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Integrated Algebra I Fall 2007 Ninth Grade Unit 3 Solving Equations

Integrated Algebra I Fall 2007 Ninth Grade Unit 3 Solving Equations. Mrs. Ribak. Table of Contents:. Lesson # 1 What are algebraic expressions and how can we evaluate them ? Lesson # 2 How do we solve an equation of the type x + a = b? Lesson # 3

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Integrated Algebra I Fall 2007 Ninth Grade Unit 3 Solving Equations

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  1. Integrated Algebra I Fall 2007Ninth GradeUnit 3 Solving Equations Mrs. Ribak

  2. Table of Contents: • Lesson # 1 • What are algebraic expressions and how can we evaluate them? • Lesson # 2 • How do we solve an equation of the type x + a = b? • Lesson # 3 • How do we solve an equation of the type ax = b? • Lesson # 4 • How do we solve equations of the type ax + b = c? • Lesson # 5 • What is meant by the distributive property?

  3. Lesson # 1 • AIM: What are algebraic expressions and how can we evaluate them? • Math Standards:AA1, AA2 • Technology Standards:3D • Time Allotted: 45 min • Students will be able to: • define, both orally and in writing, the terms: variable, algebraic expression. • describe the process of evaluating an algebraic expression • evaluate simple algebraic expressions given value(s) for the variable(s) • write verbal expressions that match given mathematical expressions

  4. Do Now: • Simplify: • 6 + 7 x 8 • (25 - 11) x 3 • 12 - (71-68)³+ 40 ÷ 2³ = 62 = 42 = 14

  5. Motivation • You are buying songs on the internet and notice a sale where; ten songs cost $1.10, twenty songs cost $2.20, and thirty songs cost $3.30. According to this observation, how much do you think forty or fifty songs will cost to download? (create a table to organize your data ) • What generalization can you make on the price songs based on the information in the table? • How can we represent this generalization in mathematical symbols? • Excel Chart $4.40 $5.50 X 1.1 X

  6. Development • Introduction to Algebra • What is a variable? • What is an algebraic expression? • What does evaluate mean? • What are some terms that represent mathematical symbols? • Algebra Vocabulary • How can we evaluate algebraic expressions? • Ex 1: Evaluate g + 5 when g = 7 • Ex 2: What is a number decreased by five if the number is thirteen?

  7. Medial Summary • What is the strategy for evaluating expressions? • Strategy • What do you do if you have more than one variable, such as, Let x = 10,  y = 4,  z = 2.   Evaluate the following expression x + yz =   10 + 4  2 = 10 + 8 = 18

  8. Procedure 112 • Let x = 10,  y = 4,  z = 2, and evaluate the following: • x² + 2(y + z) = • (x + 2)(y + z) =     • x − 3(y − z) = • x + (x + 1) + (x + 2) = • Translate into an algebraic expression: • A certain number times six.   • Thirteen less than a certain number.   • Eight more than twice a certain number. • Create your own math problem that requires translating and evaluating an algebraic expression 72 4 33 6m Y - 13 2v + 8

  9. Summary/Homework • Summary: • Write one similarity and one difference between a numerical and an algebraic expression. • Explain the steps needed to evaluate an algebraic expression. • What is wrong with the statement "Two negatives make a positive”? • Homework:

  10. Lesson # 2 • AIM: How do we solve an equation of the type x + a = b? • Math Standards:AA21, AA22 • Technology Standards: 3D • Time Allotted: 45 min • Students will be able to: • describe the ‘balance’ required for an expression to be considered an equation • define a solution of an equation, and state what is meant by solving an equation • translate verbal sentences into equations of the type x + a = b • solve equations of the type x + a = b, where a and b are rational numbers, by isolating the variable using inverses. • communicate in writing how the additive inverses facilitate solving the equation • check the solutions and write the solution set

  11. Do Now: • Simplify: • 9 + 3 - 3 = • 8 + 6 - 6 = • x + 4 – 4 = • What reappearing pattern do you notice in the above problems?

  12. Motivation • What is an Equation (55-140 sec)? • What is an equation? • What is a solution? • What is a coefficient? • Algebra Vocabulary • Brain Pop-Tim and Moby

  13. BALANCE Development • What are mathematical operations and inverse operations? ( hint: do now problems) amount Moby ate + ? amount left? = original amount 150 + x = 1200 • When we solve an equation, we work backwards or “undo” what has been done by using the inverse on both sides to maintain the balance. Addition and subtraction are inverse operations.

  14. Medial Summary • How do the additive inverses help in solving an equation? • How do you know your solution is correct? • What if we have an equation with subtraction? • Example: Noah received a paycheck on Friday. After buying some athletic shoes for $112, he had $96 left. How much was his paycheck? Words:The amount of paycheck minus amount of the shoes is $96. Variable: Let a = amount of paycheck

  15. Procedure Solve and check the following equations: • 23 = 27 + x • a + 9 = -3 • 8 + t = - 6 • y – ½ = 1/2 • Twenty-two less than a number is five. Write an equitation and find the number.

  16. Summary/Homework • Summary: • You could solve 6 = x + 9 by subtracting 9 from each side, or by subtracting 6 from each side. Describe why it is better to subtract the 9. • Homework:

  17. Lesson # 3 • AIM: How do we solve an equation of the type ax = b? • Math Standards:AA21, AA22, AA25 • Technology Standards: 3D • Time Allotted: 45 min • Students will be able to: • translate verbal sentences into equations of the type ax = b • solve equations of the type ax = b, where a and b are rational signed numbers, by isolating the variable using reciprocals. • investigate the connections between solving equations of the form ax=b and x + a=b • select and apply the appropriate isolation method for solving equations of the form ax=b as compared to solving equations of the form x + a=b • communicate in writing how the use of multiplicative inverses (reciprocals) facilitate solving the equation • check the solutions and write the solution set

  18. Do Now: • Solve and check the equations and evaluate the expression when needed. • a + 12 = 15 • 12 = t –10 • y - .5 = 2.5  Evaluate y + 7

  19. Motivation/Development • Brain Pop-Tim and Moby (skip 8 times ) • What are the inverse operations for multiplication and division? • pieces per string ● ?number of strings? = popcorn leftover 50s = 1050 • Example: Eight times a number is negative sixty-four.

  20. Medial Summary • What are some connections between solving equations of the form ax = b and x + a = b • What if we had a fraction as a coefficient? Such as • Since x is multiplied by 3/5, multiply both sides by 5/3(reciprical), to cancel off the fraction on the x.  • How does the use of multiplicative inverses (reciprocals) help in solving equations?

  21. Procedure • 20 =5d • -5d = 45 • . • Six cans of soda cost $ 4.50 . Find the cost of one can. Let x is the cost of one can.

  22. Summary/Homework • Summary: • What does it mean for a value to ‘satisfy’ an equation? • How are the words ‘root’ and ‘solution’ related to each other? • Homework

  23. Lesson # 4 • AIM: How do we solve equations of the type ax + b = c? • Math Standards:AA21, AA22, AA25 • Technology Standards: 3D • Time Allotted: 45 min • Students will be able to: • isolate the variable using properties of inverses and identities • solve equations of the type ax + b = c where a, b, and c are rational signed numbers • check the solutions and write the solution set

  24. Do Now: • Translate each problem into an equation and solve. • Eight times a number is 96. Find the number. • A number divided by 3 is 43. Find the number. • Bob drives 245 miles in 5 hours. How far does he drive in 1 hour?

  25. Motivation/Development • Nicole bought six concert tickets for a total of $113. This included a service charge of $5. How much did each ticket cost? • Translate the problem into mathematical symbols. Words: six concert tickets and a service charge of $5 was equal to $113 Variable: let t be the number of tickets 6t + 5 = 113 • How many operations are n this equation? • When we solve the equations using more than one operation. We have to undo the addition or subtraction first, then undo multiplication or division. • 6t +5 -5 = 113 -5 ( subtract 5 from each side) 6t = 108 ( Divide each side by 6.) t = 18 • Balancing equations

  26. Medial Summary

  27. Procedure • 5a + 2 = 7 • ½ x +7 = 15 • ¾ x – 4 = 17 • Kendra works part-time at a hospital and is paid by the hour. How much does she earn per hour if her pay was $93 and she worked 20 hours?

  28. Summary/Homework • Summary: • How is solving an equation like working the order of operations backwards? • Homework:

  29. Lesson # 5 • AIM: What is meant by the distributive property? • Math Standards:AN1, AA3 • Technology Standards: 3D • Time Allotted: 45 min • Students will be able to: • state the distributive property • use the distributive property to evaluate numerical expressions • use the distributive property to change the form of an algebraic expression • state the property justifying steps in a number proof

  30. Do Now: • Simplify: • 3 (4 + 8) - 5²= • 2 ( n – 6) = • -4 (9 + 3y )

  31. Motivation/Development 60 + w w • Your dinning room table top is rectangular. The table’s length is 60 in. more than its width. The perimeter of the table is 240in. Find the length and width of the table. Words: The tables length is 60 in. more than its width. perimeter of the table is 240in Variable: Let w be the table’s width Let 60 + w be the table’s length • What is the perimeter and how do you find it in a rectangle? • What equation can we form to solve this problem? 2 lengths + 2 widths = perimeter 2(60 + w) + 2w = 240 • Follow the order of operations to solve. • Distribute the 2 to get rid of the parenthesis 120 + 2w + 2w = 240 • Combine like terms 120 + 4w = 240 • Solve the two-step equation 4w = 120 w = 30 in 4. To find the length plug in the value for the width w= 30, L= 60+30 = 90in 5. Check! w 60 + w

  32. Medial Summary • Solve and write out the steps: 12n + 9(10 - n) = 111

  33. Procedure • Approximating solutions • 3(x - 5) = 6 • -3 ( 2t – 1) = 15 • Balancing mystery

  34. Summary/Homework • Summary: 1) Why is it appropriate that the word distribute is found in the name of the Distributive Law?. 2) Explain the circumstances when the distributive property is used. Illustrate you answer with an example. 3) Explain the error: -4 (x+3) = -4x + 12. • Homework:

  35. References • Association of Mathematics Assistant Principals Supervision of NYC • Teacher Tube • EdOnline • Brain Pops video • National Library Virtual Manipulative • Ask Dr. Math • Integer rap • www.purplemath.com • www.regentsprep.org • www.jmap.org

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