Six starting checks — Kenneth Cheng

TOPIC 01

Concept of a set

Build understanding of concept of a set through definitions, contrasting cases and justified applications.

What you will be able to explain

Apply concept of a set with explicit conditions.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Count the distinct elements.

A=\{3,4,3,5\}

Use the domain, units and sampling assumptions stated in the question.

Working and explanation

BUILD THE REASONING

Hint 1

Check the membership rule and count distinct objects.

Hint 2

Remove the repeated entry.

Worked solution

Check the membership rule and count distinct objects.
Calculate or simplify this relation.
$A=\{3,4,5\};\quad |A|=3$
Order and repetition do not change a set.

The requested value is 3.

Checks and common pitfalls: Order and repetition do not change a set.

Think first. Reveal a hint when the class is ready.

TOPIC 02

Quadratic functions, equations and inequalities

Build understanding of quadratic functions, equations and inequalities through definitions, contrasting cases and justified applications.

What you will be able to explain

Apply quadratic functions, equations and inequalities with explicit conditions.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Solve the quadratic inequality.

(x-3)(x-6)\le0

Use the domain, units and sampling assumptions stated in the question.

Working and explanation

BUILD THE REASONING

Hint 1

Find the roots or vertex, then check signs and endpoints.

Hint 2

The upward-opening quadratic is nonpositive between its roots.

Worked solution

Find the roots or vertex, then check signs and endpoints.
Calculate or simplify this relation.
$3\le x\le6$
The non-strict comparison includes both roots.

x∈[3,6].

Checks and common pitfalls: The non-strict comparison includes both roots.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

TOPIC 03

Function concept and representations

Build understanding of function concept and representations through definitions, contrasting cases and justified applications.

What you will be able to explain

Interpret and use function concept and representations with domain checks.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Evaluate the function at the stated input.

f(x)=x^2-3x+1;\quad f(4)

Use the domain, units and sampling assumptions stated in the question.

Working and explanation

BUILD THE REASONING

Hint 1

Identify the input domain, rule and requested output before substituting.

Hint 2

Substitute the whole input into every occurrence of x.

Worked solution

Identify the input domain, rule and requested output before substituting.
Calculate or simplify this relation.
$f(4)=(4)^2-3(4)+1=5$
Function evaluation is different from solving f(x)=0.

The requested value is 5.

Checks and common pitfalls: Function evaluation is different from solving f(x)=0.

Think first. Reveal a hint when the class is ready.

TOPIC 04

Exponents

Build understanding of exponents through definitions, contrasting cases and justified applications.

What you will be able to explain

Interpret and use exponents with domain checks.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Simplify the product.

2^{3}\cdot2^3

Use the domain, units and sampling assumptions stated in the question.

Working and explanation

BUILD THE REASONING

Hint 1

Use exponent laws only within their real-number domain.

Hint 2

Add exponents when multiplying equal bases.

Worked solution

Use exponent laws only within their real-number domain.
Calculate or simplify this relation.
$2^{3+3}=2^{6}=64$
The bases must be the same for this rule.

The requested value is 64.

Checks and common pitfalls: The bases must be the same for this rule.

Think first. Reveal a hint when the class is ready.

TOPIC 05

General angles and radian measure

Build understanding of general angles and radian measure through definitions, contrasting cases and justified applications.

What you will be able to explain

Use general angles and radian measure with explicit angle units and domains.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Write the angle as cπ radians; find c.

90^{\circ}

Use the domain, units and sampling assumptions stated in the question.

Working and explanation

BUILD THE REASONING

Hint 1

Keep degrees and radians distinct and use the definition of angular measure.

Hint 2

One straight angle is π radians.

Worked solution

Keep degrees and radians distinct and use the definition of angular measure.
Calculate or simplify this relation.
$90^{\circ}\cdot\frac{\pi}{180^{\circ}}=\frac{3}6\pi$
The numerical coefficient changes with the unit, but the geometric angle does not.

The requested value is 0.5.

Checks and common pitfalls: The numerical coefficient changes with the unit, but the geometric angle does not.

Think first. Reveal a hint when the class is ready.

TOPIC 06

Concept of plane vectors

Build understanding of concept of plane vectors through definitions, contrasting cases and justified applications.

What you will be able to explain

Connect geometric and algebraic forms of concept of plane vectors.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Find the vector magnitude.

\mathbf a=(9,12)

Use the domain, units and sampling assumptions stated in the question.

Working and explanation

BUILD THE REASONING

Hint 1

Separate magnitude, direction and position; compare vectors by displacement.

Hint 2

Use the distance from the origin in coordinate space.

Worked solution

Separate magnitude, direction and position; compare vectors by displacement.
Calculate or simplify this relation.
$|\mathbf a|=\sqrt{(9)^2+(12)^2}=15$
The magnitude is a nonnegative scalar.

The requested value is 15.

Checks and common pitfalls: The magnitude is a nonnegative scalar.

Think first. Reveal a hint when the class is ready.