4.1 College Algebra: Math 104
College Algebra (Math 104) is a course designed for students to review concepts such as real numbers, equations and inequalities, polynomials, system of equations, linear functions, quadratic functions and exponential and logarithmic function applications.
The first prompt requires students to work in a quadratic function unit. The instructor should give the students the list of topics to be included in the exam and the students themselves should create the problems. The instructor can use these exams as a review for the unit test or pick problems to put in the unit test for the class. The second prompt requires the students to pick one of the topics from the list provided by the instructor and create a lesson plan including every detail of the lesson plan and practice problems. Finally, in the third prompt, the students will create a video executing the lesson they created and present it in the class.
Student Learning Objectives
By completing these forms, students will be able to:
- Learn how to solve a quadratic equation using the Zero-Factor Property, completing the square, and the quadratic formula.
- Understand the properties of quadratic functions and how to use them to draw a graph.
- Write a quadratic model if given two x- and y-coordinates ( or the vertex and a point).
- Solve the word problems involving quadratic models.
- Find the max/min values using the vertex form of the quadratic function.
OER Activity Prompt #1: Creating an exam
Before the test, students will work in groups of three to five to create a practice exam for the chapter they just finished based on the topics covered in the chapter. Students can use their notes and the review paper while creating their practice problems.
In the process of learning the topics for the unit that the project will be assigned, after every topic, the students must come up with at least one question along with its solution based on that topic that can be put on the test they will create. The instructor will specify to the students before each unit what the most important topics are that the students will be learning, which the students will take into consideration regarding the questions they create. Once all the topics have been taught, students will have until the next class to develop their project, putting the questions they think are best on a shared document with their partners and developing well-thought-out answer choices besides the actual answer. Students will have to include five multiple choice and five short response questions in their test, working with their partners to decide on which questions are best from the questions they made for homework after every topic that was taught. The topics are suggested as the following:
1) Factoring quadratics completely
2) Solving quadratics by factoring
3) Solving quadratics by taking the square root
4) Solving quadratics by completing the square
5) Solving quadratics by using the quadratic formula
6) Writing quadratic functions in vertex form.
7) Graphing quadratic functions.
8) Writing a quadratic function given the vertex and the coordinates of a point.
9) Maximum/ minimum problems
10) Application of quadratic functions
After students hand in the tests that they created, the instructor will select tests to use to practice with the students as their review for the unit test that will be scheduled on the next class day.
- Students may not copy any problem that they have seen from the homework or classwork/notes.
- Students will make sure to include all topics on the chapter exam that their group is responsible for.
- Students have to make their own problems.
- Each student needs to solve the problems and check their answers with each other to make sure that there are no errors. (Each person in a group will be penalized as a whole for any errors in their test that they created.)
- Students will present their work to the class. (Students will present the problems they created and the solution to each problem.)
Worked Example #1: Given a vertex (1, 2) and a point (4, 8) on a parabola, find a quadratic equation in a vertex form and a standard form.
Step # 1: Write down the formula needed to be used.
y = a (x − h)2 + k
Step # 2: Substitute the coordinates of the vertex (h,k) →(1,2) and the coordinates of the given point (x, y) → (4, 8)
8 = a (4 − 1)2 + 2
Step # 3: Solve for a.
8 = a(3)^2 + 2
Step # 4: Write the vertex form of the equation of the parabola.
y = (⅔) (x − 1)2 + 2
Step # 5: Write the equation of the parabola in standard form.
y = (⅔) (x2 − 2x + 1) + 2
y =(⅔) x2 − (⁴୵₃)x + (⅔) + 2
y = (⅔) x2 − (⁴୵₃)x + (⁸୵₃)
OER Activity Prompt #2: Creating a lesson
After submitting the problem set of the quadratic unit for the exam, students will work with their groups to create two-lesson plans using two of the topics they used to make problems for their exam, one topic per lesson.
During the process of creating the lessons, students need to make sure they include the following into their lesson plan:
2) Lesson objectives
3) Warm-up questions
4) Explicit teaching (In this part students should include at least two problems with their answers and all necessary explanations, and two problems that the students need to try on their own.)
5) Formative Assessment (This is the part when the instructor checks how the students are understanding the lesson so far.)
6) Practice problems (This is the part that students need to do individually to practice what they just learned.)
7) Check for understanding
At the end of each lesson a type of assessment question should be included to check the student's understanding of the main goal of the lesson. Questions from their exams may be used in their lesson plans if applicable. However, students may not use questions from other groups’ exams in their lesson plans, and their problems must be original.
Worked Example #2: Solve the quadratic equation using completing the square method
Aim: How do we solve the quadratic equation using the completing square method?
Objectives: Students will be able to
- Identify a perfect square trinomial
- Write a perfect square trinomial as a square.
- Find the missing term of a perfect square trinomial.
- Use completing the square method to solve a quadratic equation.
- Express the answers in the simplest form.
- Solve the quadratic equation using the square root method.
(x − 2)2 = 6
- Factor the expression below. What do you notice?
x2 + 6x + 9
3) Expand the following expression:
(x − 5)2
The expressions like:
- x2 − 2x + 1 and b) 4x2 − 20x + 25
are perfect square trinomials because we can write them as squares.
- x2 − 2x + 1 = (x − 1)2
- 4x2 − 20x + 25 = (2x − 5)2
Pair-Share: Work with your partner to complete the blanks below as to complete the square.
- x2 x2 − 10x2 + ……….
- x2 + 12x +……….
Examples: Solving the quadratic equations using completing the square method:
Example # 1:
Step # 1: Add (b-2)2 to both sides of the equation to complete the square on the left side.
x2 + 2x = 3
x2 − 2x + 1 = 3 + 1
x2 − 2x + 1 = 4
Step # 2: Factor the left side as the square of a binomial.
(x+1)(x+1) = 4
(x+1)2 = 4
Step # 3: Take the square root of both sides.
Step # 4: Solve for x.
x + 1 = 2 x + 1 =-2
x = 1 x =-3
Practice problems: Solve each of the questions below using completing the square method.
- x2 − 6x + 5 = 0
- x2 + 12x = -13
- x2 - 5x + 6 = 0
Example # 2:
Step # 1: Divide both sides of the equation by the leading coefficient
2x2 + 6x = 8
x2 + 3x = 4
Step # 2: Add (b-2)2 on both sides of the equal sign
Step # 3: Complete the square on the left side of the equal sign
Step # 4: Take the square root of both sides.
# 5: Solve for x.
x = 1 and x = -4
You try! Solve the quadratic equations using completing the square method:
- 3x2 + 6x + 1 = 0 2) 2x2 - 4x + 9 = 0
Exit Ticket: Solve 2x2 - 4x + 6 = 0 by completing the square, expressing the result in the simplest radical form.
More Practice: Solve the equations below by completing the square method.
- 3x2 + 8x + 1 = 0
- x2 - 5x + 9 = 0
- 2x2 = 6x + 5
Homework: Solve by completing the square, express the result in the simplest radical form.
- x2 - 4x = 2
- 2x2 - 5x + 1 = 0
- x2 - 8x + 15 = 0
OER Activity Prompt #3: Making a video
After creating and submitting the lessons, groups should record themselves teaching their lessons. Each lesson should be a separate video. Students do not need to be seen in the videos, but their voice needs to be heard as they solve the problems and teach the concepts of each lesson. Each student must participate in the videos and must be heard at least twice in each video for each lesson.
Each group’s videos either need to be uploaded on YouTube as unlisted and they need to email their instructor the link to their videos or they can simply email the videos to their instructor.
Each group must make sure that the videos are edited properly if needed.
By Aneta Bega