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Math 6 Expressions and Equations EE Learning targets EE.D1 Model and solve problems involving whole number exponents. EE.D2 Use mathematical properties to generate and identify equivalent expressions. EE.D3 Write expressions using variables from contextual problems. EE.D4 Solve real-world problems by writing and solving equations. Write inequalities from contextual problems and graph or describe the solution. Use variables to represent two quantities in a contextual problem. Analyze the relationship between variables using graphs and tables. Vocabulary of this unit Term, Expression, Equation, Variable, Variable Structure, Coefficient, Exponent, Factors, Simplify, Solve EE.D1 Algebra Vocabulary EE.D1 Term Parts of an expression or series separated by + or Expression: Terms: More Vocabulary EE.D1 Coefﬁcient A numerical or constant quantity placed before and multiplying the variable in an algebraic expression Exponent A number indicates the operation of repeated multiplication of the base. Coefficient exponent What’s a factor? EE.D1 Factor the numbers or variables used to create a product What are the factors of: 80 125 w3 Expressions vs Equations An Expression A group of terms added or subtracted An Equation - 5x + 3x = 10 What’s the difference between an expression and an equation? EE.D1 Solving vs Simplifying Solve – Make a true statement, find a value that makes a true statement. Example: Solve for n when 3 + n = 5 Simplify - Rewrite an expression so it is more simply put. Example: 3 + 5 => 8 or 2n + 3n => 5n EE.D1 You try…. EE.D1 EE.D1 Exponents How many ways can you write these numbers using exponents? 125 27 48 81 EE.D1 Simplify: Write out factors then recombine EE.D1 Simplify: Write out factors then recombine EE.D1 Simplify: Write out factors then recombine U3D1 Independent Practice ICP GM U3D1 Must Do: *Please set yesterday’s assignment on your desk so I can check off that you did it! In your notebook on the vocabulary page, write a definition and give an example of the vocabulary we discussed so far. Term, Expression, Equation, Variable, Coefficient, Exponent, Factors, Simplify, Solve EE.D2 Use mathematical properties to generate and identify equivalent expressions. Let’s discuss EE.D2 What does an equal sign mean? EE.D2 What is an equivalent Expression? Write an expression equivalent to: 8 6+4 n+2 EE.D2 Let’s review and discuss Order of Operations 1. Calculations must be done from left to right. 2. Calculations in brackets (parenthesis) are done first. When you have more than one set of brackets, do the inner brackets first. 3. Exponents (or radicals) must be done next. 4. Multiply and divide in the order the operations occur. 5. Add and subtract in the order the operations occur. Practice Let’s Create some Equivalent Expressions Expression (y + y + y) 9(3y + n) 3(3f + g) Equivalent Expression EE.D2 Simplest Expression EE.D2 What’s the difference between n+n+n and nxnxn How would you simplify each? Are they equivalent? EE.D2 What terms can you simplify? How do you know if you can combine terms? n+4 4n + 4 4n + n 9) 5n + 9n 8) 1 - 3v + 10 Simplify expressions by combining like terms 10) 4 EE.D2 11) 35n - 1 + 46 10) 4b + 6 - 4 12) - 13) 30n + 8n 12) -33v - 49v 14) 7 15) 10x + 36 - 38x - 47 14) 7x + 31x 16) - 9. 10. 11. 12. 13. 14. 15. 16. Challenging EE.D2 EE.D2 Independent Practice EE.D2 EE.D3 Review and Continue: Use mathematical properties to generate and identify equivalent expressions. EE.D3 Write expressions using variables from contextual problems. Write an equivalent equation 3+x=5 2n + 5 = 15 EE.D3 What does it mean to solve an equation? 3n + 4 = 16 Let’s solve it…. Using a bar model, and algebra EE.D3 Solving simple equations using a bar model and algebraic manipulation 4n + 6 = 14 Let’s do bar model first. How can we write equivalent equations to help us solve? EE.D3 EE.D3 Solve this one Discuss: Is this the same as 10/4 + 6/4? Talk about misconceptions in reducing….. EE.D3 Writing Expressions and equations from contextual problems Write an expression EE.D3 Tom is 12 years old. Find his age after a) 4 years b) 7 years c) x years Write an expression EE.D3 A worker’s salary is $200 less than onethird of a manager’s salary. Find the worker’s salary when the manager’s salary is a) $7,200 b) $m Write an expression EE.D3 The price of a papaya is $2 and the price of a mango is $1. Find the total price of a) 4 papayas and 5 mangos b) x papayas and y mangos Write an expression EE.D3 The price of a cup is $5 and the price of a plate is $12. Find the total price of a) 5 cups and 6 plates b) N cups and M plates EE.D3 Report on conclusions and the reasoning behind them Do you agree? Prove and explain why you agree or disagree EE.D3 Independent Practice EE.D3 EE EE.D4 Learning Targets I can solve equations arising from a context by first drawing a model and then writing an equation to represent the context. Mathematical Practices Make sense of problems and persevere in solving them Model with mathematics Resource: Singapore 3.3-7a Review EE.D4 The price of a cup is $5.50 and the price of a plate is $6.75. Find the total price of a) 5 cups and 6 plates b) N cups and M plates Concept Review Order of operations Solve when x = 3 and n = 2 EE.D4 EE.D4 Problem solving.. Use any method to solve The average score a student earned on 11 tests was 90. He averaged a score of 88 on the first 5 tests he took, and the average of his final 5 test scores was 94. What score did the student earn on his 6th test? Draw a model, write an equation Solve EE.D4 At birth, baby George weighs 2 pounds lighter than his twin brother Harry. a) Suppose Harry’s weight is x pounds, what is George’s weight in terms of x? b) If the sum of their weight is 15 pounds, how much does each weigh? Draw a model, write an equation Solve EE.D4 A store is selling a notebook computer at $90 more than 4 times the price of a mobile phone. a) Suppose the price of the mobile phone is $m. Express the price of the notebook in terms of m. b) Given that the notebook computer costs $n, write a formula connecting m and m Draw a model, write an equation Solve Stella has $11 more than Roland. Together they have $159. How much money does Stella have? How much money does Roland have? EE.D4 Draw a model, write an equation Solve Jared has twice as many DVDs as Eric. Together they have 51 DVDs. How many DVDs does Jared have? How many DVDs does Eric have? EE.D4 Draw a model, write an equation Solve The sum of three consecutive integers is 405. What are the three numbers? EE.D4 Write an equation Solve YOU TRY! Write an equation and solve A corn plant was 4 inches tall. It grew 1.5 inches a day and is now 22 inches tall. How many days did it take? EE.D4 EE.D5 Learning Targets I can differentiate between like and unlike terms I can use the distributive law and add and subtract linear equations Mathematical Practices Make sense of problems and persevere in solving them Model with mathematics Resources: Singapore 4.1 & 4.2 – 7a Bell Work Let x = 4, y = -2, z = 0 EE.D5 Warm Up EE.D5 Draw a model, write an equation EE.D5 Kylie is addressing invitations to her party. She has already addressed 17 of them. She finishes the rest of them in 4 hours. If she addressed a total of 81 invitations, how many invitations did she address per hour? Like and unlike terms EE.D5 Review vocab: Terms, Coefficient Like terms have the same variable structure: same variables with same exponents, but the coefficient need not be the same. Constants are always “Like Terms”: 5, 24, -73 EE.D5 Simplifying by adding or subtracting like terms If two terms are “like” (have the same variable structure), you can add or subtract them, but add the coefficient in front of the variables. Any numbers without a variable (a constant), can just be added or subtracted. Examples You Try EE.D5 The Distributive law of multiplication over addition Demonstrate using area models Solve 3(4 + 5) using an area model Solve 2(4x + 5y) using an area model Solve -x(4y + 5y) using an area model EE.D5 EE.D5 The Distributive Law in general a(x + y) = ax + ay a(x - y) = ax - ay a(x + y + z) = ax + ay + az You Try; use the distributive property to remove parenthesis EE.D5 EE.D5 Harder… use distributive property and collect like terms (2x + 3y) – (3x + 2y) 4x – x(3 + 2y) + xy EE.D5 Independent Practice ICP EE.D5 EE.D6 Learning Targets I can solve equations in one variable Mathematical Practices Make sense of problems and persevere in solving them Model with mathematics Resources: Singapore 5.1 – 7a EE.D6 15 minute Skill Check EE.SC1 Model and write an equation Solve EE.D6 Sam is 4 less than twice Bill’s age. Jack is 3 years older than Bill. If the sum of their ages is 47, how old are Sam, Bill & Jack? Solving Equations Using Algebra EE.D6 When we solve an equation using algebra, we want to undo the operations on the unknown in order to find out what it is. Remember! Keep the equation balanced by doing the same thing on both sides!!! Example: Solve 3x 48 Explain the opposite of multiplication is division, etc… Solving Equations Using Algebra Solve each equation 1. 2 y 68 2. 39 w 20 p 3. 32 2 Do first problem with class, have students do 2 & 3. EE.D6 Solving Two-Step Equations What’s the first step in solving this equation? Why? How about this one? Why? p 8 10 2 EE.D6 Solving Two-Step Equations Solve each equation 1. p 68 3 2. 6 y 9 30 3 3. Do first problem with class, have students do 2 & 3. EE.D6 A word about notation Are these equivalent? What about these? EE.D6 EE.D6 Independent Practice EE.D6 EE.D7 Learning Targets I can solve equations involving parenthesis in one variable I can rearrange equations to solve. Mathematical Practices Make sense of problems and persevere in solving them Model with mathematics Resources: Singapore 5.2 – 7a Bell work EE.D7 a) The sum of 3 consecutive numbers is 234. Find the numbers a) The sum of 3 consecutive even numbers is 192. Find the numbers a) The sum of 3 consecutive odd numbers is 111. Find the numbers Review Let x = 5, y = -3, z = 0 EE.D7 Problem Solving Kylie is addressing invitations to her graduation party. She has already addressed 17 of them. She finishes the rest of them in 4 hours. If she addressed a total of 81 invitations, how many invitations did she address per hour? EE.D7 Problem Solving EE.D7 104 chickens and goats in a farm have 246 legs altogether. How many of each type of animal are there? Can us solve with algebra, a model, or some other method? Solving Multi-Step Equations Let’s review the Distributive Property for a moment… The Distributive Property says a(b + c) = Simplify 2( x 3) EE.D7 Solving Equations with the Distributive Property What should we do first to solve this equation? Why? 5(2 x 1) 15 Explain we need both sides of equation as simplified as possible to make solving easier… EE.D7 Solving Equations with the Distributive Property Solve each equation 1. 2. 3. 2 3( x 4) 15 21 7( x 4) 21 4 x 5(2 x 4) 10 Do first problem with class, have students do 2 & 3. EE.D7 A little more challenging examples…. How does the distributive property apply here? EE.D7 U4D8 Independent Practice EE.D7 EE.D7 Math 6 EE.D8 Standards Assessed: I can solve equations using algebraic manipulation I can create and solve equations arising from a context Mathematical Practices: Make sense of problems and persevere in solving them Model with mathematics EE.D8 U4D8 Practice Solving proportions EE.D8 Keep practicing! EE.D8 Write an equation and solve. Draw a model if necessary. The length of a rectangle is 50 meters. This is 6 meters more than twice the width. Find the width of the rectangle. EE.D8 You Try! Write an equation and solve. Draw a model if necessary. Twelve decreased by 8 times a number is 36. Find the number. EE.D8 You Try!! Write an equation and solve. Draw a model if necessary. JoGrandpa is 75 years old. This is nine years less than seven times the age of Jo. How old is Jo? EE.D8 You Try!! Write an equation and solve. Draw a model if necessary. JoGrandpa is 75 years old. This is nine years less than seven times the age of Jo. How old is Jo? EE.D8 Write an equation and solve A-1 Plumbing company pays Sam $45 per day, plus 10.50 for each hour he works. If he earned 566.25 last week, how many hours a day did he work. (Sam works a 5 day week, and each day he works the same number of hours) EE.D8 Write an equation and solve Lisa is bored, riding in the backseat of her parent’s SUV on their family vacation. She notices the mile markers on the interstate. She adds up three consecutive mile markers and discovers the sum is 312. Which three mile marker signs did her family just pass? Independent Practice EE.D8 EE.D9 Continue developing algebraic manipulation skills by solving inequalities Expressing a solution to an inequality EE.D9 Solutions to equations vs solutions to inequalities How do we express a solution to the equations: 2n = 10 2n + 1 = 2n + 10 2n = 2n EE.D9 Inequalities! • Vocab: An Inequality is a comparison of two values that may or may not be equal. EE.D9 The Symbols and Their Meanings Symbol Meaning a>b a is greater than b a≥b a is greater than or equal to b a<b a is less than b a≤b a is less than or equal to b EE.D9 What does it all mean!? Examples! Graph each inequality on a number line using 1. x2 2. x 5 3. x0 4. x 1 or 2 properties ws EE.D9 Solutions to an Inequality • Remember that the solution to an equation gave us a single, unique result… but for inequalities… • The solution to an inequality is a range of possibilities. – Example: The solution of x 2 allows every real number that is bigger than 2. So we get to use any possible number we can think of as long as it is greater than 2. Let’s try some…. Examples! Solve each inequality and graph the solution on a number line. 1. x 4 6 x 3 2. 2 3. x 2 1 4. 12 3x EE.D9 EE.D9 You Try! • Solve each inequality and graph the solution on a number line. x 1. 4 3 2. x 6 8 3. 11 2x 1 EE.D10 Standards Addressed Today: Attend to Precision Reason Abstractly and Quantitatively Solve Inequalities Using Algebraic Manipulation Explain the Multiplication Property by Negative Integers for Inequalities Concept Review EE.D10 • Solve each inequality and graph the solution on a number line. 1. 2. x 3 11 x 8 12 3. x 1 6 4. 7 x 35 Now for the tricky part… • Consider the inequality 4 < 12 1. Divide both sides by 2. Is the inequality still true? 2. Now divide both sides by –2. Is the inequality still true? 3. Divide by 4. 4. Divide by –4. 5. Complete questions 1-4 on your half sheet. EE.D10 Solve each inequality 1. 2 x 16 2. 8 4 y 3. x 1 3 2w 4. 12 3 EE.D10 EE.D10 Challenge! 1. 2 2(4w 7) 4. 6 2w 2(6w 7) 4w 2. 16 8 2 5. 2w 16 12 3 6. 2w 8 3 12 3. 3x 15 21 9x EE.D10 Describe a situation … that could be modeled by the inequality 3x 10 Work with a partner EE.D10 • You are building a patio. You want to cover the patio with Spanish Tile that costs $5 per square foot. You budget for the tile is $1700. How wide can you make the patio without going over budget? Independent Practice EE.D10 EE.D10 EE.D11 EE.D11 Standards: Attend to Precision Reason Abstractly and Quantitatively Solve Inequalities Using Algebraic Manipulation Solve Inequalities Arising From a Context EE.D11 Concept Review • Solve each equation or inequality. 1. 2. 3. 2 x 9 20 7 8(3 2 x) 2(8 3x) 17 8x 7 Graph the solution to the inequality. 4. 3(2 x 1) 9 5. 7 x 3x 4 9 3 6. 3x 1 4 6 EE.D11 Concept Review Write an inequality and solve. • Four times a number is at least –48. What is the number? EE.D11 Problem Solving Write an inequality and solve. • Ilona is saving up to buy a digital camera that costs $495.75. So far, she has $175 saved. She would like to buy the camera 5 weeks from now. At least how much must she save every week to have enough money to purchase the camera? Use the definition of mean to write an inequality and solve • You need a mean score of at least 90 to advance to the next round of the trivia game. What score do you need on the fifth game to advance? Problem Solving EE.D11 • For what values of x will the area of the blue region be greater than 12 square units? Try this on your own a) For what values of y will the area of the trapezoid be less than or equal to 10 square units? EE.D11 EE.D11 In Class Practice EE.D11 – ICP Unit Review Solving equations and inequalities