Alpha School · Math · built 2 October 2026
This guide covers the nine skills a Grade 7 math test asks about most. Each one starts with something you can already do, adds one new move, gives you the rule, names the trap, and then lets you practice. Between the skills are mixed checks where nothing tells you which skill to use, because that is how a test works.
Every question here is new. None of them is from a test, so you can work them freely and look at every answer.
How to use it: answer, press Check, then read the answer even when you were right. The answer tells you where each wrong option comes from, and that is what keeps you out of the trap next time. Your work is saved in this browser, so you can close the page and come back.
In this guide
What is 3 + 5? And what is 3 − 5?
3 + 5 = 8.
3 − 5 means start at 3 and move 5 to the left.
3 − 5 = −2.
You already know both. Everything in this section is that number line, used carefully.
The new part is doing the same moves with decimals and mixed numbers, inside a story, and keeping track of every piece. Nerina practices guitar 1½ hours on Monday, 2¼ hours on Tuesday and 1¾ hours on Thursday. Next week she does the same. How many hours in the two weeks?
One week: 1½ + 2¼ + 1¾.
Change to quarters: 1²⁄₄ + 2¹⁄₄ + 1³⁄₄.
Whole numbers: 1 + 2 + 1 = 4. Quarters: 2 + 1 + 3 = 6 quarters, which is 1½.
One week is 5½ hours.
Two weeks: 5½ × 2 = 11 hours.
Signs follow two short rules. Adding or subtracting: same signs, add the sizes and keep the sign; different signs, subtract the sizes and take the sign of the bigger size. Multiplying or dividing: same signs give a positive, different signs give a negative.
You will also be asked where a number lives. Natural numbers are 1, 2, 3, … Whole numbers add 0. Integers add the negatives. Rational numbers add every fraction and every decimal that ends or repeats. Each set sits inside the next, so −7 is an integer and a rational number, but not a whole number.
Adding and subtracting: same signs keep, different signs take the bigger one's sign. Multiplying and dividing: same signs positive, different signs negative.
Read the story twice and count every piece before you add.
Turning “minus a negative” into a plain minus, or “−3 − 5” into “−3 + 5”. It is tempting because the signs sit close together and the eye wants to merge them. Say the sentence out loud: negative 3, minus 5, goes further left, so it is −8.
The other trap is stopping at a subtotal. A one-week total, or a cost before the delivery fee, is sitting right there as a wrong option.
Practice
Three to warm up. Work each one on the number line in your head.
a) 5. b) −13. c) 36.
a) Different signs: 9 − 4 = 5, and 9 is bigger, so the answer is positive.
b) Minus 9 moves further left from −4. The answer is −13.
c) Same signs multiply to a positive. The answer is 36.
Dana buys 3 bags of mulch at $4.75 each and pays a $12 delivery fee. Which expression gives the total she pays?
3 × 4.75 + 12
Three bags cost 3 × 4.75. The fee is paid once, so it is added once.
3 × (4.75 + 12) charges the delivery fee three times. 3 + 4.75 + 12 forgets to multiply. 3 × 4.75 − 12 takes the fee off instead of adding it.
Two to compute.
a) A diver is at −12.5 meters and goes down another 7.25 meters. What is the new depth?
b) What is the value of −2.5 × (−6) + (−4)?
a) −19.75 meters. b) 11.
a) Going down is more negative. −12.5 − 7.25 = −19.75.
b) Multiply first. −2.5 × (−6) = 15.
Then 15 + (−4) = 11.
If you got −19 for b), the two negatives were not multiplied to a positive.
Marisol swims 2⅓ hours on Monday, 1¾ hours on Tuesday and 1½ hours on Thursday. Next week she swims the same amounts. How many hours does she swim in the two weeks?
And one about where numbers live.
11 1/6 hours. It is an integer and a rational number.
Twelfths: 2⁴⁄₁₂ + 1⁹⁄₁₂ + 1⁶⁄₁₂.
Wholes: 2 + 1 + 1 = 4. Twelfths: 4 + 9 + 6 = 19 twelfths, which is 1⁷⁄₁₂.
One week is 5⁷⁄₁₂ hours. Two weeks: 5⁷⁄₁₂ × 2 = 11²⁄₁₂, which is 11⅙.
5 7/12 hours is one week, the subtotal trap. 10 1/6 drops a whole hour in the doubling. 11 1/12 forgets to double the fraction part.
−7 has no fraction part, so it is an integer, and every integer is rational. It is not whole or natural because those sets stop at 0.
A pack of 6 granola bars costs $3. What does one granola bar cost?
3 ÷ 6 = 0.50.
One granola bar costs $0.50. That one division is the whole skill.
A table is proportional when every row gives the same answer to “y divided by x”. That number is the constant of proportionality, k, and the equation is y = kx. A cyclist's table says 4 minutes, 1.2 km and 10 minutes, 3 km.
1.2 ÷ 4 = 0.3.
3 ÷ 10 = 0.3.
Same number, so k = 0.3 and the equation is y = 0.3x.
Scale drawings and unit conversions are the same idea written as a proportion. A drawing that is 6 inches long for a 15-foot wall has a scale of 15 ÷ 6, which is 2.5 feet for every inch: “1 inch represents 2.5 feet”.
Divide y by x. If every row gives the same number, that is k, and y = kx.
For a scale or a conversion, write the two matching quantities as a fraction and keep the same order on both sides.
Dividing the wrong way round. 4 ÷ 1.2 is also a constant, but it tells you minutes per kilometer, and the question asked for kilometers per minute. Check which quantity the question puts first: “y per x” means y divided by x.
A scale read backwards (“2.5 inches represent 1 foot”) and a difference used instead of a ratio (15 − 6 = 9) are both waiting as options.
Practice
Two divisions to warm up.
a) 4.5. b) 3.
18 ÷ 4 = 4.5. 7.5 ÷ 2.5 = 3, because three lots of 2.5 make 7.5.
A hose fills 3.5 gallons in 2 minutes. Which proportion can be used to find g, the gallons it fills in 9 minutes?
3.5/2 = g/9
Gallons over minutes on both sides. Solving it: g = 3.5 × 9 ÷ 2, so g = 15.75 gallons.
2/3.5 = g/9 flips one side only. 3.5/9 = g/2 pairs gallons with the wrong minutes. 3.5 × 2 = 9g is not a proportion at all.
A table shows the cost of swim passes at a pool.
| Passes, x | Cost, y |
|---|---|
| 3 | $7.50 |
| 5 | $12.50 |
| 8 | $20.00 |
k is 2.5, so the equation is y = 2.5x.
7.50 ÷ 3 = 2.5. Check another row: 20 ÷ 8 = 2.5.
y = 0.4x divides the other way (3 ÷ 7.5). y = x + 4.5 works only for the first row, which is the sign of an additive guess. y = 2.5 + x adds k instead of multiplying by it.
Two awkward ones.
a) A model of a canoe is 6 inches long. The real canoe is 15 feet long. Which scale did the model use?
b) One mile is about 1.6 kilometers. A trail is 12.5 miles long. About how many kilometers is that?
a) 1 inch represents 2.5 feet. b) 20 kilometers.
a) Feet per inch: 15 ÷ 6 = 2.5.
1 inch represents 0.4 foot divides the other way (6 ÷ 15). 2.5 inches represent 1 foot is the scale read backwards. 1 inch represents 9 feet subtracts instead of dividing.
b) 1.6/1 = k/12.5, so k = 1.6 × 12.5 = 20.
Mixed check 1
These two questions do not say which skill they use. Decide the method first, then answer.
A recipe uses 2.5 cups of flour for 20 cookies. How many cups of flour are needed for 48 cookies?
Method: Proportions and rates
6 cups.
2.5/20 = c/48, so c = 2.5 × 48 ÷ 20 = 6.
Or the unit rate: 2.5 ÷ 20 = 0.125 cup per cookie, and 0.125 × 48 = 6.
At 6 a.m. the temperature was −3.5°C. By noon it had risen 9.25 degrees. What was the temperature at noon?
Method: Rational numbers
5.75°C.
−3.5 + 9.25: different signs, so 9.25 − 3.5 = 5.75, positive because 9.25 is bigger.
What is 10% of 80?
10% means 0.10.
0.10 × 80 = 8.
And 1% of 80 is 0.8. Every percent question is built from “percent as a decimal, times the amount”.
Percent change asks how big the change is compared with where you started. A jacket drops from $80 to $68.
Change: 80 − 68 = 12.
Divide by the original: 12 ÷ 80 = 0.15.
The price dropped by 15%.
Three cousins of the same move. A sale price is the original minus the discount amount. Simple interest is principal × rate × years, and the balance is principal plus that interest. Net worth is everything you own minus everything you owe.
Percent change: change ÷ original × 100. Simple interest: principal × rate (as a decimal) × years.
Then re-read the question: did it ask for the change, or for the new amount?
Dividing by the new amount instead of the original. 12 ÷ 68 is tempting because 68 is the number you just wrote down. The original is the one the question says “from”.
The second trap is answering the discount or the interest when the question wants the sale price or the balance, or the other way round. Both numbers are always among the options.
Practice
Three to warm up.
a) 15. b) 60. c) 0.07.
25% is a quarter: 60 ÷ 4 = 15. 0.4 × 150 = 60. 7% is 7 hundredths, 0.07, not 0.7.
A jacket was priced at $80 and is now $68. Which expression gives the percent the price was reduced by?
(80 − 68) ÷ 80 × 100, which is 15%.
(80 − 68) ÷ 68 × 100 divides by the new price and gives about 17.6%. 68 ÷ 80 × 100 is 85%, the percent that is left, not the percent taken off. 80 − 68 is the change in dollars, not a percent.
Tomas puts $500 in an account that pays 4% simple annual interest. He leaves it for 3 years.
Interest $60. Balance $560.
Interest: 500 × 0.04 × 3 = 60.
Balance: 500 + 60 = 560.
If you got 6,000, the rate went in as 4 instead of 0.04. The balance and the interest are one word apart; read which one is asked.
a) The same headphones are on sale at three stores. Store A: $45 with 20% off. Store B: $42 with 10% off. Store C: $50 with 30% off. Which store has the lowest sale price?
b) Priya owns $2,400 in savings and a bike worth $650. She owes $900 on a loan and $275 to a friend. What is her net worth?
a) Store C. b) $1,875.
a) Store A: 45 × 0.80 = 36. Store B: 42 × 0.90 = 37.80. Store C: 50 × 0.70 = 35.
The biggest percent off is not automatically the lowest price, and the lowest sticker price is not either. You have to finish all three.
b) Owns: 2,400 + 650 = 3,050. Owes: 900 + 275 = 1,175.
Net worth: 3,050 − 1,175 = 1,875.
x + 7 = 12. What is x?
Take 7 from both sides.
x = 5.
One move undoes one operation. A two-step equation is two of these, in the right order.
Solve 4x + 7 = 31.
Undo the + 7 first: 4x = 24.
Then undo the × 4: x = 6.
Writing the equation is the same skill backwards. A trophy order costs $9 per trophy plus a one-time $25 engraving fee. For t trophies the total is C = 9t + 25: the “per” number multiplies, the one-time number is added once.
An inequality uses exactly the same two steps, with one extra rule: if you multiply or divide both sides by a negative number, the sign flips. On a number line, < and > get an open circle; ≤ and ≥ get a closed one.
Undo the add or subtract first, then the multiply or divide. Dividing by a negative flips the inequality sign.
Dividing first. 4x + 7 = 31 is not x + 7 = 7.75; dividing by 4 has to happen to the 7 as well, so wait until the 7 is gone. The option built from this mistake is usually there.
On inequalities, the two traps are forgetting the flip and swapping open and closed circles. The number in the answer can be right while the picture is wrong.
Practice
Two one-step equations.
a) y = 13. b) m = 7.
Add 9 to both sides. Divide both sides by 5.
A gym charges $30 to join and then $12 each month. Which equation gives T, the total cost for m months?
T = 12m + 30
The $12 is “each month”, so it multiplies m. The $30 is paid once, so it is added once.
T = 30m + 12 swaps them and charges the joining fee every month. T = 42m adds the two numbers and multiplies both by m. T = 12m − 30 subtracts a fee you have to pay.
Solve each equation.
a) x = 9. b) x = −6.
a) Add 8: 3x = 27. Divide by 3: x = 9.
b) Take 5 away: −2x = 12. Divide by −2: x = −6.
If you got 6 for b), the sign of the −2 was dropped in the last step.
Solve 4 − 3x ≤ 19.
Closed circle at −5, shaded to the right. The values −5, 0 and 2 satisfy it.
Take 4 away: −3x ≤ 15.
Divide by −3 and flip the sign: x ≥ −5.
“≥” is a closed circle, and “greater than or equal” shades right. Shaded to the left is the forgotten flip. An open circle would be for < or >. A circle at 5 lost the negative.
Check −6: 4 − 3(−6) = 22, and 22 is not ≤ 19, so −6 is out.
Mixed check 2
These three questions do not say which skill they use. Decide the method first, then answer.
A phone plan costs $20 a month plus $0.05 for each text. One month the bill was $27.50. How many texts were sent?
Method: Equations and inequalities
150 texts.
0.05t + 20 = 27.50.
Take 20 away: 0.05t = 7.50.
Divide by 0.05: t = 150.
A coat marked $120 is on sale for 35% off. What is the sale price?
Method: Percent
$78.
Discount: 120 × 0.35 = 42.
Sale price: 120 − 42 = 78.
If you wrote 42, that is the discount, one word away from what was asked.
On a scale drawing, 2 inches represents 5 feet. A wall is 7 inches long on the drawing. How long is the real wall?
Method: Proportions and rates
17.5 feet.
2/5 = 7/f, so 2f = 35 and f = 17.5.
Or 5 ÷ 2 = 2.5 feet per inch, and 7 × 2.5 = 17.5.
A triangle has sides 3, 4 and 5. You double every side. What are the new sides?
3 × 2 = 6. 4 × 2 = 8. 5 × 2 = 10.
The new triangle is the same shape, twice the size. The number 2 is the scale factor.
Similar figures are exactly this, with the scale factor hidden. Every side of the big figure is the matching side of the small one times the same number. Triangle ABC is similar to triangle DEF. AB is 6 and its partner DE is 9. BC is 8. How long is its partner EF?
Small over big, both sides: 6/9 = 8/EF.
Cross multiply: 6 × EF = 72.
EF = 12.
Check with the scale factor: 9 ÷ 6 = 1.5, and 8 × 1.5 = 12.
Match each side to its partner. Put the small figure on top in both fractions, then cross multiply.
Adding the difference instead of multiplying. “9 is 3 more than 6, so EF is 8 + 3 = 11” feels reasonable and is wrong; similar figures grow by multiplying. The other trap is pairing a side with the wrong partner, usually the longest with the shortest because they look alike in the picture.
Practice
Two to warm up.
a) 2/3. b) x = 12.
Divide top and bottom by 3. Cross multiply: 2x = 24, so x = 12.
Triangle PQR is similar to triangle STU. PQ = 4, QR = 6, ST = 10 and TU = y. Which proportion can be used to find y?
4/10 = 6/y
PQ pairs with ST, and QR pairs with TU. Small over big on both sides.
4/10 = y/6 flips one side. 4/y = 6/10 pairs each side with the wrong partner. 4 + 6 = 10 + y treats similar figures as if they grew by adding.
Using the triangles above:
y = 15. Scale factor 2.5.
4y = 60, so y = 15.
10 ÷ 4 = 2.5. Check: 6 × 2.5 = 15.
Jonah says: “Triangle A has sides 6 and 8. In the similar triangle B the side matching 6 is 9. So the side matching 8 must be 11, because 9 is 3 more than 6.” Is Jonah right?
No, the side is 12, because every side is multiplied by 1.5, not increased by 3.
9 ÷ 6 = 1.5.
8 × 1.5 = 12.
10.5 is 6 × 1.75, a scale factor made up from the wrong pair. 13.5 is 9 × 1.5, scaling a side that was already scaled.
The diameter of a circle is 10 cm. What is the radius?
Half of 10 is 5.
The radius is 5 cm. Most circle mistakes happen before any formula, right here.
Circumference is the distance around: C = πd, or 2πr, the same thing. Area is the space inside: A = πr², and the radius is squared, not the diameter. For the 10 cm circle:
C = π × 10, which is about 31.4 cm.
A = π × 5², which is π × 25, about 78.5 cm².
Press the π key on your calculator and round once, at the very end. If a question says “closest to”, the π-key answer and the 3.14 answer are both closest to the same option.
π itself is the circumference divided by the diameter of any circle: π = C ÷ d.
A composite shape is a shape made from pieces. Add the pieces that are there and take away the pieces that are cut out.
Circumference uses the diameter: C = πd. Area uses the radius squared: A = πr². Press π, round once at the end.
Putting the diameter into the area formula. π × 10² is 314, four times too big, and it is always one of the options. The other classic is doubling the radius instead of squaring it: π × 5 × 2 is the circumference, not the area.
Practice
Three to warm up.
a) 49. b) 9. c) 13.
7 × 7 = 49. A diameter of 18 has radius 9. A radius of 6.5 has diameter 13.
A circle has a diameter of 16 inches. Which expression gives its area?
π × 8²
The radius is half of 16, so 8, and area squares the radius.
π × 16² squares the diameter and is four times too big. π × 16 and 2 × π × 8 are both the circumference.
A circle has a radius of 4 cm.
Circumference 25.1 cm. Area 50.3 cm².
C = 2 × π × 4, which is 8π. With the π key that is 25.13; with 3.14 it is 25.12. Both are closest to 25.1 cm.
A = π × 4², which is 16π. With the π key that is 50.27; with 3.14 it is 50.24. Both are closest to 50.3 cm².
12.6 is 4π, which is π times the radius. 201.1 is π × 8², the diameter squared. 37.7 is 12π, nothing this circle has.
a) A rectangular tray is 30 cm by 20 cm. Two circular holes, each with a diameter of 10 cm, are cut out of it. Which is the best estimate of the area of the tray that is left?
b) A circle has a circumference of 28.3 cm and a diameter of 9 cm. Which expression best represents the value of π?
a) about 443 cm². b) 28.3 ÷ 9.
a) Rectangle: 30 × 20 = 600.
One hole: radius 5, so π × 5², which is 25π, about 78.5.
Two holes: about 157.1 with the π key, 157 with 3.14.
Left: 600 − 157.1, about 442.9, so about 443 cm².
522 takes away only one hole. 537 takes away two circumferences (2 × 10π) instead of two areas. 757 adds the holes on.
b) π is circumference divided by diameter: 28.3 ÷ 9, about 3.14.
Mixed check 3
These two questions do not say which skill they use. Decide the method first, then answer.
A triangle with sides 9, 12 and 15 is similar to a triangle whose shortest side is 6. How long is the longest side of the smaller triangle?
Method: Similar figures
10.
Shortest pairs with shortest: 6/9 = x/15.
9x = 90, so x = 10.
Scale factor check: 6 ÷ 9 = 2/3, and 15 × 2/3 = 10. Subtracting 3 from 15 would give 12, the additive trap.
A circular rug has a diameter of 6 feet. Which is closest to the area of the rug?
Method: Circles
28.3 ft²
Radius: 6 ÷ 2 = 3. Area: π × 3², which is 9π, about 28.27 with the π key or 28.26 with 3.14.
18.8 is the circumference, 6π. 113.1 is π × 6², the diameter squared. 9.4 is 3π.
A rectangle is 5 cm by 3 cm. What is its area? And if a box has that rectangle as its base and is 4 cm tall, what is its volume?
Area: 5 × 3 = 15 cm².
Volume: 15 × 4 = 60 cm³.
Volume of any prism is “area of the base, times the height”. You just did it.
A triangular prism works the same way: find the area of the triangle first, then multiply by the length. A pyramid is one third of the prism with the same base and height. Surface area is different: it is the total of every face, which is easiest to see on the net. A triangular prism has a right-triangle base with legs 6 and 8 and hypotenuse 10, and it is 7 cm long.
Base area: ½ × 6 × 8 = 24 cm².
Volume: 24 × 7 = 168 cm³.
Surface area: two triangles plus three rectangles.
Triangles: 2 × 24 = 48.
Rectangles: 7 × 6 + 7 × 8 + 7 × 10, which is 7 × 24, so 168.
Total surface area: 48 + 168 = 216 cm².
The 168 on its own is the lateral surface area: the sides only, no bases.
Volume of a prism: base area × height. Pyramid: one third of that. Surface area: add every face on the net.
Using a side of the triangle as if it were the base area. “Base” in the formula means the whole face, so a triangle has to be ½ × base × height before anything else happens. The others: dropping the ⅓ on a pyramid, counting one triangle instead of two on the net, and giving the lateral area when the total was asked, or the reverse.
Practice
Two to warm up.
a) 30. b) 30.
½ × 10 × 6 = 30. 90 ÷ 3 = 30.
A pyramid has a base area of 36 in² and a height of 9 in. Which expression gives its volume?
⅓ × 36 × 9
36 × 9 is the prism with the same base and height, three times too big. ⅓ × 36 + 9 adds the height instead of multiplying. 36 × 9 × 3 multiplies by 3 instead of dividing.
Two volumes.
a) A rectangular prism is 12 cm by 5 cm by 4 cm.
b) The pyramid from the question above: base area 36 in², height 9 in.
a) 240 cm³. b) 108 in³.
a) 12 × 5 = 60, then 60 × 4 = 240.
b) 36 × 9 = 324, then 324 ÷ 3 = 108.
The net of a triangular prism is shown. The triangle has legs 5 cm and 12 cm and a hypotenuse of 13 cm. The prism is 8 cm long.
Total surface area 300 cm². Volume 240 cm³.
Each triangle: ½ × 5 × 12 = 30. Two of them: 60.
Rectangles: 8 × 5 + 8 × 12 + 8 × 13, which is 8 × 30, so 240.
Total: 60 + 240 = 300 cm².
240 cm² is the lateral area, the three rectangles alone. 270 counts only one triangle. 360 uses 5 × 12 for each triangle without the half.
Volume: 30 × 8 = 240 cm³. It happens to equal the lateral area here; different things, same number.
A spinner has 8 equal sections and 3 of them are yellow. How many sections are there in all? What fraction are yellow?
8 sections in all.
3 of the 8 are yellow, so the fraction is 3/8.
That fraction is the probability of landing on yellow.
Probability is the part you want over everything that could happen. To write it as a decimal, divide: 3 ÷ 8 = 0.375.
To predict from an experiment, use the fraction you observed and multiply by the number of future trials. If the spinner landed on yellow 14 times in 40 spins, then in 100 more spins:
14/40 × 100 = 35.
Expect about 35 yellows.
For two or three coins, list every outcome. Two coins give HH, HT, TH, TT. “Exactly one head” is HT and TH: 2 of the 4, so ½.
Probability = the part you want ÷ all the possibilities. To predict, multiply that fraction by the number of future trials.
Putting “what you want” over “what you don't want”. 3 yellow and 5 not-yellow tempts you into 3/5, but the bottom of a probability is everything, 8. With repeated flips, the trap is “at least” versus “exactly”: exactly two heads in three flips is HHT, HTH, THH, which is 3 of 8, not 4.
Practice
Two to warm up.
a) 0.625. b) 150.
5 ÷ 8 = 0.625. 200 ÷ 4 = 50, then 50 × 3 = 150.
A box holds 4 silver coins, 6 gold coins and 10 bronze coins. One coin is picked without looking. Which expression gives the probability it is gold?
6/20
6 gold over all 20 coins. 6/14 puts gold over not-gold. 14/20 is the probability of not-gold. 6/10 compares gold with bronze only.
Same box: 4 silver, 6 gold, 10 bronze.
P(gold) = 0.3. P(not bronze) = 0.5.
6 ÷ 20 = 0.3.
Not bronze means silver or gold: 4 + 6 = 10, and 10/20 = 0.5.
a) A spinner was spun 40 times and landed on green 15 times. Based on this, how many of the next 120 spins should land on green?
b) Two coins are flipped. What is the probability of exactly one head?
a) 45. b) 1/2.
a) 15/40 × 120 = 45.
b) Outcomes: HH, HT, TH, TT. Exactly one head is HT or TH, 2 of 4, so 1/2.
1/4 is exactly two heads. 3/4 is at least one head. 1/3 comes from thinking there are only three outcomes.
Mixed check 4
These three questions do not say which skill they use. Decide the method first, then answer.
A spinner has 10 equal sections and 4 of them are blue. If it is spun 150 times, about how many spins should land on blue?
Method: Probability
60.
4/10 × 150 = 60.
A square pyramid has a base with sides of 10 cm and a height of 12 cm. What is its volume?
Method: Volume and surface area
400 cm³.
Base area: 10 × 10 = 100.
Volume: ⅓ × 100 × 12 = 400.
1,200 is the prism, missing the ⅓.
Lin puts $800 in an account paying 3% simple annual interest for 2 years. How much interest does the account earn?
Method: Percent
$48.
800 × 0.03 × 2 = 48.
The balance would be 848; the question asked for the interest.
Here are five numbers: 2, 4, 4, 7, 8. What is the median? What is the mode?
The median is the middle number when they are in order: 4.
The mode is the most common: 4.
The range is biggest minus smallest: 8 − 2 = 6.
A box plot shows five numbers: minimum, first quartile, median, third quartile, maximum. The line inside the box is the median. The box runs from Q1 to Q3, and its width is the interquartile range, IQR = Q3 − Q1. Whisker end to whisker end is the range.
Median: 18.
IQR: 24 − 14 = 10.
Range: 30 − 10 = 20.
On a dot plot, the mode is the tallest column, and the median is the middle dot when you count from the left.
To infer from a sample to a population, multiply the fraction in the sample by the size of the whole group. If 12 of 50 students surveyed chose soccer, in a school of 800 you expect 12/50 × 800, which is 192.
Box plot: line in the box is the median, box width is the IQR, whisker to whisker is the range. Sample to population: sample fraction × whole population.
Reading the box as the range. The box is only the middle half of the data. The range needs the whisker ends. The second trap is a statement that is true for one plot but is being compared with the wrong part of the other. Write the five numbers under each plot before comparing anything.
Practice
Data: 3, 5, 6, 6, 9, 11.
Median 6. Range 8. Mode 6.
Six numbers: the middle two are 6 and 6, so the median is 6. Range: 11 − 3 = 8. The mode is 6.
A box plot of the ages of people at a chess club is shown.
24 − 14, which is 10.
The box runs from 14 to 24. 30 − 10 is the range. 18 − 14 and 24 − 18 are each only half the box.
The box plots show the number of minutes two classes spent reading one evening.
The median for Class B is greater than the median for Class A. 22 is greater than 18.
Range: Class A is 30 − 10 = 20, Class B is 28 − 12 = 16, so A has the greater range.
IQR: Class A is 24 − 14 = 10, Class B is 26 − 16 = 10, equal.
Minimum: A is 10, B is 12, so B is greater.
a) A survey of 50 randomly chosen students at a school of 800 asked for a favorite club: 12 chose soccer, 20 chose basketball and 18 chose art. Which prediction is NOT supported by the survey?
b) The dot plot shows the number of books 10 customers bought.
a) About 300 students would choose soccer. b) Mode 2.
a) Soccer: 12/50 × 800 = 192. Basketball: 20/50 × 800 = 320. Art: 18/50 × 800 = 288.
Three of the statements match those numbers. 300 for soccer does not, so it is the one not supported.
b) The tallest column is above 2. The median is also 2, the fifth and sixth dots. The mean is (3 + 10 + 6) ÷ 10, which is 1.9.
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.
A circle has a radius of 5 meters. Which is closest to its circumference?
Method: Circles
31.4 m
C = 2 × π × 5, which is 10π, about 31.42 with the π key or 31.4 with 3.14.
78.5 is the area. 15.7 is 5π, half the circumference. 62.8 uses the diameter twice.
The same circle, radius 5 meters. Which is closest to its area?
Method: Circles
78.5 m²
A = π × 5², which is 25π, about 78.54 with the π key or 78.5 with 3.14.
31.4 is the circumference from the question before: same circle, other quantity. 157.1 is 2 × 25π. 25 forgot π.
A pool drains at 4.5 gallons per minute. Which equation gives g, the gallons drained in m minutes?
Method: Proportions and rates
g = 4.5m
Gallons are 4.5 times the minutes. g = m + 4.5 adds the rate. g = m ÷ 4.5 divides the wrong way. m = 4.5g swaps the variables.
A rectangular prism is 8 cm by 6 cm by 3.5 cm. What is its volume?
Method: Volume and surface area
168 cm³.
8 × 6 = 48, then 48 × 3.5 = 168.
A box plot has a minimum of 5, a first quartile of 9, a median of 12, a third quartile of 20 and a maximum of 26. What is the IQR?
Method: Data displays
11.
20 − 9 = 11. The range would be 26 − 5 = 21.
What is the value of −8 + 3 × (−2)?
Method: Rational numbers
−14.
Multiply first: 3 × (−2) = −6.
Then −8 + (−6) = −14.
10 comes from adding first. −2 comes from treating the −6 as +6.
The price of a ticket rose from $40 to $46. By what percent did it increase?
Method: Percent
15%.
Change: 46 − 40 = 6.
6 ÷ 40 = 0.15, so 15%.
6 ÷ 46 gives about 13%, the divide-by-the-new-amount trap.
Solve 5 − 2x > 11. Which describes the graph of the solution?
Method: Equations and inequalities
Open circle at −3, shaded to the left.
Take 5 away: −2x > 6.
Divide by −2 and flip: x < −3.
“<” is an open circle shaded left. Shaded right is the missing flip. A closed circle would need ≤. A circle at 3 lost the negative.
Two rectangles are similar. The small one is 4 by 6. The large one has a short side of 10. How long is its long side?
Method: Similar figures
15.
4/10 = 6/x, so 4x = 60 and x = 15.
Adding 6 to each side would give 12, the additive trap.
A fair coin is flipped three times. What is the probability of exactly one head?
Method: Probability
3/8
Eight outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. Exactly one head: HTT, THT, TTH, which is 3 of 8.
1/8 is no heads at all. 2/3 puts the heads over the flips. 1/2 is “at most one head”, which is 4 of 8.
Read this once, cold. The first move of each skill, then a few habits.
| Skill | First move |
|---|---|
| Rational numbers | Say the signs out loud before you compute. Multiply and divide before you add. Count every piece of the story. |
| Proportions and rates | Divide y by x. Same number every row means k, and y = kx. Write proportions with the same order on both sides. |
| Percent | Change ÷ original. Then re-read: sale price or discount, balance or interest? |
| Equations and inequalities | Undo the add or subtract first. Flip the sign when you divide by a negative. |
| Similar figures | Pair each side with its partner. Small over big on both sides. Multiply, never add. |
| Circles | Write the radius down first. C uses d. A uses r². Press π, round once at the end. |
| Volume and surface area | Base area first, then × height. Pyramid gets ⅓. Surface area counts every face on the net. |
| Probability | What you want over everything. To predict, multiply the fraction by the future trials. |
| Data displays | Write the five numbers under a box plot before comparing. Sample fraction × whole population. |