Hardin Middle School Math Cheat Sheets, Study notes of Algebra

Download Hardin Middle School Math Cheat Sheets and more Study notes Algebra in PDF only on Docsity! Name: __________________________________________ 7th Grade Math Teacher: ______________________________ 8th Grade Math Teacher: ______________________________ Hardin Middle School Math Cheat Sheets You will be given only one of these books. If you lose the book, it will cost $5 to replace it. Compiled by Shirk &Harrigan - Updated May 2013 Alphabetized Topics Pages Pages  Area 32  Place Value 9, 10  Circumference 32  Properties 12  Comparing 23, 26  Proportions 22, 23, 24, 25, 26  Congruent Figures 35  Pythagorean  Theorem 36  Converting 14, 15, 23  R.A.C.E. 41  Divisibility Rules 8  Range 11  Equations 37, 38, 39  Rates 25  Flow Charts  Ratios 25, 26  Formulas 32, 33, 34  Rounding 10  Fractions 18, 19, 20, 21, 22, 23, 24  Scale Factor 35  Geometric Figures 30, 31  Similar Figures 35  Greatest Common Factor (GF or GCD) 22  Slide Method 22  Inequalities 40  Substitution 29  Integers 18, 19  Surface Area 33  Ladder Method 22  Symbols 5  Least Common Multiple (LCM or LCD) 22  Triangles 30, 36  Mean 11  Variables 29  Median 11  Vocabulary Words 43  Mode 11  Volume 34  Multiplication Table 6  Word Problems 41, 42  Order of Operations 16, 17  Percent 23, 24, 25, 26  Perimeter 32 5 Mathematic Symbols Cheat Sheet + Plus or Positive Line AS – Minus or Negative Line segment AS • * x Multiplied by Ray AS ÷ / Divided by Triangle ABC = Equal to Angle ABC ≠ NOT equal Angle B  Approximately equal to Right angle  Congruent to Perpendicular to < Is less than Perpendicular to > Is Greater than ˚ % Degree Percent ≥ Is greater than or equal to Σ Sum Square root of x ≤ Is less than or equal to π Pi (3.14.159) a/b a:b Ratio of a to be or a divided by b or the fraction a/b ! x n Factorial N th power of x (a, b) Ordered pair Infinity Multiplication Table - 30x30 11) 12|13| 14) 15|16|17| 18/19] 20| 21| 22/23] 24| 25|26|27| 28 29] 30 22 | 24 26 | 28 | 30 | 32 | 34 36 | 38 40 | 42 | 44 | 46 48 | 50 52 | 54) 56 | 5é| 60 33 | 36/39) 42) 45 | 48) Si 54/57 60/63) 66/69 72| 75 78 | 81) 84) 87) 90 44/48 | 52| 56 | 60 | 64] 68 | 72 | 76 | 80 | 84) 88 | 92 | 96 {100 104)/108)112|116)120) 55/60/65) 70 | 75 | 80/ 85 | 90 35/100 105/110 115) 120) 125)130)/135/140/145/150 66 | 72 | 78 | 84 | 90 | 96 |102/108)114)120) 126) 132/138) 144) 150/156/162/168 174/180 77 | $4) 91) 98 |105)112/119 126/133 140) 147 154)161/168)175/182)189)196/203)210) 88 | 96 |104/112)120/128/136 144/152 160) 168)176/184) 192) 200 208/216) 224/232/240 99 |108/117)126)135)144/153 162/171 180)189) 198/207 216) 225 234/243)/252|261)270) 110) 120/130) 140) 150/160/170 180/190) 200/210) 220/230 240/250 260/270) 280 290/300 | | 124 132/143/154 165/176/187 s23i208 220 231) 242 253/264| 275/286 /297| 308 319/330 12 | 24 | 36| 48| 60 72 | 84| 96 |108/120/132 156/168] 180)/192/204/216) 228/240) 252 | 264/276/288| 300/312 /324) 336) 348/360 13 | 26 | 39) 52) 65 | 76 | 91 |104)117/130/143/156 169 182)195)208/221 234)247' 260/273) 286/299 312/325 338)351)364/377/390) 14} 28 | 42 | 56) 70 | 84 | 98 |112)126/140] 154] 168/182 196 210)224)/238 252/266) 280/294 308/322 336/350 364/378)392|406)420) 15 | 30 | 45 | 60) 75 | 90 105|120/135/150/165|180|195 210| 225/240 255 270|285) 300/315 330/345 360/375 390)405)420/435/450) 16 | 32 | 48 | 64) 80 | 96 |112 128/144 160/176) 192)208/224/240 256) 272 288/304 320/336) 352/368 384/400 416/432|448 464/480) 17 | 34 | 51) 68 85 |102 119 136/153 170/187) 204)221)238/255|272 289/306 323/340 357/374/391/408|425|442/459/476/493|510 18 | 36 | $4| 72| 90 |108|126|144 162/180/198| 216/234/252|270 288/306 342/360/378|396|414/432) 450/468/486|504 522/540 19] 38 | 57] 76 | 95 |114/133]152)/171)190|209|228/247/266|285/304/323/342 361/380 399)418/437/456 475 /494/513/532/551/570 20 | 40 | 60 | 80 |100/120/140|160)180|200/220| 240/260/280/300/320/340 360|380| 490) 420)440/460)480 500/520/540/560/580/600 L| 21 | 42 | 63 | 84 105/126 147|168/189/210]231|252|273|294|315/336|357 378/399 420 441 462 483 504/525 546/567| 588 609/630 44) 66) 838 110/132 154|176|198/ 220/242 264|286)/308|330|352|374| 396|418) 440 462 apa 506 528/550 572|594/616/638/660) 23 | 46 69 | 92 |115/138/161)184/207/230/253 276/299/322| 345 368/391/414 437/460) 483/506) 529/552 $75|598/621/644 667/690 50 72 | 96 |120/144]168]192/216/240/264|286/312|336|360)/384/408/432|456/480| 504 526/552 576 600 624/648/672 696/720 75 |100 125/150 175/200) 225|250|275|300/325|350/375 400/425 450/475) 500/525)550|575|600 625|650/675| 700 725/750 26 | 52 | 78/104) 130/156 182|208)234/260/286|312|338|364|390/416|442|468|494|520|546 572|598|624| 650|676|702|728 754/780 27| 54| 81/108 135/162 189|216|243|270|297|324/351/378|405/432|459|486|513|540|567/594|/621|648|675|702 [729/756 783/810 28 | 56 | 84 |112]140|168|196|224/252/280|308|336|364/392|420/448|476 [$04 532 560/588 616/644) 672| 700) 728/756 812/840 a} 29 | 5B) 87/116 145/174 203 232/261 290/319) 348) 377) 406/435 |464/493|522 551|580/609/638|667/696|725/754/783|812 30 | 60 | 90 |120)1S0/180|210/240)270/300/330| 360|390|420|450/480/510/540 570|/600/630|660|690/720| 7S0/780|810|840 870|900) 7 Types of Numbers – Cheat Sheet Prime Number – A number that has exactly two (2) factors  Zero (0) and One (1) are neither prime nor composite because they only have one factor (itself) Composite Number – A number that has three (3) or more factors Even  Numbers ending in 0, 2, 4, 6, 8 Odd  Numbers ending in 1, 3, 5, 7, or 9 10 Place Value & Rounding Comparing & Ordering Decimals Rounding Rules Example Example 1. Underline the determined value 42.3 576.8 2. Draw an arrow to number to the right of underlined number 42.3 576.8 3. 0 – 4 = Round Down (Keep the underline number the same) a. All numbers to the left of underlined number stay the same b. Underlined number stays the same c. All numbers to the right of underlined number go to zero 4. 5 – 9 = Round Up (Underline number goes up 1) a. All numbers to the left of the underline number stay the same b. Underline number goes up 1 c. All numbers to the right of underlined number go to zero Round Down 42.3 ≈ 42.0 Round Up 576.8 ≈ 580.0 Comparing Decimal Rules 1. Line up the decimals using their decimal point ** If you do not see a decimal point, it is at the end of the number Example = 423 = 423.0 2. Fill in zeros so that all numbers have the same place value 3. Compare each number in their “lanes” (from left to right) 4. Determine greatest to least or least to greatest Billions Millions Thousand Ones . Decimals H u n d red B illio n T en -B illio n s B illio n s H u n d red -M illio n s T en -M illio n s M illio n s H u n d red - T h o u sa n d T en -T h o u sa n d s T h o u sa n d s H u n d red s T en s O n es . T en th s H u n d red th s T h o u san d th s T en -T h o u san d th s H u n d red - T h o u san d th s M illio n th s . . 11 Measures of Central Tendency: The Mean, Median, Mode, and Range When finding the measures of central tendency the first step is to place the numbers in order from least to greatest. Mean (Average): Add up a list of values in a set of data and divide by the number of values you have. 6, 4, 4, 3, 8 Step 1 Put in order from least to greatest 3, 4, 4, 6, 8 Step 2 Add up all the numbers 3 + 4 + 4 + 6 + 8 = 25 Step 3 Divide by the number of values you have 25 ÷ 5 = 5 Answer The mean is 5 Median (Middle): The middle value in a set of data when the values are written in order. If there are 2 values in the middle, find the mean of the two. 6, 4, 4, 3, 8 Step 1 Put in order from least to greatest 3, 4, 4, 6, 8 Step 2 Find the middle number **If there are an odd number of data values 3, 4, 4, 6, 8 Answer The median is 4 6, 4, 4, 3, 8, 5 Step 1 Put in order from least to greatest 3, 4, 4, 5, 6, 8 Step 2 Find the middle number **If there are an even number of data values then there will be two middle numbers 3, 4, 4, 5, 6, 8 Step 3 Find the mean of the two middle numbers 4 + 5 = 9 9 ÷ 2 = 4.5 Answer Median = 4.5 Mode (MOST): The value in a set of data that is repeated most often. A set of data could have no mode, one mode, or more than one mode. 6, 4, 4, 3, 8 Step 1 Put in order from least to greatest 3, 4, 4, 6, 8 Step 2 Find the number that occurs most often 3, 4, 4, 6, 8 Answer The mode is 4 Range: The largest number minus the smallest number 6, 4, 4, 3, 8 Step 1 Put in order from least to greatest 3, 4, 4, 6, 8 Step 2 Subtract the largest number minus the smallest number 8 - 3 Answer The Range = 5 12 Properties 1. Commutative Property  Numbers can be added or multiplied in any order and the answer is still the same. Examples: Commutative Property of Addition: 3 + 2 = 2 + 3 a + b = b + a Commutative Property of Multiplication: 5(4) = 4(5) ab = ba 2. Associative Property  When adding OR multiplying 3 or more numbers, they can be grouped in any way and the answer remains the same. Examples: Associative Property of Addition: (2 + 4) + 9 = 2 + (4 + 9) a + (b + c) = (a + b) + c Associative Property of Multiplication: (5x4)x2 = 5x(4x2) (cd)e = c(de) 3. Identity Property of Addition  When you add 0 to any number your answer is that number. Examples: 5 + 0 = 5 0 + 1,253 = 1,253 a + 0 = a 0 + b = b 4. Identity Property of Multiplication  When you multiply any number by 1 your answer is that number. Examples: 4 ∙ 1 = 4 1 x 746 = 746 1 x a = a b x 1 =b 5. Property of Zero  Any number multiplied by zero is zero. Examples: 0 x 8 = 0 52 ∙ 0 = 0 a ∙ 0 = 0 0 x b = 0 6. Distributive Property  Multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. Examples: 2(3 + 4) = 2∙3 + 2∙4 a x (b + c) = (a x b) + (a x c) 15 Metric Conversion King Henry Died By Drinking Chocolate Milk King Henry Doesn’t Usually Drink Chocolate Milk Example Convert Compare 16 Order of Operations Cheat Sheet There is a specific order in which math problems should be worked out. It is called the “order of operations.” If you do not work math problems in the correct order, you probably will get the wrong answer. It is like a step-by-step recipe to work out a math problem that will lead you to the correct answer. 1st Parenthesis & Grouping Symbols – 2ndExponents – 3rdMultiply or Divide – 4thAdd or Subtract Hint: Please guys, excuse my dear Aunt Sallly Examples:: P Parenthesis 1st Do the parenthesis and all other grouping symbols. Parenthesis: (6 + 7) Brackets: [(3 + 2) – (2-1)] Brackets usually go around a set of parenthesis. Work inside the brackets first until there is nothing left to do. Fraction Bars: = Do everything above the fraction bar, then everything below the Fraction bar, and then divide. G Grouping symbols such as brackets or a fraction bar. E Exponents 2nd Do all exponents. 23 = 2·2·2 = 8 42 = 4(4) = 16 M Multiply 3rd Multiply or divide from LEFT TO RIGHT Sometimes you multiply first, but sometimes you divide first. You decide by going left to right. 6·2 ÷ 4 18 ÷ 3·5 Multiplying comes 3 ÷ 4 Dividing comes 6 · 5 first 12 first 30 D Divide A Add 4th Add or subtract from LEFT to RIGHT Sometimes you add first, but sometimes you subtract first. You decide by going left to right. 4 + 2 – 5 7 – 3 + 3 Adding comes 6 – 5 Subtracting comes 4 + 3 first 1 first 7 S Subtract 17 Examples of using the proper order of operations: Example 1: (17 + 3) + 23 ÷ 4 ∙ 2 (17 + 3) + 23 ÷ 4 ∙ 2 20 + 23 ÷ 4 ∙ 2 20 + 8 ÷ 4 ∙ 2 20 + 2 ∙ 2 20 + 4 24 1st – parenthesis 2nd – exponents 3rd – divide 4th - multiply 5th – add Answer Example 2: 2[6 + (4 – 3)] – 5 2[6 + (4 – 3)] – 5 2[6 + 1] – 5 2[7] – 5 14 – 5 9 1st – inner parenthesis 2nd – brackets 3rd – multiply 4th - subtract Answer Example 3: – 3 + 10 - 3 + 10 – 3 + 10 3 – 3 + 10 0 + 10 0 1st – grouping symbols (above & below fraction bar) 2nd – divide 3rd – subtract 4th – add Answer Example 4: 33 – + 2 33 – + 2 33 – + 2 33 - 5 + 2 27 – 5 + 2 22 + 2 24 1st – grouping symbols (above & below fraction bar) 2nd – divide (finish the grouping symbol ) 3rd – exponent 4th – subtract (because it comes first) 5th - add Answer 20 Fraction Operations Adding & Subtracting Fractions 1. Make sure the denominators are the same. 2. If needed, we have to build each fraction so that the denominators are the same. 3. Then, we add or subtract the numerators. 4. The denominator of your answer will be the same denominator of the built-up fractions. 5. Reduce or simplify the answer, if required. Examples: To add or subtract fractions with a common denominator, you simply omit Step#1. 1/3 + 1/3 = 2/3 Note: DO NOT add or subtract denominators! When adding fractions with different denominators, we do all the steps. 1/2 + 1/3 3/6 + 2/6 = 5/6 Multiplying Fractions Here are the Rules for multiplying fractions... 1. You do not have to worry about a common denominator! 2. If possible, simplify before you multiply. 3. Multiply the numerators. 4. Multiply the denominators. 5. Simplify or reduce the resulting fraction, if possible. Examples: 3 2 x 5 4 = 15 8 Remember: You do not have to worry about a common denominator! Just multiply the numerators & then multiply the denominators!! Multiplying Mixed Numbers 1. Change the mixed numbers into improper fraction 2. If possible, simplify first. 3. Multiply the numerators. 4. Multiply the denominators. 5. If necessary, rewrite your answer as a mixed number and check to be sure it is in simplest form. Examples: 3 1 1 X 2 4 3 = Change mixed numbers to improper fractions then solve. 3 4 X 4 11 = 12 44 = 3 11 = 3 3 2 21 Dividing Fractions A Key Word to Understand Reciprocal A reciprocal of a number is when the numerator and denominator switch places. If the fraction is a mixed number, change it to an improper fraction first, then write its reciprocal. The product of any number and its reciprocal is always one. Example: The reciprocal of 4 3 is 3 4 . The reciprocal of 5 1 is 1 5 . Example of reciprocal with mixed numbers: 2 1 1 equals 2 3 and it’s reciprocal is 3 2 Steps for Dividing Fractions 1. Rewrite the division problem as a multiplication problem, but multiply by the reciprocal of the number you were dividing by. 2. Simplify before you multiply. 3. Multiply the numerators. 4. Multiply the denominators. 5. Be sure your answer in its simplified or reduced form. Change improper fraction to whole numbers or mixed numbers. Example: 2 1 ÷ 3 1 Rewrite as a multiplication using the reciprocal. 2 1 X 1 3 Now solve. 2 1 X 1 3 = 2 3 Simplified = 1½ Hints for Dividing Mixed Numbers 1. Change the mixed numbers into improper fraction 2. Rewrite the division problem as a multiplication problem, but multiply by the reciprocal of the number you were dividing by. 3. Simplify before you multiply. 4. Multiply the numerators. 5. Multiply the denominators. 6. Be sure your answer in its simplified or reduced form. Change improper fraction to whole numbers or mixed numbers. Example: 2 1 1 ÷ 2 3 2 Rewrite division problem with improper fractions. 2 3 ÷ 3 8 Now rewrite as a multiplication using the reciprocal, and solve. 2 3 X 8 3 = 16 9 22 Ladder / Slide Method Greatest Common Factor or Divisor (GCF/GCD): Highest number that divides exactly into two or more numbers Least Common Denominator or Multiple (LCM or LCD): Smallest number that is a multiple of two or more numbers Smallest Number that is a multiple of two or more denominators Simplified Fractions: Reduce a number to make as simple as possible. (Numbers only have a factor of one that is the same) Step 1: Write the two numbers in a box Step 2: Find a factor that goes into both numbers Step 3: Divide both numbers Step 4: Continue this process until both numbers only have a factor of 1 that is similar GCF/GCD Multiply the left side LCM/LCD Multiply the left side and the bottom numbers Simplified Fractions Bottom numbers become you simplified fraction 25 Ratios Rates & Proportions Ratio: A comparison between two different amounts. There are 3 ways to write ratios 8 to 3 8:3 A ratio is usually a part-to-part comparison, but it can be a part to whole comparison. Example: The score was 15 to 4. There are two parts being compared - the score of one team being compared to the score of the other team. Proportion: Two ratios that are equal to each other. Example: Proportions are used when two things are being compared and one of the parts is missing. Example: Margaret knows that she can serve 7 people with 2 cans of green beans. She will be feeling 84 people at the luncheon. How many cans of green beans will she need to buy? N = 24 cans Rate: A ratio comparing 2 amounts measured in 2 different units. Example: The ratio below is comparing minutes to kilometers. These are two different units of measurement so this ratio is a rate. Unit Rate: A unit rate is the amount for 1 item Example: The car gets 32 miles per gallon of gasoline. This is a unit rate because we are talking about 1 gallon of gasoline A proportion can be used to find a unit rate. Example: A bottle of shampoo cost $3.99 for 13.5 ounces. Find the unit rate. N = about $0.30 per ounce 26 Comparing with Fractions, Percents, Ratios, and Proportions What is being compared? Fractions: Always a part to whole comparisons. Numerator part Denominator  whole Percents: Always a part to whole comparison. The percent is the part out of 100. Example: 53% 53 is the part out of 100. The 100 represents the whole. Ratios: Usually a part to part comparisons, but may be Part to whole comparisons. - Most of the time 1 part is being compared to another part - Sometimes 1 part is being compared to the whole - You need to look at what the number represent then think…. Are these separate parts or is one a whole? Proportions: Always comparing 2 equal ratios. Used to help find a missing part when things are being compared. Example: Key Words “to” A ratio usually uses “to”. Look for 2 things being compared. “altogether” “Altogether” usually refers to a whole. “all” “All” usually refers to a whole. “total” “Total” usually refers to a whole. There are 8 girls and 12 boys in Mrs. Green’s 4th hour class. Find the ratio of boys to girls. Think: A ratio is a part to part comparison.  Ask yourself: What part are boys? 12 boys  Ask yourself: What part are girls? 8 girls  Now write your ratio with the boys first and then the girl. 12 :8 or 12 to 8 or 12/8 Find the fraction of the students that are girls. Think: A fraction is a part to whole comparison.  Ask yourself: What part are the girls? 8 girls  Ask yourself: What number represents the whole class? 20 students Find the percent of students that are girls. Think: A percent is a part to whole comparison.  Ask yourself: What part of the class are girls? 8 boys  Ask yourself: What number represents the whole class? 20 students. Think: You just found the fraction of the students.  Change the fraction to a decimal to a percent. = 0.4 = 40% \ 27 Solving Percent Problems Finding Percent of a Number -- There are 2 common ways – using a proportion or using an equation. Finding the Percent of a Number Using a Proportion Using an Equation Things you need to know: - Remember: A percent is a part to whole comparison. The part is the percent and the whole is 100. - A percent can be written as a fraction out of 100. - Things you need to know: - Remember: A percent is a part to whole comparison. The part is the percent and the whole is 100. - A percent can be written as a decimal by dividing the percent by 100. - 72% = 72 ÷ 100 = 0.72 How it works: 1. Find 25% of 68 2. Write a part to whole proportion. 3. Solve the proportion by multiplying diagonals and dividing by left- over. So, n = 17. 4. Therefore, 25% of 68 is 17. 5. Hint: The “of” in the problem “25% of 68” will usually be hooked to the number that represents the whole. How it works: 1. Find 25% of 68 2. In math “of” usually always means multiply. 3. So 25% of 68 would mean to multiply 25% by 68. 4. First, change 25% to a decimal. 25% = 25 ÷ 100 = 0.25 5. Rewrite the original problem as a multiplication problem, but multiply by the percent written as a decimal. 25% of 68 0.25 x 68 = 17 6. Therefore, 25% of 68 is 17 Other examples: 1. 11% of 840 Solve and n = 92.4 So 11% of 840 = 92.4 2. 32% of 912 Solve and n = 291.84 So, 32% of 912 is 291.84 Other examples: 1. 11% of 840 Remember: 11% = 0.11 0.11 x 840 = 92.4 So 11% of 840 = 92.4 2. 32% of 912 Remember: 32% = 0.11 0.32 x 912 = 291.84 So, 32% of 912 is 291.84 30 Geometric Figures Polygons are two-dimensional closed geometric figures formed by line segments. Two-Dimensional Figures Triangles have 3 sides and 3 angles. - The sum of the measure of the inside angles of any triangles is always 180º. - Angle + Angle + Angle = 180º Scalene Triangle Isosceles Triangle Equilateral Triangle No congruent sides or congruent angles At least 2 congruent sides and at least 2 congruent angles 3 congruent sides and 2 congruent angles Right Triangle Acute Triangle Obtuse Triangle Has a right angle (measure 90º) All angles measure less than 90º Has an angle that measures more than 90º Quadrilaterals have 4 sides and 4 angles. - The sum of the measure of the inside angles of any triangles is always 360º. - Angle + Angle + Angle + Angle = 360º Quadrilateral Parallelogram Trapezoid Any closed figure with 4 sides Opposite sides are congruent and parallesl Exacly 1 pair of parallel sides Rectangle Rhombus Square A parallelogram with 4 right angles A parallelogram with 4 congruent sides A parallelogram with 4 right angles and 4 congruent sides. (A rhombus with 4 right angles) (A rectangle with 4 equal sides.) 31 Other Common Two-Dimensional Figures Pentagon Hexagon Octagon A polygon with 5 sides and 5 angles A polygon with 6 sides and 6 angles A polygon with 8 sides and 8 angles Three Dimensional Figures A 3-dimensional figure has length, width, and height. The surfaces may be flat or curved. A 3-dimensional figure with flat surfaces is called a polyhedron. Prisms Triangular Prisms: - 5 faces (2 bases) - 9 edges - 6 vertices Rectangular Prisms: - 6 faces (2 bases) - 12 edges - 8 vertices Cubes: - 6 faces (2 bases) - 12 edges - 8 vertices Pyramids Triangular Pyramid: - 4 faces (1 base – it’s a triangle) - 6 edges - 4 vertices Rectangular Prisms: - 5 faces (1 base – it’s a rectangle) - 8 edges - 5 vertices 32 AREA (Covering) - The number of square units it takes to cover a figure or an object. PERIMETER (Distance Around)- The sum of the sides of straight sided figures. Shape Example Area Equation/Formula Perimeter Equation/Formula Rectangle A = l w P = S1 + S2 + S3 + S4 (P = 2l + 2w) Triangle A = 2 bh OR A = ½ bh P = S1 + S2 + S3 Parallelogram A = bh P = S1 + S2 + S3 + S4 Trapezoid A = ½ h(b + b) or A = h(b + b) 2 P = S1 + S2 + S3 + S4 Circle A = π r 2 Circumference C = πd or C = 2πr The Circle Circumference The distance around a circle. Radius The distance between the center of the circle and any point on the circle Diameter The distance across the circle through the center Pi π  3.14 or 7 22 Key b = base h = height l = length w = width d = diameter r = radius A = Area π  3.14 or 7 22 C = Circumference 35 Congruent and Similar Figures Understanding Congruent Figures The symbol for congruent Congruent Figures Must Have -Same Shape -Same Angles -Same Size -Same Side Lengths EXAMPLE: Triangles ABC DEF Therefore, they have the…. - Same Shape - Same Angles - Same Size - Same Side Lengths Understanding Similar Figures The symbol for similar Similar Figures Must Have -Same Shape -Same Angles -A Scale Factor* -Same Side-to-Side Ratios** EXAMPLE: Rectangles ABCD EFGH Therefore, they have the…. - Same Shape - Same Angles - A Scale Factor* - Same Side-to-Side Ratios** *So, what does Scale Factor mean? The Scale Factor is the magic number that all of the side lengths of one figure are multiplied by to get all of the side lengths of new figure. Because all of the side lengths of the smaller figure are all multiplied by 3, the scale factor is 3 or SF = 3. In similar figures the sides that are in the same position are called corresponding sides. We call the angles that are the same in similar figures, corresponding angles. **Then what are Side-to-Side Ratios? In Rectangle ABCD, if you compare the ratio of the long side to the short side, it should be equal to the ratio of Rectangle EFGH’s long side to its short side. Rectangle ABCD: Rectangle EFGH: Therefore, these rectangles have the same side-to-side ratios. Corresponding Sides and Corresponding Angles In congruent and similar figures the sides that are in the same position in both figures are called corresponding sides. The angles that are the same in both congruent figures and similar figures are called corresponding angles. EXAMPLES: In the rectangles above the short sides in rectangle ABCD corresponds with the short sides in EFGH. In the triangles above, angle A corresponds with angle D because they are both 50º. 36 Pythagorean Theorem Pythagoras was a Greek philosopher and mathematician, born in Samos in the sixth century B.C. He and his followers tried to explain everything with numbers. One of Pythagoras’s most popular ideas is known as The Pythagorean Theorem. Things you need to know: 1. Right triangles have 2 legs and a hypotenuse. - The legs are the short side. - The hypotenuse is the long side that is opposite the right angle. 2. What is the Pythagorean Theorem - The Pythagorean Theorem says that the sum of the legs squares of a RIGHT triangle equal the square of the hypotenuse. a 2 + b 2 = c 2 . 3. You can find the missing parts of a right triangle. Examples A. Find the hypotenuse. a 2 + b 2 = c 2 1. Write formula. 3 2 + 5 2 = c 2 2. Show substitutions. 9 + 25 = c 2 3. Solve. 36 = c 2 4. Find the square root of c 2 . c = 6 cm 5. The hypotenuse equals 6 cm. B. Find the missing side. a 2 + b 2 = c 2 1. Write formula. a 2 + 7 2 = 25 2 2. Look closely & then show substitutions. a 2 + 49 = 625 3. Solve. – 49 – 49 4. Subtract 49 from each side. a 2 = 576 fdd 5. Find the square root of a 2 . a = 24 m 6. The missing side is 24 m. 37 Solving Equations with Hands-On-Algebra Solving equations is all based on maintaining balance. A scale is used to represent that balance. Example 1 Example 2 1. Set up your balance scale. 4x + 5 = 2x + 13 1. Set up your balance scale. Hint: The 2 outside the parenthesis means you must do the inside of the parenthesis twice. 2(x +3) = x + 8 2. There are pawns on both sides so to maintain balance, remove 2 pawns from each side. 2. When you lay it all out it looks like this. 3. Now you are left with 2x + 5 = 13. 3. There are pawns on both sides so to maintain balance, remove 1 pawn from each side. 4. There are cubes on both side. Now remove 5 from the cubes on each side. 4. Now you are left with x + 3 = 8 5. You are now left with 2x = 8 5. There are cubes on both sides. Now remove 6 from the cubes on each side. 6. If 2 pawns equals 8, then each pawn must equal 4. So, x = 4 (Hint: 8÷2) 6. Because you have all your pawns on one side and all of your cubes on the other you are finished. You are now left with x = 2. 7. Finally check your answer if x = 4. 4x + 5 = 2x + 13 Substitute: 4(4) + 5 = 2(4) + 13 Solve: 16 + 5 = 8 + 13 21 = 21 It checks. 7. Finally check your answer if x = 2. 2(x +3) = x + 8 Substitute: 2(2 + 3) = 2 + 8 Solve: 2(5) = 10 10 = 10 It checks. 40 Inequalities Inequality Two values that are not equal (less than, greater than) < Greater than > Less than ≤ Greater than or equal to ≥ Less than or equal to ≠ Not equal Graphing Inequalities x < 4 y ≥ - 3 1. Locate the value for the variable 2. Mark the point with one of the following a. Closed Circle if symbol is ≥ or ≤ b. Open Circle if symbol is < or > 3. Determine which direction you will draw the arrow a. Left  If variable is smaller than the value b. Right  If variable is larger than the value Solving Inequalities by Adding & Subtracting Addition & Subtraction Properties of Inequality: You can add or subtract the number to each side of an inequality and the problem stays balanced. n + 3 ≤ -4 n – 14 > 10 – 3 – 3 - Undo adding by subtracting + 14 + 14 - Undo subtraction by adding n ≤ -7 n > 24 Solving Inequalities by Multiplying & Dividing Multiplication & Division Properties of Inequality: You can multiply and divide each side of the inequality by the same number, BUT you must be careful about the directions of the inequality sign. - IF you multiply or divide by a positive number the sign stays exactly how it was. - IF you multiply or divide by a negative number, the sign flips the opposite way. – 1 ≤ 7 +1 +1 ≤ 8 ≥ (8)2 n ≥ 16 1) Add 1 to each side. 2) Multiply both sides by 2. Since you are multiplying each side by a positive number, the sign stays the same. -3n + 4 > 16 – 4 – 4 -3n > 12 -3n < 12 -3 -3 n < -4 1) Subtract 4 from each side. 2) Divide both sides by -3. Since you are dividing each side by a negative number you must switch the sign from > to <. 41 Correctly Answering a Question: R Restate the question You need to restate the question so that the person reading your answer knows what the question was asked. A Answer all parts of the question. Many questions have multiple parts, be sure to read, and re- read and answer all parts of the question C Cite Evidence How do you know that this is the correct answer. Many times this can be shown in your work. E Explain Explain the process you used to get the correct answer. 42 Word Problem Cheat Sheet If you see these words in a word problem then use... Addition (Sum) Subtraction (Difference)  Add  Altogether  And  Both  How many  How much  More than  In all  Increased by  Plus  Sum  Together  Total  Are not  Change  Decreased by  How many did not have  Less than  Have left  Left over  How many more  How much more  Difference  Fewer Multiplication (Product) Division (Quotient)  By (dimensions)  Double (times two)  Triple (times three)  Each group  Group  Multiplied by  Of  Product of  Times  Twice (times two)  Each group has  Half (divide by 2)  How many in each  Share something equal  Fractions – divide by denominator  Parts  Quotient of  Separated  Split  Divided by 45 Coefficient A number used to multiply a variable Commutative Property of Addition & Multiplication Multiply or add in any order without changing the answer 3 x 6 = 6 x 3 5 + 2 = 2 + 5 Complimentary Angles Two angles that add up to 90˚ Composite Numbers Numbers that has more than two factors Example: 4, 6, 8, 9, 12 Compute To solve Cone A 3-dimensional object that has a circular base and it comes to a point Congruent Same measures (angles, length, shape, or size) Consecutive Numbers that follow each other in order without gaps 20, 21, 22, 23… Convert To change from one measurement to a different measurement 6 mm = _____ km Coordinate Graph Graph that contains an x-axis and y-axis that intersect Criterion (Criteria) Standards or rules that make something true or false If a closed figure has 5 straight sides it is a pentagon. 46 Cube Root The number multiplied by itself 3 times that gives the perfect cube (See Perfect Cube) Cylinder A 3-dimensional (3-D) shape that has two congruent and parallel round faces Deca- Prefix for tens - 10 Decade – 10 years Decagone – 10 sided figure Deci - Prefix for Tenths - 0.1 0.1 Decimal Any number including whole numbers and numbers with a decimal point. 9 or 17.5 Denominator Bottom number in a fraction Descending Ordering from biggest to smallest Diameter Distance across a circle going through the center Difference Answer to a subtraction problem Minuend – Subtrahend = Difference 8 – 5 = 3 Dilation Polygon grows or shrinks but keeps exactly the same shape (Similar Figure – must have a scale factor) 47 Distribution (Data) Data and how often (frequency) it occurs Distributive Property The number on the outside of the parentheses is distributed (multiplied) to the numbers on the inside of the parentheses Dividend Number being divided Dividend ÷ Divisor = Quotient 24 ÷ 8 = 3 Divisor Number dividing Dividend ÷ Divisor = Quotient 24 ÷ 8 = 3 Equation Problem with an equal sign Equivalent Equal Estimate (Estimation) Approximate answer (Around the same number) 3.4 ≈ 3 Evaluate Solve the problem!!!!!! Even Numbers ending in 0, 2, 4, 6, and 8 Example: 2, 12, 14, 102 Event A single incident (occurrence) Exponent Shows how many times you multiply a number Expression Problem without an equal sign 4 · 5 50 Irrational Number A decimal that cannot be written as a fraction – It goes on forever without repeating. π ≈ 3.14159… Isosceles Triangle Triangle with two equal sides and two equal angles Kite Quadrilateral with two pairs of congruent sides adjacent to each other Least Common Multiple (Denominator) (LCM/LCD) Smallest number that is a multiple of two or more numbers Smallest Number that is a multiple of two or more denominators Less Than Smaller < Linear Makes a line y = mx + b Lowest Terms See Simplify Mean Average (add all numbers together and divide by how many items there are in a set of data) Example: 5 + 5 + 8 + 12 4 Median Middle number in a set of data when the numbers are put in order from least to greatest. **If there are two middle numbers must find the mean of the two numbers** 51 Milli- Mixed Number Fraction with a whole number and a proper fraction Mode Number that occurs the most often in a set of data 3, 3, 5, 6, 6, 6, 9, 9  The mode = 6 Multiple Result of multiplying by a whole number Multiples of 3: 3, 6, 9, 12… Non-Linear Not a straight line Non-Terminating Decimal A decimal that does not end, and may or may not repeat 4.2596391142869281… Negative Number less than zero Not Equal Values are not the same amount ≠ Numerator Top number in a fraction Obtuse (Angle) Angle greater than 90˚ but less than 180˚ 52 Octagon 8-sided figure Odd Numbers ending in 1, 3, 5, 7 and 9 Operation Add, Subtract, Multiply, Divide + – x ÷ Opposite Same distance from zero but in the other direction Negative → Opposite = Positive Positve → Opposite = Negative Order of Operations The rules of which calculations come first in an expression or equation (The order we solve a problem) Please Guys Excuse My Dear Aunt Sally Ordered Pairs Two numbers written in parentheses showing the x and y coordinates Origin Where the x-axis and y-axis intersect Point = (0,0) Always start at the origin when plotting points Outlier Value that “lies” outside the other set of data **Either much larger or smaller than the rest of the data Parallel Lines that are always the same distance apart and never touch 55 Positive Numbers to the right of zero on the number line Predict Based on data make an estimation of something that might happen in the future or will be a consequence of the current data Prime A number that can be divided evenly by only one and itself Example: 2, 3, 5, 7, 11, 13, 17… Prism A solid figure that has two faces that are congruent (the same or equal) Probability The chance something will happen (the likelihood of an event taking place Product Answer to a multiplication problem Factor x Factor = Product 5 x 4 = 20 Proportion Two ratios set equal to each other Pyramid A solid object where:  Base is a polygon  Sides are triangles which meet at the top (Apex) Pythagorean Theorem Right Angle Triangle – The long side (hypotenuse) squared equals the sum of the squares of the other two sides a 2 + b 2 = c 2 56 Quadrilateral Four sided figure Qualitative Information (Data) that describes something Quantitative Information (Data) that can be counted or measured Quantity How much there is of something Quotient Answer to a division problem Dividend ÷ Divisor = Quotient 45 ÷ 9 = 5 Radius Distance from the center to the edge of a circle Random Sample A selection that is chosen randomly (by chance – no prediction) Range The difference between the lowest and highest value 5, 12, 13, 15, 24 Range = 24 – 5 = 19 Rate Ratio that compares two different quantities using different units Miles per hour $ per gallon Ratio A comparison of two quantities by division Written in 3 different ways Miles : Hour Miles to Hour Miles / Hour 57 Rational Number Number that can be made by dividing one integer by another Example: 0.5, 1.73, -15.23, 5/3 Reciprocal Number you multiply another number to get one (1) Rectangle 4 sided figure with right angles and two sets of equal sides Rectangular Prism Solid object that has six (6) sides that are all rectangles Rectangular Pyramid A solid object where:  Base is a rectangle or square  Sides are triangles which meet at the top (Apex) Reflection An image or shape as it would be seen in a mirror (reflects over an area) Regular Polygon All sides and angles are equal Repeating Decimal A fraction that when written as a decimal repeats in a pattern that goes on forever Example: 1/3 = 0.3333333… Right (Angle) Angle that is exactly 90˚ 60 Square Root The number that is multiplied by itself that gives you the perfect square. (See Perfect Square) Stem and Leaf A plot where ach data value is split into a “leaf” (usually the last digit) and a “stem” (the other digit) Example: 32 = 3 (stem) and 2 (leaf) Straight (Angle) Line - 180˚ Substitution Replacing a variable with a number Sum Answer to addition problem Addend + Addend = Sum 4 + 3 = 7 Supplementary Two angles that add up to 180 degrees 61 Surface Area Total area of a three-dimensional object See cheat sheet for formulas Table Numbers or quantities arranged in rows and columns Tax Percentage of the cost of an item added to the total cost Terminating Decimal Decimal number that has digits that stop 0.5 Transformation Moving a shape in a different position, but it will not change shape, size, area, angles or lengths. (See Rotation & Reflection) Translation Moving a shape, without rotating or flipping it (Sliding) Transversal A line that crosses at least two other lines Trapezoid Four sided figure with one pair of parallel sides 62 Tree Diagram A diagram to help you determine the probability of an event  Multiply along branches  Add along columns Unique Leading to only one result 4 + 5 = 9 Unit One – single item One Ounce Unit Rate Amount per item (One Item) Variable A letter that represents a number in an equation or expression 5 + x = 15 x is the variable Variability How close or far apart a set of data is Vertical Runs up and down Vertical Angles Vertical angles are angles that are opposite each other when two lines cross  Vertical angles are always congruent