Alpha School · Math · built 2 October 2026
This guide covers the skills a Grade 8 math test asks about, one skill at a time, in the order that builds best. Each section starts with a question you can already answer, adds one new move, gives you the rule, names the trap, then gives you a short ladder of practice.
Every question here is new. None of them is from a test, and the numbers and stories have all been changed. Between the sections are mixed checks where nothing tells you which skill to use. That is the part the test is really testing, so do not skip them.
How to use it: answer, press Check, then read the answer even when you were right. The answer shows the method and where each wrong option comes from. Your work is saved in this browser, so you can close the page and come back.
In this guide
Which is greater, 0.6 or 0.56?
Line them up by place value. 0.6 is 0.60.
0.60 has 6 tenths. 0.56 has 5 tenths.
So 0.6 is greater. You compared two decimals by giving them the same number of places.
A test list mixes fractions, percents, decimals and square roots. The move is to turn every one of them into a decimal first, then compare the way you just did.
3/8 is 3 ÷ 8, which is 0.375. 45% is 45 hundredths, which is 0.45. √20 is not a whole number, but 20 sits between 16 and 25, so √20 sits between 4 and 5, and nearer 4.5 than 4.
Scientific notation is the other half of this section. 4,600,000 is 4.6 × 106 because the decimal point jumps 6 places to land after the first non-zero digit. 0.00032 is 3.2 × 10−4 because the point jumps 4 places the other way. A number bigger than 10 gets a positive exponent; a number smaller than 1 gets a negative one.
Make every number a decimal before you compare. For scientific notation, count the jumps of the decimal point: a big number gets a positive exponent, a number less than 1 gets a negative exponent.
√20 is not 10 (half of 20) and not 400 (20 squared). It is a little under 4.5. A percent is hundredths: 45% is 0.45, not 4.5 and not 45. With negative numbers the order flips: −1.25 is less than −1.2 even though 1.25 is more than 1.2.
Practice
Write each number as a decimal. For the square root, say which two whole numbers it sits between.
a) 0.375. b) 0.45. c) between 4 and 5.
3 ÷ 8 is 0.375. 45% means 45 hundredths, so 0.45. For √20, find the perfect squares on either side: 16 and 25, so √20 is between √16 and √25, which is between 4 and 5. It is about 4.47.
Which list shows these numbers in order from least to greatest?
−5/4, −1.2, 110%, √2
As decimals: −5/4 is −1.25, −1.2 stays −1.2, 110% is 1.1, √2 is about 1.41.
On the negative side the bigger-looking number is smaller: −1.25 is less than −1.2.
The second option forgot that flip. The third treated 110% as 110 and put it last. The fourth is the right order the wrong way round, greatest to least.
A list of numbers is ordered from least value to greatest value. One number is missing.
√6
The gap needs a number bigger than 2 1/3 (about 2.33) and smaller than 2.5.
√6 is between √4 and √9, so between 2 and 3; it is about 2.45. That fits.
2.25 and √4 (which is exactly 2) are too small. 2.6 is too big. Turn every option into a decimal and test it against both neighbors.
a) The mass of a grain of pollen is about 0.00000032 gram. Written in scientific notation it is 3.2 × 10n.
b) A city has about 4,600,000 people. Written in scientific notation it is 4.6 × 10m.
a) n = −7. b) m = 6.
For a), the decimal point has to jump 7 places to the right to sit after the 3: 0.00000032 becomes 3.2. A number less than 1 gets a negative exponent, so n is −7. A common slip is counting only the zeros (6) and forgetting the jump past the 3.
For b), the point jumps 6 places to the left: 4,600,000 becomes 4.6. A big number gets a positive exponent, so m is 6.
A car travels 120 miles in 2 hours. How many miles does it travel in one hour?
120 ÷ 2 = 60.
60 miles per hour. You just found a rate: how much y changes for every one x.
Slope is that same rate read off two points. Take the points (2, 120) and (5, 300). The change in y is 300 − 120, which is 180. The change in x is 5 − 2, which is 3. Slope is 180 ÷ 3, which is 60. Rise over run.
On a graph, pick two points that sit exactly on grid corners, then count rise and run. In a table, pick any two rows. In a story like “Leo spent $36 for 4 plants, each the same price,” the rate is 36 ÷ 4, which is 9 dollars per plant, and the graph goes through (0, 0) and (4, 36).
To describe a slope in words, say “y-units per one x-unit”: 9 dollars per plant, 60 miles per hour.
Slope = change in y ÷ change in x. Subtract in the same order top and bottom. Say it as “y-units per one x-unit.”
Run over rise. If a line goes through (0, 0) and (4, 36), the slope is 36 ÷ 4, which is 9, not 4 ÷ 36. Check it: slope is what y does when x goes up by exactly 1. A line going down has a negative slope; if you subtract in a different order on top and bottom you lose the minus sign.
Practice
Two points on a line are (1, 5) and (4, 17).
a) 12. b) 3. c) 4.
17 − 5 = 12.
4 − 1 = 3.
12 ÷ 3 = 4.
The slope is 4: y goes up 4 for every 1 that x goes up.
A line passes through (3, 8) and (7, 2). Which expression gives its slope?
(2 − 8) ÷ (7 − 3)
Change in y on top, change in x underneath, both subtracted in the same order (second point minus first point). That gives −6 ÷ 4, which is −1.5. The line goes down.
The second option is run over rise. The third and fourth mix the order, so the minus sign disappears and they give +1.5, a line going up.
The graph shows the total distance a snail crawled, in centimeters (y), after x minutes. Which statement describes the slope of the line?
The snail crawls 1.5 centimeters per minute
Two clean points: (0, 0) and (4, 6). Rise is 6, run is 4.
6 ÷ 4 = 1.5.
“4 per minute” read the run as the answer. “6 per minute” read the y-value of one point. “0.67” is run over rise, 4 ÷ 6.
A candle is lit. The table shows its height.
| Minutes burning | 0 | 10 | 20 | 30 |
|---|---|---|---|---|
| Height (cm) | 18 | 15.5 | 13 | 10.5 |
−0.25
Pick two rows: (0, 18) and (10, 15.5). Change in y is 15.5 − 18, which is −2.5. Change in x is 10.
−2.5 ÷ 10 = −0.25.
The candle loses 0.25 cm every minute, so the slope is negative. 0.25 lost the sign. −2.5 is the change in height per row, forgetting that each row is 10 minutes. −4 is run over rise.
Mixed check 1
These two questions do not say which skill they use. Decide the method first, then answer.
A list of numbers is ordered from least value to greatest value. One number is missing: −2.5, ______, −1, 0.5. Which number could be the missing number?
Method: Real numbers
−7/4
The gap needs a number between −2.5 and −1. As a decimal −7/4 is −1.75, which fits.
−3 is less than −2.5. −0.5 is more than −1. 11/4 is 2.75, bigger than everything in the list.
A bakery's ovens work at a steady rate. After 2 hours they have baked 50 muffins; after 5 hours, 125 muffins. How many muffins per hour is that?
Method: Slope and rate
25 muffins per hour
Change in muffins: 125 − 50 = 75.
Change in hours: 5 − 2 = 3.
75 ÷ 3 = 25.
Check with one point: 2 hours × 25 is 50. It fits.
In y = 3x + 5, what is y when x is 0?
3 × 0 = 0.
y = 0 + 5.
y = 5.
When x is 0 the line is on the y-axis, so 5 is where the line crosses it. The number added on is always the y-intercept.
Now build the equation from a story. A plumber charges a $40 visit fee plus $25 per hour. The “per hour” number is the slope m, because it repeats for every hour. The one-time fee is b, because you pay it even when x is 0. So y = 25x + 40.
From a graph: read where the line crosses the y-axis for b, then count rise over run for m.
When b is 0 the line goes through (0, 0) and the relationship is proportional: y ÷ x is the same for every pair. That is also called direct variation, y = kx. Find k by dividing one y by its x, then multiply by the new x.
The “per” number is m. The one-time number is b. If b is 0 the relationship is proportional and y ÷ x never changes.
Swapping m and b. “$8 per shirt plus a $20 design fee” is y = 8x + 20, not y = 20x + 8. The per-number multiplies x. And a relationship with a starting amount, like a plant already 4 cm tall, is not proportional even though it is linear.
Practice
What is the y-intercept of the line?
−3
The y-intercept is where the line crosses the y-axis, the vertical one. That happens at (0, −3).
3 is where it crosses the x-axis. 1 is the slope. 0 is where the axes cross, not where the line does.
A gym charges a $30 joining fee and then $15 per month. Which equation gives the total cost y for x months?
y = 15x + 30
$15 repeats every month, so 15 multiplies x. $30 is paid once, so it is added on.
y = 30x + 15 swaps them. y = 45x adds the two numbers as if both repeated. y = 15x − 30 treats the fee as a discount.
Which situation describes a proportional relationship?
The perimeter of a square and its side length
Perimeter is 4 × side, so y = 4x: no starting amount, and y ÷ x is always 4.
Area is side × side, which is not a straight line at all. The taxi and the tree both have a starting amount ($3, 4 feet), so their lines do not pass through (0, 0).
The cost of fabric varies directly with its length. 6 yards of fabric cost $27. What is the cost of 10 yards of the same fabric?
$45
Direct variation means y = kx. Find k first.
k = 27 ÷ 6.
k = 4.5 dollars per yard.
Cost of 10 yards = 4.5 × 10.
Cost = $45.
$31 added the 4 extra yards as dollars. $60 multiplied 10 by the 6 yards. $162 multiplied 6 by 27.
In the set {(1, 4), (2, 5), (3, 6)}, what is the output when the input is 2?
The pair with 2 first is (2, 5).
The output is 5.
Every input here has exactly one output. That is what makes the set a function.
A relation is not a function when one input shows up twice with different outputs: {(2, 1), (2, 3)} fails, because the input 2 cannot be both 1 and 3. On a graph the same idea is the vertical line test: if any vertical line touches the graph twice, it is not a function of x.
A system is two lines on the same grid. Its solution is the point where they cross, written (x, y). The question often asks for only the x-coordinate or only the y-coordinate.
An equation like 12 + 3x = 5x is a system in words: the left side is one plan's total, the right side is another plan's total, and the equal sign says they cost the same.
Function: every input has one output. A repeated x with different y breaks it. System: the solution is the crossing point, read as (x, y).
Repeated y-values are fine: {(1, 4), (2, 4), (3, 4)} is a function. Only a repeated x fails. On a system question, read whether it wants the x-coordinate or the y-coordinate; both numbers are always among the options.
Practice
Which set of ordered pairs does NOT represent y as a function of x?
{(2, 1), (2, 3), (4, 5)}
The input 2 appears twice with different outputs, 1 and 3. One input, two outputs: not a function.
{(1, 5), (2, 5), (3, 5)} is the tempting wrong pick. Its outputs repeat, but every input still has just one output, so it is a function.
A graph is drawn on a coordinate grid. A vertical line can be placed so that it crosses the graph at two points.
No. A circle.
Two crossings on one vertical line means one x-value with two y-values, so it is not a function.
A circle fails the test: a vertical line through its middle touches the top and the bottom. A non-vertical straight line, and a graph that rises then falls, are each touched only once by any vertical line, so both are functions.
The two lines graphed represent a system of equations. What is the y-coordinate of the solution?
3
The lines cross at (2, 3). The solution to the system is that point.
The question asked for the y-coordinate, the second number, so the answer is 3. If it had asked for the x-coordinate the answer would be 2.
Which situation can be represented by the equation 20 + 4x = 6x?
Plan A costs $20 to join plus $4 per visit. Plan B costs $6 per visit. After how many visits do the plans cost the same?
Left side: 20 once, plus 4 for every visit x. Right side: 6 for every visit. The equal sign says the two totals match.
The second option swaps the fee and the rate. The third makes $6 a one-time fee, which would be 20 + 4x = 6, no x on the right. The last is not a comparison at all; it is 20 + 4 × 6.
Mixed check 2
These three questions do not say which skill they use. Decide the method first, then answer.
A ride-share charges $2.50 to start and then $1.75 per mile. Which equation gives the total cost y for a ride of x miles?
Method: Line equation
y = 1.75x + 2.5
$1.75 repeats every mile, so it multiplies x. $2.50 is paid once, so it is added.
y = 2.5x + 1.75 swaps them. y = 4.25x adds the two numbers as one rate. y = 1.75x forgets the start fee, which would make the ride proportional, and it is not.
Which set of ordered pairs represents y as a function of x?
Method: Functions and systems
{(3, 1), (4, 1), (5, 1)}
Inputs 3, 4 and 5 each appear once, so each has exactly one output. Repeated outputs do not matter.
Every other set repeats an input (1, 0 or 2) with different outputs.
A human hair is about 0.00058 centimeter thick. In scientific notation that is 5.8 × 10n. What is n?
Method: Real numbers
n = −4.
The decimal point jumps 4 places to the right to sit after the 5. A number less than 1 gets a negative exponent.
Solve 3x + 7 = 22.
Undo the add: subtract 7 from both sides.
3x = 15.
Undo the multiply: divide both sides by 3.
x = 5.
One move per line, and the same move on both sides.
When x is on both sides, first bring the x-terms together. Solve 5x + 12 = 7x + 2:
Subtract 5x from both sides.
12 = 2x + 2.
Subtract 2.
10 = 2x.
Divide by 2.
x = 5.
Brackets mean distribute to every term inside: 3(x + 4) is 3x + 12.
In a story where two people spend the same total, each side of the equal sign is one person's whole bill. Five notebooks and a $4 pen against three notebooks and a $10 pen is 5n + 4 = 3n + 10.
Collect the x-terms on one side by subtracting the smaller x-term from both sides, then undo the numbers. Distribute to every term in the bracket. One equation per line.
Subtracting the x-term from only one side. Distributing to only the first term: 3(x + 4) is not 3x + 4. And in a perimeter equation, every side gets added, including the ones with no x in them.
Practice
Solve 4x − 9 = 19.
x = 7.
Add 9 to both sides.
4x = 28.
Divide by 4.
x = 7.
Ana and Ben each spent the same amount at a shop. Ana bought 5 notebooks and a $4 pen. Ben bought 3 notebooks and a $10 pen. Every notebook costs the same, n dollars. Which equation can be used to find n?
5n + 4 = 3n + 10
Ana's whole bill on the left: 5 notebooks plus a $4 pen. Ben's whole bill on the right: 3 notebooks plus a $10 pen. Same total, so equal.
The second option piles everything onto two sides at random. The third pairs each person with the other person's pen. The fourth multiplies the pen price by the notebook count, which no one paid.
Now solve it. 5n + 4 = 3n + 10.
n = 3.
Subtract 3n from both sides.
2n + 4 = 10.
Subtract 4.
2n = 6.
Divide by 2.
n = 3.
Check: Ana paid 15 + 4, which is 19. Ben paid 9 + 10, which is 19.
The perimeter of the triangle is 34 units. The side lengths are given in units.
6x − 2 = 34, and x = 6.
Perimeter is all three sides added: (x + 2) + 2x + (3x − 4).
x + 2x + 3x = 6x.
2 − 4 = −2.
So 6x − 2 = 34.
6x = 36.
x = 6.
6x + 2 added the 4 instead of subtracting it. 6x − 6 subtracted the 2 as well. 5x − 2 lost one of the x-terms.
The point (3, 2) moves 4 units to the right. Where is it now?
Right changes x. 3 + 4 = 7.
The point is at (7, 2).
Right adds to x; up adds to y. Left and down subtract.
A translation written as a rule looks like (x, y) → (x + 4, y − 3): right 4, down 3.
A reflection flips one sign. Across the x-axis the y changes sign: (3, 2) → (3, −2). Across the y-axis the x changes sign: (3, 2) → (−3, 2).
A rotation of 180° about the origin flips both signs: (3, 2) → (−3, −2). A 90° counterclockwise turn swaps the coordinates and flips the new first one: (3, 2) → (−2, 3).
A dilation from the origin multiplies both coordinates by the scale factor k: (x, y) → (kx, ky). Every side length is multiplied by k, every angle stays the same, so the image is similar but not congruent. Translations, reflections and rotations keep the shape congruent.
Translate: add to x for right, add to y for up. Reflect: flip the sign of the coordinate that crosses the axis. Dilate: multiply both coordinates by k. Only a dilation changes size.
Across the x-axis it is the y that changes sign, because the point jumps over the horizontal axis. Down subtracts from y, not from x. A dilation with k = 1/2 shrinks the shape: it is still a dilation, still similar, still not congruent.
Practice
The point (−2, 5) is translated 3 units left and 2 units up. What are the coordinates of its image?
(−5, 7)
Left 3 subtracts from x: −2 − 3 = −5.
Up 2 adds to y: 5 + 2 = 7.
A triangle is translated using the rule (x, y) → (x − 6, y + 1). Which statement describes the translation?
6 units left and 1 unit up
x − 6 means x gets smaller, which is left. y + 1 means y gets bigger, which is up.
“Right” reads the minus as a plus. “Down” mixes up which coordinate is vertical. The last swaps the two numbers.
Triangle ABC will be reflected across the x-axis to make triangle A′B′C′.
B′ is (4, −1). A dilation by a scale factor of 1/2 is the transformation that is not congruent.
B is (4, 1). Reflecting across the x-axis flips the sign of y, so B′ is (4, −1). The x stays 4.
Rotations, translations and reflections move the triangle without changing its size, so the image is congruent. A dilation multiplies every side by 1/2; the image is similar, not congruent.
A rectangle has vertices (2, 4), (6, 4), (6, 8) and (2, 8). It is dilated by a scale factor of 1/2 with the origin as the center of dilation.
(3, 4), and the image side is 2 units long.
Multiply both coordinates by 1/2. 6 × 1/2 = 3. 8 × 1/2 = 4.
Side lengths are multiplied by the same factor: 4 × 1/2 = 2.
(12, 16) multiplied by 2 instead of 1/2. (5.5, 7.5) subtracted 1/2 instead of multiplying. (6, 4) halved only the y.
Mixed check 3
These two questions do not say which skill they use. Decide the method first, then answer.
Solve 2(x − 5) = 3x + 1.
Method: Solving equations
x = −11.
Distribute to both terms: 2x − 10 = 3x + 1.
Subtract 2x from both sides: −10 = x + 1.
Subtract 1: −11 = x.
Distributing to only the x gives 2x − 5 and the wrong answer −6.
The point (−4, 6) is reflected across the y-axis. What are the coordinates of its image?
Method: Transformations
(4, 6)
Across the y-axis the x changes sign and the y stays: (−4, 6) becomes (4, 6).
(−4, −6) would be the reflection across the x-axis.
What are 6² and 8²? What is their sum?
6² = 36.
8² = 64.
36 + 64 = 100.
And √100 = 10.
Squaring, adding, square-rooting: that is the whole theorem, with the names left off.
In a right triangle the two short sides are the legs and the side opposite the right angle is the hypotenuse, always the longest. The legs squared add up to the hypotenuse squared: a² + b² = c². Legs 6 and 8 give a hypotenuse of 10.
If the missing side is a leg, subtract: hypotenuse 13, leg 5, other leg is √(169 − 25), which is √144, which is 12.
Many answers are not whole numbers. √52 is between √49 and √64, so between 7 and 8; the options will say “closest to 7.2.”
To check whether three lengths could make a right triangle, square the two smaller ones, add, and see whether you get the largest one squared.
The distance between two points on a grid is a hypotenuse: the legs are the difference in x and the difference in y.
Legs squared add up to the hypotenuse squared. Missing hypotenuse: add, then square-root. Missing leg: subtract, then square-root.
Adding when you need a leg: with hypotenuse 13 and leg 5 the other leg is √(169 − 25) = 12, not √194. Forgetting to square: 5 + 12 is not 13. Forgetting the square root at the end: 169 is not a side length.
Practice
a) What is 9²?
b) What is √(81 + 144)?
a) 81. b) 15.
9 × 9 = 81.
81 + 144 = 225.
√225 = 15.
The side lengths are in centimeters.
x² = 20² − 12², and x = 16.
20 is opposite the right angle, so it is the hypotenuse. x is a leg, so subtract.
x² = 400 − 144.
x² = 256.
x = 16.
x² = 20² + 12² treats x as the hypotenuse and gives about 23.3, longer than the hypotenuse, which is impossible. x = 20 − 12 forgets to square. (20 − 12)² squares the difference instead of subtracting the squares.
A right triangle has legs of 5 inches and 9 inches. Which measurement is closest to the length of the hypotenuse?
10.3 inches
5² + 9² = 25 + 81.
c² = 106.
c = √106.
√106 is between √100 and √121, so between 10 and 11, close to 10.3.
14 added the legs without squaring. 7.5 subtracted the squares (√56) as if 9 were the hypotenuse. 106 forgot the square root.
a) What is the distance between the points (−3, 1) and (5, 7)?
b) Which set of lengths could be the side lengths of a right triangle?
a) 10 units. b) 9, 12, 15.
a) The horizontal leg is 5 − (−3), which is 8. The vertical leg is 7 − 1, which is 6.
8² + 6² = 100.
Distance = 10.
b) 9² + 12² = 81 + 144, which is 225, and 15² = 225. It works.
6, 7, 10: 36 + 49 is 85, not 100. 5, 10, 15: 25 + 100 is 125, not 225 (5 + 10 = 15 is the trap: adding the sides proves nothing). 8, 10, 12: 64 + 100 is 164, not 144.
A circle has a diameter of 10 cm. What is its radius? What is its area?
Radius is half the diameter: 10 ÷ 2 = 5.
Area = π × 5².
Area = 25π.
On the π key that is 78.54, and with 3.14 it is 78.5. Both are “closest to 78.5.”
Halving the diameter first is the move most volume questions are really testing.
A cylinder is a circle pushed up through a height: V = πr²h, base area times height.
A cone with the same base and height holds exactly one third as much: V = (1/3)πr²h.
A sphere is V = (4/3)πr³. Notice the cube: the radius is used three times.
Work with the π key, not 3.14, and round once at the very end. The options are built so that the π-key answer and the 3.14 answer are both “closest to” the same choice; the small difference never changes which option you pick.
Cylinder: base area × height. Cone: one third of that. Sphere: (4/3)πr³. Halve a diameter before anything else. π key, round once at the end.
Using the diameter as the radius makes a cylinder four times too big. Squaring in the sphere formula where it cubes. Forgetting the 1/3 on a cone. Each of these is one of the wrong options, every time, so say the formula out loud before you touch the calculator.
Practice
A sphere has a diameter of 14 inches.
a) 7. b) 49. c) 343.
14 ÷ 2 = 7.
7 × 7 = 49.
7 × 7 × 7 = 343.
The sphere formula needs the cube, 343. Squaring instead gives 49, seven times too small.
A cone has a diameter of 8 inches and a height of 9 inches. Which equation can be used to find V, the volume of the cone in cubic inches?
V = (1/3)π(4²)(9)
Halve the diameter: radius 4. Square it, multiply by the height, take one third.
(1/3)π(8²)(9) used the diameter as the radius. π(4²)(9) is the cylinder, no one third. (1/3)π(4)(9) forgot to square the radius.
Which measurement is closest to the volume of the cylinder in cubic centimeters?
942.5 cubic centimeters
V = π × 5² × 12.
V = 300π.
π key: 942.48. With 3.14: 942. Both are closest to 942.5.
3,769.9 used 10 as the radius. 188.5 forgot to square (π × 5 × 12). 314.2 took one third as if it were a cone.
A ball has a diameter of 18 centimeters. Which measurement is closest to the volume of the ball in cubic centimeters?
3,053.6 cubic centimeters
Radius = 18 ÷ 2 = 9.
V = (4/3)π × 9³.
V = (4/3)π × 729.
V = 972π.
π key: 3,053.63. With 3.14: 3,052.08. Both are closest to 3,053.6.
339.3 squared the radius instead of cubing. 24,429 used the diameter 18 as the radius. 2,290.2 is π × 729 with the 4/3 forgotten.
Mixed check 4
These two questions do not say which skill they use. Decide the method first, then answer.
A 10-foot ladder leans against a wall. Its base is 6 feet from the wall. How far up the wall does the ladder reach?
Method: Pythagorean theorem
8 feet
The ladder is the hypotenuse. The height is a leg, so subtract.
h² = 10² − 6².
h² = 100 − 36.
h² = 64.
h = 8.
Adding instead gives √136, about 11.7, which is longer than the ladder itself.
A paper cone cup has a radius of 3 cm and a height of 10 cm. Which measurement is closest to its volume in cubic centimeters?
Method: Volume
94.2 cubic centimeters
V = (1/3)π × 3² × 10.
V = 30π.
π key: 94.25. With 3.14: 94.2. Both are closest to 94.2.
282.7 forgot the one third (the full cylinder). 31.4 forgot to square the radius. 377 used 6 as the radius.
A rectangle is 5 cm by 8 cm. What is its area? A circle has a radius of 3 cm. What is its circumference?
Rectangle: 5 × 8 = 40 square centimeters.
Circumference = 2π × 3.
Circumference = 6π.
That is 18.85 on the π key, or 18.84 with 3.14.
Surface area is nothing more than adding up the areas of the faces, and these are the two faces that keep turning up.
Peel the label off a can and it is a rectangle: its width is the circumference of the base and its height is the can's height. So the lateral surface area of a cylinder is 2πr × h. The total surface area adds the two circular ends: 2πrh + 2πr².
Same idea for a rectangular prism: lateral = perimeter of the base × height; total adds the two bases.
Working backwards: if you know the total surface area of a box and its base, subtract the two bases, then divide what is left by the perimeter of the base to get the height.
Lateral = perimeter (or circumference) of the base × height. Total = lateral + two bases. Read whether the question says lateral or total.
Lateral and total differ by exactly the two bases, and both numbers are always among the options. A can's label is lateral; the whole can is total. Also keep surface area (square units) apart from volume (cubic units); πr²h is volume, not any kind of surface area.
Practice
A cylinder has a radius of 4 cm. Give each measurement to the nearest tenth.
a) 25.1 cm. b) 50.3 square centimeters.
Circumference = 2π × 4 = 8π, which is 25.13 on the π key (25.12 with 3.14).
Base area = π × 4² = 16π, which is 50.27 on the π key (50.24 with 3.14).
The circumference becomes the width of the unrolled label; the base area is one of the two ends.
A cylinder has a radius of 4 cm and a height of 10 cm. Which expression gives its LATERAL surface area in square centimeters?
2π(4)(10)
Lateral is circumference × height: 2π × 4 × 10.
2π(4)(10) + 2π(4²) is the total surface area, with the two ends. π(4²)(10) is the volume. 2π(8)(10) used the diameter as the radius.
Which measurement is closest to the TOTAL surface area of that same cylinder, radius 4 cm and height 10 cm, in square centimeters?
351.9 square centimeters
Lateral = 2π × 4 × 10.
Lateral = 80π.
Two bases = 2 × π × 4².
Two bases = 32π.
Total = 112π.
π key: 351.86. With 3.14: 351.68. Both are closest to 351.9.
251.3 is the lateral area alone (80π). 502.7 is the volume (160π). 904.8 used the diameter 8 as the radius.
A box is shaped like a rectangular prism. Its base measures 6 feet by 4 feet, and its total surface area is 188 square feet.
a) 48 square feet. b) 7 feet.
Two bases: 2 × 6 × 4 = 48.
What is left is the lateral area: 188 − 48 = 140.
Perimeter of the base: 6 + 4 + 6 + 4 = 20.
Lateral = perimeter × height, so 20h = 140.
h = 7.
A common slip is to forget there are two bases and subtract only 24, which leaves 164 and a height that is not a whole number.
What is the mean of 4, 6, 8 and 10?
4 + 6 + 8 + 10 = 28.
28 ÷ 4 = 7.
The mean is 7. Every other number in this section hangs off a mean or a total like this one.
Mean absolute deviation (MAD) asks how far the numbers typically sit from the mean. Distances from 7: 3, 1, 1, 3. Their mean is 8 ÷ 4, which is 2. All distances are positive.
On a scatterplot, a prediction comes from the trend line, not from any single dot: read the line's height at the x the question gives.
Simple interest is I = P × r × t, with the rate written as a decimal. The balance is the principal plus the interest. Compound interest pays interest on the new balance each year, so on the same rate it earns a little more than simple.
A saving plan is a line equation: amount saved so far plus the monthly amount times the months, compared with the cost.
MAD = average distance from the mean. Interest is what you earn; balance is principal plus interest. Compound beats simple at the same rate because later years earn interest on interest.
Answering the balance when the question asks for the interest, or the other way round; both numbers are always among the options. Writing 5% as 5 instead of 0.05. In MAD, the distances are all positive, and the answer is a distance, not the mean and not the range.
Practice
The numbers are 3, 5, 8 and 12.
a) 7. b) 4.
3 + 5 + 8 + 12 = 28.
28 ÷ 4 = 7.
7 − 3 = 4.
Which expression gives the simple interest earned on $600 at 4% per year for 3 years?
600 × 0.04 × 3
Principal × rate as a decimal × years. 4% is 0.04.
600 × 4 × 3 used 4 instead of 0.04 and gives $7,200, more than ten times the principal. 600 + 0.04 × 3 adds instead of multiplying. 600 × 0.04 is one year only.
Using that same account, $600 at 4% simple interest for 3 years:
a) $72. b) $672.
Interest = 600 × 0.04 × 3.
Interest = 72.
Balance = 600 + 72.
Balance = 672.
Read the question twice: “how much interest” wants 72, “what is the balance” wants 672.
a) What is the mean absolute deviation of 3, 5, 8 and 12?
b) Two savings accounts both start with $1,000 at 5% per year for 2 years. Account A pays simple interest. Account B pays interest compounded once a year. Which earns more, and by about how much?
a) 3. b) Account B, by about $2.50.
a) Distances from the mean 7: 4, 2, 1, 5. Their sum is 12. 12 ÷ 4 = 3. 7 is the mean itself, 9 is the range, 12 is the sum of the distances before dividing.
b) Simple: 1,000 × 0.05 × 2 = 100 interest.
Compound: year 1 ends at 1,000 × 1.05 = 1,050. Year 2 ends at 1,050 × 1.05 = 1,102.50. That is 102.50 interest.
102.50 − 100 = 2.50, so B earns about $2.50 more.
“By about $100” is the simple interest itself, not the difference.
Mixed run
Every skill in this guide, in no order, and nothing tells you which one to use. Name the method on your paper before you work it. That is exactly what the test asks.
Which measurement is closest to the lateral surface area of the cylinder in square centimeters?
Method: Surface area
94.2 square centimeters
Lateral = circumference × height.
Lateral = 2π × 3 × 5.
Lateral = 30π.
π key: 94.25. With 3.14: 94.2. Both are closest to 94.2.
150.8 is the total surface area (with the two ends). 188.5 used the diameter 6 as the radius. 47.1 forgot the 2 in 2πr. Compare with the next question: same can, one word different.
Same cylinder: radius 3 cm, height 5 cm. Which measurement is closest to its total surface area in square centimeters?
Method: Surface area
150.8 square centimeters
Lateral = 30π.
Two bases = 2 × π × 3².
Two bases = 18π.
Total = 48π.
π key: 150.80. With 3.14: 150.72. Both are closest to 150.8.
94.2 is the lateral area alone. 122.5 added only one base. 141.4 is the volume, 45π.
The scatterplot shows how many kilometers a runner ran each week (y) against the number of weeks of training (x), with a trend line drawn. Based on the trend, which is the best prediction of the kilometers run in week 10?
Method: Data and money
22 kilometers
Read the trend line at x = 10: it reaches 22. The trend line rises 2 for every 1 week, starting at 2, so at week 10 it is 2 + 20.
18 is the last dot, not the trend. 20 doubled the week number. 10 is the x-value itself.
What is the slope of the line shown?
Method: Slope and rate
−2
From (0, 8) to (4, 0): change in y is 0 − 8, which is −8. Change in x is 4 − 0, which is 4.
−8 ÷ 4 = −2.
2 lost the sign; the line goes down, so the slope is negative. −1/2 is run over rise. 8 is the y-intercept.
A DJ charges a fixed booking fee plus an hourly rate. A 3-hour event costs $410 and a 5-hour event costs $650. Which equation gives the cost y of an event lasting x hours?
Method: Line equation
y = 120x + 50
Two extra hours cost 650 − 410, which is 240, so the rate is 240 ÷ 2, which is 120 per hour.
Fee = 410 − 3 × 120.
Fee = 50.
y = 50x + 120 swaps fee and rate. y = 136.67x divides 410 by 3 as if there were no fee. y = 120x + 410 uses a whole event's cost as the fee.
A rectangle has a length of 2x + 1 and a width of x. Its perimeter is 38 units. What is the value of x?
Method: Solving equations
x = 6.
Perimeter is two lengths plus two widths: 2(2x + 1) + 2x = 38.
4x + 2 + 2x = 38.
6x + 2 = 38.
6x = 36.
x = 6.
Adding only one length and one width gives 3x + 1 = 38, which is not a whole number of units.
A triangle has a vertex at (−6, 9). The triangle is dilated by a scale factor of 2/3 with the origin as the center of dilation. What are the coordinates of the image of that vertex?
Method: Transformations
(−4, 6)
Multiply both coordinates by 2/3: −6 × 2/3 = −4 and 9 × 2/3 = 6.
The image is smaller because the factor is less than 1, and it keeps the same angles.
What is the distance between the points (2, −1) and (14, 4)?
Method: Pythagorean theorem
13 units
Horizontal leg: 14 − 2 = 12.
Vertical leg: 4 − (−1) = 5.
12² + 5² = 144 + 25.
d² = 169.
d = 13.
The two lines represent a system of equations. What is the x-coordinate of the solution?
Method: Functions and systems
2
The lines cross at (2, 3). The x-coordinate is the first number, 2. The y-coordinate, 3, is the other number the options would offer.
Which list shows the numbers 3/2, 140%, √3 and 1.45 in order from greatest to least?
Method: Real numbers
√3, 3/2, 1.45, 140%
As decimals: 3/2 is 1.5, 140% is 1.4, √3 is about 1.73, 1.45 stays.
Greatest to least: 1.73, 1.5, 1.45, 1.4.
The third option is least to greatest. The others put 1.5 and 1.45 or √3 in the wrong place; √3 is not 1.5, it is between √1 and √4 and closer to 1.7.
A globe is a sphere with a radius of 6 inches. Which measurement is closest to its volume in cubic inches?
Method: Volume
904.8 cubic inches
V = (4/3)π × 6³.
V = (4/3)π × 216.
V = 288π.
π key: 904.78. With 3.14: 904.32. Both are closest to 904.8.
150.8 squared the radius instead of cubing. 678.6 is π × 216 with the 4/3 forgotten. 7,238.2 used 12 as the radius.
Read this once, cold. The first move of each skill, then a few habits.
| Skill | First move |
|---|---|
| Real numbers | Turn every number into a decimal. For scientific notation, count the decimal point's jumps. |
| Slope and rate | Pick two clean points. Change in y over change in x, same order top and bottom. |
| Line equation | The “per” number multiplies x; the one-time number is added. No one-time number means proportional. |
| Functions and systems | Look for a repeated input. For a system, find the crossing point and read which coordinate is asked. |
| Solving equations | Distribute first, then subtract the smaller x-term from both sides. One line per move. |
| Transformations | Right or up adds; a reflection flips the sign of the coordinate that crosses the axis; a dilation multiplies both by k. |
| Pythagorean theorem | Name the hypotenuse (opposite the right angle). Missing hypotenuse: add. Missing leg: subtract. |
| Volume | Halve the diameter. Say the formula out loud. π key, round once at the end. |
| Surface area | Read “lateral” or “total” before you start. Total is lateral plus two bases. |
| Data and money | Interest or balance? Read the question twice. MAD is the average distance from the mean. |