Mrs. Ambre s Math Notebook

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1 Mrs. Ambre s Math Notebook Almost everything you need to know for 7 th grade math Plus a little about 6 th grade math And a little about 8 th grade math 1

2 Table of Contents by Outcome Outcome Topic Page Pre-Requisite Skills Multiplication chart 5 Place Value 6 Number Lines 6 Divisibility Rules 6 Order of Operations 6 Rounding 6 Prime Factors 1 through Properties 8 Graphs and Diagrams 9 Key Words 10 How Do I Do it? Chart 11 Fractions to Decimals 12 Improper Fractions 12 Fraction, Decimals and Percents 12 Benchmark Percents 13 Exponents 14 Least Common Multiple 18 Greatest Common Factors 18 Improper Fractions 18 Equivalent Fractions 18 Prime Factor Tree 18 Reciprocals 18 Other Skills Scientific Notation 14 Pythagorean Theorem 16 Square Root 16 Cube Root 61 Rate of Change 25 Slope 25 Transformations 27 Reflections 27 Rotation 27 Translation 27 Outcome A Integers Adding Integers 15 Subtracting Integers 15 Multiplying Integers 16 Dividing Integers 16 2

3 Outcome Topic Page Absolute Value 16 Integers 16 Outcome B Decimals Place Value 17 Estimation 17 Adding Decimals 17 Subtracting Decimals 17 Outcome C Fractions Fractions 18 Fraction Ring 19 Adding Fractions 20 Subtracting Fractions 20 Multiplying Fractions 20 Dividing Fractions 20 Percents, Fractions and Decimals 21 Percent of Change 21 Discount 21 Percent 21 Outcome D Rates and Ratios Unit Rates 22 Rates 22 Ratios 22 Proportions 22 Scale 22 Scale Factor 22 Constant of Proportionality 23 Proportional Relationships 23 Customary Conversions 24 Outcome E Equations Graphing Function Tables 25 Function Tables 25 Equations 26 Greater Than 26 Less Than 26 Inequalities 26 One-Step Equations 26 Two-Step Equations 26 3

4 Outcome Topic Page Outcome F Geometry 1 Lines 27 Angles 27 Complimentary Angles 27 Supplementary Angles 27 Right Angles 27 Acute Angles 27 Obtuse Angles 27 Vertical Angles 27 3D Shapes 28 Nets 28 Polygons 29 Outcome G Geometry 2 Perimeter 3030 Area 30 Volume 30 Surface Area 30 Diameter 30 Radius 30 Circumference 30 Pi 30 Outcome H Statistics Mean 31 Median 31 Mode 31 Outlier 31 Range 31 Scale 31 Interval 31 Outcome I Probability Combinations 32 Permutations 33 Tree Diagrams 33 Probability 34 Dependent Events 34 Independent Events 34 Theoretical Probability 34 4

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6 Place Value and Number Lines Millions Hundred Thousands Ten Thousandths Thousands Hundreds Tens Ones.5 4 Tenths Hundredths Thousandths to order numbers: Line up the decimals then compare the number in each place value (if a number has no decimal it is at the end) If the numbers are fractions, change them to decimals! (see fraction notes) To Round: Ex: Round to the nearest tenth Find the place value you re supposed to round to and underline the digit Copy all the numbers to the left of the underlined digit (those don t change) 568. Look to the right of the underlined digit. If it s 5 or higher, add one to the underlined digit, if not, keep 4 < 5 so it stays the underlined digit the same. the same Write zeroes for all the remaining digits to the right of the underlined digit Your rounded number should have the same number of digits that you started with! Divisibility Rules A number is divisible by: 10 if it ends in 0 5 if it ends in 0 or 5 2 if it ends in 0,2,4,6,8 (even numbers) 3 if all the digits are added and their total is a multiple of 3 (3,6,9,12,15,18 ) Prime Numbers are not divisible by anything (except 1 & itself) Ex:2, 3, 5, 7, 11, 13, 17, 19 Composite Numbers are composed of other numbers and have many Factors Order of Operations Please Excuse My Dear Aunt Sally P Parenthesis ( ) E Exponents 3 M/D Multiply or Divide Left to Right A/S Subtract or Add Left to Right 6

7 Prime Factors: Numbers 1 through : 2 3: 3 4: 2x2 5: 5 6: 2x3 7: 7 8: 2x2x2 9: 3x3 10: 2x5 11: 11 12: 2x2x3 13: 13 14: 2x7 15: 3x5 16: 2x2x2x2 17: 17 18: 2x3x3 19: 19 20: 2x2x5 21: 3x7 22: 2x11 23: 23 24: 2x2x2x3 25: 5x5 26: 2x13 27: 3x3x3 28: 2x2x7 29: 29 30: 2x3x5 31: 31 32: 2x2x2x2x2 33: 3x11 34: 2x17 35: 5x7 36: 2x2x3x3 37: 37 38: 2x19 39: 3x13 40: 2x2x2x5 41: 41 42: 2x3x7 43: 43 44: 2x2x11 45: 3x3x5 46: 2x23 47: 47 48: 2x2x2x2x3 49: 7x7 50: 2x5x5 51: 3x17 52: 2x2x13 53: 53 54: 2x3x3x3 55: 5x11 56: 2x2x2x7 57: 3x19 58: 2x29 59: 59 60: 2x2x3x5 61: 61 62: 2x31 63: 3x3x7 64: 2x2x2x2x2x2 65: 5x13 66: 2x3x11 67: 67 68: 2x2x17 69: 3x23 70: 2x5x7 71: 71 72: 2x2x2x3x3 73: 73 74: 2x37 75: 3x5x5 76: 2x2x19 77: 7x11 78: 2x3x13 79: 79 80: 2x2x2x2x5 81: 3x3x3x3 82: 2x41 83: 83 84: 2x2x3x7 85: 5x17 86: 2x43 87: 3x29 88: 2x2x2x11 89: 89 90: 2x3x3x5 91: 7x13 92: 2x2x23 93: 3x31 94: 2x47 95: 5x19 96: 2x2x2x2x2x3 97: 97 98: 2x7x7 99: 3x3x11 100: 2x2x5x5 7

8 Properties & Identities *The signs are all the same the problem was just written with a different order* *The original signs are all multiply or all add. The parenthesis just moved* *The problem is totally rewritten. A number is written more than once on the other side* Zero Property Words Numbers Algebra The product of 0 and any number is = 0 a 0 = 0 *Multiply by zero* 8

9 Tables & Plots Frequency Table Tells the number of times something occurs Stem & Leaf Plot *Adding zero or multiplying by 1* Arranges data by place value. The stems represent values of 10. The leaves are ones. Don t forget Line Plot Uses a number line & X s or other symbols to show the frequencies of values. Graphs Bar Graph To display countable data grouped in categories. Bars can t touch Line Graph To show change over time Plot each dot then connect the dots Double Line Graph Use double lines or bars to compare similar data All graphs must include: *A title *A scale that starts at 0 (use the lightning bolt symbol to jump up) *A label for the y axis (left side) what do those numbers stand for? *A label for the x axis (bottom) & labels for each point or bar Pie Chart Histogram Venn Diagram To compare parts to a whole A complete circle is 100% A bar graph that shows frequency of occurance of each interval. The bars touch each other and the y axis. Used to compare two sets of data. A is one thing. B is another. C is what they have in common. 9

10 Word Problem Clues Expression Word Phrases Put parts together What is the sum? How many in all? How many / how much altogether? What is the total? If its repeated addition it would be better to multiply n + 5 Add 5 to a number Sum of a number and 5 5 more than a number The total of 5 and a number A number increased by 5 A number plus 5 How much more or less How many more? Less? Find the difference. Compare two numbers: How much bigger? How much taller? Heavier? Older? n-11 Subtract 11 from a number Difference of a number and less than a number A number decreased by 11 A number minus 11 Take away 11 a number less 11 Put equal parts together How many in all? What is 12% of 3? This much per how many / how much altogether? Find the product. Each one is total is? He did it 3 times total is? 3n 3 multiplied by a number Product of 3 and a number a number times 3 Separate into equal parts Find the quotient. What would one unit be? If shared? If divided? How much Each? Per? Find the average / mean (add numbers then divide by how many numbers were given) a 7 or a 7 7 divided by a number Quotient of a number and 7 7 divided into a number 10

11 Add Subtract Multiply Divide Integers Use the Butterfly Method Fractions OR Find Common Denominators Decimals How do I do it? 11

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13 Benchmark Percents 13

14 Exponents & Scientific Notation Exponents: Scientific Notation: Scientific notation using a calculator: Enter the given number. Press the 2 nd key then the DRG key. Use the arrows to move to SCI. Hit Enter twice. (To clear memory press the on and clear button at the same) Standard notation using a calculator: Enter the given number in scientific notation. 3.2 x10 ^7 then hit Enter. The number will appear in standard notation. Positive Exponent move to the Right Negative Exponent move to the Left The Ten Trick To multiply any number by ten or multiple of 10: multiply the front digits then add on the total number of zeroes to the end 2 x 10 Think 2x1=2 then add 1 zero = x 10 Think 3x1=3 then add 2 zeroes = x 50 Think 15x5=75 then add 1 zero =750 14

15 (-3) + 5 = 2 When you subtract integers: ADD THE OPPOSITE 15

16 Pythagorean Theorem: a 2 + b 2 = c 2 (Used to find missing sides in a Right Triangle) Example: 9 15? a 2 +b 2 = c 2 b b 2 = 15 2 The sides that create the square are 81+ b 2 = 225 always the legs a & b The hypotenuse or slanty side across b 2 = 144 b = 144 b= 12 Integers (the set of positive & negative numbers) Its best just to enter in your calculator & check twice! Multiplying / Dividing: If the signs are the same: Positive. If the signs are different: Negative Adding: Your answer will always be the sign of the bigger number (not looking at positive or negative just the number) = find the difference between the two numbers = find the difference between the two numbers = add the two numbers = add the two numbers Subtracting: Add the opposite of the second number a from the square is always c. c is always the largest number of the three sides change to change to change to = = = -15 c Squares & Square Roots: To find Squares enter the number then press the x 2 key. To find Square Roots Press the Blue 2 nd key then the x 2 key to get the sign then enter the number you want to find the square root of. Perfect Squares Cube Roots 1x1x1 = 1 2x2x2 = 8 3x3x3 = 27 In your calculator use the exponent key (^) to find the cube. To find the cube root enter 3 press the 2 nd key then the (^) key and the number you want to find the cube root of Absolute Value: the distance from 0 on the number line -5 the absolute value of -5 is 5 6 the absolute value of 6 is find the value of what s inside then take its absolute value of is 2 16

17 There are three kinds of Estimation: Front End, Rounding, and Compatible Numbers. Front End Estimation: use only the whole number to estimate. Rounding: Look at the tenths place to round. If it is a 0, 1, 2, 3, or 4 you should leave the number the same. If it is 5, 6, 7, 8, or 9 you should round up. Compatible Numbers: Make the problems easier by finding close numbers that are easy to add, subtract, multiply or divide. Adding and Subtracting Decimals 1. Line Up Your Decimals 2. Like The Buttons On A Shirt. 3. Even out the Place Values with Zeros 4. Bring the Decimal Straight Down 5. Don t forget when to carry (adding) 6. And when to borrow (subtracting)

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20 Adding and Subtracting Fractions Multiplying and Dividing Fractions 20

21 Percents as Fractions 45% 45 9 = or Write the percent over 100, Use A b /c button to simplify (If number on top is bigger temporarily flip the numbers to simplify then flip them back) Ratios (fractions) as Percents At the Pet Expo, 35 out of every 50 dogs were Yellow Labs. 35 out of Divide 35 by =.7 move the decimal 2 places to the right.7= 70% of the dogs were yellow labs Decimals as Percents 0.3 = 30% Move the decimal 2 spaces to the right and add percent symbol = 145% Find the Percent 18 is what percent of 120? is 18 % x Replace is with 18 of Replace of with 120. Cross multiply to solve Find the Part 120. What number is 70% of 300? is of % 100 x is 15% of 210 is 70% of 300. Sale Price So-Fro Fabrics is having a sale. All of their fabric is 25% off. Find the sale price of fabric originally priced \$10 a yard. Find the amount of the discount. Find 25% of \$10. of means multipy Change the percent to a decimal and multiply = 2.5 Subtract the amount of the discount from the original price. \$10 \$2.50 = \$7.50 The discounted price is \$7.50 a yard Percents as Decimals 55% = 0.55 Divide percent by 100 and remove percent symbol (or just move the decimal 2 places to the left) Percent of Change The Math Club had 20 members. Now it has 30 members. Find the percent of increase. Step 1 Subtract to find the amount of change = 10 Step 2 Write a ratio that compares the amount of change to the original number of members. Write the ratio as a percent. percent of change = 10 = 20 amount of change original amount The amount of change is 10 The original amount is 20. = 0.5 or 50% Divide then write as a Percent The percent of increase is 50%. 21

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24 Customary Conversions 24

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28 3 D shapes and Nets Cylinder Picture Net Sphere Cone Square / Rectangular Prism Triangular Prism Square Pyramid Triangular Pyramid 28

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35 Index Topic Page 3D Shapes 28 Absolute Value 16 Acute Angles 27 Adding Decimals 17 Adding Fractions 20 Adding Integers 15 Angles 27 Area 30 Associative Property 8 Bar Graph 9 Benchmark Percents 13 Capacity 24 Centimeter 24 Chord 29 Circle Graph 9 Circumference 30 Combinations 32 Commutative Property 8 Compatible Numbers 17 Complimentary Angles 27 Composite Number 6 Congruent 27 Constant of Proportionality 23 Coordinate Plane 25 Cross Multiply 22 Cube Root 16 Customary Conversions 24 Dependent Events 34 Diameter 29, 30 Difference 10 Discount 21 Distributive Property 8 Dividing Fractions 20 Dividing Integers 16 Divisibility Rules 6 Equations 26 Equilateral Triangle 29 Equivalent Fractions 18 Estimation 17 Exponents 14 Topic Page Expression 10 Feet 24 Fraction Ring 19 Fraction, Decimals and Percents 12 Fractions 18 Fractions to Decimals 12 Frequency Table 9 Front End Estimation 17 Function Tables 25 Graphing Function Tables 25 Graphs and Diagrams 9 Greater Than 26 Greatest Common Factors 18 Hexagon 29 Histogram 9 How Do I Do it? Chart 11 Identity Property 8 Improper Fractions 12, 18 Inches 24 Independent Events 34 Inequalities 26 Integers 16 Interval 31 Isosceles Triangle 29 Key Words 10 Least Common Multiple 18 Length 24 Less Than 26 Line Graph 9 Line Plot 9 Lines 27 Mass 24 Mean 31 Median 31 Meters 24 Metric Units 24 Miles 24 Millimeter 24 Mode 31 Multiplication Chart 5 35

36 Topic Page Multiplying Fractions 20 Multiplying Integers 16 Nets 28 Number Lines 6 Obtuse Angles 27 Obtuse Triangle 29 Octagon 29 One-Step Equations 26 Order of Operations 6 Ordered Pair 25 Ounces 24 Outlier 31 Parallel 27 Pentagon 29 Percent 21 Percent of Change 21 Percents, Fractions and Decimals 21 Perimeter 30 Permutations 33 Perpendicular 27 Pi 30 Pie Chart 9 Place Value 6, 17 Plane 27 Point 27 Polygons 29 Pounds 24 Prime Factor Tree 18 Prime Factors 1 through Prime Number 6 Probability 34 Product 10 Properties 8 Proportional Relationships 23 Proportions 22 Pythagorean Theorem Quadrilateral 29 Quotient 10 Radius 29, 30 Range 31 Index Topic Page Rate of Change 25 Rates 22 Ratios 22 Ray 27 Reciprocals 18 Rectangle 29 Reflections 27 Right Angles 27 Right Triangle 29 Rotation 27 Rounding 6, 17 Scale (Probability) 31 Scale 22 Scale Factor 22 Scalene Triangle 29 Scientific Notation 14 Segment 27 Slope 25 Square Root 16, 29 Stem and Leaf 9 Subtracting Decimals 17 Subtracting Fractions 20 Subtracting Integers 15 Sum 10 Supplementary Angles 27 Surface Area 30 Theoretical Probability 34 Ton 24 Transformations 27 Translation 27 Tree Diagrams 33 Triangle Sum Theorem 29 Two-Step Equations 26 Unit Rates 22 Venn Diagram 9 Vertical Angles 27 Volume 30 Weight 24 Yard 24 Y-intercept 25 Zero Property 8 36

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