SHSAT Sunset

On the number line above

2018 Handbook

2018 Test B Math · Question 23 of 57

Also in 2026 Practice Test B Math, question 21; 2025 Practice Test B Math, question 23.

Question

A number line with point M on the left and point N on the right. N is labeled 1 1/3; no other coordinates are marked.

On the number line above, MN=556MN=5\frac{5}{6}. What is the position of point MM?

  • A−716-7\frac{1}{6}
  • B−412-4\frac{1}{2}
  • C4124\frac{1}{2}
  • D7167\frac{1}{6}
Show the answer

Correct answer: B

To find MM, subtract N−MN-M and set it equal to the length:

113−M=5561\frac{1}{3}-M=5\frac{5}{6}

−M=556−113-M=5\frac{5}{6}-1\frac{1}{3}

−M=556−126-M=5\frac{5}{6}-1\frac{2}{6}

−M=436-M=4\frac{3}{6}

M=−412M=-4\frac{1}{2}

Question analysis

This number line question gives the distance between points M and N as a mixed number and asks for the position of M. Students subtract fractions with unlike denominators, moving past zero into negative numbers. Direction along the line and the sign of the result are where it gets tricky.

NYS standards
  • 7.NS.1c rational number subtraction
  • 6.NS.6c positioning rational numbers and points
Grade level
Grade 7 content, by NYS standard
Compared with grade 7
At grade 7 Grade 7 students find distances on a number line as the absolute value of a difference. Doing it with mixed numbers that have unlike denominators, and crossing zero, is a typical and slightly demanding grade 7 task.
Estimated difficulty
Moderate: several steps, or a close reading with one strong distractor.
Solution
3 steps
Reading level
Question and choices: about grade 8, 17 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 distance is given and one endpoint is known, so the question is really about direction and sign. Because the drawing shows no zero, students must reason that moving 5 5/6 to the left from 1 1/3 carries M past zero. The choices pair each size with each sign, so getting the size right is only half the work; the sign decides between two choices.

  1. Reading a figure, table or graph moderate Understanding the question

    The number line labels only N, at 1 1/3, and shows no zero, so students must see from the picture that M lies left of N and work out whether it passes zero. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)

  2. Error-prone arithmetic moderate Working it out

    The subtraction 1 1/3 − 5 5/6 mixes unlike denominators and crosses zero: in sixths it is 8/6 − 35/6 = −27/6, which then reduces to a mixed number. (Brown & Burton, 1978; Junker & Sijtsma, 2001)

  3. A very close wrong answer moderate Choosing the answer

    4 1/2 and −4 1/2 are written the same except for the sign; only noticing that M lies left of N, so below 1 1/3, separates them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)

Strategy: Read N's coordinate from the number line, convert everything to sixths, and move the given distance from N toward M's side of the line.

Traps

  • Negatives and direction: Arithmetic with negative numbers, the order of a subtraction, or direction on a number line. (Moving a term across the equals sign without changing its sign is a faulty procedure.)

The wrong answers, classified

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Distance on a number line
distance=right point−left point\text{distance} = \text{right point} - \text{left point}
On a number line, numbers grow as you move right. The distance between two points is the larger number minus the smaller one. That is the right point minus the left point, so a distance is never negative.
Mixed number to fraction
abc=a×c+bca\frac{b}{c} = \frac{a \times c + b}{c}
a is the whole number part, b is the top of the fraction, and c is the bottom. Each whole holds c pieces, so a wholes hold a times c pieces. Then add the b extra pieces.

  1. Read the number line

    Point N is at 1131\frac{1}{3}. Point M is to the left of N. MN means the distance from M to N, and it is 5565\frac{5}{6}.

    You need the number at point M.

  2. Decide which way to move

    On a number line, numbers get smaller as you move left. M is left of N, so N is the right point.

    distance=right point−left point\text{distance} = \text{right point} - \text{left point}

    556=113−M5\frac{5}{6} = 1\frac{1}{3} - M

    So M is what you get when you take the distance away from N.

    M=113−556M = 1\frac{1}{3} - 5\frac{5}{6}

    The distance is bigger than 1131\frac{1}{3}, so you will pass 0. M will be negative.

  3. Change both numbers to sixths

    To subtract, the fractions need the same bottom number. Use sixths, since 3 goes into 6. Change each mixed number to an improper fraction, a fraction whose top is bigger than its bottom.

    113=1×3+13=43=861\frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{4}{3} = \frac{8}{6}

    556=5×6+56=3565\frac{5}{6} = \frac{5 \times 6 + 5}{6} = \frac{35}{6}

  4. Subtract and simplify

    M=86−356=−276M = \frac{8}{6} - \frac{35}{6} = -\frac{27}{6}

    27 sixths is 4 wholes (24 sixths) and 3 sixths more. And 36=12\frac{3}{6} = \frac{1}{2}. So M is −412-4\frac{1}{2}.

    Watch out: 4124\frac{1}{2} without the minus sign is choice C. That point would be to the right of N, not the left.

    So the answer is B, −412-4\frac{1}{2}.

Check your answer

Find the distance from your M to N. From −412-4\frac{1}{2} to 0 is 4124\frac{1}{2}, or 276\frac{27}{6}. From 0 to N is 86\frac{8}{6}. Together, 276+86=356=556\frac{27}{6} + \frac{8}{6} = \frac{35}{6} = 5\frac{5}{6}. That matches MN.

More questions like this

The original DOE question

Question 80 as printed on page 157 of the DOE handbook.
As printed in the DOE’s Specialized High Schools Student Handbook 2018–2019, page 157, question 80. Open page 157 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).