7 th Grade Math Anchor Charts FULL YEAR BUNDLE

Transcription

1 7 th Grade Math Anchor Charts FULL YEAR BUNDLE I Math

2 7 th Grade Math Anchor Charts The Number System I Math

3 Absolute Value What is absolute value? The absolute value of a number is the distance between the number and zero on a number line. Numbers that are the same distance from zero on a number line have the same absolute value. The absolute value of any number is always positive. 1. Count the distance from the number to zero on a number line. Example 1: is 8 spaces away from 0 so 8 = 8 Example 2: is 6 spaces away from 0 so 6 = 6

4 Additive Inverse What is an additive inverse? The additive inverse of a number is the opposite of that number on a number line. When a number and its additive inverse are combined the sum is Keep the original number. 2. When the original sign is *positive: make it negative* *negative: make it positive* Example 1: What is the additive inverse of - 35? The original sign is negative so the final answer will be positive. The additive inverse of - 35 is 35 Example 2: What is the additive inverse of 26? The original sign is positive so the final answer will be negative. The additive inverse of 26 is - 26

5 Adding Integers with Same Signs What is an integer? An integer is any number from the set {, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, } 1. Find the absolute value of both integers. 2. Add the two absolute values together. 3. Use the same sign as the original integers for the sign of the answer. Example 1: (- 24) 1. 7 = 7 24 = = Example 2: = = =

6 Adding Integers with Different Signs What is an integer? An integer is any number from the set {, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, } 1. Find the absolute value of both integers. 2. Subtract the smaller number from the larger number. 3. The sign of the larger number in the original integer will be the sign for the answer. Example 1: = 15 8 = = Example 2: 29 + (- 21) = = = 8 *15 is the larger number. Since it is originally negative the final answer will be negative.* 3. 8 *29 is the larger number. Since it is originally positive the final answer will be positive.*

7 Adding Integers on a Number Line What is an integer? Example 1: An integer is any number from the set {, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, } 1. Place a dot on the number line representing the first term. 2. If the integer in the second term is Example 2: 3 + (- 4) *negative* - move left/down. *positive* - move right/up the same units as the integer.

8 Subtracting Integers What is an integer? An integer is any number from the set {, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, } 1. Use the KCC Rule to rewrite the problem. K KEEP the same sign C CHANGE to addition C CHANGE to the opposite sign 2. Follow the rules for adding integers to solve. Example 1: K C C (- 3) = Example 2: (- 24) (- 24) K C C = = =

9 Subtracting Integers on a Number Line What is an integer? An integer is any number from the set {, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, } 1. Use the KCC rule to rewrite the problem using addition. K KEEP the same sign C CHANGE to addition C CHANGE to the opposite sign 2. Place a dot on the number line representing the first term. Example 1: 5-7 Example 2: - 1 (- 4) 3. If the integer in the second term is *negative* - move left/down. *positive* - move right/up the same units as the integer.

10 Multiplying Integers What is an integer? An integer is any number from the set {, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, } 1. Find the absolute value of both integers. 2. Multiply the absolute values. 3. Follow the chart to get the sign of your answer. First Term Second Term Answer Example 1: = 8 7 = = Example 2: 6 1 ( 9) 1. 6 =6 9 = =

11 Dividing Integers What is an integer? An integer is any number from the set {, - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4, } 1. Find the absolute value of both integers. 2. Divide the absolute values. 3. Follow the chart to get the sign of your answer. Example 1: 81 ( 9) = 81 9 = = Example 2: 36 ( 3) = 36 3 = =

12 Rational Numbers What is a rational number? A rational number is a number that can be written as a fraction. The root of the word rational is ratio, and we know ratios can be written in fraction form. This includes: repeating decimals terminating decimals percents whole numbers Integers Fractions Mixed numbers Rational numbers can be positive or negative. Example 1: Example 2: Example 3: Example 4: % 24

13 Mixed Numbers & Fractions to Decimals What is the difference between a mixed number and a fraction? A fraction, which can also be called a ratio, is a part of a whole. ;<=>?@AB? C>;B=D;@AB? Example 1: Turn - E into a decimal. F 1. - E is not a mixed number. F = 0.6 A mixed number is a fraction and a whole number together. 1. Turn any mixed numbers into improper fractions. 2. Divide the numerator by the denominator. Example 2: Turn 5 H I 1. 5 H I = JH I = 5.25 into a decimal.

14 Terminating Decimals to Fractions What is a terminating decimal? A terminating decimal is a decimal that ends (remainder of zero). Example 1: Turn into a fraction thousandths 2. KJF H,MMM (for thousandths you use 1,000) 3. KJF H,MMM HJF HJF F N 1. Write the place value of the last digit. 2. Use the number after the decimal as the numerator and the place value of the last digit as the denominator. Example 2: Turn 2.75 into a fraction and 75 hundredths 2. OF HMM (for hundredths you use 100) 3. Simplify the fraction. 4. The number in front of the decimal becomes the whole number in front of the fraction. 3. OF HMM 4. 2 E I JF JF E I

15 Percents to Fractions What is a percent? A percent is a fraction of the whole, 100%. Example 1: Turn 45% into a fraction 1. 45% IF Think of it as a pizza. If you had 100% of a pizza you would still have the whole thing. 2. IF HMM It can easily be written as a ratio with the given percent as the numerator over IF HMM F F P JM 1. Drop the percent symbol off of the value and use it as the numerator. 2. The denominator is 100. Example 2: Turn 210% into a fraction % 2. JHM HMM JHM 3. Simplify. 3. JHM HMM HM HM JH HM 2 H HM *When the numerator exceeds the denominator turn the fraction into a mixed number.*

16 Fractions to Percents How can I convert fractions to percents? raction ecimal ercent F D P Example 1: Turn I into a percent. F 1. I F % When converting from fractions to percents you must first convert the fraction to a decimal. Once your number is in decimal form you may then convert it to a percent. The fraction 4 5 Example 2: Turn H I is equal to 80% into a percent. 1. H I Convert the fraction to a decimal. (Use the fractions to decimals anchor chart.) % 2. Convert the decimal to a percent. (Use the decimals to percents anchor chart.) The fraction 1 4 is equal to 25%

17 Percents to Decimals Why should I use a decimal? A lot of times decimals are used in situations where an answer needs to be more precise. A key example in math would be to use a decimal when dealing with money. 1. Drop the percent symbol and replace it with a decimal. 2. Move the decimal two digits to the left. 3. Add a zero to the front when necessary. Example 1: Turn 53% into a decimal 1. 53% Example 2: Turn 127% into a decimal %

18 Decimals to Percents What is a percent? A percent is a fraction of the whole, 100%. Think of it as a pizza. If you had 100% of a pizza you would still have the whole thing. It can easily be written as a ratio with the given percent as the numerator over Move the decimal two digits to the right adding zeros when necessary. 2. If the decimal is at the end of the number you may remove it as well as any extra zeros at the beginning. If there are still digits after the decimal you must keep it in place. Example 1: Turn 0.9 into a percent *you can drop the decimal because it is at the end.* 3. 90% Example 2: Turn into a percent *you can not drop the decimal because it is not at the end.* % 3. Add a percent symbol to the end.

19 Percent Conversions Greater Than 100% Percents to decimals: 1. Drop the percent symbol and replace it with a decimal. 2. Move the decimal two digits to the left. 3. Add a zero to the front when necessary. Percents to fractions: 1. Drop the percent symbol off of the value and use it as the numerator. 2. The denominator is Simplify. *When the numerator exceeds the denominator turn the fraction into a mixed number.* Example 1: Turn 125% into a decimal % Example 2: Turn 125% into a fraction % HJF HMM HJF HMM HJF JF F 1 H JF I I

20 Percent Conversions Less Than 1% Percents to decimals: 1. Drop the percent symbol. 2. Move the decimal two digits to the left filling any empty spaces with zeros. 3. Add a zero to the front when necessary. Example 1: Turn 0.25% into a decimal % Percents to fractions: 1. Drop the percent symbol off of the value and use it as the numerator. 2. The denominator is 100 Example 2: Turn.08% into a fraction % M.MN 3. Move the decimal in the numerator to the end counting the number of digits it moves. Add the same number of zeros to the denominator. 4. Simplify M.MN HMM N HM,MMM N HM,MMM N N H H,JFM

21 Comparing Rational Numbers What is a rational number? Rational numbers are numbers that can be expressed as a decimal. This includes fractions, decimals (both repeating and terminating), percents, and integers. Any of which may be negative. > < = greater than less than equal to Example 1: Compare 1.2 to 1 H I is already in decimal form. 1 H I = < < 1 H I *both have 2 s in the tenths place so you must look at the hundredths place to compare. If 1.2 had a number in the hundredths place it would be 0. Since 5 Is bigger than 0, 1.25 is greater than 1.2.* 1. Convert rational numbers to decimal form. 2. Write a mathematical sentence using the correct symbol from above. 3. Rewrite the sentence using the original number forms. Example 2: Compare - 4 to 4 H E can be written as H E = > > - 4 H E *When looking at the tenths place the 3 is bigger than the 0. However, since both numbers are negative, this means that is a greater NEGATIVE number than is actually closer to 0 which makes it the greater number of the two.*

22 Ordering Rational Numbers When ordering rational numbers Positive numbers are greater than negatives. The negative number closer to 0 on a number line has the greater value. The positive number that is bigger has a greater value. Use place value in situations where the decimals are very close in value. Example 1: Order 3 E, 4.6, - 3 H, 380% I F 1. 3 E = 3.75 I H = F 380% = is the smallest because it is the only negative. 1. Convert rational numbers to decimal form. 2. Identify the smallest number using the rules and symbols for comparing. 3. Use process of elimination and repeat step 2 until you have ordered all the numbers. 4. List the numbers from least to greatest. 5. Rewrite using the original form. 3. Of 3.75, 4.6, and 3.8: 3.75 is the smallest. Of 4.6 and 3.8: 3.8 is the smallest. 4.6 is the greatest number , 3.75, 3.8, H F, 3E I, 380%, 4.6

23 Adding Like Fractions What are like fractions? Like fractions are fractions that have same denominators. 1. Add the numerators. (follow your rules for adding integers when there are negative numbers) Example 1: E N + H N I N E N + H N E N + H N I = H I J Example 2: - I F + J F = 4 *the numerator will be 4* 8 * the denominator will remain 8* 2. Keep the denominator the same. 3. Simplify I F + J F 2. - I F + J F = - 2 *the numerator will be - 2* 5 * the denominator will remain 5* 3. - J F *The fraction is in simplest form.

24 Subtracting Like Fractions What are like fractions? Like fractions are fractions that have same denominators. 1. Subtract the second numerator from the first. (Follow the rules for subtracting integers when there are negative numbers.) Example 1: K O E O K E O O K E O O 6-3 = 3 *the numerator will be 3* 7 * the denominator will remain 7* E *The fraction is in simplest form* O Example 2: - J F E F 2. Keep the denominator the same. 3. Simplify J F E F 2. - J F E F 3. - F F = = * the denominator will remain 5* F F = H H = 1

25 Adding Unlike Fractions What are unlike fractions? Unlike fractions are fractions that have different denominators. Example 1: : 8, 16, 24, 32, 40, 48, 56, 64, 72 9: 9, 18, 27, 36, 45, 54, 63, 72, E N 1P = JO 1P OJ & I P 1N = EJ 1N OJ 1. Find the least common denominator (LCD) 3. JO + EJ = FP OJ OJ OJ 2. Rewrite each fraction using the least common denominator. 4. FP OJ *The fraction is in simplest form. 3. Follow the steps for adding like fractions. 4. Simplify.

26 Subtracting Unlike Fractions What are unlike fractions? Unlike fractions are fractions that have different denominators. Example 1: : 2, 4, 6, 8, 10, 12 3: 3, 6, 9, 12, 15, H J 1E 1E = E K & J E 1J 1J = I K 1. Find the least common denominator (LCD) 2. Rewrite each fraction using the least common denominator. 3. Follow the steps for subtracting like fractions E I K K 3 4 = 3 + (- 4) = - 1 ih K ih K *The fraction is in simplest form. 4. Simplify.

27 Adding Mixed Numbers What are mixed numbers? Example 1: Mixed numbers are fractions with whole numbers. Example: 3 H J E HM ife HM & 2 H I P I ife HM + P I 1. Turn any mixed numbers into improper fractions : 10, 20, 30, 40, 50 4: 4, 8, 12, 16, 20, 24, 28, 32, 2. If the fractions are like fractions then skip to step ife HM 1J = ihmk 1J JM & P I 1F = IF 1F JM 3. If they are unlike fractions you must find the LCD and turn them into like fraction. 4. ihmk JM + IF = ikh JM JM 4. Add. 5. ikh JM = 3 H JM 5. Simplify and turn back to mixed numbers.

28 Subtracting Mixed Numbers What are mixed numbers? Example 1: Mixed numbers are fractions with whole numbers. Example: 3 H J 1. 4 I F JI F & 3 H E HM E JI F HM E 1. Turn any mixed numbers into improper fractions. 2. 5: 5, 10, 15, 20, 25, 30 3: 3, 6, 9, 12, 15, 18, 21, 24, 2. If the fractions are like fractions then skip to step JI F 1E = OJ 1E HF & HM E 1F = FM 1F HF 3. If they are unlike fractions you must find the LCD and turn them into like fraction. 4. OJ FM = JJ HF HF HF 4. Subtract. 5. JJ HF = 1 O HF 5. Simplify and turn back to mixed numbers.

29 Multiplying Fractions by Fractions What is a fraction? A fraction is a way to represent a PART of a whole object. For example if you had 2 H pizzas left over after a I party, this means you have: Example 1: Seth decided to start painting his house. On Monday he had E F of his house left. On Tuesday he painted E I of what was left. How much did he paint on Tuesday? E 1 E = P F I E 1 E = P F I JM 3. P JM *The answer is in simplest form* 2 whole pizzas and H of another. I 1. Multiply numerator by numerator. 2. Multiply denominator by denominator. 3. Simplify. *Refer to the Anchor Chart on GCF to help identify a divisor* Example 2: I O 1 JH EJ I 1 JH = NI O EJ I 1 JH = NI O EJ JJI NI JJI JN JN = E N

30 Multiplying Mixed Numbers by Fractions What is a mixed number? A mixed number is a combination of a whole number and a fraction H J Example 1: Solve 12 H J 1 E N = 25 JF J In math, you will need to convert mixed numbers to improper fractions and vice versa. 2. JF 1 E = OF J N HK I E 1 H E *Use division* 3. OF HK 4 HH HK 4 H F JH F *Multiply & add* 4 H F = 21 Example 2: Solve J F 1 5 H I 1. Turn mixed number into an improper fraction. 2. Multiply numerator by numerator & denominator by denominator. 3. Simplify & turn back into a mixed number H I J 1 JH = IJ F I JM IJ JM = 21 J = JH J HM JH I 2 H HM

31 Multiplying Mixed Numbers What is a mixed number? A mixed number is a combination of a whole number and a fraction. In math, you will need to convert mixed numbers to improper fractions and vice versa. I E 1 H E *Use division* Example 1: Solve 4 J E 1 6 E HM 1. 4 J E 2. 6 E HM HI 1 KE = NNJ E HM EM = = 63 HI E KE HM 4 H F JH F *Multiply & add* 4 H F = NNJ EM K = HIO K F 29 J F 1. Turn mixed numbers into improper fractions. 2. Multiply numerator by numerator & denominator by denominator. 3. Simplify & turn back into a mixed number.

32 Converting Units What is a unit of measurement? Units are the type of measurement used to measure an object. To convert between units you will need to use dimensional analysis. Customary Conversions 1 foot = 12 inches (in) 1 cup = 8 fluid ounces 1 yard = 3 feet 1 pint = 2 cups 1 mile = 5,280 feet 1 quart = 2 pints 1 pound = 16 ounces 1 gallon = 4 quarts 1 ton = 2,000 pounds 1 inch = 2.54 centimeters 60 seconds = 1 minute 60 minutes = 1 hour 24 hours = 1 day 365 Days = 1 year Example 1: Convert 6 quarts to pints quart = 2 pints K mnopqr HJ H H 1 J stuqr H mnopqr pints = 12 pints = HJ mnopqr stuqr H mnopqr Example 2: Convert 10,000 minutes to days hour = 60 minutes 1 day = 24 hours 1. Choose the conversion(s) you will use. 2. Multiply the number you are converting (over 1) by the conversion. *Make sure the common units are opposite of each other to divide out common units.* 3. Simplify HM,MMM wtunqxr H HM,MMM yznpr KM HM,MMM {o r H,IIM 1 1 H yznp KM wtunqxr H {o JI yznpr NM = HJF NM HN = HM,MMM wtu yp KM wtu = HM,MMM yp {o HIIM yp = 6 HO HN days

33 Dividing Whole Numbers by Fractions What is the keep change flip method? When dividing a fraction by another fraction, you can use the keep change flip method to turn the division problem into a multiplication problem. Example 1: Solve 6 J E K J H E K H J E K C F = K 1 E H J H J Change it H K = H J 1 K H 3. K H 1 E J = HN J = P H or 9 Keep it Flip it 1. Turn the whole number into a fraction by placing the whole number over Follow the keep change flip method. Example 2: Solve 8 J F N J H F N J H F = N 1 F H J 3. Multiply numerator by numerator & denominator by denominator. *Simplify if necessary* 3. N H 1 F J = IM J = JM H or 20

34 Dividing Fractions by Whole Numbers What is the keep change flip method? When dividing a fraction by another fraction, you can use the keep change flip method to turn the division problem into a multiplication problem. Example 1: Solve H E H I E H H I E H = H 1 H E I H J Change it H K = H J 1 K H 3. H E 1 H I = H HJ Keep it Flip it Example 2: Solve J E 3 1. Turn the whole number into a fraction by placing the whole number over Follow the keep change flip method J E E H J E E H = J E 1 H E 3. Multiply numerator by numerator & denominator by denominator. *Simplify if necessary* 3. J E 1 H E = J P

35 Dividing Fractions by Fractions What is the keep change flip method? When dividing a fraction by another fraction, you can use the keep change flip method to turn the division problem into a multiplication problem. Example 1: Solve E I H F E H I F E H I F = E 1 F I H H J Change it H K = H J 1 K H 3. E I 1 F H = HF I = 3 E I *Never leave an answer as an Improper fraction unless told otherwise.* Keep it Flip it 1. Turn both whole numbers into fractions by placing them over Follow the Keep Change Flip method. Example 2: Solve H E H K H H E K H H E K = H 1 K E H 3. Multiply numerator by numerator & denominator by denominator. *Simplify if necessary* 3. H E 1 K H = K E = J H = 2

36 Dividing Mixed Numbers by Fractions What is the keep change flip method? When dividing a fraction by another fraction, you can use the keep change flip method to turn the division problem into a multiplication problem. Example 1: Solve 5 H I J E JH J I E JH J I E = JH 1 E I J H J Keep it Change it H K Flip it = H J 1 K H 3. JH I 1 E J = KE N = 7 O N Example 2: Solve 4 H J H K 1. Turn the mixed number into an improper fraction. 1. P H J K 2. Follow the Keep Change Flip method. 3. Multiply numerator by numerator & denominator by denominator. 2. P H J K = P J 1 K H 4. If the answer is an improper fraction turn it back into a mixed number. 3. P 1 K J H = FI J = JO H = 27

37 Dividing Fractions by Mixed Numbers What is the keep change flip method? When dividing a fraction by another fraction, you can use the keep change flip method to turn the division problem into a multiplication problem. Example 1: Solve H J 1 H N H P J N H P J N = H 1 N J P H J Change it H K = H J 1 K H 3. H J 1 N P = N HN = I P Keep it Flip it 1. Turn the mixed number into an improper fraction. Example 2: Solve E F 2 E I 1. E HH F I 2. Follow the Keep Change Flip method. 3. Multiply numerator by numerator & denominator by denominator. 4. If the answer is an improper fraction turn it back into a mixed number E HH F I E 1 I F HH = = HJ FF E 1 I F HH

38 7 th Grade Math Anchor Charts Proportional Relationships I Math

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69 7 th Grade Math Anchor Charts Expressions & Equations I Math

70 Evaluating Algebraic Expressions What is an algebraic expression? An algebraic expression is a mathematical sentence that includes variables, numbers and operations. Algebraic expressions do NOT have equal signs. 1. Replace variables with the given values. (remember when a variable has a coefficient that means you must multiply) 2. Solve using order of operations. Example 1: Evaluate 3x + 2 when x = x Example 2: Evaluate 5a J 3b when a = 5 and b = a J 3b J J

71 Writing Expressions Some important math words Add: more than plus sum total increased combined multiply: times multiply product of subtract: minus less than difference decreased deduct loss divide: divided by over quotient 1. Reading the expression from left to write, circle key words and numbers. 2. Replace each key word and number with an algebraic symbol or number. Replace unknowns with variables. Example 1: three years less than two times the age of Ben. 1. three years less than two times the age of Ben B 3 *Ben is an unknown, so the variable B is used. Example 2: 78 combined with three times the first five combined with three times the first five

72 Commutative Property What is the commutative property? The commutative property is used for both multiplication and addition problems. The commutative property of multiplication states that as long as the only operation is multiplication, the order does not matter. a 1 b = b 1 a The commutative property of addition states that as long as the only operation is addition, the order does not matter. a + b = b + a Example of commutative property of multiplication: can be written as: and yield the same solution Example of commutative property of addition: can be written as: and yield the same solution

73 Associative Property What is the associative property? The associative property is used for both multiplication and addition problems. The associative property of multiplication states that as long as the only operation is multiplication, the values can be grouped in any way with parentheses. a 1 b 1 c = a 1 (b 1 c) The commutative property of addition states that as long as the only operation is addition, the values can be grouped in any way with parentheses. (a + b) + c = a + (b + c) Example of associative property of multiplication: 5 1 (7 1 4) can be written as: (5 1 7) 1 4 and yield the same solution Example of commutative property of addition: (24 + 6) + 32 can be written as: 24 + (6 + 32) and yield the same solution

74 Additive Identity Property: Additional Properties The additive Identity Property states that anytime 0 is added to a number, the answer is that number = = 25 x + 0 = x Multiplicative Identity Property: The Multiplication Identity Property states that anytime a number is multiplied by 1, the answer is that number = = 62 a 1 1 = a Multiplicative Property of Zero: The Multiplicative Property of Zero states that anytime a number is multiplied by 1, the answer is = = 0 a 1 0 = 0

75 Order of Operations What is the order of operations? The order of operations is the order in which an expression is solved. FIRST parentheses SECOND exponents THIRD multiplication and division working left to right FOURTH addition and subtraction working left to right p-e-m-d-a-s Solve E Solve 52 i10 2ƒ8 i7 5 J *Because there is no parentheses we skip to exponents* *8 replaces 2 E * *30 replaces 5 1 6* *32 replaces * *2 replaces 32 30* *Simplify the numerator and the denominator following order of operations.* *25 replaces 5 J in the numerator and 10 replaces (2+8) in the denominator.* *15 replaces and 3 replaces 10 7.* 5 *The last step when a fraction is involved is to divide the numerator by the denominator.*

76 Distributive Property What is the distributive property? a b + c = ab + ac a b c = ab ac The distributive property is the process in which the number outside of the parentheses is multiplied by each term inside the parentheses. 1. Expand the expressions by distributing the number outside of the parentheses to each term inside. 2. Simplify Example 1: 3(x + 4) 3(x + 4) 3 1 x x + 12 Example 2: 6 x x x x x 5

77 Distributive Property with Rational Coefficients What is the distributive property? a b + c = ab + ac a b c = ab ac Example 1: 1 (x + 14) 2 H J (x + 14) H J 1 x + H J = 7 1 = 7 The distributive property is the process in which the number outside of the parentheses is multiplied by each term inside the parentheses. 1. Expand the expressions by distributing the number outside of the parentheses to each term inside. 2. Simplify H J x + 7 Example 2: 1 (x 8) 4 H I (x 8) H I 1 x H I 1 8 H I x = 2 1 = 2

78 Combining Like Terms What are like terms? Like terms are terms that contain the same variables raised to the same power. Like Terms 1. Use circles, underlines, or a highlighter to identify like terms. 2. Combine like terms. Unlike Terms 3x 5x 6x 2y 2a J a J 3x J 3x Example 1: 3x 2 + 4x + 5 3x 2 + 4x + 5 3x + 4x = x = 3 x + 3 Example 1: 7x 2y 2x + 2y 4 7x 2y 2x + 2y 4 2y + 2y = 0 7x 2x = 5x 5x 4

79 Adding Linear Expressions What is a linear expression? A linear expression is an algebraic expression in which there is only one variable and it is raised to the first power. Linear Expression 3x 1. Apply the distributive property where it is necessary. 2. Simplify. 3. Combine any like terms. 4. Simplify. Nonlinear Expression 5ab 6x + 4 5y E + 2 x 5 x J 1 Example 1: 3 x 5 + 2(2x + 4) 1. 3 x 5 + 2(2x + 4) x x x x x x x + 4x x 7

80 Subtracting Linear Expressions What is a linear expression? A linear expression is an algebraic expression in which there is only one variable and it is raised to the first power. Linear Expression 3x 1. Apply the distributive property where it is necessary. 2. Simplify. 3. Combine any like terms. 4. Simplify. Nonlinear Expression 5ab 6x + 4 5y E + 2 x 5 x J 1 Example 1: 4 2x + 3 (5x 9) x + 3 (5x 9) x x x x ( 9) 8x x x x x 5x x + 21

81 Greatest Common Factor What is a factor? A factor is a number that can be multiplied by a value to get another number. The greatest common factor of two numbers is the largest factor that can applies to both numbers. 1. Create a factor tree for each monomial. 2. Write the prime factorization. 3. Circle the common factors. 4. Identify the greatest common factor. Example 1: 18xy, 45x 1. 18xy 45x 18 xy 45 x xy: x 1 y 45x: x 3. 18xy: x 1 y 45x: x x = 9x x y

82 Factoring Linear Expressions What is a factor? A factor is a number that can be multiplied by a value to get another number. The greatest common factor of two numbers is the largest factor that can applies to both numbers. 1. Create a factor tree for each term. 2. Write the prime factorization. 3. Circle the common factors. 4. Factor out the greatest common factor from each each term. (Think opposite of distributive property) Example 1: 15y + 42xy 1. 15y 42xy 2. 15y: y 42xy: x 1 y 3. 15y: y 42xy: x 1 y y = 3y 15 y 42 xy y(5 + 14x) x 2 3 y

83 One Step Equations: Addition What is a one step equation? An algebraic equation is similar to an algebraic expression. They both include variables, numbers, and operations. However, an equation has an = sign, unlike an expression. A one- step equation is an equation in which the variable can be solved for in one step. 1. Rearrange the equation so the variable comes first. 2. Create a do/undo table. 3. Apply the undo to both sides of the equation. Example 1: 5 + x = x + 5 = x + 5 = x = 7 Example 1: 4 + r = r + 4 = DO UNDO +5-5 DO UNDO r + 4 = r = 14

84 One Step Equations: Subtraction What is a one step equation? An algebraic equation is similar to an algebraic expression. They both include variables, numbers, and operations. However, an equation has an = sign, unlike an expression. A one- step equation is an equation in which the variable can be solved for in one step. 1. Rearrange the equation so the variable comes first. 2. Create a do/undo table. 3. Apply the undo to both sides of the equation. Example 1: f 8 = 3 1. f 8 = f 8 = x = 11 Example 1: 6 + x = 4 1. x 6 = 4 2. DO UNDO DO UNDO x 6 = x = 10

85 One Step Equations: Multiplication What is a one step equation? An algebraic equation is similar to an algebraic expression. They both include variables, numbers, and operations. However, an equation has an = sign, unlike an expression. A one- step equation is an equation in which the variable can be solved for in one step. 1. Create a do/undo table. 2. Apply the undo to both sides of the equation. Example 1: 5x = x = x = 3 Example 1: 2x = DO UNDO DO UNDO 1 ( 2) ( 2) 2. 2x = 14 ( 2) ( 2) x = 7

86 One Step Equations: Division What is a one step equation? An algebraic equation is similar to an algebraic expression. They both include variables, numbers, and operations. However, an equation has an = sign, unlike an expression. A one- step equation is an equation in which the variable can be solved for in one step. 1. Create a do/undo table. 2. Apply the undo to both sides of the equation. Example 1: 8 = I = I = I 1 4 x = 32 Example 1: F = F = 11 DO UNDO DO UNDO F 1 5 = x = 11

87 One Step Equations: Fraction Coefficients What is a one step equation? Example 1: E I x = 6 An algebraic equation is similar to an algebraic expression. They both include variables, numbers, and operations. However, an equation has an = sign, unlike an expression. A one- step equation is an equation in which the variable can be solved for in one step. 1. DO E x = 6 I E E I I UNDO Create a do/undo table. 2. Apply the undo to both sides of the equation. 3. Simplify 3. E I x E I = 6 E I E I x 1 I E = 6 1 I E E x 1 I = K 1 I I E H E x = JI E x = 8

88 Basic Two-Step Equations What is a two step equation? An algebraic equation is similar to an algebraic expression. They both include variables, numbers, and operations. However, an equation has an = sign, unlike an expression. A two- step equation is an equation that takes two steps to solve for the variable. 1. Create a do/undo table. 2. Apply the undo to both sides for any addition or subtraction. 3. Apply the undo to both sides for any multiplication or division. Example 1: 2x + 8 = DO 2. 2x + 8 = x = x = x = 3 UNDO

89 Two-Step Equations with Rational Numbers What is a two step equation? Example 1: 3 x 10 = 2 4 An algebraic equation is similar to an algebraic expression. They both include variables, numbers, and operations. However, an equation has an = sign, unlike an expression. 1. DO UNDO A two- step equation is an equation that takes two steps to solve for the variable. 1. Create a do/undo table. 2. Apply the undo to both sides for any addition or subtraction. 3. Apply the undo to both sides for any multiplication or division. 2. E I 3. x 10 = E I E I x = 12 x = 12 E E I I x = 12 I E = 16 Keep Change Flip

90 Multi-step Equations Method 1 What is a multi step equation? A multi- step equation is an equation that takes more than two steps to solve for the variable. 1. Create a do/undo table. 2. Divide both sides by the rational coefficient. 3. Apply the undo to both sides for any addition or subtraction. 4. Apply the undo to both sides for any multiplication or division. Example 1: H J H J DO 1 2 3x + 5 = 16 3x + 5 = 16 H J 3x + 5 = x = x = 9 UNDO Keep Change Flip

91 Multi-step Equations Method 2 What is a multi step equation? A multi- step equation is an equation that takes more than two steps to solve for the variable. Example 1: 1 2 DO x + 5 = 16 UNDO Apply the distributive property. 2. Create a do/undo table. 3. Apply the undo to both sides for any addition or subtraction. 4. Apply the undo to both sides for any multiplication or division. H J 3x + 5 = 16 H J 1 3x + H J 1 5 = 16 E J x + F J = 16 F J E J E J F J x = JO J E J x = 9 Keep Change Flip

92 Multi-step Equations Variables on Both Sides What is a multi step equation? A multi- step equation is an equation that takes more than two steps to solve for the variable. 1. Apply the distributive property if necessary. 2. Move one of the terms with a variable to the opposite side using addition or subtraction. 3. Create a do/undo chart. 4. Apply the undo to both sides for any addition or subtraction. 5. Apply the undo to both sides for any multiplication or division. Example 1: 2(4x + 8) = 10x (4x + 8) = 10x 6 8x + 16 = 10x x + 16 = 10x 6 8x 8x 16 = 2x 6 3. DO = 2x = 2x = 2x 2 2 x = 11 UNDO

93 Graphing Inequalities What do I need to know to graph an inequality? > greater than < less than greater than or equal to less than or equal to Steps to graph an inequality: 1. Create a number line with at least 5-6 dashes. 2. Start by using the solution of the inequality as the center dash. Then label the remaining dashes. 3. Correctly use either or to identify the solution on the number line. Example 1: Graph n > Example 1: Graph x If the variable represents numbers: greater than the solution then draw an arrow pointing to the right. Fewer than the solution then draw and arrow pointing to the left

94 Writing Inequalities What are some key words to look for when writing an inequality? > Greater than More than < Less than Fewer than Greater than or equal to More than or equal to Less than or equal to Fewer than or equal to At least Minimum Is no less than At most Maximum Is no more than 1. Circle all numbers and key words. 2. Choose the correct inequality symbol that will be used based on the key words. 3. Write an inequality using the correct inequality symbol and a variable for unknown information. Example 1: Sophia has less than $50 in her bank account. 1. Sophia has less than $50 in her bank account. 2. for less than we use: < 3. x < 50 Example 2: You must be at least 13 years old to go to the skating ring without an adult. 1. You must be at least 13 years old to go to the skating ring without an adult. 2. for at least we use: *This mean x represents all numbers less than 50.* *This mean x represents all numbers greater than or equal to 13.* 3. x 13

95 One-Step Inequalities: Addition What is an inequality? An inequality is similar to an equation. Instead of an equal sign, an inequality symbol is used. Inequalities symbols include: > greater than < less than greater than or equal to less than or equal to Your goal when solving inequalities is to get the variable on one side by itself. Example 1: x + 5 > is being added to x must be subtracted. 4. x + 5 > x > 3 DO UNDO If necessary, rearrange the equation so the variable comes first. 2. Identify what is being done to the variable. In this case you would be adding. 3. Identify how this can be undone. In order to undo addition you must subtract. 4. Apply the results from step 2 to BOTH sides of the equation. Example 2: 4 + r r 36 r is being added to r must be subtracted. 4. r r 29 DO UNDO + 7-7

96 One-Step Inequalities: Subtraction What is an inequality? An inequality is similar to an equation. Instead of an equal sign, an inequality symbol is used. Inequalities symbols include: > greater than < less than greater than or equal to less than or equal to Your goal when solving inequalities is to get the variable on one side by itself. 1. Identify what is being done to the variable. In this case you would be subtracting. 2. Identify how this can be undone. In order to undo subtraction you must add. 3. Apply the results from step 2 to BOTH sides of the equation. Example 1: x is being subtracted from x must be added. 3. x x 20 Example 2: x 3 < is being subtracted from x must be added. 3. x - 3 < x < 7 DO UNDO DO UNDO

97 One-Step Inequalities: Multiplication What is an inequality? An inequality is similar to an equation. Instead of an equal sign, an inequality symbol is used. Inequalities symbols include: > greater than < less than greater than or equal to less than or equal to Your goal when solving inequalities is to get the variable on one side by itself. 1. Identify what is being done to the variable. In this case you would be multiplying. 2. Identify how this can be undone. In order to undo multiplication you must divide. 3. Apply the results from step 2 to BOTH sides of the equation. Example 1: 5m < m is being multiplied by must be divided. 3. 5m < m < 8 Example 2: 12x > x is being multiplied by must be divided x > x > 12 DO UNDO DO UNDO

98 One-Step Inequalities: Division What is an inequality? An inequality is similar to an equation. Instead of an equal sign, an inequality symbol is used. Inequalities symbols include: > greater than < less than greater than or equal to less than or equal to Your goal when solving inequalities is to get the variable on one side by itself. 1. Identify what is being done to the variable. In this case you would be dividing. 2. Identify how this can be undone. In order to undo division you must multiply. 3. Apply the results from step 2 to BOTH sides of the equation. Example 1: 2 x 4 1. x is being divided by must be multiplied I x Example 1: h h is being divided by must be multiplied. 3. y F h 30 DO UNDO DO UNDO 5 1 5

99 One-Step Inequalities: Multiply & Divide Negatives What is an inequality? An inequality is similar to an equation. Instead of an equal sign, an inequality symbol is used. Inequalities symbols include: > greater than < less than greater than or equal to less than or equal to Example 1: 6 3x 1. x is being multiplied by must be divided x x DO UNDO Your goal when solving inequalities is to get the variable on one side by itself. 1. Identify what is being done to the variable using a do/undo table. 2. Undo the operation. 3. Apply the results from step 2 to BOTH sides of the equation. 4. Reverse the inequality when multiplying or dividing by a negative. Example 1: h i5 < 4 1. h is being divided by must be multiplied. 3. y F < h > - 20 DO UNDO 5 1 5

100 Two-Step Inequalities What is a two- step inequality? An inequality is similar to an equation. Instead of an equal sign, an inequality symbol is used. Inequalities symbols include: > greater than < less than greater than or equal to less than or equal to Your goal when solving inequalities is to get the variable on one side by itself. A two- step inequality takes two steps for this to happen. 1. Create a do/undo chart Example 1: 2x + 5 < DO 2. 2x + 5 < x < x < x < 8 UNDO Undo any addition/subtraction. 3. Undo any multiplication/division.

101 Two-Step Inequalities: Multiply & Divide Negatives What is a two- step inequality? An inequality is similar to an equation. Instead of an equal sign, an inequality symbol is used. Inequalities symbols include: > greater than < less than greater than or equal to less than or equal to Your goal when solving inequalities is to get the variable on one side by itself. A two- step inequality takes two steps for this to happen. 1. Create a do/undo chart 2. Undo any addition/subtraction. 3. Undo any multiplication/division. 4. Reverse the symbol. Example 1: 3x 9 < DO 2. 3x 9 < x < x < 27 ( 3) ( 3) x < 9 4. x > ( 3) UNDO ( 3)

102 7 th Grade Math Anchor Charts Geometry I Math

103 Classifying Angles What is an angle? An angle is made up of two rays that are connected at a vertex. acute angle acute angles are less than 90. vertex angle obtuse angle right angle obtuse angles are greater than 90 but less than 180. right angles are 90 angles. They can be identified by the L shape, and sometimes have a box. straight angle Straight angles are exactly 180. They can be identified by the straight line formed.

104 Vertical Angles What are vertical angles? A pair of vertical angles are formed when two lines cross. They are directly across from one another and never share a line. pair of vertical angles pair of vertical angles Vertical angles are congruent, meaning they have the same value. Steps to determine a missing measure: 1. Write an equation setting a pair of vertical angles equal to each other. 2. Solve for any missing variables, following steps to solve one and two- step equations. Example 1: Determine the value for x using the figure. 1. 3x = x 3 = 18 3 x = 6 Example 2: Determine the value for x using the figure. 1. 2x + 18 = 32 3x x = x = 14 x = x

105 Adjacent Angles What are adjacent angles? A pair of adjacent angles can be formed when two lines cross. They are directly next to one another, sharing a line. pair of adjacent angles pair of adjacent angles pair of adjacent angles pair of adjacent angles Steps to determine a missing measure: 1. If the sum of the pair of adjacent angles is known (if they form a straight line, the sum is 180 ) then an equation can be written. 2. Solve for any missing variables, following steps to solve one and two- step equations. Example 1: Determine the value for x using the figure. 1. 3x + 75 = x = x = 105 x = 35 Example 2: Determine the value for x using the figure. 4x x = x 12 = 180 4x = x = 192 x = 48 3x 75 30

106 Complementary Angles What are complementary angles? Complementary angles have a sum of 90. When two angles form a right angle, they are complementary. Example 1: Determine if the two angles are complementary or not = Yes, the angles are complementary because the sum is 90. Example 2: Determine the value for x. 1. Write an equation in the form A + B = Solve for the missing variable using your rules for one and two- step equations. 1. x + 2x = x = 90 x = 30 2x x

107 Supplementary Angles What are supplementary angles? Example 1: Determine if the two angles are supplementary or not. Supplementary angles have a sum of When two angles form a straight angle, they are supplementary. 1. Write an equation in the form A + B = Solve for the missing variable using your rules for one and two- step equations = Yes, the angles are supplementary because the sum is 180. Example 2: Determine the value for x. 4x x 4 + 3x + 9 = x + 5 = 180 7x = 175 x = 25 3x + 9

108 Classifying Triangles by Angles What is a triangle? a triangle is a figure with three sides and three angles. The three angles of a triangle ALWAYS add up to 180. acute triangle three acute (less than 90 ) angles right triangle one right (90 ) angle obtuse triangle one obtuse (greater than 90 ) angle

109 Classifying Triangles by Sides What is a triangle? a triangle is a figure with three sides and three angles. The three angles of a triangle ALWAYS add up to 180. isosceles triangle two congruent sides equilateral triangle three congruent sides scalene triangle NO congruent sides

110 Classifying Quadrilaterals What is a quadrilateral? A quadrilateral is a shape with four angles and four sides. square four equal sides & four 90 angles rectangle rhombus two sets of equal sides & four 90 angles four equal sides & two sets of equal angles

111 Classifying Quadrilaterals What is a quadrilateral? A quadrilateral is a shape with four angles and four sides. trapezoid a pair of parallel bases that are different lengths parallelogram two sets of equal sides & two sets of irregular equal angles all four angles & all four sides are different

112 Missing Angle Measure of a Triangle What is a triangle? a triangle is a figure with three sides and three angles. Example 1: Determine the missing angle measure x + 90 = The three angles of a triangle ALWAYS add up to x = x = 42 x 1. Set up an equation in the form A + B + C = Solve for the missing variable. Example 2: Determine the value of the variable. 1. x + x + 2x = x = x = 45 x 2x x

113 Area of a Triangle What is the formula for finding the area of a triangle? A = area b = base h = height A = 1 2 b 1 h 1. Identify each part of the triangle. 2. Use substitution to plug the given values into the formula above. 3. Solve for the area. Example 1: Determine the area of the given triangle. 12cm. 1. b = 20 cm. h = 12 cm. 2. A = H J 20cm A = 120cm J

114 Area of a Triangle What is the formula for finding the area of a triangle? A = area b = base h = height A = 1 2 b 1 h 1. Identify each part of the triangle. 2. Use substitution to plug the given values into the formula above. 3. Solve for the area. Example 1: Determine the area of the given triangle. 1. b = 9 ft. h = 5 ft. 5 ft. 2. A = H J ft. 3. A = 22.5 ft J

115 Area of a Quadrilateral: Squares & Rectangles What is the formula for finding the area of a square or rectangle? A = b 1 h - or- A = l 1 w Example 1: Determine the area of the given rectangle. A = area b = base h = height l = length w = width 16 in. 6 in. 1. Identify each part of the quadrilateral. 2. Use substitution to plug the given values into the formula above. 3. Solve for the area. 1. b = 16 in. h = 6 in. 2. A = A = 96 in J

116 Area of a Quadrilateral: Parallelograms What is the formula for finding the area of a parallelogram? A = b 1 h A = area b = base h = height 1. Identify each part of the quadrilateral. 2. Use substitution to plug the given values into the formula above. 3. Solve for the area. Example 1: Determine the area of the given parallelogram. 10 ft. 1. b = 14 ft. h = 10 ft. 14 ft. 2. A = ft. 3. A = 140 ft J

117 Area of a Quadrilateral: Trapezoids What is the formula for finding the area of a trapezoid? A = H J h(b H + b J ) A = area b H = top base b J = bottom base h = height 1. Identify each part of the quadrilateral. 2. Use substitution to plug the given values into the formula above. 3. Solve for the area. Example 1: Determine the area of the given trapezoid. 1. b H = 4 cm. b J = 8 cm h = 6 cm. 2. A = H J 6 cm. 4 cm. 8 cm. 1 6(4 + 8) 3. A = H cmj J A = 36cm J

118 Area of a Composite Figure What is a composite figure? A composite figure is an irregular figure that can be broken up into regular figures. 1. Break the shape up into regular figures. 2. Determine the area of each regular figure. 3. Determine the sum for all areas of the regular figures. Example 1: Determine the area of the composite figure cm 8 cm 8 cm 2. Triangle: A = H J bh A = H J A = 20 cm J Rectangle: A = bh A = A = 24 cm J 3. Total Area = 20 cm J + 24 cm J Area = 44 cm J

119 Area of a Composite Figure What is a composite figure? A composite figure is an irregular figure that can be broken up into regular figures. Example 1: Determine the area of the composite figure m 5 m 5 m 2 m 4 m 4 m 6 m 1. Break the shape up into regular figures. 2. Determine the area of each regular figure. 3. Determine the sum for all areas of the regular figures. 2. Rectangle 1: A = bh A = A = 10 m J Rectangle 2: A = bh A = A = 8 m J 3. Total Area = 10 m J + 8 m J Area = 18 m J

120 Radius and Diameter of a Circle What are radius and diameter of a circle? The radius (r) is the distance from the edge of a circle to the center. The diameter (d) is the distance from one edge to the other on a circle when crossing through the center. diameter d = 2r r = d 2 Example 1: If the diameter is 8 m. determine the radius. 1. r = C J 2. r = 4m r = N J m Example 2: If the radius is 17 cm. determine the diameter. 1. d = 2r d = cm 2. d = 34 cm J

121 Area of a Circle What is the area of a circle? The area of a circle is the space that the circle takes up. The shaded region makes up the area. The formula to determine the area is: A = πr 2 1. Identify the radius. 2. Substitute the radius into the formula for area. 3. Solve for the area. Example 1: Determine the area of a circle with a 4 cm. radius. 1. r = 4 cm. 2. A = π 1 r J A = J 3. A = A = 50.24cm J Example 1: Determine the area of a circle with a 12 m. diameter. 1. d = 12 m. so r = 6 m. 2. A = π 1 r J A = J 3. A = A = m J

122 Circumference of a Circle What is the circumference of a circle? The circumference is the total edge length of circle. The highlighted region is the circumference. The formula used is: C = 2πr - or- C = πd 1. Identify the radius or diameter. 2. Substitute into one of the formulas for circumference 3. Solve for circumference. Example 1: Determine the circumference when the radius is 4 cm. 1. r = 4 cm. 2. C = 2πr C = C = C = cm Example 1: Determine the circumference when the diameter is 12 m. 1. d = 12 m. 2. C = πd C = C = cm

123 Area of a Semi Circle What is the area of a semi circle? The area of a circle is the space that the circle takes up. A semi circle is half of a circle. The formula to determine the area of a semi circle is: A = 1 2 πr2 1. Identify the radius. 2. Substitute the radius into the formula for area. 3. Solve for the area. Example 1: Determine the area of a semi circle with a 3 cm. radius. 1. r = 3 cm. 2. A = H π 1 J rj A = H J J 3. A = H A = cmj J Example 1: Determine the area of a semi circle with a 14 m. diameter. 1. d = 14 m. so r = 7 m. 2. A = H π 1 J rj A = H J J 3. A = H A = mj J

124 Area of a Shaded Region of a Circle What is the area of a circle? The area of a circle is the space that the circle takes up. The shaded region makes up the area. The formula to determine the area is: A = πr 2 1. Determine the total area of the circle. 2. Determine the area of any unshaded region. 3. Subtract the area of the unshaded region from the area of the circle. Example 1: Determine the area of the shaded region. 1. A = πr J A = π 1 2 J A = in J 2 in. 2. The area of the unshaded region is H of the whole circle. I unshaded region = H inj I = 3.14in J = 9.42in J

125 3D Figures cube rectangular prism triangular prism cone cylinder sphere hemisphere triangular pyramid rectangular pyramid

126 Cross Sections of 3D Figures vertical slicing of a cube horizontal slicing of a cube vertical slicing of a cone horizontal slicing of a cone vertical slicing of a cylinder horizontal slicing of a cylinder

127 Cross Sections of 3D Figures vertical slicing of a rectangular prism horizontal slicing of a rectangular prism vertical slicing of a rectangular prism horizontal slicing of a hemisphere vertical slicing of a rectangular pyramid horizontal slicing of a rectangular pyramid

128 Cross Sections of 3D Figures vertical slicing of a sphere horizontal slicing of a sphere vertical slicing of a triangular prism horizontal slicing of a triangular prism vertical slicing of a triangular pyramid horizontal slicing of a triangular pyramid

129 Volume of a Rectangular Prism Formula for the volume of a rectangular prism: Example 1: Determine the volume of the rectangular prism. V = l 1 w 1 h l = length w = width h = height 1. Identify each of the dimensions. 2. Substitute the dimensions into the given formula. 1. length = 2 cm. width = 10 cm. height = 6 cm. 2. V = V = 120 cm E 6 cm. 10 cm. 2 cm. 3. Solve using order of operations.

130 Volume of a Triangular Prism Formula for the volume of a triangular prism: Example 1: Determine the volume of the triangular prism. V = B 1 h B = area of the triangular base h = height area of a triangle: A = H J 1 b 1 h 1. Identify each of the dimensions. Determine the B if necessary. 2. Substitute the dimensions into the given formula. 3. Solve using order of operations. 1. b = 5 cm. h of the triangle = 6 cm. B = H = 15 cmj J h of the prism = 9 cm. 2. V = cm. 6 cm. 9 cm. 3. V = 135 cm E

131 Volume of other Prisms Formula for the volume of a prism: V = B 1 h B = area of the base h = height 1. Identify each of the dimensions. 2. Substitute the dimensions into the given formula. Example 1: Determine the volume of the prism. 24 cm J 1. B = 24 cm J h = 5 cm. 2. V = V = 120 cm E 5 cm. 3. Solve using order of operations.

132 Surface Area of a Rectangular Prism What is surface area? The surface area is the area that covers the surface of a shape. The formula to determine the surface area of a rectangular prism is: SA = 2lw + 2lh + 2wh 1. Identify each of the dimensions. 2. Substitute the dimensions into the given formula. 3. Solve using order of operations. Example 1: Determine the surface area of the rectangular prism. 1. l = 2 cm. w = 10 cm. h = 6 cm. 2. SA = SA = SA = 184 cm J 6 cm. 10 cm. 2 cm.

133 Surface Area of a Triangular Prism What is surface area? The surface area is the area that covers the surface of a shape. To determine the surface area of a triangular prism, you must find the area of every surface. AREA FORMULAS: Triangle > A = H 1 l 1 w J Rectangle > l 1 w 1. Identify the shape of each surface, and the dimensions. 2. Determine the area of each shape. 3. add Example 1: Determine the surface area of the triangular prism. There are 2 of these in. 6 in. There are 2 of these. 12 in. 7 in. 5 in. 2. Area of each triangle: H J = 15inJ Area of rectangle 1: = 84 in J Area of rectangle 2: = 72 in J in J + 15 in J + 84 in J + 84 in J + 72 in J SA = 270in J 12 in. 6 in.

134 Surface Area of Other Prisms What is surface area? The surface area is the area that covers the surface of a shape. To determine the surface area of a triangular prism, you must find the area of every surface. AREA FORMULAS: Rectangle > l 1 w 1. Identify the shape of each surface, and the dimensions. 2. Determine the area of each shape. 3. add Example 1: Determine the surface area of the prism cm J There are 5 of these. 3 cm. 5 cm. 32 cm J There are 2 of these. 3 cm. 5 cm. 2. area of the rectangle: = 15 cm J area of the pentagon = 32 cm J SA = 139 cm J

135 Volume of a Pyramid What is a pyramid? A pyramid is a 3D figure with a flat base, and triangular sides that meet at a point. V = 1 3 Bh Example 1: Determine the volume of the triangular pyramid. 1. h = 7ft B = 24 ft J 7 ft B = area of the base & h = height 1. Identify each of the dimensions. 2. Substitute the dimensions into the given formula. 2. V = H E V = V = 56 ft E 24 ft J 3. Solve using order of operations.

136 Surface Area of a Pyramid What is surface area? The surface area is the area that covers the surface of a shape. To determine the surface area of a triangular prism, you must find the area of every surface. AREA FORMULAS: Triangle > A = H 1 l 1 w J Rectangle > l 1 w 1. Identify the shape of each surface, and the dimensions. 2. Determine the area of each shape. 3. add Example 1: Determine the surface area of the pyramid. 8 m 1. 4 m There are 4 of these. 4 m 4 m 8 m 4 m 2. area of the rectangle: = 16 m J area of the triangle: H = 16 mj J SA = 80m J

137 Volume of a Composite 3D Figure What is a composite figure? A composite figure is an irregular shape made up of regular shapes. Example 1: Determine the volume of the composite figure. 6 in shape 1: rectangular prism 3 in 3 in 3 in 1. Identify each of the regular shapes that make up the composite figure. 2. Identify the dimensions of each regular shape. 3. Determine the volume of each regular figure. 4. Add the volumes together. 6 in shape 2: rectangular prism 6 in 3 in 3 in 3 in shape 1: V = = 54 in E shape 2: V = = 54 in E Total volume = 108 in E 6 in

138 Surface Area of a Composite Figure What is a composite figure? A composite figure is an irregular shape made up of regular shapes. Example 1: Determine the surface area of the composite figure. shape 1: rectangular prism 3 in 3 in 6 in 3 in 3 in 3 in 1. Identify each of the regular shapes that make up the composite figure. 2. Identify the dimensions of each regular shape. 3. Determine the surface area of each regular figure and subtract the area where the figures overlap. 4. Add the surface areas together. 6 in shape 2: rectangular prism 6 in 3 in 3 in 3 in overlap area shape 1: SA = = 90 in J 90 in J 9 in J = 81 in J shape 2:SA = = 90 in J 90 in J 9 in J = 81 in J Total surface area = 162 in J 6 in 9 in J 3 in 3 in

139 7 th Grade Math Anchor Charts Statistics I Math

140 Mean What is the mean? Another word for mean is the average value of a set of numbers. Mean, median, and mode are all ways to find the center of a set of data; however, they do not always yield the same number. 1. Find the sum of all numbers in a data set. 2. Count how many numbers are in the data set. 3. Divide the sum by the amount of numbers involved. Example 1: Find the mean of the following numbers: 18, 12, 37, 16, 31, = There are 6 numbers in the data set = 23 Example 2: What was Kevin s average math test score if his scores were: 89, 94, 72, 86, = There area 5 numbers in the data set = 83.6

141 Median What is the median? The median is the number directly in the middle of a data set when the numbers are ordered from least to greatest. When there are two middle numbers in a data set, the mean of those two numbers is used for the median. 1. Order the set of numbers from least to greatest. 2. Cross out the number farthest to the left and farthest to the right. Example 1: Find the median of the following numbers: 18, 12, 37, 16, 31, , 16, 18, 24, 31, , 16, 18, 24, 31, , 16, 18, 24, 31, = = 21 Example 2: Find the median of the following numbers: 76, 51, 84, 68, 59, 64, , 59, 64, 68, 76, 76, Repeat step two until there is only one or two numbers remaining in the center. 4. If there are two numbers remaining you must find the mean of those numbers , 59, 64, 68, 76, 76, , 59, 64, 68, 76, 76, 84 The median is 68.

142 Mode What is the mode? Mode is the number that appears the most frequently in a data set. If there are multiple numbers that appear several times in a set of numbers it is possible to have more than one mode. It is also possible to not have a mode at all if there are no numbers that appear more often. Of the three different methods to find the center value, mode tends to be the one used the least. 1. Order the set of numbers from least to greatest. 2. Identify the number(s) that appear most frequently. Example 1: Identify the mode of the following numbers: 19, 12, 18, 16, 11, , 12, 16, 18, 18, is the mode because it occurs more frequently than the other numbers. Example 2: Identify the mode of the following numbers: 27, 31, 27, 35, 26, , 27, 27, 31, 31, Because both 27 and 31 are listed twice, and there aren t any other numbers that occur more frequently, they are BOTH the mode.

143 Range What is the range? The range is the difference between the greatest and smallest numbers of a set of date. 1. Identify the greatest number. 2. Identify the smallest number. 3. Find the difference between the two numbers identified. Example 1: Determine the range of the data set: , 12, 37, 16, 31, = 25 The range is 25. Example 2: Determine the range of the data set: 89, 94, 72, 86, = 22 The range is 22.

144 Measures of Central Tendency What are measures of central tendency? Measures of central tendency include mean, median and mode. Solution: Mean based on 5 test scores: (OH ƒ PJ ƒ NO ƒ FI ƒ OH) Mean based on 6 test scores: F = 77 You may be asked questions about all measures of central tendency. Below is an example of a question and to the right is the example solved. Example: 1. Paul scores a 71%, 92%, 87% 54%, and 71% on his math tests this year. If Paul scores a 94% on his 6 th test, which measure of central tendency changes the most? (OH ƒ PJ ƒ NO ƒ FI ƒ OHƒ PI) Median based on 5 test scores: K = , 71, 71, 87, Median based on 6 test scores: 54, 71, 71, 87, 92, Mode based on 5 test scores: 54, 71, 71, 87, Mode based on 6 test scores: 54, 71, 71, 87, 92, The value of the median changes the most.

145 When to use Mean, Median, or Mode How do I choose between mean, median, and mode? Mean: best to use when there aren t any outliers present. Median: best to use when there are outliers but no large gaps within the data. Mode: best to use when there are numbers occurring much more frequently than others. Example 1: Would you use mean, median, or mode for the following data set? 24, 86, 10, 31, 27, 18, 12 Because there is an obvious outlier (86) the best type of measure would be to find the median. Example 2: Would you use mean, median, or mode for the following data set? 45, 67, 45, 8, 45, 56, 49 Although there is an obvious outlier (8), the number 45 occurs more than any other number. Therefore, the best type of measure would be to find the mode. Example 3: Would you use mean, median, or mode for the following data set? 89, 87, 91, 76, 90, 86 There are no outliers, and all of the numbers are only listed 1 time. Therefore, the best type of measure would be to find the mean.

146 Quartiles What are measures of variation? After a set of data is ordered from least to greatest (or greatest to least) measures of variation are used to examine the data by separating it into quartiles. Measures of Variation include: Quartiles: the four equal parts a set of data is divided into. First Quartile: the median of the values less than the median of the whole data set. Third Quartile: the median of values greater than the median of the data set. Interquartile Range: The distance between the first and third quartiles. Range: The distance between the biggest and smallest number in the data set. Example 1: Find the measures of variation for the data. Average High Temperature in New Orleans in (rounded) January 62 February 65 March 72 April 78 May 85 June 89 July 91 August 91 September 87 October 80 Range: biggest- smallest = 29 Median: 62, 65, 65, 71, 72, 78, 80, 85, 87, 89, 91, = = 79 First Quartile: 62, 65, 65, 71, 72, = = 68 November December Third Quartile: 80, 85, 87, 89, 91, 91 Interquartile Range: = = = 88 62, 65, 65, 71, 72, 78, 80, 85, 87, 89, 91, 91 Quartile 1 median Quartile 3

147 Mean Absolute Deviation What is mean absolute deviation? The mean absolute deviation of a set of numbers is the average (mean) of the distance from each number in the data set and the mean of all the numbers. 1. Find the mean of the set of numbers. 2. Find the distance from the mean to each number in the data set. Make a list. 3. Find the mean of the list of numbers compiled in step 2. Example 1: Find the mean absolute deviation. Rollercoasters at Theme Parks = = Distance from mean to each number = = = = = = = = = = = = = = = HK O

148 Data Distribution What are the different ways data can be distributed? After being plotted, the distribution of data can an appear many different ways: Data can appear very evenly or uniformly distributed. Data can appear in a cluster when the values are all very close together. There may appear to be small, or even sometimes large, gaps within a data distribution. This is due to no data to plot. Some graphs appear to have obvious peaks in data while others do not. Steps to describe data distribution: 1. Examine the graph and make observations using the information listed above. Example 1: Describe the data distribution Frequency 10 Number of Pets Per Student Pets The numbers in the data are all pretty close to one another, or clustered together. There does not appear to be any gaps in the data. There is an obvious peak or maximum at the 1. Example 2: Describe the data distribution x x x x x x x x x Frequency The numbers are all close together other than a small gap between 18 and 21. There is an obvious peak at the 17.

149 Box and Whisker Plot What are box and whisker plots? A box plot, also known as a box and whisker plot, is a number line using measures of variations to display the data distribution. The labeled parts of a box plot include a lower and upper extreme, first and third quartiles, and a median. Outliers may also be plotted, as well. In a box plot, 50% of the data is plotted within the box, and 25% extends out from both sides making up the remaining 50%. 1. List the data from least to greatest. Example 1: Create a box plot of the numbers: 27, 46, 36, 22, 49, 31, 35, 29, , 27, 29, 31, 35, 36, 42, 46, There are no outliers. lower extreme: 22 upper extreme: 49 median: 35 first quartile: 28 third quartile: Identify any outliers, the lower extreme (smallest number excluding outliers), the upper extreme (greatest number excluding outliers), the median, and the first and third quartiles. 3. Create a number line and plot the values identified in step Draw a box around the first and third quartiles. 5. Draw lines extending from the box to both extremes and connect the median to the box using a vertical line

150 Unbiased Sampling What is an unbiased sample? An unbiased sample is a set of data that is chosen without any biased. Each member of the population set has an equally likely chance to be chosen. An unbiased sample can be found from doing a simple random sample or a systematic random sample. In a simple random sample each member of the population is randomly chosen. In a systematic random sample each member of a population is selected by following a specific interval. (Ex: Every 3 rd person) Example of an Unbiased Sample: Avery conducts a survey of students who wear glasses in her school. She prints a list of every student on campus and chooses every 10 th person to determine if they wear glasses or not. Example of a Biased Sample: Darrin is doing a poll to determine the favorite hangout spot for students in his school. He goes to the mall to conduct the poll and questions 100 students.

151 Probability What is probability? Probability is the likelihood or chance of an event occurring. To determine the probability of an event us the formula: Example 1: What is the probability of pulling a green marble out of a bag with 1 green, 1 yellow, 1 blue, 1 purple and 1 orange marble. 1. There is 1 favorable outcome because there is only 1 green marble. P(event) = number of favorable outcomes number of possible outcomes 2. There are 5 possible outcomes because there are 5 different marbles % 25% 50% 75% 100% 1 3. P(green marble) = H F 1 5 = 0.2 = 20% impossible unlikely equally likely or unlikely likely certain The chance of drawing a green marble from the bag is 20%, which is unlikely.

152 Complement What is a complement? Two events are complementary when the probability of each combines to make 1 or 100%. Example 1: Determine the probability of not rolling a 4 or 6. P(4,6) + P(not a 4,6) = 1 For example, the probability of rolling a 2 on a die, and the probability of NOT rolling a 2 on a die combine to make 100% probability. There are no other options, than to roll a 2 or not to roll a P 4,6 = P(not a 4,6) = = 1 1. Set up an equation. 2. Determine the missing value. 3. Identify the probability. P not a 4,6 = I K = 4 6 = The probability of not rolling a 4 or 6 is % which isa likely event.

153 Theoretical Probability What is theoretical probability? The theoretical probability is what should happen in an event. For example, when flipping a coin you should land on tails 50% of the time. The theoretical probability is not always the same as what actually happens. 1. Identify the number of favorable outcomes. 2. Identify the number of possible outcomes. Example 1: Rhonda rolls a six sided dice 10 times and the results are listed below. Determine the theoretical probability of rolling a 5. roll result There is only one five on the dice, so the favorable outcome is one. 2. There are six different options on the dice so the possible outcome is six. 3. Put the information from step 1 and 2 into the formula. Ä@ÅB?@ÆÇ> ÈBÉÉDÆÇ> 3. H K 4. Determine the percent %

154 Experimental Probability What is experimental probability? The experimental probability is what actually happens during an experiment. The experimental and theoretical probability are often times different values. 1. Identify the number of actual outcomes. 2. Identify the total number of trials in the experiment. 3. Put the information from step 1 and 2 into the A?D@ÇÉ 4. Determine the percent. Example 1: Rhonda rolls a six sided dice 10 times and the results are listed below. Determine the experimental probability of rolling a Rhonda only rolled a five one time. 2. There were a total of 10 trials. 3. roll result H HM 4. = 10%

155 Simulations What is a simulation? A simulation is a type of generator for an experiment that models the actions of a situation. The resulting probability of a simulation is generated by running trials. Example: A computer program generates the results of two coins being thrown 50 times. The results are show below. outcome TT TH HT HH frequency A) Determine the theoretical probability of tossing two heads. B) Determine the experimental probability of tossing two heads. C) Is this an accurate simulation? A) There are four possible outcomes and only one of those will get you two heads. The theoretical probability of tossing two heads is 25%. B) The simulation generated 50 results and of those the coins both landed on heads 11 times. The experimental probability is 22%. C) The difference between the theoretical probability and the experimental probability of the simulation is 3%. This is a very small amount, so the simulation is accurate.

156 Tree Diagram What is a tree diagram? A tree diagram is a method of determining the probability of an event. It is a diagram of all the possible outcomes. Example 1: A coin and an 8 sided dice with numbers 1-8 listed on the sides are each tossed. Determine the total number of outcomes. COIN DICE SAMPLE SPACE 1. Create a tree diagram. 2. Determine the total number of outcomes. heads tails heads, 1 heads, 2 heads, 3 heads, 4 heads, 5 heads, 6 heads, 7 heads, 8 tails, 1 tails, 2 tails, 3 tails, 4 tails, 5 tails, 6 tails, 7 tails, 8 There are 16 different outcomes.

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