9.2.4A Represent & Solve: Quadratic & Exponential
Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities.
For example: A diver jumps from a 20 meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits the surface of the water, the resulting quadratic equation has a positive and a negative solution. The negative solution should be discarded because of the context.
Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations.
Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from rational numbers to real numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients.
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.
Overview
Standard 9.2.4 Essential Understandings
In this standard, students learn to model real-world mathematical situations using linear, quadratic, exponential and nth root functions. In order to do so, they need to understand the characteristics of each type of function: linear functions have a constant rate of change; quadratic functions have a linear rate of change (the rate of change of the rate of change, or the 2nd difference, is constant); and exponential functions have a rate of change that is proportional to the value of the function (the output is multiplied by a constant factor).
Students need to be able to choose from a variety of methods to solve these functions, whether symbolically or graphically. Once solutions are found, students need to be able to determine the reasonableness of an answer given the real-world context for the function - a particular solution may not be applicable in the original context.
Students need to understand the real number system, including the subsets of natural and whole numbers, integers, rational and irrational numbers, and that many of these numbers were invented to solve equations. They need to know how the real number system fits into the non-real complex number system, and how operations in the non-real complex number system correspond to those of the real number system. Students need to understand how a solution in the non-real complex number system is graphed, and what that solution means in terms of a given context for a quadratic equation.
Students need to be comfortable in using a graphing calculator or other graphing utilities to find solutions and describe characteristics of linear, quadratic and exponential functions. They should know how to use the table and graphing features to check solutions for these functions.
All Standard Benchmarks
9.2.4.1
Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities.
Example: A diver jumps from a 20 meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits the surface of the water, the resulting quadratic equation has a positive and a negative solution. The negative solution should be discarded because of the context.
9.2.4.2
Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations.
9.2.4.3
Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from rational numbers to real numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients.
9.2.4.4
Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which parts of the boundary are included in and excluded from the solution set using solid and dotted lines.
9.2.4.5
Solve linear programming problems in two variables using graphical methods.
9.2.4.6
Represent relationships in various contexts using absolute value inequalities in two variables; solve them graphically.
Example: If a pipe is to be cut to a length of 5 meters, accurate to within a tenth of its diameter, the relationship between the length x of the pipe and its diameter y satisfies the inequality | x - 5| ≤ 0.1y.
9.2.4.7
Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods.
Example: The equation $\sqrt{x-9}=9\sqrt{x}$ may be solved by squaring both sides to obtain x - 9 = 81x, which has the solution x = - $\frac{9}{80}$.
However, this is not a solution of the original equation, so it is an extraneous solution that should be discarded. The original equation has no solution in this case.
Another example: Solve $\sqrt[3]{-x+1}=-5$.
9.2.4.8
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.
Benchmarks Group A - Quadratic and Exponential Equations
9.2.4 Algebra
9.2.4.1
Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities.
Example: A diver jumps from a 20 meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits the surface of the water, the resulting quadratic equation has a positive and a negative solution. The negative solution should be discarded because of the context.
9.2.4.2
Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations.
9.2.4.3
Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from rational numbers to real numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients.
9.2.4.8
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.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Represent real-world situations using a mathematical model (table, graph, or equation) and use this representation to make sense of the situation and solve the problem. They should be flexible in selecting the representation they use and in selecting the solution strategy;
- Know how to solve some quadratic equations by factoring. They should be able to explain how the zero product property is used with factors;
- Solve all quadratic equations that can be written with integer coefficients by completing the square. They should be able to explain why each step in the solution process works;
- Solve a quadratic equation by putting it into standard form and using the quadratic formula. Students should be able to use the quadratic formula for any quadratic equation;
- Solve quadratic equations with solutions that are real numbers by graphing. The graph of an equation involves treating the left side and right side of the equation each as a function, and the solutions of the equation are the intersections of the graphs of the functions. If the graphs do not intersect, then the students should know that the solution set is not a real number. For example,
\begin{matrix}
\underbrace{x^{2}-3x} &= &\underbrace{10} \\Y_{1}
& &Y_{2}
\end{matrix}
The solutions are x = 5 or x = -2;
- Solve quadratic equations using the graphing feature or the table feature of graphing utilities. Irrational solutions can only be approximated using most graphing utilities. Many Computer Algebra Systems (CAS) can find exact solutions to any quadratic equation. The standard does not specify if students should be able to use a CAS to solve an equation;
- Solve quadratic inequalities by plotting the solutions to the equality (critical points) on a number line and then checking intervals using test points. They may select any method they want to find the critical points;
- Solve quadratic inequalities using a graphing approach. For example,
\begin{matrix}
\underbrace{x^{2}-3x} &\geq &\underbrace{10} \\Y_{1}
& &Y_{2}
\end{matrix}
The students should be able to explain that the solution set the inequality above includes all values of x where the value of Y1 is greater than or equal to the value of Y2.
- Use context to explain whether a solution to an equation is relevant.
For example: A diver jumps from a 20 meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits the surface of the water, the resulting quadratic equation has a positive and a negative solution. The negative solution should be discarded because of the context. A student should be able to set up the equation $-49t^{2}+3t+20$, given the equation $h=-49t^{2}+vt+h_{0}$. They should be able to solve the equation correctly to get t = -1.74 seconds or t = 2.35 seconds. Although both of the values of t are solutions, the equation only works in this situation for values of t greater than or equal to 0 seconds. Students should be able to explain that t = -1.74 does not make sense in this situation;
- Solve exponential equations using numerical or graphical methods. Solving an exponential function using numerical methods include solving the equation by inspection (i.e., knowing that the solution to 2t +1 = 32 is t = 4 since 32 = 25) or using guess and check with a calculator (i.e., approximating the solution for 2x = 9 by entering in 23.1and judging whether to raise or lower the guess for x based on this result). The graphical method involves finding the intersection points when the expressions on either side of the equation are graphed as two functions. The solution set is the intersection point.
\begin{matrix}
\underbrace{2^{x}} &= &\underbrace{9} \\Y_{1}
& &Y_{2}
\end{matrix}
Both of these methods will provide students with approximate solutions;
- Use a graphing technology to find an approximate solution for cases involving exponential functions. Students should be able to use an appropriate tool to solve equations graphically;
- Expand their knowledge of the number system as they solve more sophisticated equations in grades K to 12. The development of the number system often occurred as a result of solving more difficult equations. When children first learn about whole numbers, they are asked to solve equations of the form $3+\square=7$ and 13 = c + 5. The solutions of these equations are whole numbers. When students begin to think about solving equations like 9 + d = 5 and 7 - 10 = e they need to expand their number system to include negative numbers. Students expand their number system to include rational numbers when they try to make sense of solutions of equations involving multiplication and division in equations like 9 • f = 6 and g = 3 ÷ 4;
- Know that the set of integers are all that are needed to solve equations like x2 = 9 but not enough to solve equations like x2 = 7. They should be able to explain that some quadratic equations have solutions that are rational and some that are irrational. Eventually students should know that some quadratic equations like x2 = -4 have no real solution and realize that their number system has to be expanded to include complex numbers in order to solve the equation.
Work from previous grades that supports this new learning includes:
- Represent real-world and mathematical situations using equations and inequalities involving linear expressions. They can solve equations and inequalities symbolically and graphically;
- Interpret solutions in the original context;
- Learn to identify and name instances of the commutative properties of multiplication and addition, associative property of multiplication and addition, and the distributive property over addition. They should be able to use these properties along with the order of operations to flexibly simplify expressions involving numerals and variables;
- Know and be able to use the addition, subtraction, multiplication and division properties of equality using rational numbers to generate equivalent equations. Students should be able to identify and define equivalent equations;
- Multiply multi-digit whole number showing an understanding of place value;
- Represent relationships in various contexts with equations and inequalities involving the absolute value of a linear expression. They can solve such equations and inequalities and graph the solutions on a number line;
- Represent relationships in various contexts using systems of linear equations. They can solve systems of linear equations in two variables symbolically, graphically and numerically;
- Understand that a system of linear equations may have no solution, one solution, or an infinite number of solutions. They can relate the number of solutions to pairs of lines that are intersecting, parallel, or identical. They can check whether a pair of numbers satisfies a system of two linear equations in two unknowns by substituting the numbers into both equations;
- Use the relationship between square roots and squares of a number to solve problems;
- Represent geometric sequences using equations, tables, graphs and verbal descriptions, and have used these representations to solve problems.
NCTM Standards
Understand patterns, relations, and functions:
- Generalize patterns using explicitly defined and recursively defined functions;
- Understand relations and functions and select, convert flexibly among, and use various representations for them;
- Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
- Understand and perform transformations, such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions;
- Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions;
- Interpret representations of functions of two variables.
Represent and analyze mathematical situations and structures using algebraic symbols:
- Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations;
- Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency - mentally or with paper and pencil in simple cases and using technology in all cases;
- Use symbolic algebra to represent and explain mathematical relationships;
- Use a variety of symbolic representations, including recursive and parametric equations, for functions and relations;
- Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
Use mathematical models to represent and understand quantitative relationships:
- Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;
- Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts;
- Draw reasonable conclusions about a situation being modeled.
Analyze change in various context:
Approximate and interpret rates of change from graphical and numerical data.
Common Core State Standards (CCSM)
A-SSE (Seeing Structure In Expressions) Write expressions in equivalent forms to solve problems.
-
- A-SSE.3.Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
§ A-SSE.3.a. Factor a quadratic expression to reveal the zeros of the function it defines.
§ A-SSE.3.b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
§ A-SSE.3.c. Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
o A-SSE.4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.
- A-CED (Creating Equations) Create equations that describe numbers or relationships.
- A-CED.1.Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
o A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
o A-CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
· A-REI (Reasoning with Equations and Inequalities) Understand solving equations as a process of reasoning and explain the reasoning.
o A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
o A-REI.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
· A-REI (Reasoning with Equations and Inequalities) Solve equations and inequalities in one variable.
o A-REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
o A-REI.4. Solve quadratic equations in one variable.
§ A-REI.4.a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.
§ A-REI.4.b. Solve quadratic equations by inspection (e.g., for x2 = 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 complex solutions and write them asa ± bi for real numbers a and b.
- A-REI (Reasoning with Equations and Inequalities) Represent and solve equations and inequalities graphically.
- 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 line).
- 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.
- F-LE (Linear, Quadratic, & Exponential Models)Construct and compare linear, quadratic, and exponential models and solve problems.
- F-LE(Linear, Quadratic, & Exponential Models) Interpret expressions for functions in terms of the situation they model.
- F-LE.5.Interpret the parameters in a linear or exponential function in terms of a context.
Misconceptions
Student Misconceptions and Common Errors
- Students often struggle to create a mathematical model for a real-world situation.
- Students incorrectly apply the distributive property to multiply polynomials.
(e.g., (3a + b)2 = (3a)2 + (b)2).
- Students will omit 0 from the solution set. For example, when students are asked to solve the equation 2x2 - 10x = 0, they may factor to get 2x(x - 10) = 0, then either divide both sides of the equation by 2x or just focus on the factor (x - 10) to state that the solution is only x = 10.
- Many students who solve quadratic equations by taking the square root of both sides of the equation will lose one of the solutions. For example,
$(x+2)^{2}=9\rightarrow \sqrt{(x+2)^{2}}=\sqrt{9}\rightarrow x+2=3\rightarrow x=1$. - Students will turn an expression into an equation by setting it equal to zero and solving. For example, factor the trinomial m2 + m - 20.
Student writes:
m2 + m - 20
(m + 5)(m - 4)
(m + 5)(m - 4) = 0
m = -5 or m = 4
- Students may find non-real complex numbers to be difficult to simplify, or may not understand what a non-real solution represents in a given quadratic context.
- Students may not take the real-world context for a quadratic relationship into account when giving solutions.
Vignette
In the Classroom
Students need to make sense of problems involving quadratic functions. In the vignette below, the problem involves drug testing which is very common in athletics and is an increasing trend in many occupations. Students need to identify the variables involved in the situation and think about how the dependent variable is changing with the independent variable. Once they identify the variables, they begin to build a model. Area models are often useful with quadratic models. You will notice that the students struggle with the interpretation of the solution. The teacher needs to make sure to discuss how the solution relates to the original situation.
The students are asked to read the following problem and write down a sentence with a blank that answers the question.
Drug Testing
There are over 100,000 drug tests given to athletes at a cost of over $30 million. In order to save money, some teams are trying to devise ways to cut the number of tests given to the student athletes but still test everyone. The University of Minnesota decided to combine the samples of two athletes together and run the test on the pooled sample. Will this strategy reduce the number of tests the University gives to its student athletes?
Whole class discussion:
Noam: I wrote: "The pooling strategy will _______ the number of tests given to student athletes." But this is easy because the answer is yes because they only have to give half the number of tests.
Teacher: So you would put "yes" in the blank in your sentence.
Noam: Yes, no, I mean I would put "reduce in half" because it would cut the number of tests in half.
Michele: I agree.
Matt: I am not sure that would work. What happens if the pooled sample came back positive? Then they would have to test again because they would not know who was positive.
Michele: I agree.
Nick: You agree with everyone. If a [pooled] sample came back positive, then you would have to test each person individually, so you would actually test three times for two people. This plan stinks!
Michele: I do not agree with you, Nick. If a sample comes back positive, then you may only have to test one of the two people. If that test comes back negative, then you know the next person is positive and do not have to do another test.
Nick: That is still two or three tests per sample of two. I still say you should do individual tests and it will be less complicated and it still costs the same.
Teacher: There are three possibilities for the number of tests for a sample of two people. There could be one test, two tests, or three tests as long as we keep track of whom we test second. So should we pool the samples?
Noam: It all depends on how many people would test positive. If nobody uses drugs, then the total number of tests you would have to run would be half. If all people used the drug, then we would have to run three tests for each pair and that would cost more than testing individually.
Teacher: The percent of people using a drug does influence the number of tests the university would have to do. What percents would it make sense to pool groups of two students together and when would it not? Please work in your groups to find a solution to this problem.
After 15 minutes, students presented their thinking.
Matt: I decided to find the number of tests required, given the percent of people who would test positive. I used a rectangle to represent the two people.
The p means that the person is positive. The 1 - p means the person is negative. We decided that the average number of tests given is the same as the expected value of this rectangle. We found the area of each little rectangle and multiplied it by the number of tests for each region.
average #tests = 3∙p2 + 2∙p(1 - p) + 3∙p(1 - p) + 1∙(1 - p)2
But we do not know what to do with this.
Michele: We wanted to find when the average number of tests would be equal to two. We only have to solve the equation:
3∙p2 + 2∙p(1 - p) + 3∙p(1 - p) + 1∙(1 - p)2 = 2
3p2 + 2p - 2p2 + 3p - 3p2 + 1 - 2p + p2 = 2
-p2 + 3p + 1 = 2
-p2 + 3p - 1 = 0
We tried to use factoring but had to use the quadratic formula. We got $p=\frac{-3\pm \sqrt{5}}{-2}$.
Nick: What does that mean?
Noam: You need to simplify the radical to say what the value is. I put it in my calculator and got p = 2.61 and p = 0.38.
Matt: The value of p has to be between zero and one because p is a percent of people. So does this mean we will save money if 38% of the people are known to use the drug?
Samantha: No at 38-percent the average number of people who test positive is two and it would not matter if they pooled the tests or just tested each person individually.
Jesse: I just made a graph and it shows that the average number of tests is less than two when the percent of people who test positive is less than 0.38.
So we think that the University of Minnesota will save money if the percent of people who they think take the drug is less than 38%. Otherwise, they should test the students individually.
Resources
Teacher Notes
- Many of the mistakes that students make when working with quadratic equations involve the distributive property. Students need ways to make sense of the distributive property in order to apply it correctly when using variables. The mistake (3a + b)2 = (3a)2 +(b)2is common and involves "distributing" the exponent of 2. Mistakes like this should be brought out and discussed in class. Some teachers emphasize the definition of exponents by showing that (3a + b)2 means (3a + b)(3a + b). Some teachers substitute numbers into the expression to show that exponents do not distribute over multiplication.
- All students should be able to draw an area model to make sense of the distributive property when these mistakes occur. Students began making these drawings in third grade when they created array and area models to demonstrate multiplication of two numbers. When they multiplied multi-digit numbers in fourth grade they learned to draw a picture to explain how place-value works. In high school students should be able to explain how calculating the same area in two different ways shows how the distributive property works.
|
|
|
|
area model for (3a + b)(3a + b) |
area model for 9a2 + 6ab + b2 |
|
(3a +b) (3a +b) = 9a2 + 6ab +b2 |
|
- The concept of an equivalent equation is very important in high school. The idea of equivalent equations was introduced to students in middle school as they learned to solve linear equations using properties of equations. An equivalent equation is any equation that has the same solution set. When students show their work vertically as is modeled by many textbooks and teachers the implication is that each equation is equivalent. Mistakes made by students as they solve equations often can be detected by looking for places where there are some non-equivalent equations.
|
Sample work with student reasons: |
|
|
2x2 - 10x = 0 |
Original equation. |
|
2x(x - 10) = 0 |
Factored. |
|
(x -10) =0 |
Divided by 2x on each side. |
|
x = 10 |
added 10 to each side. |
The graphing method provides students with an opportunity to see where the equations are equivalent and what action caused the equations to be non-equivalent.
|
Sample work with student reasons: |
Graph |
||
|
step 1 |
2x2 - 10x = 0 |
Original equation |
|
|
step 2 |
2x(x-10) = 0 |
Factored |
|
|
step 3 |
(x - 10 ) = 0 not equivalent to equation in step 2 |
Divided by 2x on each side. |
|
|
step 4 |
x = 10 |
Added 10 to each side. |
|
The step that caused the mistake is step 3 where the division by 2x occurred. This mistake and the graph provides teachers with an opportunity to discuss the idea of the limitations of the division property of equality and the reason that the solution x = 0 was lost was due to division by zero. Matching the steps the students use with the equation with the resulting actions on the graphs provides a way to help students make sense of the concept of equivalent equations.
- In middle school, students learn the addition, subtraction, multiplication and division properties of equality. They use statements like "What you do to one side you do to the other" to explain why these properties work. Taking the square root of each side of an equation fits this notion of doing the same thing to both sides when solving equations like (x + 2)2 = 9 but as students become more sophisticated in the functions they apply to both sides of an equation, there are subtle limitations that they need to be aware of. The equations $(x+2)^{2}=9\rightarrow$ $ \sqrt{(x+2)^{2}}=\sqrt{9}\rightarrow$ $ x+2=3\rightarrow x=1$ are not all equivalent. The first equation in the string has two solutions, x = 1 or x = -5 while the last only has one. Solving an equation means finding all the values that make the equation true, not just one. One way to help students see where the mistake occurs is to look at graphs of each equation as described above. Another approach is to use a computer algebra system that has a show steps feature. Wolfram Alpha has a show steps feature. Students can use the program to solve the equation (x + 2)2 = 9 and see the steps the program used to solve the equation.
The student who made the mistake described above and the teacher can look at the steps the program used and compare the steps with what the student did. The teacher can have a richer discussion with the class about the types of functions that can be applied to both sides of an equation and still maintain equivalent equations. This approach can provide more meaning than simply reminding the student to always remember to put a ± in front of the radical when you take the square root of both sides of an equation.
Smokey Bear Takes Algebra
This activity asks students to connect linear, quadratic and exponential functions to make sense of a forest fire danger index.
Additional Instructional Resources
John and Betty's Journey into Complex Numbers
This is a children's book-on-the-web from Australia, designed to introduce complex numbers in story form in a way that is intuitive and enjoyable for students. John, Betty and their dog Trevor solve a series of problems designed to introduce integers, fractions, surds, imaginary numbers, complex numbers, Argand diagram; vectors, multiplication in polar form, relating polar and Cartesian form, De Moivre's theorem; and the Mandelbrot set.
Complex Numbers: Introduction from Purplemath
This website does an excellent job of explaining what complex numbers are: how to add, subtract, multiply, divide and simplify complex numbers, and how complex numbers are used to solve and graph quadratic equations with imaginary roots.
Kennedy, P., and St. John, D. (2006). Chapter 2, Complex Numbers and Matrices. In Navigating through Number and Operations in Grades 9-12. Reston, VA: National Council of Teachers of Mathematics.
This chapter provides activities to enable students to demonstrate fluency with addition and multiplication of complex numbers, understand different representations of complex numbers, and to determine the properties of the complex number system and to compare them with other number systems.
Algebra 2: Complex Numbers
Students look for patterns to determine how the calculator adds, subtracts, multiplies and divides complex numbers. In each problem, students examine completed examples and discuss a process with a partner, then complete additional problems to confirm or refine the process.
irrational numbers:
real numbers:
imaginary numbers:
complex numbers:
quadratic equation:
roots of a function:
zero product property:
Zero product property from www.hotmath.com)
factoring:
completing the square:
Completing the Square from www.hotmath.com )
quadratic formula:
exponential functions:
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
- Do students understand the difference between an equation and an expression?
- Do students think of the expression $\frac{1+\sqrt{5}}{2}$ as a number? Are they able to find a decimal approximation?
- Are students allowed to struggle with problems as they solve them?
- Are students dependent upon the teacher too much to set up problems?
Materials
- Campe, K. D. (2011). Do it right: Strategies for implementing technology. Mathematics Teacher, 104 (8), 621-625.
- Hodges, T. E., & Conner, E. (2011). Reflections on a technology-rich mathematics classroom. Mathematics Teacher, 104 (6), 432-438.
- Common Core State Standards (http://www.corestandards.org/the-standards/mathematics)
- Ruddell, M.R., & Shearer, B.A. (2002). "Extraordinary," "tremendous," "exhilarating, "magnificent": Middle school at-risk students become avid word learners with the vocabulary self-collection strategy (VSS). Journal of Adolescent & Adult Literacy, 45, 352-363.
Assessment
1.
DOK Level: 2 / Cognitive Level: Understanding
Answer: D
Source: MCAII Item Sampler 11th Grade Mathematics http://education.state.mn.us/MDE/Accountability_Programs/Assessment_and_Testing/Assessments/MCA/Samplers/index.html
2. Select the equation that has two complex solutions.
A. 3x2 = -7
B. 7 - x2 = 2x
C. $\frac{1}{x+3}=17$
D. (x + 5) (x - 2) = 0
E. (x + 5) (x - 2) = -6
DOK Level: 3 / Cognitive Level: Knowledge
Answer: A
3. What are all values of x2 such that x + 7x ≥ 0?
a) -6 ≤ x ≤ -1
b) -6 ≤ x ≤ 1
c) 1 ≤ x ≤ 6
d) x ≤ -6 or x ≥ -1
e) x ≤ 1 or x ≥ 6
DOK Level: 3 / Cognitive Level: Hard (30.06% correct)
Answer: D
Source: NAEP Grade 12, 2005, Question M4-16 (http://nces.ed.gov/nationsreportcard/itmrlsx/detail.aspx?subject=mathematics)
4. The number of bacteria present in a laboratory sample after t days can be represented by 500(2)t. What is the initial number of bacteria present in this sample?
a) 250
b) 500
c) 750
d) 1000
e) 2000
DOK Level: 3 / Cognitive Level: Hard (33.07% correct)
Answer: b
Source: NAEP Grade 12, 2005, Question M12-14 (http://nces.ed.gov/nationsreportcard/itmrlsx/detail.aspx?subject=mathematics)
5. A car costs $20,000. It decreases in value at the rate of 20 percent each year, based on the value at the beginning of that year. At the end of how many years will the value of the car first be less than half the cost?
Answer: ________ years
Justify your answer.
DOK Level: 3 / Cognitive Level: Hard (30.65% correct)
Answer: 4 years with correct work shown or explanation given. An answer of 3.1063 (or some rounded value of that to the nearest tenth or smaller) with correct work is acceptable. The correct work could show a sequence of correct computations and/or the results of those computations. For example, since a calculator is available for this question, the student may just show the amounts $16,000; $12,800; $10,240; and $8,192 at the ends of the first 4 years, respectively.
Source: NAEP Grade 12, 2005, Question M12- 12
(http://nces.ed.gov/nationsreportcard/itmrlsx/detail.aspx?subject=mathematics)
Differentiation
- Students who struggle should be encouraged to make more connections between numerical values and specific points on the graph.
- Students need to be taught specifically how to read a mathematics textbook. One pre-reading method for students reading a mathematics textbook on their own is to use an Anticipation Guide. Good readers activate prior knowledge prior to reading, while poor readers start reading without any prior preparation. Students would use a guide similar to the example shown below prior to reading a section in their mathematics textbook on quadratic equations.
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Anticipation Guide Quadratic Equations Directions: Before you read the text below, put a check mark in the column labeled "Me" next to any statement that you think is true. After reading the text, compare your opinions about those statements with the information in the text. |
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1. Every quadratic equation has a solution. |
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2. A solution to an equation is the value of the variables that make the equation true. |
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3. The shape of the graph of a quadratic function is a line. |
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4. Quadratic equations are always equal to 0. |
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5. (3x -2)(x + 4) = 117 is a quadratic equation. |
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This guide helps all students to become better readers of their textbooks and learn to anticipate the content they will be reading.
- Students should be able to connect quadratic and exponential functions as they try to figure out all of the real values of x that satisfy the equation: $(x^{2}-5x+5)^{x^{2}-9x+20}=1$
The solution is $x\epsilon \left \{ 1,2,3,4,5 \right \}$.
- National Debt and Wars
In this extension activity from NCTM's Illuminations, students will collect information about the national debt, plot the data by decade and determine whether an exponential curve is a good fit for the data. Then student groups will determine and compare common traits and differences in changes in the national debt in three major eras: the CivilWar, WorldWar I and WorldWar II.
Parents/Admin
Administrative/Peer Classroom Observation
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Students are: |
Teachers are: |
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being flexible in choosing a method to solve a quadratic equation. |
encouraging students to select the method for solving equations that works best. |
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deciding when to use a graphing utility or other technology to solve a problem.
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modeling how to use technology and discussing when use is appropriate. |
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making connections between and among representations as they solve problems.
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helping students make sense of key ideas by connecting representations. |
Parent Resources
- Algebra Tiles
This virtual manipulative helps students make connections between an area model and symbols, make sense of multiplying using the distributive property, as well as factoring.