1 / 127

Warm Up Add or subtract. 1. 6 + 104 2. 12(9) 3. 23 – 8 4.

Warm Up Add or subtract. 1. 6 + 104 2. 12(9) 3. 23 – 8 4. Multiply or divide. 5. 324 ÷ 18 6. 7. 13.5(10) 8. 18.2 ÷ 2. 108. 110. 15. 18. 6. 135. 9.1. Objectives. Translate between words and algebra. Evaluate algebraic expressions.

iorwen
Download Presentation

Warm Up Add or subtract. 1. 6 + 104 2. 12(9) 3. 23 – 8 4.

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Warm Up Add or subtract. 1. 6 + 104 2. 12(9) 3. 23 – 8 4. Multiply or divide. 5. 324 ÷ 18 6. 7.13.5(10) 8. 18.2 ÷ 2 108 110 15 18 6 135 9.1

  2. Objectives Translate between words and algebra. Evaluate algebraic expressions.

  3. A variable is a letter or a symbol used to represent a value that can change. A constant is a value that does not change. A numerical expressioncontains only constants and operations. An algebraic expression may contain variables, constants, and operations.

  4. Writing Math These expressions all mean “2 times y”: 2y 2(y) 2•y (2)(y) 2 x y (2)y

  5. Example 1: Translating from Algebra to Words Give two ways to write each algebra expression in words. A. 9 + r B. q – 3 the difference of q and 3 the sum of 9 and r 9 increased by r 3 less than q C. 7mD. j  6 the product of m and 7 the quotient of j and 6 m times 7 j divided by 6

  6. Check It Out! Example 1 Give two ways to write each algebra expression in words. 1b. 1a. 4 - n 4 decreased by n the quotient of t and 5 n less than 4 t divided by 5 1c. 9 + q 1d. 3(h) the sum of 9 and q the product of 3 and h 3 times h q added to 9

  7. To translate words into algebraic expressions, look for words that indicate the action that is taking place. Put together, combine Find how much more or less Separate into equal groups Put together equal groups

  8. Example 2A: Translating from Words to Algebra John types 62 words per minute. Write an expression for the number of words he types in m minutes. m represents the number of minutes that John types. Think: m groups of 62 words 62 · m or 62m

  9. Example 2B: Translating from Words to Algebra Roberto is 4 years older than Emily, who is y years old. Write an expression for Roberto’s age y represents Emily’s age. y + 4 Think: “older than” means “greater than.”

  10. Example 2C: Translating from Words to Algebra Joey earns $5 for each car he washes. Write an expression for the number of cars Joey must wash to earn d dollars. d represents the total amount that Joey will earn. Think: How many groups of $5 are in d?

  11. To evaluatean expression is to find its value. To evaluate an algebraic expression, substitute numbers for the variables in the expression and then simplify the expression.

  12. Check It Out! Example 3 Evaluate each expression for m = 3, n = 2, and p = 9. a.mn mn = 3 · 2 Substitute 3 for m and 2 for n. = 6 Simplify. b. p – n Substitute 9 for p and 2 for n. p – n = 9 – 2 = 7 Simplify. c. p ÷ m p ÷ m = 9 ÷ 3 Substitute 9 for p and 3 for m. = 3 Simplify.

  13. Example 4a: Recycling Application Approximately eighty-five 20-ounce plastic bottles must be recycled to produce the fiberfill for a sleeping bag. Write an expression for the number of bottles needed to make s sleeping bags. The expression 85s models the number of bottles to make s sleeping bags.

  14. Example 4b: Recycling Application Continued Approximately eighty-five 20-ounce plastic bottles must be recycled to produce the fiberfill for a sleeping bag. Find the number of bottles needed to make 20, 50, and 325 sleeping bags. Evaluate 85s for s = 20, 50, and 325. To make 20 sleeping bags 1700 bottles are needed. 85(20) = 1700 To make 50 sleeping bags 4250 bottles are needed. 85(50) = 4250 To make 325 sleeping bags 27,625 bottles are needed. 85(325) = 27,625

  15. Helpful Hint A replacement set is a set of numbers that can be substituted for a variable. The replacement set in Example 4 is {20, 50, and 325}.

  16. 2 2 2 2 3 5 Warm Up Simplify. 1. –4 –|4| 2. |–3| 3 Write an improper fraction to represent each mixed number. 6 2 14 55 3. 4 4. 7 3 7 3 7 Write a mixed number to represent each improper fraction. 12 24 5. 6. 5 9

  17. Objectives Add real numbers. Subtract real numbers.

  18. All the numbers on a number line are called real numbers. You can use a number line to model addition and subtraction of real numbers. Addition To model addition of a positive number, move right. To model addition of a negative number move left. Subtraction To model subtraction of a positive number, move left. To model subtraction of a negative number move right.

  19. 11 10 7 6 5 4 3 2 1 9 8 0 Example 1A: Adding and Subtracting Numberson a Number line Add or subtract using a number line. –4 + (–7) Start at 0. Move left to –4. To add –7, move left 7 units. + (–7) –4 –4+ (–7) = –11

  20. -3 -1 0 1 2 3 4 6 8 9 -2 7 5 Check It Out! Example 1a Add or subtract using a number line. –3 + 7 Start at 0. Move left to –3. To add 7, move right 7 units. +7 –3 –3 + 7 = 4

  21. The absolute value of a number is the distance from zero on a number line. The absolute value of 5 is written as |5|. 5 units 5units - - - - - 6 5 - 1 0 1 2 3 4 5 6 4 3 2 |–5| = 5 |5| = 5

  22. When the signs of numbers are different, find the difference of the absolute values: Example 2A: Adding Real Numbers Add. Use the sign of the number with the greater absolute value. The sum is negative.

  23. Example 2B:Adding Real Numbers Add. y + (–2) for y = –6 y + (–2) = (–6) + (–2) First substitute –6 for y. When the signs are the same, find the sum of the absolute values: 6 + 2 = 8. (–6) + (–2) –8 Both numbers are negative, so the sum is negative.

  24. Check It Out! Example 2b Add. –13.5 + (–22.3) –13.5 + (–22.3) When the signs are the same, find the sum of the absolute values. 13.5 + 22.3 –35.8 Both numbers are negative so, the sum is negative.

  25. Two numbers are opposites if their sum is 0. A number and its opposite are on opposite sides of zero on a number line, but are the same distance from zero. They have the same absolute value.

  26. A number and its opposite are additive inverses. To subtract signed numbers, you can use additive inverses. Subtracting 6 is the same as adding the inverse of 6. Additive inverses 11 – 6 = 5 11 + (–6) = 5 Subtracting a number is the same as adding the opposite of the number.

  27. Example 3A:Subtracting Real Numbers Subtract. –6.7 – 4.1 –6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1. When the signs of the numbers are the same, find the sum of the absolute values: 6.7 + 4.1 = 10.8. = –10.8 Both numbers are negative, so the sum is negative.

  28. Example 3B:Subtracting Real Numbers Subtract. 5 – (–4) 5 − (–4) = 5 + 4 To subtract –4 add 4. Find the sum of the absolute values. 9

  29. First substitute for z. , add . To subtract Rewrite with a denominator of 10. Example 3C:Subtracting Real Numbers Subtract.

  30. When the signs of the numbers are the same, find the sum of the absolute values: . Example 3C Continued Write the answer in the simplest form. Both numbers are negative, so the sum is negative.

  31. 1 2 –3 To subtract add . When the signs of the numbers are the same, find the sum of the absolute values: = 4. 1 2 3 1 2 1 2 3 + Check It Out! Example 3b Subtract. 4 Both numbers are positive so, the sum is positive.

  32. Check It Out! Example 4 What if…?The tallest known iceberg in the North Atlantic rose 550 feet above the oceans surface. How many feet would it be from the top of the tallest iceberg to the wreckage of the Titanic, which is at an elevation of –12,468 feet? Minus elevationof the Titanic –12,468 elevation at top of iceberg 550 – 550 – (–12,468) To subtract –12,468, add 12,468. 550 – (–12,468) = 550 + 12,468 Find the sum of the absolute values. = 13,018 Distance from the iceberg to the Titanic is 13,018 feet.

  33. Warm-Up Add or subtract. –2 2. –5 – (–3) 1. –2 + 9 7 Add or subtract. 3. –23 + 42 19 4. 4.5 – (–3.7) 8.2 5. 6. The temperature at 6:00 A.M. was –23°F. At 3:00 P.M. it was 18°F. Find the difference in the temperatures. 41°F

  34. Objectives Multiply real numbers. Divide real numbers.

  35. When you multiply two numbers, the signs of the numbers you are multiplying determine whether the product is positive or negative. 3(5) Both positive 15 Positive 3(–5) One negative –15 Negative –3(–5) Both negative 15 Positive This is true for division also.

  36. Multiplying and Dividing Signed Numbers

  37. Multiplying and Dividing Signed Numbers –18 ÷ 2 = –9

  38. Example 1: Multiplying and Dividing Signed Numbers Find the value of each expression. A. The product of two numbers with different signs is negative. –5 B. The quotient of two numbers with the same sign is positive. 12

  39. First substitute for x. Example 1C: Multiplying and Dividing Signed Numbers Find the value of the expression. The quotient of two numbers with different signs is negative.

  40. Check It Out! Example 1a and 1b Find the value of each expression. 1a. 35  (–5) The quotient of two numbers with different signs is negative. –7 1b. –11(–4) The product of two numbers with the same sign is positive. 44

  41. Two numbers are reciprocals if their product is 1. A number and its reciprocal are called multiplicativeinverses. To divide by a number, you can multiply by its multiplicative inverse. Dividing by a nonzero number is the same as Multiplying by the reciprocal of the number.

  42. Multiplicative inverses 1 10 10 ÷ 5 = 2 10 ∙ = = 2 5 5 Dividing by 5 is the same as multiplying by the reciprocal of 5, .

  43. Helpful Hint You can write the reciprocal of a number by switching the numerator and denominator. A whole number has a denominator of 1.

  44. and have the same sign, so the quotient is positive. Example 2A: Dividing by Fractions Example 2 Dividing by Fractions Divide. To divide by , multiply by . Multiply the numerators and multiply the denominators.

  45. Write as an improper fraction. To divide by , multiply by . and have different signs, so the quotient is negative. Example 2B: Dividing by Fractions Divide.

  46. Write as an improper fraction. To divide by multiply by . Check It Out! Example 2c Divide. The signs are different so the quotient is negative.

  47. No number can be multiplied by 0 to give a product of 1, so 0 has no reciprocal. Because 0 has no reciprocal, division by 0 is not possible. We say that division by zero is undefined.

  48. 1 3 · 0 = 0 Properties of Zero Multiplication by Zero The product of any number and 0 is 0. WORDS 0(–17) = 0 NUMBERS ALGEBRA a · 0 = 0 0 · a = 0

More Related