Terrell played 5 computer games

2024 Handbook

2024 Test A Math · Question 41 of 57

Question

Terrell played 5 computer games and earned a mean score of 8 points per game. If his mean score for the first 4 games was 7 points per game, how many points was his score in the fifth game?

Show the answer

Correct answer: C

If the mean number of points for the first four games is 7, the total number of points earned in those games is 28.

$4 \times 7 = 28$

If the mean number of points for all five games is 8, the total number of points earned is 40.

$5 \times 8 = 40$

To find the number of points in the fifth game, find the difference between 40 and 28.

$40 - 28 = 12$

Question analysis

This mean, or average, word problem says Terrell averaged 8 points over 5 computer games and 7 points over the first 4, and asks for his score in the fifth game. Students turn each average into a total and compare the totals. Combining the two averages directly, without going through totals, is the tempting shortcut that fails.

NYS standards: 6.SP.5c range, center, and overall pattern
6.SP.3 measures of center and variation
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: At grade 7 Computing a mean is grade 6 content. Working backward from two means to the totals they stand for, then to one missing score, is a common grade 7 test task, and it comes with a tempting shortcut.
Estimated difficulty: Moderate: several steps, or a close reading with one strong distractor.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: Turning words into math moderate
Something you must already know minor
Solution: 3 steps
Reading level: Question and choices: about grade 4, 35 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 question rewards turning both means into totals, 40 points for five games and 28 for four, after which one subtraction finishes it. Averaging 7 with the unknown score is a quick shortcut that gives a listed choice, but it weighs four games as one. Checking that 28 plus the answer, divided by 5, gives 8 catches that error and the near miss next to the answer.

Turning words into math moderate Understanding the question
Each mean has to become a total before the fifth game can be found: "a mean score of 8 points per game" over 5 games means 40 points in all. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Something you must already know minor Working it out
Students must know that a total is the mean times the number of games, and that means over different numbers of games cannot simply be averaged. (Anderson & Krathwohl, 2001)

Strategy: Turn each mean into a total by multiplying by its number of games, then subtract the four-game total from the five-game total.

Traps

Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.

The wrong answers, classified

A Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reveals a misconception Expected pull: strong
Averages the four-game mean with the fifth score, (7 + x) ÷ 2 = 8, so x = 9, as if the first four games counted once.
B Counting the ends (Brown & Burton, 1978)
Reveals partial knowledge Expected pull: moderate
Sees the fifth game must lift the mean 1 point above 7 but spreads that over only 4 games, 7 + 4 = 11, instead of all 5.
D Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)
Reveals a surface strategy Expected pull: weak
No common working gives 14, which is twice the four-game mean and matches no total or difference, so it mostly draws guesses.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Total from a mean: $\text{total} = \text{mean} \times \text{number of games}$; The mean, or average, is what each game would score if the total were shared out equally. So the total is the mean times the number of games. Use it to turn each mean back into a total. Totals can be subtracted, but means over different numbers of games can’t.

Sort out the question
Terrell played 5 games. His mean score for all 5 was 8 points per game. His mean for the first 4 games was 7 points per game. The mean, or average, is the total split equally among the games. You need his score in game 5.
Find the total for all 5 games
Turn the mean for all 5 games into a total. Multiply the mean by the number of games.
$\text{total} = \text{mean} \times \text{number of games}$
$\text{total} = 8 \times 5 = 40$
He scored 40 points in all 5 games together.
Find the total for the first 4 games
Use the same formula with the mean of 7 and 4 games.
$\text{total} = 7 \times 4 = 28$
He scored 28 points in the first 4 games.
Subtract to find game 5
The 40 points include the first 4 games and game 5. Take away the first 4 games to leave game 5.
$40 - 28 = 12$
He scored 12 points in game 5. Watch out: don’t average 7 and the fifth score as if they were two games. That gives 9, choice A. The 7 stands for 4 games, not 1. Watch out: adding 4 to 7 gives 11, choice B. Each of the first 4 games was 1 point below 8. So game 5 needs 8 plus 4 more points.
So the answer is C, 12 points.

Check your answer

Try scores of 7, 7, 7, 7 and 12. They add up to $28 + 12 = 40$ , and $40 \div 5 = 8$ . That is the mean for all 5 games, so 12 is right.

The original DOE question

Question 98 as printed on page 87 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 87, question 98. Open page 87 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).