Reasoning with Equations and Inequalities


 Brendan Lane
 3 years ago
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1 Instruction Goal: To provide opportunities for students to develop concepts and skills related to solving quadratic equations using the square root property. The quadratic equations will have two solutions, one solution, or no real solutions. Common Core Standards Algebra: Solve equations and inequalities in one variable. AREI.4. Solve quadratic equations in one variable. Solve quadratic equations by inspection (e.g., for = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives comple solutions and write them as a ± bi for real numbers a and b. Student Activities Overview and Answer Key Station 1 Students will be given a number cube. Students will use the number cube to populate quadratic equations that can be solved by using the square root property. They will realize that quadratic equations solved by the square root property have answers which are the +/ of the same number. Answers 1. Answers will vary. Make sure they add the constant to both sides of the equation.. Answers will vary. Make sure they take square root of both sides of the equation. 3. Answers will vary. Make sure they find the +/ square root of the constant. 4. Two, because the eponent tells you that there are two solutions. 5. Answers will vary. Make sure that students first add the constant to both sides of the equation, then take the square root of both sides, then add the constant from the left side to the right side of the equation. 6. Answers will vary. Make sure that students first add the constant to both sides of the equation, then take the square root of both sides, then add the constant from the left side to the right side of the equation. 7. Two solutions, because the square root of a number is the +/ of the number; the square of a positive is a positive and the square of a negative is also a positive Walch Education Algebra I Station Activities for Common Core Standards
2 Station Students will be given 1 inde cards with the following written on them: a = 1; a = ; a = 4; a = 5; b = 0; b = 0; b = 0; b = 0; c = 16; c = 5; c = 36; c = 49 Instruction Students work together to set up quadratic equations and solve them using the square root property. Answers 1. Answers will vary. Possible answer: a = 1, b = 0, and c = 16, so 16 = 0 = 16 = ± 4. Answers will vary. Possible answer: a =, b = 0, and c = 36, so 36 = 0 = 36 = 18 = ± 3 3. Answers will vary. Possible answer: a = 5, b = 0, and c = 5, so 5 = 5 = 5 = ± 5 4. Answers will vary. Possible answer: a = 4, b = 0, and c = 49, so 4 = 49 = = ± = ± 4 5. two solutions, because you find the square root Algebra I Station Activities for Common Core Standards 011 Walch Education
3 Station 3 Instruction Students will be given a deck of playing cards that contain only the numbers 10. Students will draw numbers and use them to populate quadratic equations. They will solve the quadratic equations using the square root property. They will realize that the square root of a negative number yields no real solutions. They will also realize that the square root of 0 yields only one solution. Answers 1. Answers will vary. Make sure students subtract the constant from both sides of the equation.. negative, because of subtraction 3. You can t find the square root of a negative number in the real number system. 4. no real solutions 5. Answers will vary. Make sure students subtract the constant from both sides of the equation. 6. It cancels out, so the righthand side of the equation is = ± 0 = 0 8. ± 0 = = 6 3 = 0 = = 0 5 = 15 = 5 = no real solutions Station 4 Students will be given five inde cards with the following written on them: 7 ; 5 3 = ; 36 ; = ±6 ; = Students will arrange the inde cards in order to demonstrate how to find the solution to a quadratic equation that is not given in standard form. Students will indicate their strategy and acknowledge the square root property. Then they will solve quadratic equations not given in standard form using the square root property Walch Education Algebra I Station Activities for Common Core Standards
4 Answers = = = ±6 6. square root property 7. Answers will vary = = 144 = ± = = 0 = = = 8 = 4 = no real solutions Instruction Materials List/Setup Station 1 Station number cube 1 inde cards with the following written on them: a = 1; a = ; a = 4; a = 5; b = 0; b = 0; b = 0; b = 0; c = 16; c = 5; c = 36; c = 49 Station 3 deck of playing cards that contain only the numbers 10 Station 4 five inde cards with the following written on them: 7 ; 5 3 = ; 36 ; = ±6 ; = Algebra I Station Activities for Common Core Standards 011 Walch Education
5 Discussion Guide Instruction To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to debrief the station activities. Prompts/Questions 1. How many solutions does a quadratic equation have?. If the standard form of a quadratic equation is a + b + c = 0, which letter (a, b, or c) should be equal to 0 in order to use the square root property? 3. If = a negative number, how many solutions does the equation have in the real number system? 4. If = 0, how many solutions does the equation have? 5. Why does the square root property yield answers that are the ± version of the same number? Think, Pair, Share Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class. Suggested Appropriate Responses 1. two solutions. b = 0 3. no real solutions 4. one solution, which equals zero 5. Because there are two possible solutions, a positive number can be squared to equal a positive number, and a negative number can be squared to yield that same positive number. For eample, 3 squared is 9, and so is ( 3) squared. Possible Misunderstandings/Mistakes Not realizing that there are two solutions when = a positive number Not realizing that there are no real solutions when = a negative number Not realizing that there is only one solution when = 0 Incorrectly combining like terms before solving an equation Walch Education Algebra I Station Activities for Common Core Standards
6 Station 1 At this station, you will find a number cube. Roll the number cube and write the number in the first bo below. Roll it again, and write the second number in the second bo. = 0 As a group, solve this quadratic equation by finding the value of. 1. What is the first step you take to solve this equation?. What is the second step you take to solve this equation? 3. What is the solution to this equation? 4. How many solutions did you find? Eplain your answer. Given: ( ) = 0 Roll the number cube and write the number in the bo above. Repeat this process until each bo contains a number. 5. Solve this equation. Show your work. continued 6 Algebra I Station Activities for Common Core Standards 011 Walch Education
7 6. What strategy did you use to solve this equation? 7. How many solutions did you find? Eplain your answer Walch Education Algebra I Station Activities for Common Core Standards
8 Station At this station, you will find 1 inde cards with the following written on them: a = 1; a = ; a = 4; a = 5; b = 0; b = 0; b = 0; b = 0; c = 16; c = 5; c = 36; c = 49 Place the inde cards in three piles, organized by their letter: a, b, and c. For each problem, work as a group by choosing a card from each pile and writing it in the form shown. For eample, if you draw the cards a = 4, b = 0, and c = 16, inserting them into the form a + b c results in this equation: = 16 Then solve each equation for using the square root property. Show your work. 1. a + b c = 0. a + b c = 0 continued 8 Algebra I Station Activities for Common Core Standards 011 Walch Education
9 3. a + b = c 4. a + b = c 5. How many solutions did you find for each problem? Eplain your answer Walch Education Algebra I Station Activities for Common Core Standards
10 Station 3 At this station, you will find a deck of playing cards that contain only the numbers 10. Place the cards in a pile. As a group, select one card. Write the number on the face of the card in the bo below. + = 0 1. What is the first step you take to solve for?. Is the constant now positive or negative? Eplain your answer. 3. How does the sign of the constant affect finding the square root of? 4. What is the solution for? Select another card from the pile. Write the number on the face of the card in each bo below. + = 5. What is the first step you take to solve for? 6. What happens to the constant? continued 30 Algebra I Station Activities for Common Core Standards 011 Walch Education
11 7. Based on your new constant, what is the value of? 8. Why does only have one value? Work together to solve problems 9 and 10. Show your work = = Walch Education Algebra I Station Activities for Common Core Standards
12 Station 4 At this station, you will find five inde cards with the following written on them: 7 ; 5 3 = ; 36 ; = ±6 ; = Shuffle the cards. As a group, arrange the cards to show how to find the solution to 5 3 = Write the steps on the lines below What property did you use to find the solutions for? 7. What strategy did you use to arrange the cards? As a group, solve problems Show your work = continued 3 Algebra I Station Activities for Common Core Standards 011 Walch Education
13 = = Walch Education Algebra I Station Activities for Common Core Standards
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