Writing Prompts Intermediate Algebra. M. E. Waggoner, Simpson College Indianola, Iowa Updated March 30, 2016


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1 Writing Prompts Intermediate Algebra M. E. Waggoner, Simpson College Indianola, Iowa Updated March 30, 2016
2 Contents Chapter 1: Real Numbers and Expressions... 3 Chapter 2: Linear Equations and Inequalities... 6 Chapter 3: Lines... 8 Chapter 4: Systems of Linear Equations Chapter 5: Polynomials Chapter 6: Rational Functions Chapter 7: Solving Quadratic Equations Chapter 8: Zeros of polynomials... 17
3 Chapter 1: Real Numbers and Expressions 1.1. A friend wrote 1 > x > 5 as the answer to a problem. Gently explain to your friend what is wrong with her work Explain why 0 3 is equal to 0. Explain why 3 0 is not defined Compare and contrast the evaluation of 5 2 and ( 5) You and your friend were comparing homework and here is the work you have: (2.345) 3 (1.33)(π) = (2.35) 3 (1.33)(3.14) = (12.98)(1.33)(3.14). = Here is the work your friend has: (2.345) 3 (4 / 3)(π) = Which one of you has the better answer and why? How can you avoid this error in the future? 1.5. What personal tools do you use to help you distinguish between the commutative property and the associative property? Answer this question with a short paragraph including examples List the order of operations. Explain how you remember this. Why is an order of operations agreement important? 1.7. Compare and contrast the associative and the commutative properties Compare and contrast the associative and the distributive properties A friend of yours had the following work on his paper x = 5x Gently explain your friend's error and give him some pointers for avoiding this error in the future Explain what it means to "express x in terms of y." a) Why does the order of x and y are important when we say "the difference of x and y" but the order doesn't matter when we say "the sum of x and y"? b) For which of these statements is the order important? (i) The product of x and y (ii) The quotient of x and y Explain what is meant by (, 7] Explain the relationships of the following. x 3 [3, ) x is greater than or equal to 3 3
4 1.13. Always true, sometimes true or never true: If a is a real number, then 2 a = a. xx, The absolute value of x is defined to be x =. To understand this definition, xx, < 0 you must believe that x = x for negative values of x. Using x= 3 as an example, explain why x = ( 1)x produces the same result as taking the absolute value of x. [SM] A friend of yours had the following work on her paper = 6 Gently explain your friend's error and give her some pointers for avoiding this error in the future Calculate each of the following Use the pattern to help explain the value of If a = 5 and b = 3 the following equation is correct. a b 2 = 4 Now, subtract the equation a = 5 from this equation. Is the equation still correct? How does this help us understand how to evaluate b 2? Write three "real life" but distinctly different expressions that would be converted mathematically to 3x A friend of yours evaluates a+ 2b when a = 2 and b = 5 and gets 12. Your friend argues that the results of absolute values are always positive. Gently explain to your friend the correct answer and why their argument is faulty How many different ways can 1 x be described in words? Compare and contrast the following 2 phrases. 3 is less than x a number is 3 less than x Why do we need an Order of Operations Agreement? Give an example of a mathematical expression that could be miscalculated if the Order of Operations Agreement is not followed Compare and contrast ( 2 ) 4 and Agreement should say about 2 34? 4 ( 3 ) 2. What do you think the Order of Operations
5 1.24. A friend is having trouble simplifying x x. Help the friend by showing several examples that are easy for them and showing how x x is simplified in the same way A friend of yours has the following work on her paper. 3x+ 2 2x 3 = 3x+ 2 2x 3 ( ) = 5x 1 Gently explain to your friend the mistake she made and give the student some helpful points to avoid this mistake in the future Investigate the validity of this statement: x x x Which of the following are properties of absolute values and which are not true for all values of a and b? For the statements that are not true for all values of a and b, can you make the statement a true for all a and b by changing the equation to an inequality? ab = a b a+ b = a + b a b = a b a b a = b
6 Chapter 2: Linear Equations and Inequalities 2.1. Compare and contrast the following two problems. In addition to talking about the appearance of these two problems, also talk about the steps used to complete the two problems. Problem1: Simplify 3( x 2) 2( x+ 5). Problem 2: Solve 3( x 2) 2( x + 5)= A friend of yours has the following work on his paper. 3x 7 5x +13 3x 5x x 20 x 10 Gently explain to your friend the mistake in his work and give the student some helpful pointers to avoid this mistake in the future How many solutions are there to x 2 = 5? How many solutions are there to x 2 > 5? Explain Compare and contrast coin and stamp problems Compare and contrast a mathematical expression and a mathematical equation Compare and contrast equations and inequalities Compare and contrast mixture and motion problems Is 3 a solution to x = 3? Discuss all the solutions to x = 3. Is 3 a solution to x = 3? Discuss all the solutions to x = What formula do you need to know for motion problems? Explain what each variable means in this equation. Give an everyday example to illustrate your explanation Is 4 is a solution of 4y 5 = 3y? Is 4 is a solution of 4y 5 3y? In general, explain what it means for a number to be a solution to an equation. In general, explain what it means for a number to be a solution to an inequality Find all solutions to x 3 = 5 using trial and error. Keep a table of your guesses and other useful information. Discuss the process Find all solutions to = using trial and error. Keep a table of your guesses and x+ 2 x 5 other useful information. Discuss the process.
7 2.13. Discuss how to solve 3 < 5 2x 1 as is and compare that to solving 3 < 5 2x 1 by breaking it into 2 separate inequalities Consider the following calculations. Explain what is happening in each step along the way. Is 5 = 8? Where did the calculations go wrong and why? 5x+ 15= 2x+ 3( 2x+ 5) 5x+ 15 = 2x+ 6x+ 15 5x+ 15 = 8x+ 15 5x= 8x 5= A friend of yours has the following work on his paper. 4x = 2 x = 2 Gently explain to your friend the mistake in his work and give the student some helpful pointers to avoid this mistake in the future Chose a positive number. Multiply both sides of 2 < 4 by the positive number you chose. Is the statement still true? Chose a negative number. Multiply both sides of 2 < 4 by the negative number you chose. Is the statement still true? Describe your observations What does it mean to "solve an equation"? If I told you x = 4 is the solution to an equation what does that mean? What does "linear equation" mean and how can you differentiate a linear equation from other equations? How many answers do most linear equations have? When do linear equations not have a solution?
8 Chapter 3: Lines 3.1. Given 2 points in the plane, what is your preferred method for finding the equation of a line that passes through those 2 points? First, describe your method in general. Then, make up an example using your method and annotate it Consider this data that describes the position in height above ground (in meters) of a helicopter at specific times (in minutes). Plot this data on a coordinate system. Choose a scale for each axis so that the points in this list fill the graph and explain your choice. Do you think this data is linear or not? Time Position x y 3.3. a) Explain why c and d are the xintercept and yintercept, respectively, of + = 1. c d x y b) How are the xintercept and yintercept related to c and d in + = 2? [FWG] c d c) Generalize a) Can a function have more than one xintercept? b) Can a function have more than one yintercept? c) What do the intercepts say about the graph of a function? 3.5. A friend of yours in class did the following work. P1 ( 3, 2) P2 ( 3,5) 5 ( 2) 7 m = = 3 ( 3) 6 Gently and correctly, explain to your friend what their mistake is and give them pointers for organizing their work so they can avoid this error in the future On a sheet of graph paper draw a coordinate system that fills the page. a) Using one color start by graphing 5 points where x > 0 and y > 0. Make a list of these points. b) Make a new list of points from the list you made in part a) by making all the ordinates negative. Using a second color, graph these points on your coordinate system. c) Make a new list of points from the list you made in part a) by making all the abscissas negative. Using a third color, graph these points on your coordinate system. d) Make a final set of points from the list you made in part a) by making both the abscissa and the ordinate negative. Using a fourth color, graph these points on your coordinate system. e) Explain geometrically the relationship between the points (x, y) and (x, y). f) Explain geometrically the relationship between the points (x, y) and ( x, y). g) Explain geometrically the relationship between the points (x, y) and ( x, y).
9 3.7. When graphing a line by plotting points, why is it a good practice to plot three points instead of two? 3.8. Give a real life example of a relation that is a function. Give a real life example of a relation that is not a function. Explain What are the units of the slope of line if the units of x are seconds and the units of y are feet? Do a dimensional analysis of y = mx + b Write up to 5 distinct questions that I might ask on an exam about the following information. P 1 ( 3, 2) P 2 3,5 ( ) Write up to 5 distinct questions that I might ask on an exam about the following information. 3x + 2y = Consider the following analogies. Graphs are to equations like pictures are to memories. Stephanie DeSloover Graphs are to equations as a diagram is to a basketball play. Jessica Montag Graphs are to equations as maps are to directions. Mike Miller Graphs are to equations like notes are to music. Vista Kalipa and Mandy Carlson Choose one of these analogies and explain it more completely. It would be good to list reasons why the analogy makes sense, as well as, some reasons why the analogy is not perfect Answer these questions and supply a graph of the pool table with the path of the ball. The yaxis, the xaxis, the line x = 6, and the line y = 12 determine the four sides of a 6 by 12 rectangle. Imagine that this rectangle is a pool table. There are pockets at the four corners and at the points (0,6) and (6,6) in the middle of each of the longer sides. When a ball bounces off one of the sides of the table, it obeys the "pool rule," the slope of the path after the bounce is the negative of the slope before the bounce. Your pool ball starts at (3,8). You hit it toward the yaxis, along the line with slope 2. a) Where does the ball hit the yaxis? b) If the ball is hit hard enough, where does it hit the side of the table next? And after that? And after that? c) Does it ultimately return to (3,8)? d) Would the ball return to (3,8) if the slope had been different from 2? e) What is special about the slope 2 for this table? If the points (2, 4) and ( 1, 6) are on the line L, find another point on L. Explain and illustrate your method.
10 3.15. Consider the following functions. 3+ x f( x) = g(x) = 2x +1 h(x) = 4 x 1 Evaluate each function at x = 2. Evaluate each functions at x = 1. Discuss your observations and compare the process of evaluating each function Explain what the points on a graph of an equation in 2 variables represent Explain how to graph an equation of a straight line using 2 points. Explain how to graph an equation of a straight line using the x and yintercepts How do we know whether an equation can be graphed with a straight line or not. Give examples of equations that can be graphed with a straight line and some that cannot Explain why y = mx + b is called the slopeintercept form of the line Compare and contrast the pointslope form of the line with the slopeintercept form of the line Consider a line y = mx + b. What happens to the graph of this line if we change the value of b? What happens to the graph of this line if we change the value of m? Below is the graph of the function f(t) that represents your height about ground at time t. Write an explanation of what you are doing to make the graph look like it does. [MR] f(t) Discuss the equations of horizontal and vertical lines. Why do these lines give students problems? How can you recognize when a line is horizontal or vertical when given 2 points? How can you recognize a horizontal or vertical line from their equations? Give recommendations for finding the equations of horizontal and vertical lines.in various information Give the definition of the slope of a line. Explain why the slope will be the same no matter what 2 points you choose from the line. [St] t
11 3.25. Describe what the line looks like in each of following cases. Explain each case. a) A line with a small, negative slope. b) A line with a large, positive slope. c) A line with a small, positive slope. d) A line with a large, negative slope. [St] e) A zero slope. f) A slope of 1.
12 Chapter 4: Systems of Linear Equations 3x 2y = What does it mean for a point to be a solution to? How do you determine x + 2y = 6 if (0, 1) is a solution to this system? 4.2. Compare and contrast the strategies you would use to solve each of the following systems of equations. 3x + 2y = 2 2x 6y =15 4 x + 3y = 30 x = 4y What would the graph of y = 4x + 8 look like? The graph of 2x 6y =15? Explain the 2x 6y =15 relationship of the graphs of these equations and the solution of y = 4x In how many different ways can 2 lines in the plane be oriented? What would the equations of each of these orientations look like? What do we call each of these orientations? 4.5. In how many different ways can 2 planes in space be oriented? What would the equations of each of these orientations look like? What do we call each of these orientations? 4.6. Describe your method for solving the following system of equations. 3x + y 2z = 2 x + 2y + 3z =13 2x 2y + 5z = Consider the following story problem. Explain how to solve this using a single equation in 1 unknown. Explain how to solve this using 2 equations in 2 unknowns. Compare and contrast these 2 methods. A coin bank contains only nickels and dimes. The total value of the coins in the bank is $2.50. The number of nickels is 3 times the number of dimes. How many dimes are there? 4.8. We have covered several sections about solving linear systems of equations. Explain what "linear system of equations in 2 unknowns" means? What would a nonlinear system of equations in 2 unknowns look like? Briefly explain your strategy for solving a linear system of equations. Explain why that strategy may or may not work for a nonlinear system of equations.
13 Chapter 5: Polynomials 5.1. Explain the exponent rules for multiplication ( a n a m n ) and exponentiation (( a ) m ). 3 5 Explain why we can calculate ( 3a b ) 3 using the exponentiation rule and we can 3 calculate the same thing as ( 3ab 3 5 ) ( 3ab 3 5 )( 3ab 3 5 )( 3ab 3 5 ) = using the multiplication rule and we get the same thing both ways Consider the process of adding the polynomials 3x 4 2x 2 + x 4 and 5x 3 5x + 6. Explain how to do this using a vertical format and a horizontal format. Discuss your preference of method and why that method is better for you Consider the process of multiplying the polynomials 2x 2 + x 4 and x 2 5x + 6. Explain how to do this using a vertical format and a horizontal format. Discuss your preference of method and why that method is better for you Explain to another student how to multiply two trinomials. Use an annotate example to help illustrate your method Explain to another student how to multiply three binomials. Use an annotate example to help illustrate your method Explain when you add exponents and when you subtract them in polynomial operations. Give examples of when you add the exponents, when you subtract the exponents, and of when you can t combine the exponents at all Show why x 2 x 3 2 is calculated differently than ( x ) 3 by calculating both and explaining the calculations Explain what a negative exponent means and how to use it. Give examples Explain why we cannot combine like terms in the expression 3x 2 + 2x 1. Explain why x 2 y and xy 2 are not like terms. Give other examples of terms that are not alike and can't be combined Here is a question from an exam and the answer the student gave: Question: Multiply ( x + 3) 2. Answer: x Give the correct answer to this problem and explain the mistake the student made What is the rule for factoring the difference of squares? Write an argument to convince a friend that this rule is correct Why can we not factor a sum of squares?
14 5.13. Explain your factoring strategy for factoring binomials Explain your factoring strategy for factoring polynomials with 4 terms Consider the two problems below. Problem A: Factor 2x 2 3x 5. Problem B: Solve 2x 2 3x 5 = 0. Compare and contrast the solutions to these two problems Give advice to another student that will help them calculate 23x 2 3x+ 2 4x 2 + x 5 correctly every time. ( ) ( ) You probably remember hearing me say, " Factoring is great because you can always check your work." By creating and annotating your own example, explain what this means Compare and contrast the factoring of a z + 2z and a ( b + 3) + 2( b + 3) You can approach factoring trinomials by trial and error only, but it better to go into the process with a plan. Explain your strategy for factoring trinomials How many factors does x( x 2)( x 3) x( x 2)( x 3) 0 + have? How many solutions should + = have? What are these solutions? Give advice to someone to help him or her not miss any of the solutions If P(x) is a polynomial of degree 3 and Q(x) is a polynomial of degree 4, what can be said about P + Q? What can be said about PQ? If P(x) is a polynomial of degree 3 and Q(x) is a polynomial of degree 3, what can be said about P + Q? What can be said about PQ? Compare and contrast the process of "multiplying 2 polynomials" and "factoring a polynomial" Compare and contrast the process of "dividing 2 polynomials" and "factoring a polynomial" Compare and contrast the process of "dividing 2 integers" and "dividing 2 polynomials." Explain how to factor y 4 1 completely. Create another example that looks different but can be factored in the same way Investigate the validity of this statement: If polynomial px ( ) = x + x x + 1 then p(x) is a
15 Chapter 6: Rational Functions 6.1. What is a rational function? How do they compare to polynomial functions? 6.2. How do we determine the values excluded from the domain of a rational function? 6.3. When is a rational expression in simplest form? Explain the process of finding the simplest form of a rational expression Compare and contrast adding rational expressions and multiplying rational expressions Compare and contrast multiplying rational expressions and dividing rational expressions A friend of yours in class did the following work. 3 x 3+ x + = Gently and correctly, explain to your friend what their mistake is and give them pointers for organizing their work so they can avoid this error in the future Investigate the validity of this statement: The quotient of any two functions is a rational function.
16 Chapter 7: Solving Quadratic Equations 8.1. Solve the following 2 quadratic equations (giving exact values). Justify the methods you used to solve each equation. x 2 2x = 0 x + x 2 = Write up to 5 distinct questions that I might ask on an exam about the following information. x 2 2x = When we solve x 2 x 6 = 0 we get x = 3 and x = 2. What does it mean for 3 and 2 to be solutions to this equation? Why are there 2 solutions instead of just 1? 8.4. Describe the strategies for solving quadratic equations What does "quadratic equation" mean and how can you differentiate a quadratic equation from other equations? How many answers do most quadratic equations have? When do linear equations not have that number of solutions? 8.6. Solve x 2 = 4 by factoring, by finding square roots, and by using the quadratic formula. For this particular equation, what is your preference and why.
17 Chapter 8: Zeros of polynomials 9.1. What do the following 4 questions have in common? a) Find the zeros of f(x) = x 3 3x 1. b) Find the values of x for which x 3 3x = 1. c) Solve the equation x 3 3x 1 = 0. d) Find the xintercepts of f(x) = x 3 3x What is the xintercept of the graph of a parabola? How many xintercepts can the graph of a parabola have? 9.3. What is the yintercept of the graph of a parabola? How many yintercepts can the graph of a parabola have?
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