Let x be an odd number

2024 Handbook

2024 Test A Math · Question 23 of 57

Question

Let x be an odd number. In terms of x, what is the sum of the two even numbers closest to x?

A $x$
B $2x$
C $2x - 2$
D $2x - 4$

Show the answer

Correct answer: B

If x is odd, then the two even numbers closest to x are $(x - 1)$ and $(x + 1)$ . Find the sum of these two expressions:

$(x - 1) + (x + 1) = 2x$

Question analysis

An algebra question about odd and even numbers: x is odd, and students write the sum of the two even numbers closest to x in terms of x. It tests representing neighboring integers with expressions and adding them. Deciding exactly which even numbers are closest to an odd number is where errors start.

NYS standards: 6.EE.2a writing algebraic expressions
7.EE.1 linear expressions with rational coefficients
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Grade 7 students write and combine linear expressions; here the expressions describe neighboring integers, a familiar idea from consecutive-integer problems. Two short steps and a quick number check put it at typical grade 7 demand.
Estimated difficulty: Straightforward: two steps or one careful reading.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: Variables instead of numbers moderate
Turning words into math minor
Solution: 2 steps
Reading level: Question and choices: about grade 3, 26 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The algebra is a single addition, so the work is in naming the right two numbers. An odd x sits between two even numbers, one step away on each side, and the minus one and plus one cancel. Wrong choices come from picking both neighbors below x, from treating x as even, or from giving the middle of the pair; a test value catches each of them.

Variables instead of numbers moderate Understanding the question
Here x stands for any odd number, so the two even neighbors must be written as x − 1 and x + 1; trying x = 7, with neighbors 6 and 8, makes it concrete. (Koedinger & Nathan, 2004; Enright, Morley & Sheehan, 2002)
Turning words into math minor Understanding the question
The phrase "the two even numbers closest to x" means one just below and one just above x, not two in a row on the same side. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)

Strategy: Substitute a real odd number such as 7 for x, find its two closest even numbers and their sum, then see which expression gives that sum.

Traps

Misreads the question: Misses or misreads a condition or a word in the question: "at least", "each", "remaining", "not".
Answers a different question: A choice is right for another quantity: the other unknown, the rest instead of the part, the part instead of the whole.

The wrong answers, classified

A Answers a different question (King et al., 2004; Hegarty, Mayer & Monk, 1995)
Reveals partial knowledge Expected pull: weak
Finds the number midway between the two even neighbors, x itself, which is their mean rather than their sum.
C Misreads the question (Hegarty, Mayer & Monk, 1995)
Reveals a slip Expected pull: weak
Treats x as if it were even and adds it to the even number below it, x + (x − 2), ignoring that x is odd.
D Misreads the question (Hegarty, Mayer & Monk, 1995)
Reveals a slip Expected pull: moderate
Takes the two even numbers just below x, x − 1 and x − 3, instead of the closest one on each side.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Neighbors of a number: $x - 1 \text{ and } x + 1$; x is any whole number. The number just below it is x − 1, and the number just above it is x + 1. Whole numbers take turns being odd and even. So if x is odd, both of its neighbors are even.
Combining like terms: $x + x = 2x$; Like terms have the same letter, such as x and 2x. To add them, add the numbers in front. x by itself means 1x, so x + x is 2x, which is two x’s.

Sort out the question
x is an odd number. You need the sum of the two even numbers closest to x. Sum means the result of adding. In terms of x means write the answer using the letter x.
Try a real number first
Pick an odd number, like 7. The even numbers closest to 7 are 6 and 8, one on each side.
$6 + 8 = 14$
Notice that 14 is $2 \times 7$ .
Write the neighbors with x
Odd and even numbers take turns. So both numbers right next to an odd x are even.
$x - 1 \text{ and } x + 1$
For x = 7, these are 6 and 8. Watch out: take one even number below x and one above. Using two even numbers below x, $x - 1$ and $x - 3$ , gives $2x - 4$ , choice D.
Add the two neighbors
$(x - 1) + (x + 1) = x + x - 1 + 1 = 2x$
The −1 and the +1 cancel, and $x + x = 2x$ . Watch out: x is the number in the middle, not the sum, so choice A is wrong. So the answer is B, $2x$ .

Check your answer

Try another odd number, 11. Its even neighbors are 10 and 12, and $10 + 12 = 22$ . The answer 2x gives $2 \times 11 = 22$ . They match.

The original DOE question

Question 80 as printed on page 82 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 82, question 80. Open page 82 of the handbook (PDF) The handbook letters this question’s choices E–H; here they are A–D (E = A, F = B, G = C, H = D).