WRITING AND SOLVING EQUATIONS. GRADE 8 or 9 ALGEBRA

Transcription

1 WRITING AND SOLVING EQUATIONS GRADE 8 or 9 ALGEBRA Shannon Seaver 7-12 Math Clearbrook-Gonvick School Shannon@clearbrook-gonvic.k12.mn.us Jessica Strom 7-12 Math Win-E-Mac School jstrom@win-e-mac.k12.mn.us 1

2 Executive Summary: This Unit covers the following Minnesota state standards: Understand that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable. Use functional notation, such as f(x), to represent such relationships For example: The relationship between the area of a square and the side length can be 2 expressed as f ( x) = x. In this case, f (5) = 25, which represents the fact that a square of side length 5 units has area 25 units squared. Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount For example: Uncle Jim gave Emily $50 on the day she was born and $25 on each birthday after that. The function f ( x ) = x represents the amount of money Jim has given after x years. The rate of change is $25 per year Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another Evaluate algebraic expressions, including expressions containing radicals and absolute values, at specified values of their variables. For example: Evaluate πr 2 h when r = 3 and h = 0.5, and then use an approximation of π, to obtain an approximate answer Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships. For example: For a cylinder with fixed radius of length 5, the surface area A = 2π(5)h + 2π(5) 2 = 10πh + 50π, is a linear function of the height h, but it is not proportional to the height. Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used. For example: The equation 10x + 17 = 3x can be changed to 7x + 17 = 0, and then to 7x = -17 by adding/subtracting the same quantities to both sides. These changes do not change the solution of the equation. Another example: Express the radius of a circle in terms of its circumference. 2

3 Represent and solve problems in various contexts using linear and quadratic functions For example: Write a function that represents the area of a rectangular garden that can be surrounded with 32 feet of fencing, and use the function to determine the possible dimensions of such a garden if the area must be at least 50 square feet Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. This unit gradually progresses through writing expressions, simplifying expressions, writing equations and then solving equations of varying difficulty. It uses manipulatives called Algeblocks and also a few extra activities. The students should have already worked with the Algeblocks when learning about adding, subtracting, multiplying and dividing integers. The homework problems for the unit are from Algebra 1 by McDougal Littell (2004). Note: It is very important that the students are making the connections between what they are doing with the Algeblocks and what they are doing on paper. They should work in partners, one will work with the Algeblocks and the other will do the paper work, then they should switch jobs. Sample MCA questions from the 11 th grade test follow that students should be able to answer after this unit. Note: this unit is for a beginning Algebra student. There is much more that the students will learn before this test that is built on the content of this unit: 7. Tickets for a concert cost $15.00 each plus $1.50 each for handling charges. The shipping fee for an order of any number of tickets is $4.00. Which 3

4 equation could be used to determine the cost, C, of any number of tickets, t? A. C= 20.50t B. C= 17.00t C. C=16.50t+4.00 D. C=15.00t Yia wrote the equation to represent the P=0.25n-(0.05n+1) school store s weekly profit, P, from sales of n pencils. Which equation is equivalent to Yia s equation? A. P=0.20n 1 B. P=0.20n+1 C. P=0.125n2 1 D. P=0.0125n The yearly growth of a tree is modeled by concentric circles in a cross-sectional cut through the tree trunk as shown. The one-year-old tree has a diameter of 2.5 centimeters. Which equation models the diameter of the tree, d, in terms of the age of the tree in years, n? A. d= 2.5+2(n 1) B. d= 2.5n+2 C. d= 2.5+2n D. d= 2.5n A family is carpeting two rectangular rooms. They have chosen carpeting that costs the same amount per square yard for each room. A 12-foot by 15-foot carpet for the bedroom costs $600. If the dimensions of the living room are 20 feet by 18 feet, what will it cost to carpet the living room? A. $624 B. $720 C. $1,000 D. $1,200 4

5 WRITING AND SOLVING EQUATIONS FOR GRADE 9 ALGEBRA Table of Content 1. Writing expressions (2 days) a. Algeblocks b. Pictures c. Abstract 2. Simplifying expressions (1 day) 3. Writing equations (1 day) a. Algeblocks b. Pictures c. Table d. Graphs (calculators) e. Abstract 4. Solving equations (12 days) a. One step (3 days) b. Two steps (2 days) c. Multiple steps (1 day) d. Distribution and multiple steps (3 days) e. Variables on both sides/multi-step (3 days) 5

6 Writing Expressions (Day 1 and 2) Lesson Objective: Students can write expression given a real-life situation Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Materials: Boxes and candy Algeblocks Handouts Writing Expressions and Writing Expressions Homework I have who has? game cards Launch: Boxes of candy problem. (Need two boxes with same number of candies inside and also extra candy). Give one box to student A and one box to student B. Explain to the class that there is the same number of candies in each box (no tricks). Then give student A 3 more pieces of candy. Ask the class how many pieces of candy student A and student B have. You should make a table, draw a picture, and write the expression using a variable. Question: What if student A ate a piece of candy, now how many pieces of candy does the student have? -- show a picture, write an expression Question: What if I gave student B two more boxes, now how many pieces of candy does student B have? Show a picture and write an expression Get out the Algeblocks Have the students model the candy boxes problem with Algeblocks Explore: 1. Writing Expressions Handout 2. I have, who has game. Pass out cards that have an expression on one side and a question about a written equations on the other to each student. Example: I have 2n+3 who has the sum of a number and 4? Share: Have students share their answers to the handout with the class before getting their homework. 6

7 Summarize: Make sure that everyone in each group can write an expression using variables and use Algeblocks to demonstrate expression. Homework: Writing Expressions Homework sheet 7

8 Writing Expressions 1. The Rosemary family is going to the movies. It costs $4 per person to get in. Each person is also going to get candy. The candy is $1.50 each. Step 1: Identify what the variable would be in this problem. Step 2: Use Algeblocks to represent this problem. Draw your picture below Step 3: Write the expression representing this problem using a variable. Step 4: Make a table. On the left side, put your variable, on the right side put your expression. Fill in the table with various values for the variable. What do the numbers on the right side of the table stand for? 8

9 Writing Expressions Handout pg.2 2. I am twice as old as Amanda plus 3 more years. Step 1: Identify what the variable would be in this problem. Step 2: Use Algeblocks to represent this problem. Draw your picture below Step 3: Write the expression representing this problem using a variable. Step 4: Make a table. On the left side, put your variable, on the right side put your expression. Fill in the table with various values for the variable. What do the numbers on the right side of the table stand for? What do the numbers on the left side represent? 9

10 Writing Expressions Homework Sammy bought 5 bags of pencils for math class. His friend Todd didn t buy any pencils and needed to borrow 3 pencils from Sammy. How many pencils did Sammy have? Step 1: Identify what the variable would be in this problem. Step 2: Use Algeblocks to represent this problem. Draw your picture below Step 3: Write the expression representing this problem using a variable. Step 4: Make a table. On the left side, put your variable, on the right side put your expression. Fill in the table with various values for the variable. What do the numbers on the right side of the table stand for? 10

11 Simplifying expressions (Day 3) Lesson Objective: Students can write and simplify expressions Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount. Materials: Chocolates for Me Handout Algeblocks Simplifying Expressions Handout Launch: Ted brought his favorite math teacher a box of chocolates. Mandi brought the teacher 2 boxes of chocolates. Kyle meant to bring the teacher a box of chocolates, but got hungry during his 3 rd hour class and ate all but 2 chocolates. If all the boxes had the same number of chocolates, how many chocolates did the students bring their favorite math teacher? Explore: Give the students the Chocolates for Me handout. The students will use Algeblocks to represent the problem, draw pictures, write the expressions abstractly, and make a table of possible answers. Then the students will use the Algeblocks to simplify the expression and also show how they did so on paper. Share: Have students share what they did on the board. Summarize: 1. Emphasize how to use Algeblocks and paper work need to make sure the students do both simultaneously with a partner to see the connection. 2. If time allows play Are we like terms? Hand everyone an index card with a term on it. (examples: 2xy, 5xy, 3x, 5y etc) Have them find a group of people with whom they are like terms. Launch: Simplifying Expressions Handout 11

12 Explore: Students work together on handout using Algeblocks. Share: Have students share answers Summarize: Clear up any questions students may have. 12

13 CHOCOLATES FOR ME Ted brought his favorite math teacher a box of chocolates. Mandi brought the teacher two boxes of chocolates. Kyle meant to bring the teacher a box of chocolates, but got hungry during his 3 rd hour class and ate all but 2 chocolates. If all the boxes had the same number of chocolates, how many chocolates did the students bring their favorite math teacher? Use Algeblocks to represent this problem. Draw a picture of your Algeblocks below. Write an expression for the problem. Simplify your Algeblocks and draw a picture below of your result. Simplify your expression from above. How does this compare to what you did with your Algeblocks? Make a table where one column represents Ted s chocolates, Mandi s chocolates, Kyle s chocolates, and the last column represents how many chocolates the teacher got. 13

14 Simplifying Expressions Handout For each of the following problems, work with a partner. You need to represent each expression using Algeblocks and draw a picture of the Algeblocks. Then, do any simplifying you can with the Algeblocks and show this with your picture and the expression (as demonstrated in class). (Use the +/- mat with the Algeblocks) 1. 3x +5 +2x x +3 x 3. 2x + 3x + x + 3 x x + (-2x) 14

15 15

16 Writing equations (Day 4) Lesson Objective: Students can write an equation and understand the meaning of = Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Materials: Algeblocks Graph paper Writing Equations Handout Launch: Timmy forgot how many frogs he put in his neighbors mailboxes. He knows he put the same number of frogs in 4 of his neighbors mailboxes. He has 1 left over. He started with 17 frogs. Explore: As a class, explore the above problem with Algeblocks, a picture, a table, a graph, and an equation with variable(s). Make sure to emphasize that there is an equal sign and what it means for Algeblocks and the meaning of equations in general. Give the students the Writing Equations handout. Share: Have the students share their answers to problems that would be useful for all to see. Make sure to talk about the last question. Summarize: Give the students the following problem to write as an equation. If they want to use Algeblocks, they can. Timmy s sister Amanda saw him putting frogs into their neighbors mailboxes. To make up for it, she decides to make brownies for them. If she split the number of brownies between all 5 houses evenly, they would each get 4 brownies. 16

17 Writing Equations Handout 1. My sister bought 3 bags of marbles, but on her way to the marble tournament 5 of them fell out. When she got there she had a total of 7 marbles. a. Use Algeblocks to represent this equation. You will need to use the equation mat. Draw the picture below and write the equation with variables. 2. Circle the equations below: 3x +7 = 51x 6/7 = 12x -8 52x > 15x Is it possible for 3x + 1 = 3x? 17

18 Solving One Step Equations with Adding or Subtracting (Day 5) Lesson Objective: Students can write equations and solve them by using addition and subtraction Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships Represent and solve problems in various contexts using linear and quadratic functions Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables. Materials: Boxes with candy Algeblocks Paper and Pencil Candy Handout Solving Equations (addition and subtraction) Launch: Let us return back to the candy example. I gave student A a box of candy and 3 extra pieces. I noticed that student A now has 8 pieces of candy. How many candies are in the box? Explore: Candy Handout: The handout will force the students to use Algeblocks, draw a picture and show how they are solving the equation step-by-step. It is very important that students make this connection. Partner them up, have one manipulate the Algeblocks and the other write and manipulate the equation. Then they should switch roles and do this problem again. 18

19 Share: Have the groups share what they did with the class. Explore: Give the students the Solving Equations (addition and subtraction) Handout and have them work on it using Algeblocks. Share: Have the students share some of their stories and solutions. Summarize: Make sure students are doing their manipulations correctly. Homework: p135 (22-27,30,33,38) 19

20 CANDY HANDOUT Recall the box of candy problem. Student A got a box of candy and 3 extra pieces. I noticed that Student A now has 8 pieces of candy. How many candies are in the box? Use algeblocks and the equation mat to show this situation. Draw the picture below. = Write the equation. 20

21 Candy Handout pg. 2 Now manipulate the Algeblocks to get one variable by itself and show what you did by drawing the related pictures below. = Start back at the beginning. This time, while you manipulate the Algeblocks, have your partner do the same thing to the equation that you wrote. You should both end up with the same answer. Switch roles and start from the beginning again. 21

22 SOLVING EQUATIONS (addition and subtraction) Instructions: For each of the problems below, represent the equation using Algeblocks. In partners, one person will manipulate the Algeblocks and the other will manipulate the equation. Then switch roles and do each problem again. 1. x + 5 = x = = x 6 4. x + (-4) = = -n (-4) 6. x 9 = = b 6 8. R (-2) = x = Pick one of the problems above and make a story out of it. 22

23 Solving One-Step Equations Using Division (Day 6) Lesson Objective: Students can write and solve equations that involve solving by division Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships Represent and solve problems in various contexts using linear and quadratic functions Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables. Materials: Algeblocks Solving Equations (division) Handout Paper and Pencil Prelaunch: Go over any questions from homework. Launch: Arthur wants to buy three decks of cards. He pays a total of $12. How much is each deck of cards? Explore: Work with students to show this using Algeblocks. Talk about the meaning of solving for the unknown. You want to see one variable, you have three, so you need to split into 3 groups (both sides). This is the idea of division. Go over how to show this with equations and Algeblocks together. Also explore how you could solve this using a table. 23

24 Have students work on the Solving Equations (division) handout with Algeblocks. Share: Have students share their solutions and answers. Summarize: Emphasize the connection between the Algeblocks and the equation solution. Homework: p142 (24-31) 24

25 SOLVING EQUATIONS (division) Instructions: For each of the problems below, represent the equation using Algeblocks. In partners, one person will manipulate the Algeblocks and the other will manipulate the equation. Then switch roles and do each problem again. 1. 4x = x = x = x = = 5x 6. -4x = /2x = x = = 3x 10. Ed bought a bunch of packs of bubble gum. Each pack had 5 pieces of gum. He now has a total of 35 pieces of gum. How many packs did Ed buy? Write and equation and solve the same as above. 25

26 Solving One-Step Equations using Multiplication (Day 7) Lesson Objective: Student can write and solve equations involving multiplication Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships Represent and solve problems in various contexts using linear and quadratic functions Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables. Materials: Algeblocks Solving Equations (multiplication) Handout Prelaunch: Go over any questions from homework. Launch: You bring 5 friends home after school and find that you mom just made cookies. You give the same amount of cookies to each of your 5 friends. In the end, each friend got 2 cookies. How many cookies did you start with? Explore: Talk as a class about how to show this problem using Algeblocks, pictures and equations. Give each student the Solving Equations (multiplication) handout. Share: Have students share solutions to handout. 26

27 Summarize: Make sure the students understand the connection between the Algeblocks and the equation solution. Homework: p142 (33-39) 27

28 SOLVING EQUATIONS (multiplication) Instructions: For each of the problems below, represent the equation using Algeblocks. In partners, one person will manipulate the Algeblocks and the other will manipulate the equation. Then switch roles and do each problem again. x = x = 3. x 4 = f = 4 5. x = = d 6 7. x + 3 = 5 8. x ( 2) = x = 6 x 10. 5x = = p = 2 = 4 28

29 29

30 Solving Two-step Equations (Days 8 & 9) Lesson Objective: Students will be able to write and solve problems involving two-steps and understand the order in which to do so Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols Use linear equations to represent situations involving a constant rate of change, including proportional and non-proportional relationships Represent and solve problems in various contexts using linear and quadratic functions Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used. Materials: Algeblocks Solving Two-step Equations Handout Prelaunch: Go over questions from homework Launch: Franny has been collecting egg cartons. To display them she puts them on a shelf. She has 3 full shelves and then one extra. If she has collected a total of 10 egg cartons, how many fit on a shelf. 30

31 Explore: Have students show this problem using Algeblocks, equations, pictures and a table. Share as a class what the various solutions should look like. Have the students work on the Solving Two-Step equations Handout. Share: Have students share their stories and also their solutions to various problems. Summarize: Discuss if there is more than one way to solve these problems, as in the order you can do the inverse operations. Make sure students understand the connection between the Algeblocks and equation solution. Homework: p148 (16-24) 31

32 SOLVING TWO STEP EQUATIONS HANDOUT Instructions: For each of the problems below, represent the equation using Algeblocks. In partners, one person will manipulate the Algeblocks and the other will manipulate the equation. Then switch roles and do each problem again. x 1. 5 x 1 = = 7 3. x + 4 = 6 2 x = 2x = x ( 2) = = 3x x 3 = x + 1 = Come up with a story that would make sense for the following problem. Then solve it like you did with the above problems. 2 x + 7 = 15 32

33 Solving Multi-step equations (Day 10) Lesson Objective: Students will be able to solve equations that involve multiple steps, i.e. combining like terms. Students understand the different orders they can solve them Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables. Materials: Solving Multiple Steps Handout Algeblocks Paper and pencil Prelaunch: Answer any homework questions Launch: Give the students a handout Explore: Have students complete the handout with Algeblocks and equation solutions. Share: Have the students share their solutions Summarize: Talk about the different orders you can solve some of the equations and which way seems to be faster. Homework: p148 (25-28,42,43) challenge problem #50. 33

34 SOLVING MULTIPLE STEPS HANDOUT Instructions: For each of the problems below, represent the equation using Algeblocks. In partners, one person will manipulate the Algeblocks and the other will manipulate the equation. Then switch roles and do each problem again x 3x 8 = x 3 + 4x = x 6 = x 2 + 2x ( 3) = x = x 10 = = x + x 8. x + 2 x + 3x 5x x =

35 Solving equations with distribution and multiple steps (Day 11) Lesson Objective: Students will be able to write equations involving distribution from a real world problem and solve Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount. For example: Uncle Jim gave Emily $50 on the day she was born and $25 on each birthday after that. The function f ( x ) = x represents the amount of money Jim has given after x years. The rate of change is $25 per year Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used. For example: The equation 10x + 17 = 3x can be changed to 7x + 17 = 0, and then to 7x = -17 by adding/subtracting the same quantities to both sides. These changes do not change the solution of the equation. Another example: Express the radius of a circle in terms of its circumference Represent and solve problems in various contexts using linear and quadratic functions. For example: Write a function that represents the area of a rectangular garden that can be surrounded with 32 feet of fencing, and use the function to determine the possible dimensions of such a garden if the area must be at least 50 square feet Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context Justify steps in generating equivalent expressions by identifying the properties used. Use substitution to check the equality of expressions for some particular values of the variables; recognize that checking with substitution does not guarantee equality of expressions for all values of the variables Sketch graphs of linear, quadratic and exponential functions, and translate between graphs, tables and symbolic representations. Know how to use graphing technology to graph these functions. Materials: Algeblocks Karen s brother and his thieving friends handout Sally is how old? Handout 35

36 Launch: Karen s brother and his thieving friends problem. Karen walked into her room the other day and saw her brother and his friends each eat one of her Laffy Taffy s. Apparently they had been eating her stash of Laffy Taffy s for a while because they each had the same amount of wrappers in front of them, plus the one they had just eaten. If there are four thieving boys in Karen s room and 16 Laffy Taffy s are gone, how many did each boy eat? Explore: Give the students the handout Karen s brother and his thieving friends and instruct the students, in groups, to look at this problem in multiple ways, including: Algeblocks, pictures, tables, graphs, and abstract equations. Share: Have the groups share their solutions and methods. Summarize: Discuss with the class how you would solve this problem abstractly and how that relates to the other methods they were also supposed to use. Maybe have someone demonstrate the Algeblocks as you work out the problem. Give the students the handout Sally is how old? to have completed for the next day. They should work together if there is time in class. 36

37 KAREN S BROTHER AND HIS THIEVING FRIENDS Problem: Karen walked into her room the other day and saw her brother and his friends each eat one of her Laffy Taffy s. Apparently they had been eating her stash of Laffy Taffy s for a while because they each had the same amount of wrappers infront of them, plus the one they had just eaten. If there are four thieving boys in Karen s room and 16 Laffy Taffy s are gone, how many did each boy eat? 1. Represent this problem using Algeblocks. Draw it below. 2. Represent this problem using a picture. 3. Use #1 or 2 to help you write an equation for this problem and solve. Have your partner record as you manipulate the Algeblocks. 4. Make a table and graph of this problem. Does your solution match you solution to #3? 37

38 SALLY IS HOW OLD? 1. Problem: Mary is twice the sum of Sally s age and 2 years. If Mary is 12, how old is Sally? Use Algeblocks and a picture to represent this problem. Then write it as an equation. Use the Algeblocks to help you solve the problem. For problems 2 5, use Algeblocks to help you solve the problem. Make sure to record your steps. 2. 2(x+3) + 5 = (x-3) + 3(x+4) = (x+2) = (x+2) 3(x+1) = -7 38

39 Solving equations with distribution and multiple steps (Day 12) Lesson Objective: Students will demonstrate understanding of the distribution concept and how to solve the equations. MN Standards the same as Day 11 Materials: Algeblocks Pre-Launch: Have students compare their work from the previous day s handout in their groups. Go over the answers as a class. Launch: Have students create a problem that uses distribution to solve. Explore: In their groups they should share their problems with each other. In partners they should solve each problem using algeblocks and abstract solutions. Share: Each group should pick a favorite problem and share it with the class. The groups should then work on solving all of the problems. Summarize: Have each group show their work to their favorite problem on the board and make sure each group got the same solution. Discuss any differences and what that might mean. Homework: From McDougal Littell; Algebra 1 pg 148 (30, 33, 37-40) 39

40 Solving equations with distribution and multiple steps (Day 13) Lesson Objective: Students will demonstrate mastery of solving equations with the distributive property. MN State standards are the same as day 11. Materials: Algeblocks Launch: Have students get out their homework and the algeblocks. Explore: Have the students check their answers with others in their groups and then show how to do the same steps using algeblocks on problems 33 and 37. Share: Have one or two groups share their process with the class. Summarize: Address anything that my be misunderstood or missing. 40

41 Solving equations with variables on both sides (Day 14) Lesson Objective: The students will be able to solve equations with multi-steps and variables on both sides. MN state standards are the same as Day 11. Materials: Wedding Problem Handout Is it Fair? Handout Algeblocks Launch: Wedding problem My friend got married this summer and she asked a few of her friends to come over and help put together the favors for on the tables. We were supposed to put Hershey s kisses into boxes. She handed out kisses to everyone and some boxes. I noticed after a few minutes that two of us had filled one box each and each had one kiss left over. Three others had filled one box each and each had 2 kisses left over. When we put it all together and used the kisses to fill more boxes, we had 9 boxes filled. How many kisses did we put in each box (assume each box got the same number of kisses). Explore: Give the students handout Wedding Problem and instruct the students, in groups, to look at this problem in multiple ways, including: Algeblocks, pictures, tables, graphs, and abstract equations. Share: Have the groups share their solutions and methods. Summarize: Discuss with the class how you would solve this problem abstractly and how that relates to the other methods they were also supposed to use. Maybe have someone demonstrate the Algeblocks as you work out the problem. Give the students the handout Is it Fair? to have completed for the next day. They should work together if there is time in class. 41

42 WEDDING PROBLEM My friend got married this summer and she asked a few of her friends to come over and help put together the favors for on the tables. We were supposed to put Hershey s kisses into boxes. She handed out kisses to everyone and some boxes. I noticed after a few minutes that two of us had filled one box each and each had one kiss left over. Three others had filled one box each and each had 2 kisses left over. When we put it all together and used the kisses to fill more boxes, we had 9 boxes filled. How many kisses did we put in each box (assume each box got the same number of kisses). 1. Represent this problem using Algeblocks. Draw it below. 2. Represent this problem using a picture. 42

43 (More on back ) 3. Use #1 or 2 to help you write an equation for this problem and solve. Have your partner record as you manipulate the Algeblocks. 4. Make a table and graph of this problem. Does your solution match you solution to #3? 43

44 IS IT FAIR? 1. Problem: When we got back from the 4 th of July parade, Toby, Ella and I compared how many bubble gums we each had. Toby had twice as many bubble gums as I had plus 1. Ella on the other hand had four times as many bubble gums as me. If she eats four, Ella will have the same number of bubble gums as Toby and I together. How many bubble gums do I have? How many bubble gums does Toby have? How many bubble gums does Ella have? Use Algeblocks and a picture to represent this problem. Then write it as an equation. Use the Algeblocks to help you solve the problem. For problems 2-5 use Algeblocks to help you solve the problem. Make sure to record your step. 2. 6x + 22= -3x x + 20 = 5x 4. x+2 = 3x x + 7 = 4x

45 Solving equations with variables on both sides (Day 15) Lesson Objective: The students will be able to solve equations with multi-steps and variables on both sides. MN state standards are the same as Day 11. Materials: Algeblocks Pre-Launch: Have students compare their work from the previous day s handout in their groups. Go over the answers as a class. Launch: Have the students write a problem that has variables on both sides. Explore: In their groups they should share their problems with each other. In partners they should solve each problem using algeblocks and abstract solutions. Share: Each group should pick a favorite problem and share it with the class. The groups should then work on solving all of the problems. Summarize: Have each group show their work to their favorite problem on the board and make sure each group got the same solution. Discuss any differences and what that might mean. Homework: From McDougal Littell; Algebra 1 pg 157 (18, 23, 26, 27, 33) challenge problem 24 45