Course Name: Geometry/Math II Unit # 2 Unit Title: Quadratic Equations
Enduring understanding (Big Idea): Every quadratic equation can be solved using the Quadratic formula. The discriminant of a quadratic equation determines whether the equation has two real roots, one real root or no real roots.Essential Questions:
- How do you solve quadratic equations using square roots?
- How do you solve quadratic equations by factoring?
- How do you solve quadratic equations by inspections?
- How do you solve quadratic equations using the Quadratic Formula?
- How do you solve a system of linear and quadratic equations with two variables algebraically and by graphing?
BY THE END OF THIS UNIT:
Students will know…
- The Quadratic Formula
- Rules of exponents – (focus on negative, zero and fractional exponents
- Types of Factoring (excluding sum or difference of cubes)
- Special products
- Solve quadratic equations by factoring
- Solve quadratic equations by using square roots
- Solve quadratic equations by inspection
- Solve quadratic equations by using the quadratic formula
- Solve Quadratic Equations by Graphing
- Recognize non-real solutions
- Extend properties of exponents to rational exponents
Unit Resources:
Algebra 1 Textbook: Enrichment 9-6
George DPI Tasks –
BrowseFrameworks?math9-12.aspx
CCSS-M Included: A-REI.4 b , A-REI.7, A-REI.10, A-REI.11
Prior Course Knowledge: Student have already covered factoring in Unit 1.
Test Specification Weights for the Common Exams in Common Core Math II:
Standards / Constructed
Response% / Multiple-
Choice %
(Algebra) / Category
Percentage
(Algebra)
A-REI / 3% to 7% / 23% to 27% / 29% to 33%
Putting it all together: Remember solving quadratic equations by completing the square and complex roots are not covered in this unit. These will be covered in Math III. / Mathematical Practices in Focus:
- Make sense of problems and persevere in solving them
- Reason abstractly and quantitatively
Abbreviation Key:
CC – Common Core Additional Lessons found in the Pearson online materials.
CB- Concept Bytes found in between lessons in the Pearson textbook.
ER – Enrichment worksheets found in teacher resources per chapter
Suggested Pacing:
Algebra 1 Textbook:
Sec. 9-3 Solving Quadratic Equations
CB pg 554 Finding Roots
Sec. 9-4 Factoring to solve Quadratic Equations
Sec. 9-6 The quadratic Formula and the Discriminant
Sec. 9-8 Systems of Linear and Quadratic Equations
CC – 18 Solving systems
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
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Course Name: Geometry/Math II Unit # 2 Unit Title: Quadratic Equations
CORE CONTENT
Cluster Title: Solve equations and inequalities in one variableStandard: A-REI.4b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, the quadratic formula and factoring, as appropriate to the initial form of the equation.
Concepts and Skills to Master:
- Solve quadratic equations by inspection (graphing), taking square roots (e.g., for x2 = 49), the quadratic formula and factoring, as appropriate to the initial form of the equation.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:- Square Root
- Perfect Square Trinomials
- Factoring Quadratic Equations
- Zeros, Roots, Solutions
Academic Vocabulary:
Square Roots, factoring, Quadratic Formula perfect-square trinomial, zeros, solutions, discriminant
Suggested Instructional Strategies:
- They factored in the previous unit, so they will use this skill to solve for zero
- When solving by inspection students should be able to identify the number of real roots, their value and if there is no real root.
- They will not be completing the square to solve quadratic equations.
- When using the Quadratic Formula they will not be solving problems with a negative discriminant. These problems have “no real solution”.
Algebra 1 Textbook Correlation: 9-3, 9-4, 9-5, 9-6
Sample Assessment Tasks
Skill-based task:
Ex. Find the solution to the following quadratic equations:
a. x2 – 7x -18 = 0
b. x2 = 81
c. x2- 10x + 5 = 0 / Problem Task:
Ex. Ryan used the quadratic formula to solve an equation andx was his result.
a. Write the quadratic equation Ryan started with.
b. Simplify the expression to find the solutions.
c. What are the x-intercepts of the graph of this quadratic function?
CORE CONTENT
Cluster Title: Solve systems of equationsStandard: A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in tow variables algebraically and graphically.
For example the points of intersection between the line y = -3x and the circle y2 + x2= 3
Concepts and Skills to Master:
Solve a system containing a linear equation and a quadratic equation in two variables (conic sections possible) graphically and symbolically. Add context or analysis
SUPPORTS FOR TEACHERS
Critical Background Knowledge:Students should be able to graph linear and quadratic equations.
Students should be able to solve a system of equations by substitution
Academic Vocabulary:solution, intersection, substitution
Suggested Instructional Strategies:
Review solving systems of equations both linear by graphing and substitution. Then introduce the quadratics. / Resources:
Algebra 1 Textbook Correlation: 9-8, CC-18
learnzillion.com
CCSSMath.org
Sample Assessment Tasks
Skill-based task:
Solve the following system graphically and symbolically:
x2 + y2 = 1
y = x
The Yummy Gummy Pie Factory (a delicious but fattening industry) has modeled its daily output to be 10x2 + x – 3 = y. Meanwhile, the consumption of its product is modeled by y = 15x. Find where the positive point of intersection occurs.
- (A)(0, 0)
- (B)(0.5, 6)
- (C)(1.58, 23.8)
- (D)(π, 0)
Your evil math teacher (we're only half kidding) has decided to make your life really difficult by giving you an extra hard problem to solve. Your teacher has given you y = -10x + 20 and 3x2 + 2y2 = 3, for which you must find the points of intersection but you have to do it without a graph.
- (A)(0, 0) and (0, 0)
- (B)(0, 1)
- (C)No points of intersection
- (D)(-1.33, 1.43), (1.33, -1.43)
Did your evil math teacher make you do more work than you had to? Before we dive headfirst into calculating the points, we can use the discriminant to figure out if there are any points of intersection to begin with. Substituting our y value, we get 3x2 + 2(-10x + 20)2 = 3, which simplifies to 203x2 – 800x + 797 = 0. With our newborn a, b, and c values (aren't they adorable?), we can find the discriminant by calculating b2 – 4ac, which will give us -1,287,164. It's a negative number, so there are no points of intersection. Our graph below proves this as well:
CORE CONTENT
Cluster Title: Represents and solve equation and inequalities graphicallyStandard: A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a straight line)
Concepts and Skills to Master:
- The solution to an equation in two variables can be represented by an ordered pair
- All points on the graph of a two-variable equation are solutions
SUPPORTS FOR TEACHERS
Critical Background Knowledge:- Graphing and solving equations of two- variables
- Substituting values into an equations to find solutions
Academic Vocabulary:
Coordinate plane, two-variable equations, substitution, ordered pair
Suggested Instructional Strategies:
Give the students an equation of two variables and have them construct a table of solutions(no calculators). Then they can plot their solutions on the coordinate plane. Let them compare their solutions to the graph of the calculator. / Resources:
Algebra 2 Textbook Correlation: 4-2, 4-3, 4-4
Sample Assessment Tasks
Skill-based task:
Which of the following points lie on the graph of the equation -5x + 2y = 20?
a. (4, 0) b. (0, 10) c. (-1, 7.5) d. (2.3, 5)
How many solutions does this equation have? Justify your conclusion / Problem Task:
If a graph for our equation shows that the line doesn't pass through a certain point than we can assume that:
(A)The equation, when solved for that point, will be true
(B)The equation, when solved for that point, will be false
(C)The equation, when solved for that point, will be neither true nor false
(D)None of the above
Explanation:We have to understand that the equation and the graph of the equation are inseparable! They are different (written and visual) representations of the same idea. Hence, if one is false, the other one will be false as well.
CORE CONTENT
Cluster Title:Represents and solve equation and inequalities graphicallyStandard: A-REI.11 Explain why thex-coordinates of the points where the graphs of the equationsy=f(x) andy=g(x) intersect are the solutions of the equationf(x) =g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases wheref(x) and/org(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Concepts and Skills to Master:
- That an equation and its graph are just two different representations of the same thing.
- Depending on the equations there might be one solution, or more, or none at all.
- Correct answer(s) can be arrived through graphing the functions and plotting their intersection points, creating a table of x and f(x) values, and solving for x algebraically when f(x) = g(x).
SUPPORTS FOR TEACHERS
Critical Background Knowledge:- Graphing equations
- Plotting point on the coordinate plane
Academic Vocabulary:
Coordinate plane, intersection, f(x)and g(x)
Suggested Instructional Strategies:
Show students that the solution to every equation can be found by treating each side of the equation as separate functions that are set equal to each other, f(x) = g(x). Allow y1 = f(x) and y2 = g(x) and find their intersections(s). The x-coordinate of the point of intersection is the value at which these two functions are equivalent, therefore the solution(s) to the original equation. / Resources:
Algebra 2 Textbook Correlation: CB 4-4, 9-8, CC-18
Shmoop Quiz
Sample Assessment Tasks
Skill-based task:
1. At which point do the two equations 3x + 5 = y + 4x and y = x2 intersect?
- (A)(1.8, 3.2)
- (B)(-2.8, 7.8)
- (C)(0, 5)
- (D)Both (A) and (B)
We can solve the answer by simplifying our first equation to y = 5 – x and then setting the two equations equal to one another. That means x2 = 5 – x and simplifies to x2 + x – 5 = 0 and we can use the quadratic formula to solve for x. The two values for x are 1.8 and -2.8, with the y values at 3.2 and 7.8, respectively. If we graph the two functions, we can see that these answers make sense.
2. Find the intersection point of y = log x and y = 3x.
- (A)(1, 1)
- (B)(3, 2)
- (C)(7, 1)
- (D)There is no intersection
If we graph the two equations, it will be clear that the two functions never intersect. Plugging in every point into both equations and seeing that none work will give the same logical conclusion.
/ Problem Task:
1. John and Jerry both have jobs working at the town carnival. They have different employers, so their daily wages are calculated differently. John’s earnings are represented by the equation p(x) = 2x and Jerry’s by g(x) = 10 + 0.25x.
a. What does the variable x represent?
b. If they begin work nest Monday, Michelle told them Friday would be the only day they make the same amount of money. Is she correct in her assumption? Explain your reasoning.
c. When will Jerry earn more money than John? When will John earn more money than Jerry? During what day will their earnings be the same? Justify your conclusions?
2. When estimating the intersection of two lines on a graph, you can get a precise answer. Is this statement true or false?
- (A)True, since graphs are always exactly correct when it looks as if the line is specifically at one point
- (B)False, because one can never be completely sure if the graph is correct since it depends highly on the sensitivity of the graph
- (C)False, since graphs offer limited visibility and it's usually impossible to see the entire function
- (D)Both (B) and (C)
Using a graph to find the point of intersection is highly reliant on what area of the graph you're looking at, how much you zoomed in, how sensitive the scale is, and so on. Hence, it is always an approximation until you confirm the intersection using mathematical means!
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
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