Integers Integers are the set of positive whole numbers, their opposites, and zero. …-6, -5. -4. -3. -2. -1. 0, -. 1, 2, 3, 4, 5, 6… a gain of 5 pounds = 5 sea level = 0 a debt of $50 = -50 Opposites are two numbers that are the same distance from 0 on a number line but in opposite directions The opposite of 0 is 0. The opposite of -5 is 5; opposite of 3 is -3. The absolute value of a number is its distance from 0 on a number line. The symbol for absolute value is n . Opposite numbers have the same absolute value. Since -17 is 17 units from 0, -17 = 17. Since 76 is 76 units from 0, 76 = 76. Adding Integers Same signs: The sum of two positive integers is positive. The sum of two negative integers is negative. 3 + 7 = 10 -3 + (-7) = -10 Different signs: Find the absolute value of each integer. Subtract the lesser absolute value from the greater. The sum has the sign of the integer with the greater absolute value. 4 + (-9) 4 = 4, -9 = 9 Find the absolute values. 9 – 4 = 5 Subtract the absolute values. 4 + (-9) = -5 The sum is negative. -6 + 7 -6 = 6, 7 = 7 Find the absolute values. 7 – 6 = 1 Subtract the absolute values. -6 + 7 = 1 The sum is positive. Subtract Integers (1104) Use a number line to find 4 – (-4). Start at 0. Move forward 4 units for 4. For subtraction, turn around. Move backwards 4 units for -4. Stop at 8. 4 – (-4) = 8 You subtract an integer by adding its opposite. (KO+) Keep the first integer and add the opposite of the second integer. -8 – (-10)
*Multiply Fractions Multiply the numerators; multiply the denominators. Simplify. Multiply Mixed Numbers Write the mixed numbers as improper fractions. Multiply. Divide Fractions KFC – Keep the first fraction; flip the second to write its reciprocal; change the sign to multiplication. Divide Mixed Numbers Write the mixed numbers as improper fractions. Divide. Classifying Rational Numbers Absolute value of Rational Numbers Ratios A ratio is a comparison of two number by division. Ratios can be written 3 ways. *Proportions A proportion is an equation that states that Rate A rate is a ratio involving two quantities in different units. A unit rate is the rate for one unit of a given quantity.
Independent and Dependent Classifying Triangles A triangle is a closed figure made of three line segments that meet only at their endpoints. The sum of the measures is 180. Triangles can be classified by their angles and by their congruent sides. Three lengths form a triangle Learn the Triangle Inequality Theorem. This theorem simply states that the sum of two sides of a triangle must be greater than the third side. If this is true for all three combinations, then you will have a valid triangle. You'll have to go through these combinations one by one to make sure that the triangle is possible. You can also think of the triangle as having the side lengths a, b, and c and the theorem being an inequality, which states: a+b > c, a+c > b, and b+c > a. For this example, a = 7, b = 10, and c = 5. a + b > c = 17 > 5 a + c > b = 12 > 10 b + c > a = 15 > 7 7, 10, 5 make a triangle Triangles Relationships with sides and angles In a : The shortest side is always opposite the smallest The longest side is always opposite the largest interior angle Classifying Quadrilaterals A polygon with four sides is called a quadrilateral. Some quadrilaterals have special names.