Why it matters on the HSPT
Two to four Mathematics questions come with a graph or a table. The arithmetic in them is easy, which is exactly why the points are lost: students read the wrong bar, skip the scale, or answer a slightly different question than the one asked. Each kind of display brings its own few question shapes, and knowing them in advance turns these into some of the fastest points on the section.
The rules
Read the frame before the numbers. Every graph question starts the same way. Read the title, the axis labels, and the units. Check what one grid line is worth on the scale; it is often 5 or 10, not 1. Then read the question and underline the exact quantity it wants: a difference, a total, a percent, or a category name.
Pie charts. A pie chart shows how a whole is split into categories, as percents or as counts. If the slices are percents, they add to 100, and a count for one category is that percent of the total: 18 percent of 200 cars is 36 cars. If the slices are counts, the percent for one slice is its count over the total: 50 of 250 students is one fifth, which is 20 percent. The biggest slice is the mode. A pie chart suits categories, so a question that asks which data belongs on one wants an answer like "types of businesses in a town," not heights or temperatures.
Bar graphs. Each bar's height is read against the scale on the side. Questions ask for the tallest or shortest bar, the difference between two named bars, the sum of several, or the mean of all of them. A double bar graph shows two groups side by side for each category, with a key saying which shade is which. Read the key first, then treat each pair of bars like two questions.
Change from the previous period. When bars or points run across time, "the greatest increase" is not the tallest bar. It is the biggest jump from one period to the next. Work along the graph and write the change between each neighbor pair, with a sign. The largest positive number is the greatest increase; the largest negative number is the greatest decrease.
Line graphs. A line graph tracks something over time, with time along the bottom. Read each point against the scale exactly as you read a bar. The same question shapes appear: difference between two days, mean of the week, range, the day with the biggest rise. The slope of a segment tells you the direction of change; a steep drop is a big decrease.
In this graph the range is 10 minus 0, the mean is 35 divided by 7, which is 5, and the greatest increase from the day before is Sunday, up 7 from Saturday, even though Friday is the tallest point.
Frequency tables. A frequency table lists groups in one column and counts in the other. The group with the biggest count is the mode. A cutoff question, such as "how many earned 9 dollars or less," means adding every row on one side of the cutoff. Check whether the cutoff value itself belongs in the group. "Fewer than 8" excludes 8; "8 or fewer" includes it. The total of all the rows is the number of items surveyed, which you need for any percent.
One step, then stop. Nearly every graph question is read a number, then do one operation. If you find yourself doing three operations, reread the question; you have probably answered a different one.
Worked examples
Example 1
Using the pie chart, what percent of the 250 students chose basketball?
- A.20 percent
- B.25 percent
- C.50 percent
- D.5 percent
Show answer
Answer: A: 20 percent
The slices are counts, so the percent is the basketball count over the total: 50 over 250 simplifies to 1 over 5, which is 20 percent. Choice C, 50, is the count itself read as a percent, the most common trap on a pie chart of counts. Choice B would need the total to be 200.
Example 2
A pie chart shows the colors of 200 cars: blue 23 percent, red 12 percent, yellow 18 percent, green 47 percent. How many cars are yellow?
- A.18
- B.36
- C.82
- D.94
Show answer
Answer: B: 36
Eighteen percent of 200 is 0.18 times 200, which is 36. Faster: 10 percent of 200 is 20, so 18 percent is 20 plus 16, which is 36. Choice A reads the percent as a count. Choice D is the number of green cars.
Example 3
The bar graph shows how many books four students read. How many more books did Sam read than Leo and Ivy combined?
- A.3
- B.8
- C.11
- D.12
Show answer
Answer: A: 3
Leo and Ivy together read 8 plus 9, which is 17. Sam read 20, so the difference is 3. Choice D, 12, is Sam minus Leo alone, the trap for reading only one of the two names in the question.
Example 4
The line graph shows a town's rainfall in inches for six months: January 4, February 7, March 3, April 4, May 10, June 0. In which month was the increase from the previous month the greatest?
- A.February
- B.May
- C.June
- D.January
Show answer
Answer: B: May
Write the change for each month after January: February plus 3, March minus 4, April plus 1, May plus 6, June minus 10. The biggest positive change is May, up 6. May is also the tallest point here, but check anyway; the tallest point and the biggest jump are often different months. January has no previous month on the graph.
Example 5
A frequency table lists hourly pay for 26 workers. Rows: 5.00 to 7.00 dollars, 2 workers; 7.01 to 9.00 dollars, 8 workers; 9.01 to 11.00 dollars, 10 workers; 11.01 to 13.00 dollars, 6 workers. How many workers earn 9.00 dollars or less per hour?
- A.8
- B.10
- C.16
- D.18
Show answer
Answer: B: 10
Nine dollars is the top of the second row, and "or less" includes it, so add the first two rows: 2 plus 8 is 10. Choice A is the second row alone. Choice C adds the rows above the cutoff instead of below it. Choice D adds the two largest rows, an answer to a question that was not asked.
Example 6
Which set of data is best shown on a pie chart?
- A.the heights of the players on a team
- B.the kinds of pets owned by students in a class
- C.the daily high temperature for a month
- D.the prices of houses sold in a city
Show answer
Answer: B: the kinds of pets owned by students in a class
A pie chart shows how a whole splits into categories. Kinds of pets are categories, and the whole is the class. Heights, temperatures, and prices are measurements that can take any value, so they belong on a bar graph, a line graph over time, or a number line, not in slices.
Example 7
The bar graph shows quarterly sales in thousands. What is the mean of the four values?
- A.10
- B.11
- C.12
- D.13
Show answer
Answer: B: 11
Add the four bars: 8 plus 14 plus 10 plus 12 is 44. Divide by 4: 11. Choice C, 12, is the value of the last bar and the median of the sorted values. Choice A, 10, is the third bar. Do not trust a familiar-looking number until you have done the division.
Common mistakes
Trusting that one grid line is one unit. Read the scale. A bar that reaches the third line on a scale that counts by 5 is 15, not 3.
Reading the count as the percent. On a pie chart of counts, a slice of 50 is not 50 percent unless the total is 100. Divide by the total every time.
Picking the tallest bar for the greatest increase. Increase is about the jump from the period before. Write the changes down; do not eyeball them.
Answering for one name when the question lists two. "Leo and Ivy combined" means add them before you compare.
Ignoring the cutoff word. "Fewer than" and "or fewer" put the boundary group on different sides. Decide which before you add.
Practice
Graph questions live mainly in the math problem solving format, with a few in arithmetic. The Quantitative section also draws bar graphs into geometric comparison items. The data and probability lesson covers the mean, median, mode, and range you compute from the values you read here, and the percents lesson covers the ten percent shortcut used in the pie chart examples.