A student claims, “If two numbers are both divisible by 6, then their sum is divisible by 12.” Which choice best evaluates the claim?

False, because the sum is always divisible by 6 but not always by 12

True, because divisibility by 6 implies divisibility by 12

False, because divisibility by 6 implies the sum is prime

Which statement best represents a valid mathematical proof strategy at the secondary level?

Using a general argument that applies to all cases

Checking the answer with a calculator

NBPTS National Board Certification Mathematics: Adolescence and Young Adulthood

Free NBPTS National Board Practice Questions

10 sample questions for the NBPTS National Board Certification Mathematics: Adolescence and Young Adulthood, with the correct answer and an explanation for each. No account needed. Try each one before you open the answer.

Sections covered on the exam: Contexts for Mathematics, Algebra and Functions, Data Analysis and Statistics, Discrete Mathematics, Numbers and Operations, Geometry, Trigonometry, Calculus.

Question 1 · Contexts for Mathematics
A student claims, “If two numbers are both divisible by 6, then their sum is divisible by 12.” Which choice best evaluates the claim?
- A.True, because 6 + 6 = 12
- B.True, because divisibility by 6 implies divisibility by 12
- C.False, because the sum is always divisible by 6 but not always by 12
- D.False, because divisibility by 6 implies the sum is prime
Show answer and explanation
Answer: C. False, because the sum is always divisible by 6 but not always by 12
If a = 6m and b = 6n, then a + b = 6(m + n), so the sum is always divisible by 6. It is divisible by 12 only when (m + n) is even. Counterexample: 6 and 18 are both divisible by 6, but 6 + 18 = 24 (divisible by 12) works; however 6 and 12 give 18 (not divisible by 12). So the claim is false; the correct statement is “sum is divisible by 6,” not necessarily 12.
Question 2 · Numbers and Operations
A price increases from $80 to $92. What is the percent increase?
- A.10%
- B.12%
- C.20%
- D.15%
Show answer and explanation
Answer: D. 15%
Increase = 92 − 80 = 12. Percent increase = 12/80 = 0.15 = 15%.
Question 3 · Geometry
What is the distance between points (1, 2) and (4, 6)?
- A.3
- B.4
- C.√25
- D.5
Show answer and explanation
Answer: D. 5
Distance = √[(4−1)² + (6−2)²] = √(9 + 16) = √25 = 5.
Question 4 · Discrete Mathematics
How many outcomes are possible when flipping a coin and rolling a six-sided die?
- A.8
- B.12
- C.6
- D.36
Show answer and explanation
Answer: B. 12
The Fundamental Counting Principle applies. There are 2 outcomes for the coin and 6 for the die, giving 2 × 6 = 12 outcomes.
Question 5 · Data Analysis and Statistics
In a normal distribution, approximately what percentage of data falls within one standard deviation of the mean?
- A.68%
- B.50%
- C.90%
- D.95%
Show answer and explanation
Answer: A. 68%
By the empirical rule, about 68 percent of data lie within one standard deviation of the mean in a normal distribution.
Question 6 · Contexts for Mathematics
Which statement best represents a valid mathematical proof strategy at the secondary level?
- A.Showing one numerical example
- B.Restating the problem in words
- C.Using a general argument that applies to all cases
- D.Checking the answer with a calculator
Show answer and explanation
Answer: C. Using a general argument that applies to all cases
A proof must establish truth for all cases within a domain. General arguments using definitions, properties, or algebraic structure meet this requirement. Single examples do not constitute proof.
Question 7 · Numbers and Operations
A quantity decreases from 200 to 150. What is the percent decrease?
- A.15%
- B.25%
- C.20%
- D.50%
Show answer and explanation
Answer: B. 25%
Decrease is 50. Percent decrease = 50/200 = 0.25 = 25%.
Question 8 · Geometry
What is the slope of the line passing through (−2, 4) and (3, −6)?
- A.−1
- B.−2
- C.2
- D.10
Show answer and explanation
Answer: B. −2
Slope = (−6 − 4)/(3 − (−2)) = −10/5 = −2.
Question 9 · Numbers and Operations
A password consists of 2 letters followed by 3 digits. How many possible passwords are there if repetition is allowed?
- A.6,760
- B.26² · 10³
- C.26³ · 10²
- D.36⁵
Show answer and explanation
Answer: B. 26² · 10³
There are 26 choices for each letter and 10 choices for each digit. By the counting principle, total outcomes are 26² · 10³.
Question 10 · Data Analysis and Statistics
Which sampling method is most likely to produce biased results?
- A.Voluntary response sampling
- B.Stratified sampling
- C.Systematic sampling
- D.Simple random sampling
Show answer and explanation
Answer: A. Voluntary response sampling
Voluntary response samples overrepresent individuals with strong opinions, leading to bias.

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