Mrs. Ronda Garchow- 7th Grade Mathematics Math Boot Camp Survival Guide Entries
Procedures for the Survival Guide • Each day of Math Boot Camp, the student will write entries into his or her own Survival Guide Mini-Book. • WRITE SMALL SO THAT YOU CAN FILL UP THE PAGES WITH THE NECESSARY ENTRIES!!! • Whenever you see this book symbol, it means that you should write it down. • This Friday, the teacher will look at your Survival Guides and give you a grade. • The Survival Guides are NOT to be removed from the classroom until the teacher says so. • WRITE FAST…YOU WILL ONLY BE GIVEN 5 MINUTES PER SLIDE TO COPY THE INFORMATION.
Rational Numbers • Rational numbersare numbers that can be written as a fraction or a ratio where the numerator and the denominator can be any integer except that zero cannot be in the denominator. • Example:_2_ or _5_ 3 1
How to Add orSubtract Rational Numbers • You must find a common denominatorbefore adding or subtracting fractions. Example: _4_ + _5_ = _8_ + _5_ = _13_ 5 10 10 10 10 Simplify this answer in your survival guide to a mixed number instead of an improper fraction. • Now try it yourself… _7_ - _5_ = 4 6
How to Multiply Rational Numbers • To multiply rational numbers, you simply multiply the numerators straight across and then then denominators straight across. Then reduce in order to simplify. • Example: _4_ x _5_ = _20_ = _2_ 5 10 50 5 Solve and simplify your answer in your survival guide by changing it from an improper fraction to a mixed number. • Now try it yourself… _7_x _5_ = 4 6
How to Divide Rational Numbers • To divide rational numbers, you must first change the operation from division to multiplication. Next you must turn the second term into its reciprocal by flipping it upside down. Then multiply straight across the numerators and the denominators. • Example: _4_ ÷ _5_ = _4_ x _10_ = 40_= 1 15 = 1 3 5 10 5 5 25 25 5 Solve and simplify your answer in your survival guide by changing it from an improper fraction to a mixed number. • Now try it yourself… _7_÷ _5_ = 4 6
Integers • Integers are whole numbers that can be written on a number line including positive, negative, zero. • On a number line, when adding a positive integer, move to the right and when adding a negative integer move to the left. • negative # positive # -5 -4 -3 -2 -1 0 1 2 3 4 5 • Now try it…Plot the points -4.5 and 2.5 on the line.
How to Add IntegersWith LikeandUnlikeSigns For Like Signs: When adding two positive integers, the sign of the sum is always positive. 2 + 2 = 4 When adding two negative integers, the sign of the sum is always negative. -2 + -2 = -4 For Unlike Signs: • When adding a positive and a negative integer, you must first find the difference and the sign of the sum is the sign of the number that is the greatest distance from zero on the number line (or greatest absolute value). • 4 + -2 = (4 – 2 = 2) or -5 + 3 = ( 5 – 3 = -2) because we keep the sign of the number with the greatest absolute value or in this case -5.
How to Add IntegersWith LikeandUnlikeSigns Now try it yourself… For Like Signs: 15 + 20 = -31 + -8 = For Unlike Signs: 15 + -20 = -31 + 8 = -12 + 12 = NOTE: When we add a number’s opposite sign, the answer is always the same and we call that the additive inverse of a number. Inverse means “opposite” in math language.
How to SubtractIntegersWith LikeSigns For Like Signs: • Subtracting two negativeintegers is similar to adding two integers with opposite signs. • -5 – (-3) = -5 + 3 = -2 (Change! Change! Refers to changing the subtraction symbol into addition and changing the sign of the following integer that is being subtracted.) • Subtracting a positive integer from a positive integer is similar to adding two integers with opposite signs. • 9 – 5 = 4 or 9 + (-5) = 4 NOTE: Subtracting integers is the same as adding the opposite of the original number.
How to SubtractIntegersWith UnlikeSigns For Unlike Signs: • Subtracting a positive integer from a negative integer is similar to adding two negative integers. • -8 – 5 = -13 or -8 + -5 = -13 (After you Change! Change!) • Subtracting a negative integer from a positive integer is the same as adding two positive integers. • 7 – (-4) = 7 + 4 = 11 (After you Change! Change!) NOTE: Anytime you see a subtraction symbol with integers, always Change Change!!!
How to Multiply IntegersWith LikeandUnlikeSigns For Like Signs: • The product that results from multiplying two positive integers is always positive. 3 x 3 = 9 • The product that results from multiplying two negative integers is always positive. -3 x -3 = 9 For Unlike Signs: • The product that results from multiplying a negative integer and a positive is always negative. -3 x 3 = -9 • The product that results from multiplying an oddnumber of negative integers is always negative. -2 x -2 x -2 = -8 • The product that results from multiplying an even number of negative integers is always positive. -2 x -2 x -2 x -2 = 8
How to DivideIntegersWith LikeandUnlikeSigns For Like Signs: • The quotient that results from dividing two positive integers is always positive. 9 ÷ 3 = 3 • The quotient that results from dividing two negative integers is always positive. -9 x -3 = 3 For Unlike Signs: • The quotient that results from dividing a negative integer and a positive is always negative. -9 x 3 = -3 Now try it for yourself… 4 x -8 = -7 x -11 = 56 ÷ -8 = -25 ÷ -5=
Number Properties- Commutative Commutative Property works for both addition and multiplication. It allows you to move the numbers around and still achieve the same sum or product. Examples: 15 + 3 = 18 and 3 + 15 = 18 24 x 2 = 48 and 2 x 24 = 48 ½ x 30 = 15 and 30 x ½ = 15
Number Properties- Associative Associative Property works for both addition and multiplication. It allows you to move the parenthesis or “grouping” around and still achieve the same sum or product. Examples: (2 + 3) + 8 = 13 and 2 + (3 + 8) = 13 5 x (4 x 3) = 60 and (5 x 4) x 3 = 60 (¼ x 60) x 2 = 30 and ¼ x (60 x 2) = 30
Number Properties- Identity Identity Property works for both addition and multiplication. It allows you to add zero to any number or multiply any number by 1 and get the same number as a solution. Examples: 27 + 0 = 27 56 x 1 = 56
Number Properties- Distributive Distributive Property works for both multiplication for addition and subtractionanddivision foradditionandsubtraction. It allows you to expand an expression in order to solve it. Examples: 2(3 + 9) = (2 x 3) + (2 x 9) = 24 3(6 – 4) = (3 x 6) – (3 x 4) = 6 6 + 9 = 6 + 9 = 5 3 3 3 8 – 4 = 8 – 4 = 2 2 2 2
Number Properties- Additive Inverse Additive Inverse allows for you to add a number’s opposite or inverse and get a zero every time. This is used often when solving equations. Examples: -543 + 543 = 0 70 + -70 = 0 ½ + -½ = 0
Number Properties- Multiplicative Inverse Multiplicative Inverse allows for you to add the reciprocal of a number and always get a one for the solution. This property is also used in solving equations. Examples: 3 x 8 = 1 8 3 13 x 1 = 1 1 13 529 x 3 = 1 3 529
Combining Like Terms You can simplify expressions by combining all of the like terms. There are some terms that you can combine and some that you cannot. You can combine any two numbers together. You cannot combine a variable and a number, but you can combine two or more variables’ coefficients if the variables are the same letter. (A coefficient is the number being multiplied by the variable.) Examples: 3b + 4c + 5b = 8b + 4c 5 + 12p + 35 = 12p + 40 ½k + 20k = 10k
Simplify the Following Algebraic Expressions Now Try it Yourself −10v + 6v = 35n − 1 + 46 10x + 36 − 38x − 47 = You can also use the distributive property first and then combine like terms in order to simplify. Example: −2(7 − n) + 4 = (-2 x 7) – (-2 x n) = -14 – (-2n) = -14 + 2n
Math Language The following words are used commonly in math. Knowing the language can help you determine the meaning of word problems. Examples: • addition, sum, combine, join together, add to, plus, more than, total of • subtraction, difference, take away, minus, less than, fewer than, decreased by • multiply, times, of, product, double, triple • divide, part of, quotient, per, into, out of, ratio of, split, cut up, share equally, half, parts • equals, is, are, was, were, will be, yields, solution, equivalent, is the same as • inverse, opposite • reciprocal, reverse, flip • inequality, less than, greater than, is less than or equal to, is greater than or equal to
Equation Word Problem Practice Now try it for yourself: • Sara sold half of her comic books and then bought 9 more. She now has 17. How many did she begin with? • Oceanside Bike Rental Shop charges 18 dollars plus 7 dollars an hour for renting a bike. Tom paid 60 dollars to rent a bike. How many hours did he pay to have the bike checked out ? • The quotient of 35 and a number is 7. What is that number? • The difference between 20 and the sum of 2 combined with a number is 10. What is that number?
Algebraic Expressions and Equations Equations always have an = sign. If the equals sign is not there, then it is called an algebraic expression NOTan equation. Example of an algebraic expression:
Solving Equations • 1. Begin by performing the distributive property if you need to. • 2. Next combine any “like terms” you may have. • Example: • 2(2x + 3) = 5 + 4 (2 x 2x) + (2 x 3) = 5 + 4 4x + 6 = 9
REMEMBER THAT WHATEVER INVERSE OPERATION YOU DO, YOU MUST DO IT ALSO TO THE OTHER SIDE OF THE EQUATION (OR EQUAL SIGN). REMEMBER THAT WHATEVER INVERSE OPERATION YOU DO, YOU MUST DO IT ALSO TO THE OTHER SIDE OF THE EQUATION (OR EQUAL SIGN). REMEMBER THAT WHATEVER INVERSE OPERATION YOU DO, YOU MUST DO IT ALSO TO THE OTHER SIDE OF THE EQUATION (OR EQUAL SIGN). Solving Equationscontinued • 3. Begin using inverse operations with the term that DOES NOT have a variable next to it. • 4. Use inverse operations to “UNDO” whatever operation is being done in the original problem. Inverse Operations are OPPOSITE operations. • Example: Remember that whatever inverse operation you do, you MUST do it on the other side of the equals sign in order to balance the equation.
Inequalities andEquations Inequalities and equations are almost the same, but there are significant differences between them. Equations have an equals sign. Inequalities have a less than, greater than, less than or equal to, or greater than or equal to sign.
Solving Inequalities Inequalities and equations are solved almost entirely the same way. There is only ONE BIG difference. The ONE BIG difference is that when you have to divide both sides of an inequality by a negative number, you must reverse the inequality symbol in your answer.
Graphing Inequalities Graphing inequalities on a number line is easy if you understand that solutions to inequalities are more than one number AND if you know when to use a closed circle and when to use an open circle.
Congruent Figures Geometry Congruent figures in geometry have the same angle measurements and the same side lengths. They are identical in appearance. Example: ABC is congruent toLMN
Similar Figures Geometry Similar figures in geometry have the same angle measurements but their side lengths are not the same. However, their side lengths are proportional to one another. Example: ABC is similar toDEF
Proportional Sides in Similar Figures How can you tell if the side lengths are proportional? First, you set up a proportion using corresponding sides of each and then cross multiply. If the answers are the same, then they are proportional and also similar. Example: 7 = 3.5 Next Cross Multiply 8 4 7 x 4 = 8 x 3.5 28 = 28 The triangles are similar.
Scale Factor in Similar Figures Scale factor is the number you multiply by in order to get the measurements of the new figure.You can use the proportional lengths of similar figures to calculate the scale factor. Example: 3.5 x 2 = 7 Scale Factor = 2 4 x 2 = 8
Identifying PossibleTriangles Triangle measurements are either unique (or just one), not possible, or more than one triangle. If the measurements are angles that add up to 180 degrees, you can make more than one triangle out of the measurements that are similar. Example: These triangles are different sizes but have the same angle measurements.
Identifying PossibleTriangles (continued) Triangles that measure some side lengths are not possible with certain angle measurements. For instance, some side lengths can only create a right triangle. Example: The measurements 3, 4, and 5 can only create a right triangle with two acute angles.
Identifying PossibleTriangles (continued) Triangle Sum Theorem states that two sides of any triangle must add up to more than the 3rd side. If that is not true of a given set of measurements, then the triangle is not possible. Now You Try… Which of the following measurements is not possible for the 3rd side of a triangle with side lengths 5 inches and 7 inches long? A. 3 inches B. 6 inches C. 11 inches D. 12 inches
Angle Relationships Vertical Angles- pairs of opposite angles formed by intersecting lines. Vertical angles are congruent.
Angle Relationships(continued) Complementary Angles - two angles with measures that add up to 90°.
Angle Relationships(continued) Supplementary Angles - two angles with measures that add up to 180°.