The order of operations tells us the exact order to solve math — just like a recipe tells you which step comes first. We use the word PEMDAS to remember it.
Think of it like getting dressed: you put on SOCKS before SHOES, and SHIRT before JACKET. Math has the same "rule of dress."
①No parentheses, no exponents. Look for × or ÷ first.
②2 × 4 = 8
③Now add: 3 + 8 = 11
✓ Answer: 11
Problem 01Order of Operations
What is 5 + 3 × 2 ?
🎉 Correct!
Multiply first: 3 × 2 = 6, then add: 5 + 6 = 11. PEMDAS says M before A!
Problem 02Parentheses First
What is (4 + 2) × 3 ?
🎉 Correct!
Parentheses first: (4 + 2) = 6, then multiply: 6 × 3 = 18. The parentheses scream "DO ME FIRST!"
Unit 02Integers & Negative Numbers
Core Concept
Integers include all whole numbers AND their negative versions: … −3, −2, −1, 0, 1, 2, 3 … Adding a negative is the same as subtracting. Subtracting a negative is the same as adding.
Imagine a number line as a football field. Moving right = gaining yards (+). Moving left = losing yards (−). Two negatives = double-reverse, which means you go forward!
Key Rules
+ × + = + | − × − = + | + × − = −
Worked Example
Solve: −3 + (−5)
①Both are negative → add their absolute values: 3 + 5 = 8
②Keep the negative sign.
✓ Answer: −8
Problem 03Adding Integers
What is −7 + 3 ?
🎉 Correct!
Different signs → subtract: 7 − 3 = 4. Bigger number (7) is negative, so answer is −4. Like losing 7 yards but gaining 3: you're still 4 behind!
Problem 04Multiplying Integers
What is (−4) × (−3) ?
🎉 Correct!
Negative × Negative = Positive! 4 × 3 = 12, and two negatives cancel out → +12. "Two wrongs make a right" — in math!
Unit 03Fractions
Core Concept
A fraction is just a division problem written differently. 34 means "3 divided by 4" — or 3 pizza slices out of 4 total. To add fractions, you need the same denominator (bottom number). To multiply fractions, just multiply top × top and bottom × bottom.
Fractions are like comparing apples to apples. You can't add thirds + quarters until you convert them to the same "fruit" (common denominator)!
Fraction Rules
ab × cd = a×cb×d
|
ab + cb = a+cb
Worked Example
Solve: 23 + 13
①Same denominator (3) ✓ — just add the tops: 2 + 1 = 3
②Keep the bottom: 33 = 1
✓ Answer: 1
Problem 05Multiplying Fractions
What is 23 × 34 ?
🎉 Correct!
Top × Top = 2 × 3 = 6. Bottom × Bottom = 3 × 4 = 12. So 6/12 = 1/2 (divide both by 6).
Problem 06Adding Fractions (Different Denominators)
What is 14 + 12 ?
🎉 Correct!
Convert 1/2 to 2/4 (same denominator!). Then 1/4 + 2/4 = 3/4.
Unit 04Decimals & Percents
Core Concept
A percent always means "out of 100." The word "percent" literally means per cent (Latin for per hundred). To convert a percent to a decimal, just divide by 100 (move the decimal point 2 places left).
If your score is 75%, that means you got 75 out of every 100 questions right. If there were only 20 questions, scale it down proportionally!
Percent Formula
Percent = PartWhole × 100
Worked Example
What is 25% of 80?
①Convert 25% to decimal: 25 ÷ 100 = 0.25
②Multiply: 0.25 × 80 = 20
✓ Answer: 20
Problem 07Percent of a Number
What is 50% of 60?
🎉 Correct!
50% = 0.5. So 0.5 × 60 = 30. Also: 50% just means "half," and half of 60 is 30!
A variable is just a letter standing in for an unknown number — like a mystery box labeled x. An expression combines numbers, variables, and operations. Evaluating means swapping the letter for a number and calculating the answer.
Think of x as a placeholder on a menu: "Price = x + $2 tip." Once you know what you ordered (x = $10), plug it in: $10 + $2 = $12 total!
Evaluating Expressions
Substitute the value → Then calculate
Worked Example
If x = 5, what is 3x + 2?
①Replace x with 5: 3(5) + 2
②Multiply first: 15 + 2
③Add: = 17
✓ Answer: 17
Problem 09Evaluating Expressions
If y = 4, what is 2y − 3 ?
🎉 Correct!
Plug in 4: 2(4) − 3 = 8 − 3 = 5. Multiply before you subtract — PEMDAS!
Problem 10Combining Like Terms
Simplify: 3x + 2x + 5
🎉 Correct!
3x + 2x = 5x (like terms with x combine). The +5 stays alone. Result: 5x + 5.
Unit 06Solving One-Step Equations
Core Concept
An equation is like a balanced scale. Whatever you do to one side, you must do to the other side to keep it balanced. To solve for x, "undo" what was done to it using the opposite operation.
If x + 5 = 12, imagine there's a bag of apples (x) plus 5 loose apples = 12 total. Remove 5 from both sides: the bag has 7!
Golden Rule of Equations
Do the SAME thing to BOTH sides
Worked Example
Solve: x + 7 = 15
①Subtract 7 from both sides: x + 7 − 7 = 15 − 7
②x = 8
③Check: 8 + 7 = 15 ✓
✓ Answer: x = 8
Problem 11Solving Addition Equations
Solve for x: x + 9 = 20
🎉 Correct!
Subtract 9 from both sides: x = 20 − 9 = 11. Check: 11 + 9 = 20 ✓
Problem 12Solving Multiplication Equations
Solve for x: 4x = 28
🎉 Correct!
Divide both sides by 4: x = 28 ÷ 4 = 7. Check: 4 × 7 = 28 ✓
Unit 07Ratios & Proportions
Core Concept
A ratio compares two quantities. A proportion says two ratios are equal. Cross-multiplication is the fastest way to solve a proportion: multiply diagonally across the equals sign.
If a recipe needs 2 cups flour for 12 cookies, how much for 36 cookies? It's the same ratio — just scaled up! That's a proportion.
Cross Multiplication
If ab = cd , then a × d = b × c
Worked Example
Solve: 34 = x12
①Cross multiply: 3 × 12 = 4 × x
②36 = 4x
③Divide both sides by 4: x = 9
✓ Answer: x = 9
Problem 13Proportions
Solve: 25 = x15
🎉 Correct!
Cross multiply: 2 × 15 = 5 × x → 30 = 5x → x = 6. Or notice the bottom went from 5 to 15 (×3), so the top goes from 2 to 6 (×3)!
Problem 14Ratio Word Problem
In a class of 30 students, the ratio of girls to boys is 2 : 3. How many girls are there?
🎉 Correct!
Total parts = 2 + 3 = 5. Each part = 30 ÷ 5 = 6. Girls = 2 × 6 = 12.
Unit 08Geometry — Area & Perimeter
Core Concept
Perimeter = the total distance around the outside of a shape (like putting a fence around a yard). Area = how much space is inside a shape (like the amount of carpet you need for a room).
Perimeter = walking the entire border of a park. Area = how much grass is inside the park. Very different things!
Rectangle Formulas
Perimeter = 2(l + w) | Area = l × w
Worked Example
Rectangle: length = 8, width = 5. Find Area and Perimeter.
①Area = 8 × 5 = 40 square units
②Perimeter = 2(8 + 5) = 2(13) = 26 units
✓ Area = 40, Perimeter = 26
Problem 15Area of Rectangle
A rectangle has length = 9 and width = 6. What is the area?
🎉 Correct!
Area = l × w = 9 × 6 = 54 square units. Imagine 9 rows of 6 tiles — that's 54 tiles!
Problem 16Area of Triangle
A triangle has base = 10 and height = 6. What is the area? Hint: Area of triangle = 12 × base × height
🎉 Correct!
Area = ½ × 10 × 6 = ½ × 60 = 30. A triangle is exactly half a rectangle!
Unit 09Exponents & Powers
Core Concept
An exponent tells you how many times to multiply a number by itself. So 23 means 2 × 2 × 2 (not 2 × 3!). The little raised number is called the power or exponent.
If you fold a piece of paper in half once, you get 2 layers. Fold it again → 4 layers. Fold it 10 times → 210 = 1,024 layers! That's the power of exponents.
Exponent Definition
an = a × a × a … (n times)
Worked Example
Calculate 34
①34 means 3 × 3 × 3 × 3
②3 × 3 = 9, then 9 × 3 = 27, then 27 × 3 = 81
✓ Answer: 81
Problem 17Computing Exponents
What is 25 ?
🎉 Correct!
25 = 2×2×2×2×2 = 4→8→16→32. Not 2×5 (that's 10)!
Problem 18Exponents in Expressions
Evaluate: 32 + 42
🎉 Correct!
32 = 9 and 42 = 16. Then 9 + 16 = 25. (This is actually related to the Pythagorean theorem — cool!)
Unit 10Inequalities
Core Concept
An inequality is like an equation, but instead of "=" it uses <, >, ≤, or ≥. You solve it the same way as an equation — with one important exception: if you multiply or divide by a negative number, FLIP the inequality sign!
Think of < and > as a hungry alligator's mouth — it always opens toward the BIGGER number. So 3 < 7 means the alligator eats the 7!
Inequality Signs
< less than | > greater than | ≤ at most | ≥ at least
Worked Example
Solve: x + 3 > 8
①Subtract 3 from both sides: x > 8 − 3
②x > 5
③Any number bigger than 5 works (like 6, 7, 100 …)
✓ Answer: x > 5
Problem 19Solving Inequalities
Solve: 2x < 10. What is the solution?
🎉 Correct!
Divide both sides by 2 (positive, so no flip!): x < 10 ÷ 2 → x < 5. Any number below 5 makes this true!
Problem 20Inequality Word Problem
You have $20 and each snack costs $3. How many snacks can you buy at most? (Set up: 3n ≤ 20)
🎉 Worksheet Complete! 🎉
3n ≤ 20 → n ≤ 6.67. Since you can't buy a fraction of a snack, the max is 6 snacks (costs $18, leaving $2 change).