Mental math
32² = (30 + 2)² = 900 + 120 + 4 = 1,024. You used one friendly square instead of long multiplication.
Today's big idea
What does (a + b)² mean?
The tiny ² means “multiply by itself.” So (a + b)² means (a + b) × (a + b). The brackets matter: first add a and b, then square the whole result.
Example: (5 + 3)² = 8² = 64.
(a + b)² = a² + 2ab + b²
Why is there a 2ab? Because the large square contains two identical a-by-b rectangles.
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Build the formula yourself
Change a and b. The large square always breaks into four pieces: a², ab, another ab, and b².
Interactive area model
Move the sliders and watch the pieces change while the identity remains true.
a²
ab
ab
b²
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Why is this useful?
Identities help you calculate quickly, simplify complicated expressions, design spaces, and later solve equations in science and engineering.
Design and area
If a square garden grows by b metres on both sides, this identity tells you exactly how much new area appears.
Future science
Algebra lets scientists express patterns in motion, electricity, chemistry, space, and almost every technology.
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What if the sign changes?
Patterns have close relatives. Compare these three identities and notice which middle sign changes.
(a+b)² = a²+2ab+b²Square of a sum
Add two matching ab rectangles.
(a−b)² = a²−2ab+b²Square of a difference
The middle part is subtracted twice.
(a+b)(a−b) = a²−b²Difference of squares
The +ab and −ab pieces cancel each other.
a(b+c) = ab+acDistributive law
a gets multiplied by every term inside the brackets.
What about negative numbers? The algebra still works. The area picture uses positive lengths, while signed numbers extend the same pattern beyond physical shapes.
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Starter formulas worth knowing
Understand these before memorizing them. Draw a picture, try small numbers, and explain each one in your own words.
Rectangle: A = l × wArea
How many unit squares cover the rectangle.
Triangle: A = ½bhTriangle area
A triangle is half of a matching rectangle.
Circle: C = 2πrCircumference
The distance around a circle; π is about 3.14.
Circle: A = πr²Circle area
How much flat space is inside a circle.
a² + b² = c²Pythagorean theorem
Connects the three sides of a right triangle.
average = total ÷ countMean average
Share the total equally among all the values.
speed = distance ÷ timeSpeed
How much distance is covered in each unit of time.
percent = part ÷ whole × 100Percentage
Shows a part as an amount out of 100.
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Ideas that make algebra easier
Algebra is arithmetic with names for numbers. These habits help you reason instead of guessing.
Equations are balances
In x + 4 = 11, both sides have equal value. Subtract 4 from both sides to keep the balance and discover x = 7.
A variable is a placeholder
A letter can represent an unknown number or a number that changes. In p = 4s, the perimeter p changes when side length s changes.
Like terms combine
Three apples plus two apples make five apples, so 3x + 2x = 5x. But 3x + 2 cannot combine because the terms are different kinds.
Order of operations
Brackets first, then powers, multiplication and division, then addition and subtraction. Operations at the same level go left to right.
Coordinates make pictures
A point (3, 2) means move 3 units across and 2 units up from the origin. Equations can create lines and curves on a graph.
Patterns suggest rules
The sequence 3, 6, 9, 12 has nth term 3n. Test a rule on several positions before deciding it always works.
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Fractions, ratios, and chance
These connected ideas appear in recipes, maps, sports, experiments, money, and data science.
a/b + c/b = (a+c)/bSame-denominator fractions
Add the counted parts while keeping the size of each part unchanged.
a:b = ka:kbEquivalent ratios
Multiplying both parts by the same non-zero number keeps the comparison equal.
P = wanted ÷ possibleSimple probability
When outcomes are equally likely, probability compares favourable outcomes with all outcomes.
range = maximum − minimumSpread of data
The range is one quick way to describe how far a data set stretches.
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Catch these common mistakes
Finding and explaining an error is one of the strongest ways to learn mathematics.
Is (a + b)² equal to a² + b²?
Usually no. Expanding gives a² + 2ab + b². For a = 2 and b = 3, the correct result is 25, not 13.
Does 2 + 3 × 4 equal 20?
No. Multiplication comes first, so 2 + 12 = 14. Brackets would be needed to make (2 + 3) × 4 = 20.
Can we divide both sides of an equation by zero?
No. Division by zero is undefined, so it cannot be used as a valid balancing step.
If x² = 25, is x only 5?
There are two solutions: 5 and −5, because both 5² and (−5)² equal 25.