Two-dimensional figures are sorted into categories by their properties. Every square is a rectangle, a rhombus, a parallelogram and a quadrilateral, because it has all of their properties. Triangles are named by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). Tree and Venn diagrams show these groups.
The coordinate plane has two number lines that meet at right angles at the origin (0, 0). An ordered pair (x, y) names a point: start at the origin, move x units right along the x-axis, then y units up. In the first quadrant, both numbers are zero or greater. The order of the numbers matters.
Volume is the amount of space inside a solid figure, measured in cubic units. For a rectangular prism, multiply length times width times height: V = l × w × h. You can also multiply the area of the base by the height: V = B × h. For a solid made of two prisms, add the two volumes.
A line plot shows data as X marks above a number line. To solve problems, count the X’s at each fraction, add or subtract the fractions, and divide to share equally. If three X’s sit above 1/2, those measurements total 3 × 1/2 = 1 1/2. Equal sharing means total divided by the number of items.
To convert units, multiply or divide by the conversion factor. Going from a larger unit to a smaller unit, multiply: 3 ft = 3 × 12 = 36 in. Going from a smaller unit to a larger unit, divide: 36 in = 36 ÷ 12 = 3 ft. Metric conversions use powers of 10, so move the decimal point.
Dividing with unit fractions follows two patterns. A unit fraction divided by a whole number gets smaller: 1/3 ÷ 4 = 1/12, because each piece is cut again. A whole number divided by a unit fraction gets bigger: 4 ÷ 1/3 = 12, because each whole holds 3 pieces. Check every answer with multiplication.
Multiplication as scaling means that multiplying a number by a factor stretches or shrinks it. A factor greater than 1 makes the product larger than the other factor. A factor equal to 1 keeps it the same. A factor less than 1, such as 3/4, makes the product smaller.
To find the area of a rectangle with fractional sides, multiply the length by the width. You can also tile a unit square with small equal tiles and count how many tiles the rectangle covers. For example, a rectangle 3/4 foot by 2/3 foot covers 6 tiles of 1/12 square foot, so its area is 1/2 square foot.
To multiply fractions, multiply the numerators, multiply the denominators, and simplify. To multiply a fraction by a whole number, write the whole number over 1. For mixed numbers, first change them to improper fractions. An area model shows why: the overlap of the shaded parts is the product.
A fraction is a division problem: the numerator divided by the denominator. So 3/4 means 3 divided by 4, which is how much each person gets when 3 items are shared equally by 4 people. When the numerator is larger, the quotient is a mixed number: 7 divided by 3 is 7/3, or 2 1/3.
To add or subtract fractions with unlike denominators, find a common denominator, rename each fraction as an equivalent fraction with that denominator, then add or subtract the numerators. Simplify the result and write improper fractions as mixed numbers. Compare with benchmark fractions such as 0, 1/2, and 1 to check that the answer is reasonable.
To multiply decimals, multiply the numbers as whole numbers, then put the decimal point so the product has as many decimal places as both factors together. To divide by a decimal, multiply the dividend and divisor by 10, 100, or 1,000 until the divisor is whole, then divide as usual.
To add or subtract decimals, line up the decimal points so digits with the same place value are in the same column. Write zeros as placeholders if needed, add or subtract from the right, regroup when necessary, and bring the decimal point straight down. Estimate first to check that the answer makes sense.
To divide a whole number by a two-digit number, subtract easy multiples of the divisor from the dividend and add up how many you used. These are partial quotients. Estimate first with compatible numbers, keep going until what is left is smaller than the divisor, and write that leftover as the remainder.
To multiply multi-digit whole numbers, use the standard algorithm. Multiply the top number by each digit of the bottom number, writing a placeholder zero for each place after the ones. These results are partial products. Add the partial products to find the answer, and estimate first so you can check that the answer is reasonable.
Decimals are numbers with digits to the right of the decimal point: tenths, hundredths and thousandths. To read, write and compare them, use place value. Compare digits from left to right. To round, look at the digit just to the right of the rounding place: 5 or more rounds up, and 4 or less stays the same.
A power of 10 is 10 multiplied by itself, written with an exponent: 10 to the third power is 10 × 10 × 10 = 1,000. Each place value is 10 times the place to its right. Multiplying by a power of 10 moves the decimal point right, and dividing moves it left, by the exponent.
A numerical pattern is a list of numbers made by a rule, such as start at 0 and add 3. When two patterns follow different rules, the terms in the same position are called corresponding terms. Write them as ordered pairs, like (3, 6), and plot them on a coordinate plane to see how the patterns are related.
The order of operations is the set of rules that tells you which part of an expression to calculate first. Work inside parentheses, brackets and braces first, then multiply and divide from left to right, then add and subtract from left to right. Following the same order gives everyone the same answer.