Integers are whole numbers that can be positive, negative, or zero. Learning to add, subtract, multiply, and divide them is the foundation for everything else in Grade 7 math.
📍 The number line
Integers live on a number line. Numbers to the right of zero are positive; numbers to the left are negative.
... −3 −2 −1 0 1 2 3 ...
The absolute value of a number is its distance from zero, and it's always positive or zero.
|−5| = 5
|5| = 5
|0| = 0
➕➖ Adding & subtracting integers
Same sign
Add the numbers and keep the sign.
−3 + (−4) = −7
5 + 6 = 11
Different signs
Subtract the smaller from the larger, keep the sign of the bigger number.
−8 + 3 = −5
9 + (−4) = 5
Subtracting a negative is the same as adding a positive: 7 − (−2) = 7 + 2 = 9
✖️➗ Multiplying & dividing integers
Same sign → positive
−4 × −3 = 12
−10 ÷ −2 = 5
Different sign → negative
−4 × 3 = −12
10 ÷ −2 = −5
🖼️ See it
−3 + 5 = 2. Adding a positive moves right; adding a negative moves left.
A negative exponent means take the reciprocal and make the exponent positive.
a−n = 1 ÷ an
2−3 = 1 ÷ 23 = 1 ÷ 8 = 0.125
🖼️ See it
Work down the list. Operations on the same step are done from left to right.Each step down divides by 2, so a negative exponent means a reciprocal: 2⁻² = 1/2² = 1/4.
Dividing by a fraction means multiplying by its reciprocal (flip it). Add or subtract fractions only after finding a common denominator.
✏️ Quick check · 1 of 16
Divide two fractions
What is 3/4 ÷ 1/2?
Dividing by a fraction means multiplying by its reciprocal: flip 1/2.
✏️ Quick check · 2 of 16
Add two fractions
What is 1/3 + 1/4?
Use a common denominator of 12.
✏️ Quick check · 3 of 16
Multiply two fractions
What is 2/5 × 5/6?
Multiply the tops and the bottoms, then simplify.
✏️ Quick check · 4 of 16
Decimal to fraction
What is 0.35 written as a fraction in simplest form?
0.35 is 35 hundredths. Divide top and bottom by 5.
✏️ Quick check · 5 of 16
Subtract two fractions
What is 5/6 − 1/3?
Change 1/3 into sixths first.
✏️ Quick check · 6 of 16
Fraction to decimal
What is 7/8 written as a decimal?
Divide 7 by 8.
✏️ Quick check · 7 of 16
Multiply two fractions
What is 1/2 × 2/3?
Multiply across, then simplify.
✏️ Quick check · 8 of 16
Subtract fractions
What is 5/8 − 1/4?
Change 1/4 into eighths first.
✏️ Quick check · 9 of 16
Decimal to fraction again
What is 0.125 written as a fraction in simplest form?
0.125 is 125 thousandths. What is 1000 ÷ 125?
✏️ Quick check · 10 of 16
Add fractions again
What is 4/9 + 2/9?
Same denominator: add the tops, then simplify.
✍️ Type your answer · 11 of 16
Find the sum.
What is 2/3 + 5/6?
Enter an exact answer, without rounding.
Change 2/3 into sixths, add, then write the answer as a decimal.
✍️ Type your answer · 12 of 16
Find the product.
What is 5/6 × 3/10?
Enter an exact answer, without rounding.
Multiply across, simplify, then write the answer as a decimal.
✏️ Quick check · 13 of 16
Subtract fractions
What is 5/6 − 1/3?
Change 1/3 into sixths.
✏️ Quick check · 14 of 16
Decimal to fraction
Write 0.6 as a fraction in simplest form.
0.6 is six tenths.
✍️ Type your answer · 15 of 16
Add the fractions. Enter a fraction or decimal.
What is 3/4 + 1/8?
Enter an exact answer, without rounding.
Change 3/4 into eighths.
✍️ Type your answer · 16 of 16
Multiply the decimals.
What is 2.4 × 0.5?
Enter an exact answer, without rounding.
Multiplying by 0.5 is the same as halving.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
What is 1/2 × 1/4?
Question 2 of 5
What is 3/5 as a decimal?
Question 3 of 5
What is 2 ÷ 1/2?
Question 4 of 5
What is 0.75 + 0.5?
Question 5 of 5
Which is larger, 5/8 or 0.6?
Lesson 4 · Relationships
⚖️ Ratios & Proportions
A ratio compares two quantities. A proportion says two ratios are equal — and that lets us solve for missing values.
🔗 What is a ratio?
A ratio compares two amounts. It can be written with a colon or as a fraction.
3 : 4 or 3/4
If a fruit bowl has 3 apples and 4 oranges, the ratio of apples to oranges is 3:4.
🚗 Unit rates
A unit rate compares a quantity to one unit of another — like speed or price per item.
240 km in 4 hours
= 240 ÷ 4 = 60 km per hour
🧩 Solving proportions
Two equal ratios form a proportion. Solve for the unknown by cross-multiplying.
x/4 = 6/8
x × 8 = 6 × 4
8x = 24
x = 3
🖼️ See it
Check: 12 + 8 = 20, and 12 : 8 simplifies to 3 : 2.
📝 Worked examples
Simplify: 12 : 18 = 2 : 3 (divide both by 6)
Proportion: 3/4 = x/20
4x = 60 → x = 15
Unit rate: 150 km in 3 h = 50 km/h
Two ratios are equivalent if their cross products are equal: for a/b = c/d, ad = bc.
Keep units in the same order. "Boys to girls = 3 : 5" and "girls to boys = 5 : 3" are different ratios.
✏️ Quick check · 1 of 16
Solve the proportion
Solve for x: x/4 = 6/8
Simplify 6/8 first, or cross-multiply: 8x = 4 × 6.
✏️ Quick check · 2 of 16
Simplify a ratio
Simplify the ratio 20 : 30
Divide both numbers by their greatest common factor.
✏️ Quick check · 3 of 16
Find a unit rate
A car travels 180 km in 3 hours. What is its speed in km per hour?
Unit rate means per 1 hour: divide the km by the hours.
✏️ Quick check · 4 of 16
Solve another proportion
Solve for x: 3/x = 9/12
Cross-multiply: 9x = 3 × 12.
✏️ Quick check · 5 of 16
Simplify another ratio
Simplify the ratio 15 : 25
Both numbers divide by 5.
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Unit price
5 apples cost $2.50. What is the price per apple?
Divide the cost by the number of apples.
✏️ Quick check · 7 of 16
Simplify a third ratio
Simplify the ratio 8 : 12
Both numbers divide by 4.
✏️ Quick check · 8 of 16
Solve another proportion
Solve for x: 4/5 = x/20
How do you get from 5 to 20? Do the same to 4.
✏️ Quick check · 9 of 16
Another unit rate
You type 240 words in 4 minutes. What is your typing rate in words per minute?
Divide the words by the minutes.
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Scale a ratio
A recipe uses flour and sugar in a ratio of 3:2. If you use 12 cups of flour, how much sugar do you need?
How many times 3 is 12? Multiply the 2 by the same number.
✍️ Type your answer · 11 of 16
Solve for x.
x/6 = 10/12 — what is x?
Enter an exact answer, without rounding.
Simplify 10/12 to sixths.
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A car travels 150 km using 6 litres of fuel.
How many km per litre does it get?
Enter an exact answer, without rounding.
km per litre
Divide the km by the litres.
✏️ Quick check · 13 of 16
Simplify
Simplify the ratio 15 : 25.
Find the greatest common factor.
✏️ Quick check · 14 of 16
Proportional?
Which pair of ratios is equivalent?
Simplify the second fraction.
✍️ Type your answer · 15 of 16
Solve the proportion 5/8 = x/40.
What is x?
Enter an exact answer, without rounding.
What do you multiply 8 by to get 40?
✍️ Type your answer · 16 of 16
The ratio of boys to girls is 3 : 5. There are 24 boys.
How many girls are there?
Enter an exact answer, without rounding.
Find the value of one part first.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
What is 8 to 12 in simplest form?
Question 2 of 5
Solve x/6 = 10/15.
Question 3 of 5
Three notebooks cost $4.50. How much does one cost?
Question 4 of 5
Paint is mixed blue : yellow = 2 : 3. With 10 units of blue, how much yellow?
Question 5 of 5
A car travels 240 km in 4 hours. What is its unit rate?
Lesson 5 · Relationships
💯 Percentages
"Percent" means "per hundred." Percentages show up everywhere — discounts, grades, tips, and statistics.
🔄 Percent, decimal, fraction
Percent
Decimal
Fraction
50%
0.5
1/2
25%
0.25
1/4
10%
0.1
1/10
75%
0.75
3/4
To turn a percent into a decimal, divide by 100 (move the decimal point two places left).
🧮 Finding a percent of a number
part = percent × whole
20% of 150
= 0.20 × 150
= 30
🔍 Finding what percent one number is of another
Some percent questions work backwards — they give you the part and the whole, and ask you to find the percent.
percent = (part ÷ whole) × 100
27 is what percent of 90?
= (27 ÷ 90) × 100
= 0.3 × 100
= 30%
Don't mix these up: "What is 27% of 90?" means multiply (90 × 0.27 = 24.3). "27 is what percent of 90?" means divide (27 ÷ 90 = 0.3 → 30%). If the sentence already has a "%" in it, multiply. If it's asking you to find the percent, divide the part by the whole.
📈 Percent increase & decrease
A $40 jacket is 25% off.
Discount = 0.25 × 40 = $10
New price = 40 − 10 = $30
🖼️ See it
Multiply a decimal by 100 to get a percentage; divide a percentage by 100 to get a decimal.
📝 Worked examples
Part of a number: 25% of 80 = 0.25 × 80 = 20
Find the percent: 15 of 60 → 15 ÷ 60 = 0.25 = 25%
Find the whole: 30 is 20% of ? → 30 ÷ 0.20 = 150
Convert: 0.35 = 35% 3/5 = 0.6 = 60%
Shortcut: 10% is one tenth, 50% is half, 25% is a quarter. Build other percentages from these.
✏️ Quick check · 1 of 16
Find a percent of a number
What is 20% of 150?
Find 10% of 150, then double it.
✏️ Quick check · 2 of 16
Decimal to percent
What is 0.4 written as a percent?
Multiply the decimal by 100.
✏️ Quick check · 3 of 16
Find another percent
What is 15% of 80?
Find 10% and 5% of 80, then add them.
✏️ Quick check · 4 of 16
Percent discount
A $50 shirt is 20% off. What is the new price?
Find 20% of $50 and take it off.
✏️ Quick check · 5 of 16
Find another percent
What is 30% of 90?
Find 10% of 90, then multiply by 3.
✏️ Quick check · 6 of 16
Fraction to percent
What is 3/4 written as a percent?
Divide 3 by 4, then multiply by 100.
✏️ Quick check · 7 of 16
Find another percent
What is 50% of 84?
50% is a half.
✏️ Quick check · 8 of 16
Percent to decimal
What is 65% written as a decimal?
Divide the percent by 100.
✏️ Quick check · 9 of 16
Percent increase
An $80 item increases in price by 25%. What is the new price?
Find 25% (a quarter) of $80 and add it on.
✏️ Quick check · 10 of 16
Find the percent
18 is what percent of 24?
Divide 18 by 24, then multiply by 100.
✍️ Type your answer · 11 of 16
Find the value.
What is 35% of 200?
Enter an exact answer, without rounding.
Find 10% of 200, multiply by 3, then add 5% of 200.
✍️ Type your answer · 12 of 16
27 is what percent of 90?
Enter the percent (just the number, no % sign).
Enter an exact answer, without rounding.
%
Divide 27 by 90, then multiply by 100.
✏️ Quick check · 13 of 16
Fraction to percent
What is 3/5 as a percent?
Divide, then multiply by 100.
✏️ Quick check · 14 of 16
10% shortcut
What is 10% of 450?
Move the decimal point one place left.
✍️ Type your answer · 15 of 16
Find the part.
What is 40% of 250?
Enter an exact answer, without rounding.
Convert 40% to 0.40.
✍️ Type your answer · 16 of 16
Find the percent.
What percent is 18 out of 24?
Enter an exact answer, without rounding.
%
Divide the part by the whole.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
What is 50% of 90?
Question 2 of 5
Write 0.08 as a percent.
Question 3 of 5
12 is 15% of what number?
Question 4 of 5
A quantity goes from 40 to 50. What is the percent increase?
Percentages show up constantly in real life — from the tax added at checkout to the interest a bank pays you, and even in how scientists measure the accuracy of a result.
💵 Sales tax & tips
Sales tax and tips both work the same way: multiply the original amount by the rate to find the tax/tip amount, then add it to the original for the total.
Total = Original + (Original × Rate)
A $40 item with 8% tax:
Tax = 40 × 0.08 = $3.20
Total = 40 + 3.20 = $43.20
💰 Simple interest
I = P × r × t
P = $500, r = 0.04, t = 3 years
I = 500 × 0.04 × 3 = $60
🔍 Percent error
Percent error measures how far off a measurement is from the actual value.
Percent Error = |experimental − actual| ÷ actual × 100
Use MAD with the mean and the IQR with the median. The IQR is less affected by extreme values (outliers).
✏️ Quick check · 1 of 10
What does MAD measure?
Mean absolute deviation (MAD) measures:
Absolute deviation = how far each value is from the mean.
✍️ Type your answer · 2 of 10
Data: 2, 4, 6, 8. The mean is 5.
What is the mean absolute deviation (MAD)? (average of |2-5|, |4-5|, |6-5|, |8-5|)
Enter an exact answer, without rounding.
The distances are 3, 1, 1 and 3. Find their average.
✏️ Quick check · 3 of 10
What does IQR measure?
The interquartile range (IQR) measures:
Quartiles cut off the bottom and top quarters of the data.
✍️ Type your answer · 4 of 10
Data (sorted): 1, 3, 5, 7, 9, 11, 13. The lower quartile (Q1) is 3, and the upper quartile (Q3) is 11.
What is the interquartile range (IQR)?
Enter an exact answer, without rounding.
IQR = Q3 − Q1.
✍️ Type your answer · 5 of 10
Data: 10, 20, 30, 40. The mean is 25.
What is the mean absolute deviation (MAD)? (average of |10-25|, |20-25|, |30-25|, |40-25|)
Enter an exact answer, without rounding.
The distances are 15, 5, 5 and 15. Find their average.
✍️ Type your answer · 6 of 10
Data (sorted): 2, 4, 6, 8, 10, 12, 14. The lower quartile (Q1) is 4, and the upper quartile (Q3) is 12.
What is the interquartile range (IQR)?
Enter an exact answer, without rounding.
IQR = Q3 − Q1.
✏️ Quick check · 7 of 10
IQR
How is the interquartile range found?
It uses the quartiles.
✏️ Quick check · 8 of 10
What MAD tells us
A larger MAD means the data is:
MAD measures spread.
✍️ Type your answer · 9 of 10
Q1 = 12 and Q3 = 20.
What is the IQR?
Enter an exact answer, without rounding.
Subtract Q1 from Q3.
✍️ Type your answer · 10 of 10
Data: 4, 6, 8, 10. The mean is 7.
What is the MAD?
Enter an exact answer, without rounding.
Find each distance from 7, then average them.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Which measure of spread is less affected by outliers?
Question 2 of 5
Data: 1, 2, 3, 4, 5. What is the MAD?
Question 3 of 5
Q1 = 5 and Q3 = 17. What is the IQR?
Question 4 of 5
Which data set has the larger MAD?
Question 5 of 5
Q1 is the median of:
Lesson 15 · Applied Fractions
🪐 Rates with Fractions
A rate compares two fractional quantities. To find a unit rate — the amount per 1 whole unit — divide the first fraction by the second.
➗ Dividing to find a unit rate
Just like with whole numbers, a unit rate answers: "how much of the first thing for 1 of the second thing?" Divide fraction 1 by fraction 2 — flip the second fraction and multiply.
unit rate = quantity 1 ÷ quantity 2
📖 Worked example
Jamal reads 4/5 of a book every 2/3 of a week, at a constant rate. How many books does he read per week?
Dividing the distance by the time gives the rate: (½) ÷ (¼) = 2 km per hour, and the line passes through the origin.
📝 Worked examples
1/2 km in 1/4 hour: (1/2) ÷ (1/4) = 1/2 × 4 = 2 km per hour
3/4 cup per 1/2 batch: (3/4) ÷ (1/2) = 3/2 = 1.5 cups per batch
A unit rate is the amount per one unit. Divide the first quantity by the second, even when both are fractions.
Divide by a fraction = multiply by its reciprocal.
✏️ Practice · 1 of 10
Francine uses 2/3 cup of pineapple juice for every 1/3 cup of orange juice to make a smoothie.
Enter the number of cups of pineapple juice Francine uses for 1 cup of orange juice.
Enter an exact answer, without rounding.
cups of pineapple juice
Divide the pineapple amount by the orange amount: (2/3) ÷ (1/3).
✏️ Practice · 2 of 10
Darcy harvests 8¾ acres of corn every 5/6 of an hour. Darcy harvests corn at a constant rate.
How many acres does she harvest per hour?
Enter an exact answer, without rounding.
acres per hour
Write 8¾ as 35/4, then divide by 5/6 (multiply by 6/5).
✏️ Practice · 3 of 10
A recipe uses 3/4 cup of sugar for every 1/2 cup of butter.
Enter the number of cups of sugar used for 1 cup of butter.
Enter an exact answer, without rounding.
cups of sugar
Divide the sugar by the butter: (3/4) ÷ (1/2).
✏️ Practice · 4 of 10
A hose fills 2/5 of a pool every 1/4 of an hour, at a constant rate.
Enter the fraction of the pool filled per hour.
Enter an exact answer, without rounding.
pools per hour
Divide 2/5 by 1/4: multiply by 4.
✏️ Practice · 5 of 10
Mateo reads 3½ pages every 1/3 of a minute, at a constant rate.
Enter the number of pages he reads per minute.
Enter an exact answer, without rounding.
pages per minute
Write 3½ as 7/2, then divide by 1/3 (multiply by 3).
✏️ Practice · 6 of 10
A printer prints 5/8 of a page every 1/6 of a minute, at a constant rate.
Enter the number of pages it prints per minute.
Enter an exact answer, without rounding.
pages per minute
Divide 5/8 by 1/6: multiply by 6.
✏️ Quick check · 7 of 10
Unit rate meaning
A unit rate tells you:
The "unit" means 1.
✏️ Quick check · 8 of 10
Pick the rate
2 pages are read in 1/4 minute. Which gives pages per minute?
Divide pages by minutes.
✍️ Type your answer · 9 of 10
A cyclist rides 2/3 of a km in 1/6 of an hour.
What is the speed in km per hour?
Enter an exact answer, without rounding.
km/h
Flip the second fraction and multiply.
✍️ Type your answer · 10 of 10
A printer prints 3/4 of a page in 1/2 minute.
How many pages per minute?
Enter an exact answer, without rounding.
pages per minute
Flip the second fraction and multiply.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
A train travels 3 km in 3/4 hour. What is its speed?
Question 2 of 5
A car uses 1/4 litre in 1/2 km. How many litres per km?
Question 3 of 5
$6 buys 3/4 kg of cheese. What is the price per kg?
Question 4 of 5
A worker paints 2 walls in 1/4 day. How many walls per day?
Question 5 of 5
Which unit rate is the fastest?
Lesson 16 · Relationships
📈 Proportional Relationships
Two quantities are proportional when their ratio always stays the same. That constant ratio is called the constant of proportionality, and it lets us write a simple equation connecting them.
⚖️ What makes a relationship proportional?
If y is proportional to x, dividing y by x always gives the same number, called k, the constant of proportionality.
y = kx
x = 2, y = 6 → k = 6 ÷ 2 = 3
x = 4, y = 12 → k = 12 ÷ 4 = 3
x = 6, y = 18 → k = 18 ÷ 6 = 3
Since k stays the same (3) for every pair, this relationship is proportional: y = 3x.
🔍 Spotting a non-proportional relationship
x = 1, y = 5 → k = 5
x = 2, y = 11 → k = 5.5
x = 3, y = 18 → k = 6
The ratio y/x is different every time, so this relationship is not proportional — it doesn't fit the form y = kx.
A quick test: a proportional relationship's graph is always a straight line that passes through (0, 0).
🧮 Using the equation
Once you know k, you can find y for any x (or x for any y) by substituting into y = kx.
y = 4x
When x = 9:
y = 4 × 9 = 36
🖼️ See it
A proportional relationship is a straight line through the origin; the gradient (2) is the constant of proportionality.
📝 Worked examples
A proportional relationship has the form y = kx.
k = y ÷ x is the constant of proportionality.
(2, 6) is on the graph → k = 6 ÷ 2 = 3 → y = 3x
y = 2.5x at x = 8 → y = 20
Test for proportional: the graph is a straight line through (0, 0), and every y ÷ x gives the same k.
✏️ Quick check · 1 of 10
Spot the proportional equation
Which equation represents a proportional relationship?
A proportional equation has the form y = kx, with nothing added.
✍️ Type your answer · 2 of 10
A table shows: when x = 2, y = 6; when x = 4, y = 12; when x = 6, y = 18.
Enter the constant of proportionality (k).
Enter an exact answer, without rounding.
Divide y by x for any row.
✏️ Quick check · 3 of 10
Identify k from an equation
In the equation y = 7x, what is the constant of proportionality?
In y = kx, k is the number multiplying x.
✍️ Type your answer · 4 of 10
A proportional relationship is described by y = 4x.
What is y when x = 9?
Enter an exact answer, without rounding.
Multiply 4 by 9.
✏️ Quick check · 5 of 10
Compare two tables
Table A: x = 1, 2, 3 → y = 5, 10, 15. Table B: x = 1, 2, 3 → y = 5, 11, 18. Which table is proportional?
Is y ÷ x the same in every row?
✍️ Type your answer · 6 of 10
The cost of pencils is proportional to the number bought. 4 pencils cost $2.
Enter the constant of proportionality (the cost per pencil, in dollars).
Enter an exact answer, without rounding.
dollars per pencil
Divide the cost by the number of pencils.
✏️ Quick check · 7 of 10
Find k
A proportional relationship includes the point (2, 10). What is k?
Divide y by x.
✏️ Quick check · 8 of 10
Graph shape
The graph of a proportional relationship is:
Think about y = kx at x = 0.
✍️ Type your answer · 9 of 10
A relationship is y = 4x.
What is y when x = 7?
Enter an exact answer, without rounding.
Substitute x = 7.
✍️ Type your answer · 10 of 10
A table shows x = 3 when y = 12, in a proportional relationship.
What is y when x = 10?
Enter an exact answer, without rounding.
Find k first.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Which equation shows a proportional relationship?
Question 2 of 5
(4, 6) is on a proportional graph. What is k?
Question 3 of 5
Notebooks cost $2.50 each. Which equation gives the cost c of n notebooks?
Question 4 of 5
Which point must be on every proportional graph?
Question 5 of 5
y = 0.5x. What is y when x = 12?
Lesson 17 · Geometry
🗺️ Scale Copies & Drawings
A scale copy is a resized version of a figure that keeps the same shape. Every length changes by the same scale factor, and every angle stays exactly the same.
🔍 What is a scale factor?
The scale factor is the ratio of a length in the copy to the matching length in the original.
scale factor = copy length ÷ original length
Factor > 1
The copy is larger than the original.
Factor < 1
The copy is smaller than the original.
📏 Scaling every side the same way
To make a scale copy, multiply every side length by the scale factor.
Maps and blueprints use a scale like 1 cm = 5 km to represent real-world distances in a small drawing.
Scale: 1 cm = 5 km
Map distance: 8 cm
Real distance = 8 × 5 = 40 km
🖼️ See it
Lengths are multiplied by the scale factor k; the area is multiplied by k². Doubling the sides makes the area four times as big.
📝 Worked examples
Scale 1 cm : 5 m, drawing length 4 cm → 4 × 5 = 20 m
Scale factor 3: every length is ×3, every area is ×9
(4 → 12 is a scale factor of 12 ÷ 4 = 3)
Area scales by the square of the scale factor. Doubling the sides makes the area 4 times larger.
✍️ Type your answer · 1 of 10
A rectangle has sides 4 and 6. Its scale copy uses a scale factor of 2.5.
What is the length of the side corresponding to 4 in the scale copy?
Enter an exact answer, without rounding.
Multiply the original side by the scale factor.
✏️ Quick check · 2 of 10
Scale a triangle
A triangle with sides 3, 4, 5 is scaled by a factor of 1/2. What are the new side lengths?
Halve every side.
✍️ Type your answer · 3 of 10
An original figure has a side length of 12. Its scale copy has the corresponding side length of 3.
What is the scale factor (copy ÷ original)?
Enter an exact answer, without rounding.
Divide the copy length by the original length.
✏️ Quick check · 4 of 10
Effect of a large scale factor
A scale factor greater than 1 makes a copy:
Multiplying by more than 1 makes things bigger.
✍️ Type your answer · 5 of 10
On a map, the scale is 1 cm = 5 km. Two cities are 8 cm apart on the map.
What is the actual distance between the cities, in km?
Enter an exact answer, without rounding.
km
Each centimetre is 5 km: multiply 8 by 5.
✍️ Type your answer · 6 of 10
A model car is built at a scale of 1:24. The real car is 4.8 meters long.
How long is the model, in meters?
Enter an exact answer, without rounding.
meters
Divide the real length by 24.
✏️ Quick check · 7 of 10
Scale factor
A side of 4 is scaled to 12. What is the scale factor?
New divided by original.
✏️ Quick check · 8 of 10
Area
If a figure is scaled by a factor of 3, its area is multiplied by:
Area uses two dimensions.
✍️ Type your answer · 9 of 10
A scale drawing uses 1 cm for 8 m.
How many metres is 3.5 cm?
Enter an exact answer, without rounding.
ft
Multiply by 8.
✍️ Type your answer · 10 of 10
A rectangle is scaled by a factor of 3.
By what factor does the area change?
Enter an exact answer, without rounding.
Square the scale factor.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
The scale is 1 : 100. A drawing length is 7 cm. What is the real length?
Question 2 of 5
A figure is scaled by 1/2. A side of 10 becomes:
Question 3 of 5
A scale factor of 2 makes the area:
Question 4 of 5
On a map 1 cm = 10 km. How far is 3.5 cm?
Question 5 of 5
In a scaled copy, corresponding angles are:
Lesson 18 · Data
🎯 Theoretical vs. Experimental Probability
Probability can be predicted using pure reasoning, or measured by actually running an experiment — and the two don't always perfectly agree.
🔢 Theoretical probability
Theoretical probability = number of favourable outcomes ÷ total possible outcomes, calculated using known facts (like knowing a fair coin has 2 equally likely sides) before running any trials.
🎲 Experimental probability
Experimental probability = number of times an event actually happened ÷ total number of trials, based on real data. As the number of trials increases, experimental probability tends to get closer to theoretical probability.
Tree Diagram · Flipping a Coin Twice
📝 Worked examples
Theoretical P(heads) = 1/2
100 flips, 46 heads → experimental P = 46/100 = 0.46
Roll a die 60 times: expect 60 × 1/6 = 10 sixes
More trials, closer results. The experimental probability usually moves closer to the theoretical probability as you repeat more times.
✏️ Quick check · 1 of 10
What is theoretical probability?
Theoretical probability is calculated using:
Theoretical means what should happen, worked out before any trials.
✍️ Type your answer · 2 of 10
A fair 6-sided die is rolled once.
What is the theoretical probability of rolling an even number? (as a decimal)
Enter an exact answer, without rounding.
Three of the six numbers are even.
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What is experimental probability?
Experimental probability is calculated using:
Experimental means based on what actually happened.
✍️ Type your answer · 4 of 10
A coin is flipped 50 times, landing on heads 28 times.
What is the experimental probability of heads? (as a decimal)
Enter an exact answer, without rounding.
Divide the number of heads by the number of flips.
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More trials, more accuracy
As the number of trials in an experiment increases, the experimental probability tends to:
The more you repeat an experiment, the more it evens out.
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A spinner has 4 equal sections numbered 1-4, spun 40 times, landing on 3 a total of 12 times.
What is the experimental probability of landing on 3? (as a decimal)
Enter an exact answer, without rounding.
Divide the number of 3s by the number of spins.
✏️ Quick check · 7 of 10
More trials
What happens to experimental probability as the number of trials grows?
Think about flipping a coin thousands of times.
✏️ Quick check · 8 of 10
Theoretical
A spinner has 4 equal sections. What is P(landing on section 1)?
Divide favourable by total.
✍️ Type your answer · 9 of 10
Out of 50 trials, an event happened 20 times.
What is the experimental probability (as a decimal)?
Enter an exact answer, without rounding.
Divide the successes by the trials.
✍️ Type your answer · 10 of 10
A fair coin is flipped 80 times.
How many heads are expected?
Enter an exact answer, without rounding.
Multiply trials by the probability.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
What is the theoretical P(even number) on a six-sided die?
Question 2 of 5
12 of 30 marbles drawn were red. What is the experimental P(red)?
Question 3 of 5
A spinner lands on 1 with probability 1/4. About how many times in 100 spins?
Question 4 of 5
Which statement is true?
Question 5 of 5
What is the theoretical P(rolling 5 or 6)?
Lesson 19 · Data
🎰 Compound Probability
A compound event combines two or more simple events, like flipping a coin and rolling a die. Counting every possible outcome — the sample space — is the key to finding its probability.
🌳 Sample spaces
The sample space is the list of every possible outcome of a compound event.
Flip a coin, then roll a die.
Coin outcomes: Heads, Tails
Die outcomes: 1, 2, 3, 4, 5, 6
Sample space: (H,1), (H,2), (H,3), (H,4), (H,5), (H,6),
(T,1), (T,2), (T,3), (T,4), (T,5), (T,6)
✖️ The counting principle
Instead of listing every outcome, just multiply the number of choices for each event.
total outcomes = choices₁ × choices₂
Coin (2 outcomes) × Die (6 outcomes)
= 2 × 6 = 12 total outcomes
🎯 Probability of independent events
When two events don't affect each other (independent events), multiply their individual probabilities.
Four equally likely outcomes, so P(two heads) = 1/4, which is ½ × ½. For independent events, multiply the probabilities.
📝 Worked examples
Two coins, both heads: 1/2 × 1/2 = 1/4
Coin and die: P(heads and 6) = 1/2 × 1/6 = 1/12
Sample space size: 2 × 6 = 12 outcomes
3 coin flips: 2 × 2 × 2 = 8 outcomes
Independent events: multiply the probabilities. The size of the sample space is the product of the choices at each step.
✍️ Type your answer · 1 of 10
A coin is flipped and a 6-sided die is rolled.
How many outcomes are in the sample space?
Enter an exact answer, without rounding.
outcomes
Multiply the outcomes for the coin by the outcomes for the die.
✏️ Quick check · 2 of 10
Apply the counting principle
A shop has 3 shirt colours and 4 trouser colours. How many outfits are possible?
Each shirt can go with every pair of pants: multiply.
✍️ Type your answer · 3 of 10
Two coins are flipped.
What is the probability that both land on heads?
Enter an exact answer, without rounding.
P(heads) × P(heads).
✏️ Quick check · 4 of 10
Count another sample space
A bag has a red and a blue marble. A spinner has 3 equal sections (A, B, C). How many outcomes are there when picking a marble colour and spinning?
Multiply the marble choices by the spinner choices.
✍️ Type your answer · 5 of 10
A die is rolled twice.
What is the probability of rolling an even number both times?
Enter an exact answer, without rounding.
P(even) on one roll is 1/2. Multiply for two rolls.
✏️ Quick check · 6 of 10
Define independent events
Which best describes independent events?
Independent means one result has no effect on the other.
✏️ Quick check · 7 of 10
Sample space
How many outcomes are there when you flip a coin and roll a die?
Multiply the outcomes of each.
✏️ Quick check · 8 of 10
Multiply
The events are independent. How do you find P(A and B)?
"And" usually means multiply.
✍️ Type your answer · 9 of 10
A coin is flipped 3 times.
How many outcomes are in the sample space?
Enter an exact answer, without rounding.
Multiply the two choices three times.
✍️ Type your answer · 10 of 10
Two fair coins are flipped. Enter your answer as a decimal.
What is the probability of two heads?
Enter an exact answer, without rounding.
Multiply 1/2 by 1/2.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
A spinner with 4 sections is spun and a coin is flipped. How many outcomes?
Question 2 of 5
P(heads and a 3) with a coin and a die is:
Question 3 of 5
Two dice are rolled. P(both even) is:
Question 4 of 5
Someone has 3 shirts and 4 pairs of trousers. How many outfits?
Question 5 of 5
What is P(not rolling a 6) on one die?
Lesson 20 · Data
📊 Comparing Populations & Samples
We can't always survey everyone, so we use a random sample to represent a whole population — then compare groups and make careful inferences from the data.
🎲 Random samples
A random sample gives every member of the population an equal chance of being chosen, which makes it likely to represent the whole population well.
Representative
Randomly picking names from the full list of students.
Biased
Only asking members of one club, or only volunteers.
📏 Comparing variability
When comparing two groups of data, a smaller range (or spread) means the data points are more tightly clustered together — less variability.
Group X scores: 78, 80, 79, 81 (tightly clustered)
Group Y scores: 60, 95, 70, 90 (spread out)
Group X has less variability than Group Y.
🔢 Making inferences from a sample
Use a sample's percentage to estimate a count for the entire population.
Sample: 50 students, 20 walk to school
Percent who walk = 20 ÷ 50 = 0.40 = 40%
Whole school: 600 students
Estimated walkers = 0.40 × 600 = 240
🖼️ See it
A random sample is more likely to represent the population; a bigger sample is usually more reliable.
📝 Worked examples
Team A mean 8, Team B mean 6, both with MAD 2.
Difference in means = 2, which is 2 ÷ 2 = 1 MAD.
Sample inference:
15 of 50 students prefer pizza → 15/50 = 30%
For 400 students, predict 0.30 × 400 = 120
Random samples represent a population better than convenience samples. A larger sample gives a more reliable estimate.
✏️ Quick check · 1 of 10
Choose a representative sample
Which sampling method is most likely to produce a representative sample of a school's students?
Every student should have the same chance of being picked.
✍️ Type your answer · 2 of 10
Class A's test scores have a mean of 82. Class B's test scores have a mean of 76.
What is the difference between the two means?
Enter an exact answer, without rounding.
points
Subtract the smaller mean from the bigger one.
✏️ Quick check · 3 of 10
Interpret the range
A smaller range in a data set means the data is:
A small range means the values are close together.
✍️ Type your answer · 4 of 10
In a random sample of 50 students, 20 said they walk to school.
Based on this sample, what percent of students walk to school?
Enter an exact answer, without rounding.
%
Divide 20 by 50, then multiply by 100.
✍️ Type your answer · 5 of 10
A school has 600 students. A random sample estimates that 25% of students walk to school.
About how many students in the whole school walk to school?
Enter an exact answer, without rounding.
students
Find 25% (a quarter) of 600.
✏️ Quick check · 6 of 10
Compare two dot plots
Group X's data values are clustered tightly together. Group Y's data values are spread far apart. Which group has less variability?
Tightly clustered values vary less.
✏️ Quick check · 7 of 10
Good sample
Which sample best represents all students in a school?
Avoid choosing only one type of student.
✏️ Quick check · 8 of 10
Skewed data
Which measure of centre is best for skewed data with outliers?
Outliers pull the mean.
✍️ Type your answer · 9 of 10
Two groups have means 15 and 9, and both have a MAD of 3.
The difference in the means is how many MADs?
Enter an exact answer, without rounding.
Subtract the means, then divide by the MAD.
✍️ Type your answer · 10 of 10
In a random sample of 50 students, 15 prefer pizza.
About how many of 400 students would be predicted to prefer pizza?
Enter an exact answer, without rounding.
Find the sample proportion first.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
A random sample is better than a convenience sample because it is:
Question 2 of 5
Which has the greater typical value if Group A median = 70 and Group B median = 64?
Question 3 of 5
A bigger sample usually gives:
Question 4 of 5
Overlapping box plots suggest that the two groups:
Question 5 of 5
Means 20 and 10 with a MAD of 5. The difference is:
Final Challenge
🏆 30-Question Challenge Quiz
One question from every topic you just learned. Pick an answer for each question, then hit Check My Answers.
0 / 30
💪 Keep practicing — go back through the lessons and retake the quiz!
Q1 · Integers
−6 − (−9) = ?
Q2 · Fractions
2/3 + 1/6 = ?
Q3 · Ratios
Simplify the ratio 12 : 16
Q4 · Percentages
45 is what percent of 60?
Q5 · Algebraic Expressions
Evaluate 4x − 2 when x = 5
Q6 · Equations
Solve: 5x − 4 = 16
Q7 · Angles
What angle is supplementary to 110°?
Q8 · Geometry
Find the area of a rectangle with length 8 and width 5.
Q9 · Statistics
Find the median of: 4, 8, 2, 10, 6
Q10 · Probability
Rolling a standard die, what is P(rolling a 4 or a 5)?
Q11 · Integers
Which integer is the greatest?
Q12 · Order of Operations & Negative Exponents
What is 100 − 3 × 4²?
Q13 · Fractions & Decimals
Which is larger, 5/8 or 0.6?
Q14 · Ratios & Proportions
A car travels 240 km in 4 hours. What is its unit rate?