Identity Property of Multiplication: 2a. Show the Identity Property of Multiplication with 23 2b. Show the Identity Property of Multiplication with 3b Multiplicative Property of Zero: 3a. Show Multiplicative Property of Zero with 43.581 3b. Show the Multiplicative Property of Zero with 4xy Commutative Property of Addition: 4a. 4.38 + 2.01 is the same as: 4b. x + z is the same as: Commutative Property of Multiplication: ab = ba 5a. is the same as: 5b. 6k is the same as: Associative Property of Addition: (a + b) + c = a + (b + c) 6a. (1.8 + 3.2) + 9.5 is the same as: 6b. (x + 1) + 9 is the same as: Associative Property of Multiplication: 7a. (2.6 ∙ 5.4) ∙ 3.7 is the same as: 7b. (wh)l is the same as: Use the listed property to fill in the blank. Multiplicative Inverse: 8a. 3 ( ) = 1 8b. ( ) = 1 Additive Inverse: a + (–a) = 0 9a. + = 0 9b. + –x = 0 Name the property demonstrated by each statement. 10. 3 + 2 + 7 = 3 + 7 + 2 11. 5 + (5 + 4) + 6 = (5 + 5) + (4 + 6) 12. 25 + (–25) = 0 13. (2/5)(5/2) = 1 (x + 3) + y = x + (3 + y)
3.1c Class Activity: Algebraic Expressions Write two different numeric expressions for the context below: Maria bought 5 apples for $0.35 each. How would the expression change if she spent $0.40 on each apple? What if you didn’t know how much each apple cost, how could you write an expression? For each context, four algebraic expressions are offered. Make a conjecture about the correctness of the expression. Then, evaluate it for the given value, and explain why the expression did or didn’t work for the given context. Ryan bought 3 CDs for x dollars each and a DVD for $15. Write an expression of how much money Ryan spent. Expression Correct expression? Evaluate x = 7 Did it work? Why or why not? a. 3 + x + 15 b. 15x + 3 c. 15 + x + x + x d. 3x + 15 I started with 12 jellybeans. Sam ate 3 jellybeans and then Cyle ate y jellybeans. Write an expression for how many jellybeans were left. Expression Correct expression? Evaluate y = 6 Did it work? Why or why not? a. 12 – 3 – y b. 12 – (3 – y)
For #1-16, model each expression (draw Algebra Tiles representations of the variables and numbers). Then simplify each expression. 8x + 2x + 2y + 7y 3z + 2x + x 5x – 9x + 2w w –6 + 4x + 9 – 2x + 6t – 2t –3 + 3x + 11 – 5x – 2k – k 9x – 12 + 12 – 9x 1 + x + 5x – 2 + 4k – 7k –4x – 5x + 12m – 2m + 6q – 10q
5x + 3x + 5y 1 + 3x + x 3x – 5x + 4y + 3y 7m – 2x – 9m + 4x –4x – 5 + 2x 5y – 4x + 5x + 10y –4x – 5m + x + 7x – 12m x – 6x + 14y + 8y – 2x 9. 2x + x + 2y + x + 1 10. x + 3y + x + 2 + y + 1 11. 12. 13. –2x + y – 3y + y + x 14. 15. 5x – y + 2 – 4x + 2y + x + 2 16. 3.1f Class Activity: Vocabulary for Simplifying Expressions In groups of 2 or 3 students, consider the following expressions: a) 2x + 5 + 3y, b) 2x + 5 + 3x, and c) 2x + 5x + 3x. How are these expressions similar? How are they different?
Parts of an Algebraic Expression: Use the diagrams to create definitions for the following vocabulary words. Be prepared to discuss your definitions with the class. Terms Constants There are five terms in this expression. The terms are x, –4, 2y, 3, and –5y The constants are –4 and 3. (Recall that subtracting is like adding a negative number.) Coefficients Like Terms The coefficients are 1, 2, and –5 2y and –5y are like terms. –4 and 3 are also like terms. Terms: Constant: Coefficient: Like Terms: 2. Identify the terms, constants, coefficients, and like terms in each algebraic expression. Expression Terms Constants Coefficients Like Terms 4x, –x, 2y, –3 –3 4, –1, 2 4x, –x 3. Identify the terms, constants, coefficients, and like terms in each algebraic expression. Expression Terms Constants Coefficients Like Terms a. 2y + 5y – 6x + 2 b. 3x + 2x – (2/3) c. d.
23. 24. 25. 26. 27. 28. 3.1f Additional Practice Simplify each expression. 2x – y + 5x 5y + x + y 2y + 8x + 5y – 1 10b + 2 – 2b 8y + x – 5x – y 9x + 2 – 2 –2x – 6 + 3y + 2x – 3y 6m + 2n + 10m 7b – 5 + 2b – 3 2a – 3 + 5a + 2 8x + 5 – 7y + 2x 4y + 3 – 5y – 7
On the number line below, show a length of x units. Then make the length of 3x units. AREA MODELS for MULTIPLICATION (2 dimensional representation of multiplication) 25 2 50 2 × 25 5 × 10 5(3 + 7) 5(5 + 5) or 52 + 52 Create your own representation of 50: Can you draw a area model representation of 3x?
b. Write 3(2x – 5) in three different ways. How can you use what we’ve learned about integers and what we know about writing expressions with parentheses to rewrite expressions that have “ – “ in the groupings? Write the expression below in an equivalent form. If necessary, draw a model. 4. 3(x + 2) 5. 2(3x + 5) 6. 3(x + 1) 7. 4(3x – 1) 8. 2(3 + x) 9. 3(3x – 2) 10. 11. 12. 0.2(7x – 9) In sixth grade, you talked about order of operations, what is the order of operations and how is it related to what you’ve been doing above? 3.1g Homework: Multiplying Groups/Distributive Property Simplify each of the following. Draw a model or explain the meaning of the expression to justify your answer. 5(x + 1) 2. 2(3x + 2) 3. 4(x + 3)
4. 2(3x – 1) 5. 3(2x – 3) 6. 5(x – 1) 7. 8. 9. 10. 11. 12. The expressions 2(5x – 3) can be written and 10x – 6 OR 10x + (–6). Write the following expressions in two different ways as the example shows: 4(3x – 5) 16. 2(7x – 3) 3.1h Classwork: Modeling Backwards Distribution (Factoring) Review: below is a review of modeling multiplication with an array. 1 3 2 3 3 3
What is the opposite of “forward three steps then turn right”? In math, what symbol do we use that means opposite? Using the logic above, what do you think each of the following means? –x –(x) (x + 1) (1 x) a) Consider the number line below. The location of x and (x + 1) are identified on the line. Locate and label –x and –(x + 1). b) Consider the number line below. The location of (1 – x) is identified on the line. Locate and label –(1 – x). What does “ – ” in front of a set of parentheses tell us? On the line below the location of the quantity x + 1 is indicated. Locate 3(x + 1) Locate 3(x + 1) Activity: Suppose Juanita has $1200 in her savings account and once a week for 5 weeks she withdraws x dollars for bills and $10 for entertainment. Write an expression for the amount of money she now has in her savings account. In groups of 2 or 3 students, simplify each of the following. Be ready to justify your answer.
1. 4(x + 1) 2. – (x + 2) 3. – (3x + 2) 4. – 2(x + 3) 5. – 3(x – 2) 6. 2(3 – x) 7. – 2(5 – 3x) 8. – (4 – 3x) 9. – 4(2x + 3) In groups of 2 or 3 students, simplify the following exercises. Justify your answer. 10. 3x + 5 – x + 3(x + 2) 11. 3x + 5 – x – 3(x + 2) 12. 2(x – 1) + 4x – 6 + 2x 13. – 2(x – 1) + 4x – 6 + 2x 5 + 2(x – 3) 7x – 2(3x +1) 6x – 3 + 2x – 2(3x + 5) –9x + 3(2x 5) + 10
(5 – 3x) – 7x + 4 –(4x – 3) – 5x + 2 9 – 8x – (x + 2) 15 – 2x – (7 – x) 3.1i Homework: More Simplifying Simplify the following expressions: 3(2x + 1) – 3(2x + 1) – 3(2x – 1) – (x + 4) – (x – 4) – (4 – x) – 2(4x – 3) – 5(3x + 2) – 2.7(2x – 5)
Ana earned x dollars at her summer job. She put 50% of her money into savings and spent 40% of the rest on a new outfit. Write an expression for the amount of money she has left. Marcela earns money working in her dad’s office over the summer. If she spent 40% of her money on a new phone and 50% of the rest on new clothes for school, write an expression for how much of her money she spent. 3.1l Self-Assessment: Section 3.1 Consider the following skills/concepts. Rate your comfort level with each skill/concept by checking the box that best describes your progress in mastering each skill/concept. Corresponding sample problems, referenced in brackets, can be found on the following page. Skill/Concept Beginning Understanding Developing Skill and Understanding Practical Skill and Understanding Deep Understanding, Skill Mastery Recognize and explain the meaning of a given expression and its component parts. [1, 2, 3, 4] I struggle to understand the meaning of parts of a given expression. I can recognize the different parts of a given expression and match them with their meaning. I can recognize the different parts of a given expression and can explain their meaning in my own words. I can explain the meaning of a given expression and its parts in my own words. I can also write a meaningful expression given a context. Recognize that different forms of an expression may reveal different attributes of the context. I can’t tell when two expressions are really different forms of the same I can recognize that two expressions are different forms of the same expression, but I I can recognize that two expressions are different forms of the same expression, and I can explain I can write two expressions that are different forms of the same expression, and I
[5, 6] expression. struggle to see why I would write it in a different form. how the different forms reveal different attributes of the context. can explain how the different forms reveal different attributes of the context. Combine like terms with rational coefficients. [7 (rational c & d)] I struggle to combine like terms. I can combine like terms with a model. I can combine like terms with integer coefficients without a model. I can combine like terms with rational coefficients. Use the Distributive Property to expand and factor linear expressions with rational numbers. [8 c,d] I know what the Distributive Property is, but I struggle to use it. I can use the Distributive Property to expand linear expressions with integers such as Problem 8a. I can use the Distributive Property to expand linear expressions with rational numbers such as Problems 8a and 8b. I can use the Distributive Property to expand and factor linear expressions. Recognize properties of arithmentic and use them in justifying work when manipulating expressions. [10] I know there are properties, but I have a hard time recognizing them in work. I can name properties when I see them, but struggle to use them to justify manipulations. I can use properties to justify manipulations with expressions. I can use properties to justify manipulations with expressions and can use them to critique the arguments of others. Write numeric and algebraic expressions to represent contexts. I struggle to translate contexts to numeric and algebraic expressions. Most of the time I can translate contexts to numeric and algebraic expressions. I can translate contexts to numeric and algebraic expressions. I have no trouble translating contexts to a variety of numeric and algebraic expressions. Sample Problems for Section 3.1 Given the following situation, match the parts of the expression to the parts of the context. Luis went to a soccer game with some friends. He bought two sodas for $1.50 each and four giant candy bars for $2.25 each. How much did he spend? Explain the different parts of the following expression: Matthew likes to buy movies from a DVD club. He pays $5 per month plus $2 per movie. If he buys six movies in June, how much did he pay for the month of June? Write an expression to match the following situation. Explain the different parts of the expression.
Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Distributive Property of Multiplication over Addition Additive Inverse Multiplicative Inverse Use properties to justify steps: Example: Jane wants to find the sum: 3 + 12 + 17 + 28. She uses the following logic, “3 plus 17 is 20, and 12 plus 28 is 40. The sum of 20 and 40 is 60.” Why is this okay? The table below shows how to justify her thinking using properties for each step. Statement Step Justification 3 + 12 + 17 + 28 No change, this is where she started. This expression was given. 3 + 17 + 12 + 28 The 17 and the 12 traded places. Commutative Property of Addition (3 + 17) + (12 + 28) Jane chose to add the numbers in pairs first, which is like inserting parentheses. Associative Property of Addition 20 + 40 Jane found the sums in the parentheses. Jane is now following the Order of Operations. 60 Add, so . . . 3 + 12 + 17 + 28 = 60 1) The expression 3(x – 4) + 12 has been written in five different ways. State the property that allows each change.
= x – 1 = 6 Consider the equation: x – 3 = 5. What value of x makes this equation true? Justify: Draw a model of x – 3 = 5 below, then use it to illustrate the algebraic steps needed to solve. = x – 3 = 5 = Use a model to solve 8 = 7 + m. Write the algebraic procedure you followed to solve. = 8 = 7 + m
Additional Practice 18) 7 = m – 9 23) 40 = 5y 19) 12 = w + (4) 24) 5 = 2x + 9 20) 4k = 20 25) 4x + 7 = 5 21) 18 = 2n 26) 8 = 3m + 10 22) 3x = 15 27) 7z + 1 = 15 3.2a Homework: Model and Solve Equations Model and solve each equation below. Use the key below to model your equations. 1. x – 6 = –9 = x 6 = 9 x + 6 = 9 + 6 +6 Additive Inverse x = 3
= = = 3.2b Class Activity: More Model and Solve One- and Two-Step Equations Draw a model, justify your steps and then check your answer. The first one is done for you. Model/Draw the Equation What are the solving actions? Record the steps using Algebra Check solution in the equation. 1. x + 5 = 8 Add –5 to both sides. 5 is the additive inverse of 5. This step isolates the x. True, so the solution is correct.
7. 7 = 3x – 2 8. –5 = –3 + 2x 9. 7 + x/2 = –3 10. –3 = x/2 + 2 3.2b Homework: More Model and Solve One- and Two-Step Equations Model/Draw the Equation What are the solving actions? Record the steps using Algebra Check solution in the equation.
Connect arithmetic solution processes that do not use variables to algebraic solution processes that use equations. I do not understand the connection between arithmetic solution processes that do not use variables to algebraic solution processes that use equations. I can identify the connection between arithmetic solution processes that do not use variables to algebraic solution processes that use equations. I can explain the connection between arithmetic solution processes that do not use variables to algebraic solution processes that use equations. I can explain the connection between arithmetic solution processes that do not use variables to algebraic solution processes that use equations using the vocabulary of mathematical properties as appropriate. Use the properties of arithmetic to make an argument and/or critique the reasoning of others when solving algebraic equations. [3] I struggle to understand the reasoning of algebraic equations. I can tell if my reasoning or another’s reasoning to a solution is correct or not, but I struggle to explain why. I can tell if my reasoning or another’s reasoning to a solution is correct or not, and I can use properties to explain why. I can tell if my reasoning or another’s reasoning to a solution is correct or not, and I can use properties to explain why. If a solustion is incorrect, I can also explain with properities how it needs to be changed to show a correct reasoning and solution. Sample Problems for Section 3.2 Solve each of the following equations with or without a model.
Kevin is trying to understand percents, but often gets confused. He knows that in his school 55% of the students are girls and 45% are boys. It seems to him that there are 10% more girls than boys in his school. Is he correct? Explain. Ana Maria says 3/4 of the students in her class like rap music. Marco says that is 3/4% of the class. Is Marco correct? 3.3d Self-Assessment: Section 3.3 Consider the following skills/concepts. Rate your comfort level with each skill/concept by checking the box that best describes your progress in mastering each skill/concept. Corresponding sample problems, referenced in brackets, can be found on the following page. Skill/Concept Beginning Understanding Developing Skill and Understanding Practical Skill and Understanding Deep Understanding, Skill Mastery Use variables to create equations that model word problems. [1] I struggle to begin writing an equation that models a word problem. I can draw a model that represents a word problem. I struggle to use that model to create an equation. I can write an equation that represents a word problem if I draw a model first. I can write an equation that models a word problem. Solve word problems leading to linear equations. [1] I struggle to solve word problems leading to linear equations. I can usually write an equation to solve a word problem leading to a linear equation, but I struggle using that equation to get a solution. I can solve word problems leading to linear equations. I can solve word problems leading to linear equations. I can explain the solution in context. Recognize and explain the meaning of a given equation and its component parts when using percents. [2] I struggle to understand the meaning of parts of a given equation when using percents. I can recognize the different parts of given equation and match them with their meaning when using percents. I can recognize the different parts of a given equation when using percents and can explain their meaning in my own words. I can explain the meaning of a given equation and its parts in my own words. I can also write an equation given a context. Solve multi-step real-life percent problems involving calculations with positive and negative rational I struggle to solve real-life percent problems I can usually write an equation to solve a I can solve percent problems I can solve real-life percent problems
numbers in a variety of forms. [3] involving positive and negative rational numbers. percent problem, but I struggle using that equation to get a solution. involving positive and negative rational numbers. involving positive and negative rational numbers. I can explain the solution in context. Convert between forms of a rational number to simplify calculations or communicate solutions meaningfully. [3c] I struggle to convert between all forms of a rational number. I can convert between forms of a rational number, but I struggle to know how to use that to solve equations or simplify calculations. I can convert between forms of a rational number to simplify calculations, solve equations, and communicate solutions meaningfully. I can convert between forms of a rational number to simplify calculations, solve equations, and communicate solutions meaningfully. I can explain why a particular form communicates the solution meaningfully. Sample Problems for Section 3.3 Write an equation to represent each of the following word problems. Solve each problem. Express answers as fractions or decimals when appropriate. Chloe has twice as many cats as her sister has dogs. Her brother has two turtles. Together, they have five pets. How many of each pet do they have? Brian is three times older than Sydney. The sum of the ages of Brian and Sydney is eight. How old is Brian? Samantha is 8 and two-third years older than Jason. The sum of their ages is 23 years. How old is Jason? Explain the meaning of the following equation that matches the given situation: I go to a department store with a coupon for 30% off any one item. The pants that I want are on sale for 40% off. What was the original price if I pay $21? Write an equation to represent each of the following real-life percent problems. Solve each problem. Express answers as fractions or decimals when appropriate.