Ratios, Rates & Percent 🎯
Compare quantities, build ratio tables, find unit rates, and work with percent as "per 100". Standards 6.RP.A.1–3
💡 What is a ratio?
A ratio compares two quantities. If a basket has 3 apples and 5 oranges, the ratio of apples to oranges is 3 : 5 (read as "three to five"). Order matters — oranges to apples would be 5 : 3.
You can write ratios three ways: with a colon 3 : 5, as a fraction 3/5, or with words: "3 to 5".
💡 Ratio tables show equivalent ratios
A ratio table lists pairs of numbers that all have the same ratio. To get the next row, you multiply both columns by the same number.
💡 A unit rate is "how much for ONE"
A rate compares two different kinds of quantities (like dollars and pounds). A unit rate tells you how many of the first quantity go with exactly 1 of the second.
To find a unit rate, divide: unit rate = total ÷ number of units.
💡 Percent means "out of 100"
The word percent literally means per hundred. So 45% means 45 out of 100, or the fraction 45/100, or the decimal 0.45.
To find a percent of a number, multiply by the decimal form: 30% of 80 = 0.30 × 80 = 24.
Fractions, Decimals & Number Theory 🔢
Divide fractions, work with decimals, and find GCF & LCM. Standards 6.NS.A.1, 6.NS.B.2–4
💡 To divide by a fraction, multiply by its reciprocal
Keep–Change–Flip: Keep the first fraction, change ÷ to ×, and flip the second fraction upside down.
Why does this work? Dividing by ½ asks "how many halves fit?" — the answer is 2× as many. That's exactly what multiplying by 2 (the reciprocal of ½) does.
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
💡 Place value is everything
For + and −: line up the decimal points. For ×: multiply as if there were no decimals, then count total decimal places in both factors and place the point that many spots from the right. For ÷: shift the decimal point in both numbers the same amount to make the divisor a whole number.
3.6 × 0.4 — multiply 36 × 4 = 144. Total decimal places = 1 + 1 = 2. Answer = 1.44.
💡 Two powerful number tools
The GCF (greatest common factor) of two numbers is the largest number that divides both exactly. The LCM (least common multiple) is the smallest number they both go into.
Use GCF to simplify fractions. Use LCM to find common denominators or to solve "when will two events line up" problems.
Multiples of 4 = 4, 8, 12, 16, 20… Multiples of 6 = 6, 12, 18… Smallest in both = LCM = 12.
Rational Numbers & Coordinate Plane 🌡️
Negative numbers, absolute value, and the four-quadrant coordinate plane. Standards 6.NS.C.5–8
💡 Numbers live on a line — in both directions
Negative numbers sit to the left of 0 on the number line. They represent things like debt, temperatures below freezing, or depth below sea level. The opposite of a number has the same distance from 0 but the other sign — the opposite of 7 is −7, and the opposite of −3 is 3.
Absolute value |x| is a number's distance from 0 — it is always 0 or positive. So |−8| = 8 and |5| = 5.
To compare two numbers, whichever is further right on the number line is greater. So −2 > −7 because −2 is closer to 0.
|−45| = 45 dollars — the absolute value gives the magnitude even though the number is negative.
💡 The coordinate plane has four quadrants
Each point is written as an ordered pair (x, y). The sign pattern tells you the quadrant:
Q1 (+, +) top-right · Q2 (−, +) top-left · Q3 (−, −) bottom-left · Q4 (+, −) bottom-right.
Reflecting a point across the x-axis flips the sign of y. Reflecting across the y-axis flips the sign of x. So (3, 5) reflected across the x-axis is (3, −5).
Expressions, Equations & Geometry 📐
Write & solve one-step equations, use exponents, and find area & volume. Standards 6.EE.A–B, 6.G.A.1–2
💡 Variables stand for unknown numbers
A variable is a letter like x or n that represents a number. An expression like 3x + 5 is a rule you can evaluate once you know the value of x. If x = 4, then 3(4) + 5 = 17.
Exponents are a shortcut for repeated multiplication: 5³ = 5 × 5 × 5 = 125. The small number is the exponent; the big number is the base.
Order of operations (PEMDAS): Parentheses → Exponents → Multiply/Divide (left to right) → Add/Subtract (left to right).
2 + 3 × 4² = 2 + 3 × 16 = 2 + 48 = 50 (exponent first, then multiply, then add).
💡 Do the opposite — to BOTH sides
An equation is like a balanced scale. To solve for the variable, apply the inverse operation to both sides so the equation stays balanced.
Inverses: + ↔ − and × ↔ ÷.
x + 9 = 17. Inverse of +9 is −9. Subtract 9 from both sides: x = 17 − 9 = 8.
💡 Key formulas you need
Triangle area: A = ½ × base × height. The height is perpendicular (at a right angle) to the base.
Rectangular prism volume: V = length × width × height. Use the same units throughout.
Mean (average): add all values, then divide by how many values there are. The mean summarizes a data set with a single "center" number.
A = ½ × 10 × 6 = 30 cm².