13.3 In-Context Math Questions and Graphics

About 30% of Math questions on the PSAT 8/9 are in context: a short science, social studies, or everyday story, often with a table or graph. The remaining items look more like 'pure' algebra or geometry. Context questions still test Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry — they just dress the skill in a paragraph.

PSAT 8/9 practice questionsPractice questions with detailed explanations

Read the question first

A dense paragraph can contain three measurements you do not need. The efficient order is:

1. Read the last sentence (or the actual question) first. What quantity is asked, and in what unit?
2. Skim the story for that quantity. Circle the numbers that belong to it.
3. Cross out extras. A humidity percent, a rest break, or a glove purchase is only relevant if the question uses it.
4. Plan the math. Ratio, percent, linear model, area, or probability — one of the skills from earlier chapters.
5. Compute, then sanity-check. If a cooling liquid starts at 86°C, a temperature of 400°C after 12 minutes is not reasonable.

This is the opposite of 'translate every sentence into algebra before you know the ask.' Translation is useful; undirected translation wastes time in a 35-minute module of 22 on-screen questions (20 operational and 2 pretest).

Worked example: extra numbers

Story. A community garden charges a $12 seasonal fee plus $3 per plant. Priya plants 7 tomato plants. She also buys a pair of gloves for $9. The garden is open 6 days a week.

Question. What is Priya's cost for the seasonal fee and the plants?

Asked quantity: garden cost without gloves. Extra: $9 gloves and 6 days.

Cost = 12 + 3(7) = 12 + 21 = $33.

If you add the gloves you get $42, which answers a different question ('all of Priya's spending'). If you treat 6 as if it were the plant count, you get 12 + 18 = $30. Both are plausible-looking and both are wrong.

Same numbers, different ask. 'Priya's total spending, including gloves' is 33 + 9 = $42. Always match the object of the sentence.

Worked example: science cooling model

A lab notebook says a solution cools according to

C = −2t + 86

where t is minutes after the burner is turned off and C is temperature in degrees Celsius. The lab also records that room humidity is 40% and that the beaker holds 250 mL.

Question 1. What is the modeled temperature after 12 minutes?

Humidity and volume do not appear in the formula. Substitute t = 12:

C = −2(12) + 86 = −24 + 86 = 62°C.

Question 2. After how many minutes is the modeled temperature 50°C?

50 = −2t + 86

−36 = −2t

t = 18 minutes.

The algebra is a one-variable linear equation, the same skill as Linear Equations in One Variable. The context tells you that t cannot be negative in this experiment and that C should fall as t rises because the slope is −2.

A related science graphic might plot C against t as a line with slope −2 and vertical intercept 86. Reading a value from that graph is the same substitution; just confirm that the axis units are minutes and °C, not hours and °F.

Worked example: social studies table

A civics project surveys 80 students about a school-year proposal.

Question. Among students who support the proposal, what fraction are 9th graders?

This is conditional probability from Probability and Conditional Probability: given support, so the denominator is 46, not 80.

28/46 = 14/23.

A rushed student answers 28/80 (joint count over the grand total) or 28/40 (support among ninth graders). Reading the phrase 'among students who support' is the whole problem.

Question. How many more 9th-grade supporters are there than 8th-grade supporters?

28 − 18 = 10 students. No probability needed. Graphics questions often toggle between a count, a difference, a percent, and a conditional fraction in the same table.

Question. What percent of all 80 students oppose the proposal?

34/80 = 0.425 = 42.5%. That is an ordinary percent, not a 'given' item.

Informational graphics: title, units, axes

Math graphics on this exam include tables, bar graphs, line graphs, and scatterplots. Reading and Writing also uses quantitative graphics; the Math versions usually require a calculation, not only a citation. Command-of-evidence work with figures on the verbal side is a different section; here you compute.

Checklist every time a figure appears:

Scale trap. A bar graph of weekly rainfall has a vertical axis from 2.0 to 5.0 inches, not from 0. May shows a bar at 2.5 and June at 5.0. June's bar looks more than twice as tall as May's because of the cropped axis, but 5.0 is exactly twice 2.5, and the difference is 2.5 inches. Read the numbers, not only the ink.

Worked graphic. Rainfall in inches: March 3.5, April 4.0, May 2.5, June 5.0.

• June minus May: 5.0 − 2.5 = 2.5 inches
• Mean of the four months: (3.5 + 4.0 + 2.5 + 5.0) / 4 = 15.0 / 4 = 3.75 inches
• Median: ordered 2.5, 3.5, 4.0, 5.0, so (3.5 + 4.0)/2 = 3.75 inches

Center-and-spread reminders live in One-Variable Data. Percents of a total live in Percentages. Scatterplot estimates live in Two-Variable Data. In-context items freely mix those tools with algebra.

Unit trap. A line graph labels distance in kilometers, but one sentence in the stem mentions a 500-meter spur trail. 500 meters is 0.5 km. Adding 500 to a kilometer total is off by a factor of 1,000. Convert before you combine.

Mix of algebra and PSD in one story

Story. A bike-share kiosk charges $8 to unlock plus $3 per hour. A city report also lists that 120 riders used the kiosk on Tuesday and that 15% of Tuesday riders were students.

Question A. Maya rents a bike for 4 hours. What is her rental charge?

8 + 3(4) = $20. The 120 riders and 15% are unused.

Question B. How many Tuesday riders were students?

0.15 × 120 = 18 students. Maya's hourly rate is unused.

Question C. At the same $8 + $3h price, a rider pays $26. How many hours is that?

26 = 8 + 3h

18 = 3h

h = 6 hours.

One paragraph, three skills: linear model, percentage, solving for the input. The test will usually ask only one of them. Your job is to ignore the other two numbers.

Rates and unit conversions from Ratios, Rates, Proportions, and Units show up the same way. A runner finishes a 5-kilometer loop in 24 minutes, rests 6 minutes, then runs another 5 kilometers in 26 minutes. If the question asks for average running speed for the two loops, distance is 10 km and time is 24 + 26 = 50 minutes = 50/60 = 5/6 hour, so speed is 10 ÷ (5/6) = 12 km/h. The 6-minute rest is extra unless the question includes it in total elapsed time (then time is 56 minutes and the speed is lower).

Timing and checking

Math modules are 35 minutes with 22 questions on screen. In-context items can look long; they are not automatically harder. Often the computation is one substitution once you know the target.

• If the figure is a scatterplot, use a given linear model as in two-variable data.
• If the figure is a two-way table with 'given' or 'among,' use conditional probability.
• If the story is a rate or a conversion, use ratios and units.
• Enter student-produced responses without a percent sign, comma, or dollar sign.

Guessing after you eliminate a unit-mismatched choice is still better than leaving a multiple-choice item blank: the PSAT 8/9 has no wrong-answer penalty.

PSAT 8/9 practicePractice questions with detailed explanations