 < 13

• A. x > -4 (correct)
• B. x < -4
• C. x > 4
• D. x < 4

Answer: A

Why: Subtract 5 to get -2x < 8. Divide by -2 and reverse the inequality, giving x > -4.

Why B tempts people

This is the no-flip answer. Dividing by a negative reverses the inequality.

Why C tempts people

This reverses the direction but loses the negative endpoint. Eight divided by negative two is negative four.

Why D tempts people

This misses both the negative endpoint and the direction flip.

Check your understanding

Which graph matches x >= -3?

• A. closed circle at -3, shade right (correct)
• B. open circle at -3, shade right
• C. closed circle at -3, shade left
• D. open circle at -3, shade left

Answer: A

Why: The equality part includes -3, so the endpoint is closed. Greater values are to the right, so shade right.

Why B tempts people

This shades the correct direction but uses an open endpoint. The equality part means the endpoint is included.

Why C tempts people

This includes the endpoint but shades the wrong direction. Greater values live to the right.

Why D tempts people

This misses both the endpoint type and the shading direction.

Eliminate the wrong options

Which notation matches: at least 10 and less than 18?

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. 10 <= t < 18
• B. 10 < t <= 18
• C. t <= 10 or t > 18
• D. t < 10 and t >= 18

Survives elimination: A

Why: At least 10 means 10 is included. Less than 18 means 18 is not included. The word and creates the overlap 10 <= t < 18.

Check your understanding

Which notation matches: at least 10 and less than 18?

• A. 10 <= t < 18 (correct)
• B. 10 < t <= 18
• C. t <= 10 or t > 18
• D. t < 10 and t >= 18

Answer: A

Why: At least 10 means 10 is included. Less than 18 means 18 is not included. The word and creates the overlap 10 <= t < 18.

Why B tempts people

This reverses which endpoint is included. At least includes 10; less than excludes 18.

Why C tempts people

This describes outside the interval, not the values between 10 and 18.

Why D tempts people

This asks for values below 10 and at least 18 at the same time, which is impossible.

Check your understanding

If f(x) = 2x^2 - 3, what is f(-2)?

• A. 5 (correct)
• B. -11
• C. 1
• D. -7

Answer: A

Why: Substitute -2 for x: f(-2) = 2(-2)^2 - 3. The square gives 4, so the output is 8 - 3 = 5.

Why B tempts people

This treats the negative input as staying negative through the square. Parentheses matter: (-2)^2 = 4.

Why C tempts people

This likely comes from forgetting the square and doing 2(-2) - 3 incorrectly.

Why D tempts people

This comes from evaluating 2(-2) - 3, which ignores the exponent.

Eliminate the wrong options

What value is excluded from the domain of p(x) = 5 / (x + 2)?

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. x = -2
• B. x = 2
• C. x = 0
• D. x = 5

Survives elimination: A

Why: The denominator cannot be zero. Set x + 2 = 0 and solve to get x = -2, so -2 is excluded.

Check your understanding

What value is excluded from the domain of p(x) = 5 / (x + 2)?

• A. x = -2 (correct)
• B. x = 2
• C. x = 0
• D. x = 5

Answer: A

Why: The denominator cannot be zero. Set x + 2 = 0 and solve to get x = -2, so -2 is excluded.

Why B tempts people

This uses the opposite sign. The denominator is zero when x is negative two.

Why C tempts people

Zero is allowed because the denominator would be two, not zero.

Why D tempts people

The numerator does not decide the excluded input in this problem.

Put in order

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

1. Hint: one problem needs the flip rule
2. Solve the three prompts
3. Check each result

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 inequality divides by a negative number.

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 2 master procedure · How this hour will run · Equation answers versus inequality answers · Graphing inequality endpoints · The one special inequality rule. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.