8 th Grade Algebra 5 Day Lesson Plan. *Computer* *Pan Scale* *Algebra Tiles* *Equation Mat* *TI-83 Plus/ TI-73* Karen Kmiotek
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1 8 th Grade Algebra 5 Day Lesson Plan *Computer* *Pan Scale* *Algebra Tiles* *Equation Mat* *TI-83 Plus/ TI-73* Karen Kmiotek
2 Objectives Students will be able to solve equations by using algebra and the properties of equality, graphs and tables on the calculator, and an equation mat and algebra tiles. Students will experiment with various models to solve equations. Students will work with others to solve equations. Students will be able to present mathematical concepts using mathematical language. NYS Key Ideas: 1A, 1B Mathematical Reasoning 2A Number & Numeration 3A, 3E - Operations 4B, 4D, 4E, 4F-Modeling/ Multiple Representation 5E - Measurement 7E Patterns/ Functions NCTM Standards: Number & Operations Algebra Problem Solving Communication Representation 2
3 Resources Lynch, Chicha and Eugene Olmstead. Math Matters Book 2- An Integrated Approach, South-Western Educational Publishing, Chapter 5, pp Ellis, Wade, Kathleen Hollowell, Paul Kennedy, and James Schultz. Algebra 1, Holt, Rinehart and Winston Publishing, Chapter 3, pp Cuevas, Gilbert J., editor. Navigating Through Algebra in Grades 3-5. Reston, Va: National Council of Teachers of Mathematics, Panbalance applet. (Uses Internet Explorer) Mathematics TI-83 Plus/ TI-73 Application, Algebra 1Chapter 2 3
4 Materials Pan Scale Chips Algebra Tiles Equation Mat TI-83 Plus/ TI-73 Computers Overhead Unit Calculator Overhead Screen 4
5 Overview of Unit Day 1- Properties of Equality -Opening Activity: Students will experiment with a pan scale and equality -Main Activity: A lesson on the Properties of Equality will be given using the TI-83 Plus/TI-73 calculator -Closing Activity: Students will use Navigations Series applet to further experiment with pan scales Day 2- One Step Equations, Addition and Subtraction -Opening Activity: Show and discuss cartoon on overhead Show equality on pan scales using the calculator application on the TI-83 Plus -Main Activity: Use calculator program to see that equations can be solved by graphing both sides of the equations, and by viewing a table. -Closing Activity: Students will experiment with solving word problems. Day 3- One Step Equations, Multiplication and Division -Opening Activity: In groups, students will determine solutions by graphing equations and reading a table. -Main Activity: Students will be assigned a group and a method for which they must prepare a presentation on solving equations. -Closing Activity: Students will do presentations in their groups. Day 4- Two Step Equations -Opening Activity: In groups, students will determine the equation that is modeled by the equation mat and algebra tiles. -Main Activity: Examples will be done on solving two-step equations by using an equation mat. Then the students will have to solve the equations they found in the opening activity. -Closing Activity: Students will have a choice of three games to play on their calculators to review all the properties of equality. Day 5- Review and Assessment -The first half of class will be spent reviewing for the test with the internet. -The students will take a test for the second half of the period. 5
6 Day 1 Objectives: Students will experiment with pan scale and equality. Students will be able to use the properties of equality to find equivalent expressions. Opening Activity: In groups of two, students will experiment with a pan scale and weighted chips. From what they see by placing the chips on both sides, they will make conjectures about equality. Main Activity: I will present all of the properties of equality to the students on the overhead. Students will follow through the program on their calculators. When two expressions are equal, you can add the same number to each expression and the resulting sums will be equal. This is called the addition property of equality. F or all real number s a, b, and c, if a = c, then a + b = c + b. - For example, if x 6 = 10, then x = or x = 16 When two expressions are equal, you can subtract the same number from each expression and the resulting differences will be equal. This is called the subtraction property of equality. F or all real number s a, b, and c, if a = c, then a b = c b. - For example, if x + 1 = 4, then x = 4 1 or x = 3 6
7 You can also multiply two equal expressions by the same number. When you do this, the resulting products will be equal. This is called the multiplication property of equality. F or all real number s a, b, and c, if a = c, then ab = cb. - For example, if 2x = 10, then 1/2(2x) = 1/2(10) or x = 5 Finally, you can divide two equal expressions by the same number. When you do this, the resulting quotients will be equal. This is called the division property of equality. For all real number s a, b, and c, with b 0, if a = c, then a = c. b b - For example if 3x = 27, then 3x = 27 or x = Next on the calculator, the students will see that we can use these properties to solve one and two step equations. 7
8 Closing Activity: Using the Algebra Navigations Series Pan Balance applet on the computer, the students will experiment with the pan scale and different weighted shapes. They will explore the differences in the weights of four different shapes. As they play along they will have to be thinking about the relationships of the weights between the different shapes. After going through all possibilities, the students will be asked to record the chart on a separate piece of paper. The students will then compare their data with the rest of the class. Homework: 8
9 Do a write up on what you observed from the pan scale model. Make sure to include the method you used to determine the relationships between the different weights, what the relationships are (list from heaviest to lightest), and what your feelings were as you placed the shapes on the scale. Also, using your previous knowledge and the Properties of Equality, come up with two examples that demonstrate each property. Sample Student Work: In order to determine the relationships between the weights of the shapes, I began by placing one shape on the right hand side of the scale and a different shape on the left hand side. I kept adding one shape to the side that was higher until the sides were balanced. I did this for each combination of shapes. After doing all six experiments, I made a chart on which shape was heavier in each case. I then compared all six charts to determine that the heaviest shape was the diamond, then the circle, then the square, and lightest was the upside down triangle. Sometimes I felt frustrated while placing the shapes on the scale because one side would always be heavier than the other and I had to keep going back and forth to make the scale balance. But most of the time I thought this was pretty cool and helps me visually understand equality. Examples on Equality + 6x+2 = 4x-5, 6x+2+10 = 4x x+1 = 2x-6, -x+1+1 = 2x x+7 = x+1, 10x+7-2 = x x-5 = 3x-6, x-5-11 = 3x-6-11 * x+2 = 4x, 2(x+2) = 2(4x) * 7x+1 = x-2, 6(7x+1) = 6(x-2) / 3x = 21, 3x = / x+4 = 2x-1, x+4 = 2x Overhead Transparency 9
10 Addition Property of Equality For all real number s a, b, and c, if a = c, then a + b = c + b. Subtraction Property of Equality For all real number s a, b, and c, if a = c, then a b = c b. Multiplication Property of Equality For all real number s a, b, and c, if a = c, then ab = cb. Division Property of Equality For all real number s a, b, and c, with b 0, if a = c, then a = c. b b 10
11 Day 2 Objectives: Students will be able to solve equations by using various methods. Students will experiment with different models on solving equations. Students will be able to determine solutions by reading graphs. Opening Activity: The following cartoon will be presented on the overhead The students will have a couple of minutes to study the cartoon, then, as a class, we will have a discussion about it. After discussing the cartoon, I will show the students the pan scale example on the calculator on solving equations. 11
12 Main Activity: Students will use their Algebra 1 Chapter 2 APPS on their TI-83 Plus/TI-73. We will look at Where is the solution? As the students follow on their own calculators, I will go through the program on the overhead. After seeing how the program works, the students will have to create their own graphs and table. I will have a handout showing the steps on how to create a graph and table on the calculator. By using the handout, the students will have to do other examples on their own. 12
13 Student Handout Use a graph and table to determine the solution to 4x 6 = 2x. 1) Write two separate equations for each side y = 4x 6 and y = 2x. Put both equations into y = 2) To look at the table press 2 nd Graph. To find the solution, look for the intersection point where Y1=Y2. 3) Graph the equation. 4) On the TI-83 Plus, press Calculate, which is 2 nd Trace. Go to 5: Intersection because the intersection point will give us the solution to the equation. 13
14 5)Place the cursor on the first curve where you think the intersection point is. 6)Do the same for the second curve. 7)Next the calculator will ask you for a guess on the intersection point. 8)Press enter and the calculator will give you the exact intersection point. 14
15 Closing Activity: For the last part of class the students will be introduced to word problems. They will be given a word problem and in pairs, they will have to set up an equation. After the equation has been found, they will have to enter it in their calculators to find the solution, by either using a table or the graph. Question: The school needs $1500 to take the students on a field trip. So far the students have raised $ How much more money must the students raise? Answer: The students need to raise another $ Homework: For 1-5 A) Solve each equation by using either the Addition Property of Equality or the Subtraction Property of Equality. Check each solution B) Graph each equation. Use the graph and table to verify your answer. Sketch each graph and circle the point of intersection. 1) 7.2 = x ) r - 13 = ) 24 = h 53 4) d = ) 5 = x For 6 10, Create an equation for each situation. Then solve the equation. 6) The school football team needs new uniforms. So far they have raised $ towards a goal of $5000. How much more money must the football team raise? 7) Ten more than a certain number equals negative seven. What is the original number? 15
16 8) After Maria spent $27.97 at the store, she had $18.42 left. How much money did Maria have originally? 9) Bill has saved $132 towards the purchase of a new mountain bike. The bike costs $487. How much more money must Bill save before he can buy the bike? 10) A banquet room can hold 430 people. If 508 people show up for a party. How many people are going to be turned away? Looking Ahead Problems based on the next day s lesson 11) -3c = 27 12) 2 m = ) 5n = ) A case of bottled water weights 18 lbs and contains 24 bottles. How much does each bottle weigh? 15) Lisa deposits the same amount of money every month into her bank Homework Answer Key: 1) x = account. At the end of four months she has saved $368. How much did she deposit each month? 2) r =
17 3) h = 77 4) d = ) x =.5 6) x = 5000, The football team must raise another $ ) x + 10 = -7, The original number is 17. 8) x = 18.42, Maria had $46.39 originally. 9) x +132 = 487, Bill must save another $ ) 508 x = 430, 78 people are going to get turned away. 11) c = -9 12) m = 18 13) n = ) 18 = 24w, Each bottle weighs.75lbs. 15) 4x = 368, Lisa deposited $92 each month. 17
18 Day 3 Objectives: Students will be capable of determining solutions from graphs. Students will work with others to solve equations. Students will be able to present mathematical concepts using mathematical language. Opening Activity: Using the TI-83 Plus or the TI-73, the students will have to graph equations. From the graph they will have to determine the solution. They will be graphing equations involving the multiplication and division properties of equality. In groups of two, the students will take the equations off the overhead and first have to separate it into two equations. One student will graph the first equation and the other will graph the second. Looking at both of their graphs, the students will have to guess the point of intersection. After making a reasonable guess, both students will enter both equations in their calculators and see if their solution was correct. The students will have to do two problems: 1) 4x = -64, answer x = -16 2) x/7 = 12, answer x = 84 Main Activity: The class will be divided into groups of four. Each group will be assigned a method to solve equations. A couple groups will use the equation mat and algebra tiles. They will lie out the tiles to represent the equation. The equation to the left of the equal sign is placed in the 1 st box and the equation to the right of the equal sign is placed in the 2 nd box. The students will have to add in and take away tiles to solve the problem. Some groups will graph the equations in their calculatorand some will use tables, and a few groups will use the properties of equality to solve their equations algebraically. The groups will get about 15 minutes to work to solve three 18
19 equations using their assigned method. Each group will be assigned different equations. Closing Activity: The groups will be asked to choose one of their problems to present. Each group will present their problem to the class using their assigned method. The students will either use the calculator overhead screen to show their graphs of the equations or their table, an overhead equation mat and algebra tiles to show the steps involved in solving the equation, or the overhead or chalkboard to write out the problem using algebra and the properties of equality. Here are examples using each method. Calculator, graphs: x = Equation Mat: 3x = -15 = = Properties of Equality:.075 =.025x Calculator, tables: 4x + 2 = -14 x = =.025x use division property of equality x =3 19
20 Homework: For 1-5 A) Solve each equation by using either the Multiplication Property of Equality or the Division Property of Equality. Check each solution. B) Use your equation mat and algebra tiles to verify your solution. 1) 12y = 84 2) 16 = 4y 3) 9m = 0 4) 12 = 2t 5) 4x = -20 For 6-10, Create an equation for each situation. Then solve the equation. 6) If one book costs $4.82, how many books can Jeffrey buy with $20.00? 7) Natalie s family wants to make a 400 mile drive in 8 hours. What speed must they average for the trip? 8) A pizza is cut into 8 pieces. Each piece contains 10 pepperoni. How many pepperoni are on the entire pizza? 9) What is the price of one pen if Lynda can buy a package of 12 for $3.84? 10) Jen makes three times Tracie s weekly salary working at a restaurant. If Tracie makes $120 a week working as a cashier, how much money does Jen make per week at the restaurant? Looking Ahead Problems based on the next days lesson 11) 5x 3 = 17 12) x + 4 = 3x 2 13) Jack bought a pair of jeans that cost $32 and 5 shirts. His total bill, before tax, came to $122. What was the cost of one shirt? 14) A long distance phone company charges $4.99 per month plus $0.10 per minute. How many minutes were used in a month if the monthly bill is $16.99? 20
21 15) The number of cars sold at the local car dealership in July was 3 more than twice the number of cars sold in June. If 71 cars were sold in July, how many cars were sold in June? Homework Answer Key: 1) y = -7 2) y = -4 3) m = 0 4) t = 6 5) x = 5 6) 4.82x = 20, Jeffrey can buy 4 books 7) 8x = 400, 50 miles per hour 8) 1 x = 10 8 There were 80 pepperoni on the entire pizza. 9) 12x = 3.84, Each pen cost $ ) 1 x = Jen makes $360 per week. 11) x = 4 12) x = 3 13) x = 122, $18 per shirt 14) x = 16.99, 120 minutes 15) 3 + 2x = 71, 34 cars sold in June. 21
22 Main Activity groups and problems Group 1: Method: Algebra Answers 1) k/4 = 6 1) k = 24 2) 3x = 21 2) x = 7 3) 10x 4 =16 3) x = 2 Group 2: Method: Calculator, graphs 1) -8 = y/-9 1) y = 72 2) 5r = 60 2) r = 12 3)10x + 8 = 18 3) x = 1 Group 3: Method: Calculator, tables 1) 7 = f/4 1) f = -28 2) 49 = -7a 2) a = -7 3)-8y + 2 = 26 3) y = -3 Group 4: Method: Equation Mat and Algebra Tiles 1) 9 = 3x 1) x = 3 2) -12 = 4n 2) n = -3 3) 7 = 2z 7 3) z = 0 Group 5: Method: Calculator, graphs 1) x/-3 = 8 1) x = -24 2) 9x = 45 2) x = 5 3) 3x 7 = 20 3) x = 9 22
23 Day 4 Objectives: Students will be able to use an equation mat and algebra tiles to solve equations. Students will work together to solve equations. Students will be able to solving equations by using calculators. Opening Activity: Students will be given five diagrams on the overhead. In groups of two, the students will work together to write the equation modeled by the diagram. Main Activity: After all of the groups have created equations for all the diagrams, we will work together on solving two-step equations using the equation mat. The students will stay in their groups, but will be watching the overhead as we solve a couple of examples. Example 1: x = x x = 6 = = Example 2: 4x + 2 = 10 x = 2 = = 23
24 Example 3: 5x + 5 = 2x + 2 x = -1 = = After seeing these examples done, the students will work in their groups on the five equations that they found from the previous group activity. They will be asked to solve each equation. Closing Activity: With their TI-83 Plus or TI-73, the students will be given the remaining class time to practice solving equations. The students will be able to play either Solve It, Beam Dale Up, or Free Fall. Each game can be played at either the silver or gold level. The students will be encouraged to play at the gold level. Solve It! Solve It! Lets the students determine which step should be done first. Each step gives you several choices to choose from; either to add, subtract, multiply, or divide by a certain number. 24
25 Beam Dale Up Beam Dale Up requires students solve the equation in order to get Dale back to his space ship. If the student is correct, Dale will go into his ship and if they are wrong, Dale will fall out. Free Fall Free Fall gives the students a certain time period to solve the equation. The quicker the students get the solution, the more points they receive. If they do not 25
26 solve the equation correctly before it reaches the bottom, then it will SPLAT! All wrong equations will accumulate at the bottom. All of these games will help the students review solving one step and two step equations. Homework: For 1-5 Solve each two-step equation by using your equation mat. 1)2x 2 = 4x + 6 2)5h 7 = 2h + 2 3) 1 = -4c - 9 4)5a 10 = 0 5)12 = -3 5x For 6-10, Create an equation for each situation. Then solve the equation. 6)Sandy is looking to buy some office supplies. Two different suppliers offer the same box of pens at the same price per box. However, supplier A gives a $2.00 discount on 4 boxes of pens. For the same total cost as supplier A, supplier B will sell 3 boxes of pens and charge $4.00 shipping. How much does a single box of pens cost? 7)The three angles inside a triangle are 2a, a, and a+10. What is the measure of each angle? 8)Twice a number, increased by 8, is the same as three times the number, decreased by 11. What is the number? 9)Dan wants to hire a painter to paint his house. Painters Plus charges $360 plus $12 per hour. The House Painters charge $279 plus $15 per hour. Determine the number of hours for which the two costs would be the same. 10)Heather has a total of 3 tests in her history class. She scored a 79 and 88 on her first and second tests. What score does Heather need on her third test so that her average will be exactly 85? 26
27 Review for Chapter Test: Check List: -Addition Property of Equality -Subtraction Property of Equality -Multiplication Property of Equality -Division Property of Equality -One-Step Equations -Two-Step Equations Make sure you know how to solve equations by: -Algebra -Graphs and tables on the Calculator -Equation mat and algebra tiles Homework Answer Key: 1)x = -4 2)h = 3 3)c = -2 4)a = 2 5)x = -3 6)4x 2 = 3x + 4, Each box costs $6.00 7)2a + a + (a+20) = 180, The angles are 80, 40, and 60 in corresponding order. 8)2x + 8 = 3x 11, The number is 11. 9) x = x, 27 hours 10) x = 85 3 Heather needs an
28 Overhead Transparency for Opening Activity In groups of two, determine the equation modeled by each diagram. 1) 2) = = 3) 4) = = 5) = 28
29 Answer Key for Opening Activity 1) 3x + 4 = 2x + 10, x = 6 2) 2x + 4 = -8 x = -6 3) 2x + 12 = 5x x = 4 4) 4x 1 = 2x + 5 x = 3 5) -6x = -5x + 3 x = -3 29
30 Day 5 Objectives: Students will be able to use the internet as a review on solving equations. Opening Activity: The students will spend about the first 15 minutes of class reviewing for the test. They will use the internet web site Here they can play with any algebra related link. Some examples are the Mini Quiz or Rags to Riches. Both games ask the student to solve equations by either the one-step or two-step method. On the following page is an example of how Rags to Riches is played. Main Activity: The students will spend the remaining time taking a test on solving equations. 30
31 Rags to Riches Easy Algebra Equations Select your answer...think carefully... For every correct answer, you get one step closer to $1,000,000! $1,000,000 $500,000 $250,000 $128,000 $64,000 $32,000 $16,000 $8,000 $4,000 31
32 2.85y - 7 = 12.85y - 2 A 1/2 B -5/10 C -1/2 D none of these Hints Available: Give me a hint! 32
33 Name Test Solving Equations Date Solve each equation algebraically (2 points each). Check the solution (1 point). Must show all work for full credit. 1) d = ) x = ) 2.75x = 154 4) x = -1.3 Show 2 different ways of solving each equation. Choose from algebra, your equation mat and algebra tiles, or the calculator- show graphs or show tables. (4 points each) 5) 12g g + 3 = 29 6) 5m 7 = -6m 29 7) x = 2.5x ) 1 n + 8 = 3n 4 4 For 9, sketch the graph and fill in the table.(8 points) 9) x + 4 = x X Y1 Y From the table, what is the value of x? 33
34 Answer Key Chapter Test Solving Equations Solve each equation algebraically (2 points each). Check the solution (1 point). Must show all work for full credit. 1) d = 11.9 d = 3.2 2) x = -1.4 x = ) 2.75x = 154 x = 56 4) x = -1.3 x = -5 Show 2 different ways of solving each equation. Choose from algebra, your equation mat and algebra tiles, or the calculator- show graphs or show tables.(4 points each) 5) 12g g + 3 = 29 g = -2 6) 5m 7 = -6m 29 m = -2 7) x = 2.5x x = 1 8) 1 n + 8 = 3 n n = For 9, sketch the graph and fill in the table.(8 points) 9) x + 4 = x X Y1 Y From the table, what is the value of x? -8 34
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