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Information for MLCS
Kathleen Almy, Heather Foes Rock Valley College Emails: [email protected]
Blog: http://almydoesmath.blogspot.com
Blog contains video, pilot updates, presentations, and more.
Mathematical Literacy for College Students (MLCS) 3 – 6 semester hours Prerequisite: Appropriate placement or prealgebra with a grade of “C” or better The goal of developmental mathematics education is to provide students with the necessary skills and understanding required to be successful in college level mathematics. It is not necessarily intended to be a repeat of high school mathematics. While many programs and initiatives have been developed to improve the state of developmental education, part of the problem lies in the content and courses taught. Mathematical Literacy for College Students (MLCS) is a new course being developed at the national level by AMATYC’s New Life for Developmental Mathematics. Its origins are related to Quantway, funded by the Carnegie Foundation. MLCS is an alternative path to certain college level math courses or further algebra. It integrates numeracy, proportional reasoning, algebraic reasoning, and functions with statistics and geometry as recurring course themes. Throughout the course, college success components are integrated with the mathematical topics. The course focuses on developing mathematical maturity through problem solving, critical thinking, writing, and communication of mathematics. Content is developed in an integrated fashion, increasing in depth as the course progresses. Upon completion of the course, students will be prepared for a statistics course or a general education mathematics course. Students may also take intermediate algebra upon completion if they choose to pursue STEM courses.
MLCS provides an alternative to beginning algebra, creating multiple pathways for the developmental students. However, it is more rigorous than beginning algebra to ensure students are prepared for a college level math course upon successful completion. It allows students to potentially complete their developmental math and college level math requirement for an Associate in Arts degree in one year total (one semester each), working toward the goal of improving college completion rates. It promotes 21st century skills to prepare students for both the workplace and future coursework. Further, it does not diminish requirements for non-STEM college level math courses but instead creates appropriate paths to these courses with the same level of intensity and complexity as the current path through intermediate algebra. The course has college level expectations and coursework but with a pace and instructional design intended for the adult, developmental learner. This strategy emulates the approach taken by the Common Core Standards and aligns with them as well.
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MLCS Course Description and Objectives
Mathematical Literacy for College Students is a one semester course for non-math and non-science majors integrating numeracy, proportional reasoning, algebraic reasoning, and functions. Students will develop conceptual and procedural tools that support the use of key mathematical concepts in a variety of contexts. Throughout the course, college success content will be integrated with mathematical topics.
Prerequisite: Appropriate placement or prealgebra with a grade of “C” or better COURSE OUTCOMES
1. Apply the concepts of numeracy in multiple contexts. 2. Recognize proportional relationships and use proportional reasoning to solve problems. 3. Use the language of algebra to write relationships involving variables, interpret those relationships, and solve
problems. 4. Interpret and move flexibly between multiple formats including graphs, tables, equations, and words. 5. Demonstrate student success skills including perseverance, time management, and appropriate use of
resources. 6. Develop the ability to think critically and solve problems in a variety of contexts using the tools of mathematics
including technology.
COURSE OBJECTIVES
Upon successful completion of this course, the student will be able to:
Numeracy
1. Demonstrate operation sense and the effects of common operations on numbers in words and symbols. 2. Demonstrate competency in the use of magnitude in the contexts of place values, fractions, and numbers written in
scientific notation. 3. Use estimation skills. 4. Apply quantitative reasoning to solve problems involving quantities or rates. 5. Demonstrate measurement sense. 6. Demonstrate an understanding of the mathematical properties and uses of different types of mathematical
summaries of data. 7. Read, interpret, and make decisions based upon data from line graphs, bar graphs, and charts. Proportional reasoning
8. Recognize proportional relationships from verbal and numeric representations. 9. Compare proportional relationships represented in different ways. 10. Apply quantitative reasoning strategies to solve real-world problems with proportional relationships. Algebraic reasoning
11. Understand various uses of variables to represent quantities or attributes. 12. Describe the effect that changes in variable values have in an algebraic relationship. 13. Construct and solve equations or inequalities to represent relationships involving one or more unknown or variable
quantities to solve problems.
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Functions
14. Translate problems from a variety of contexts into a mathematical representation and vice versa. 15. Describe the behavior of common types of functions using words, algebraic symbols, graphs, and tables. 16. Identify the reasonableness of a linear model for given data and consider alternative models. 17. Identify important characteristics of functions in various representations. 18. Use appropriate terms and units to describe rate of change. 19. Understand that abstract mathematical models used to characterize real-world scenarios or physical relationships
are not always exact and may be subject to error from many sources. Student success
20. Develop written and verbal skills in relation to course content. 21. Evaluate personal learning style, strengths, weaknesses, and success strategies that address each. 22. Research using print and online resources. 23. Apply time management and goal setting techniques.
Mathematical success
24. Develop the ability to use mathematical skills in diverse scenarios and contexts. 25. Use technology appropriately including calculators and computers. 26. Demonstrate critical thinking by analyzing ideas, patterns, and principles. 27. Demonstrate flexibility with mathematics through various contexts, modes of technology, and presentations of
information (tables, graphs, words, equations). 28. Demonstrate and explain skills needed in studying for and taking tests.
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Implementation Options
MLCS is a 3 – 6 credit hour course depending on the depth and breadth desired.
1. Replacement Model: Use MLCS to replace beginning algebra.
2. Augmented Model: Use MLCS to create a non-STEM alternative to beginning algebra that provides sufficient
preparation for statistics or liberal arts math.
3. Supplemental Model: Use MLCS lessons for problem solving sessions in an Emporium model (lab-based traditional redesign.), engaging all students and moving beyond skills alone.
4. High School Model: Use MLCS lessons for 4th year high school course to develop college readiness and help
students place into college level math.
1. Each time the number of hours increases by 1, the test score increases by 4 .Each time the number of hours increases by 2, the test score increases by 8 .Each time the number of hours increases by 6, the test score increases by 24 .Use each of these answers to write a ratio in the form
increase in test scoreincrease in number of hours
= 41
2. Since the ratios of increase in number of hours studied to increase in test score are always 4 , we can say there is a linear relationship between the number of hours studied and the test score.
3. Write the slope with units and interpret it.Slope is 4 point/hour studied. For every hour studied, a student will gain 4 points on the final exam.
4. Write a formula for the test score, T, based on the number of hours, h, a student studies.
T = 4h + 62
5. Where is the slope in the table? In the equation?The slope is the rate of change in the table. It is the coefficient of the independent variable in the equation.
6. Where is the number 62 in the table? In the equation?It was the score associated with 0 hours studied. It is the constant in the equation.
Instructor note: We are using patterns and inductive reason-ing here. Formal equation writing comes in Cycle 4. If students need a hint, remind them that the scores start at 62 and increase by 4. This idea is very useful for writing linear equations.
On a comprehensive final exam, a professor asked each student to list the number of hours spent studying for the test. She chose a few of the students’ results and compiled them into a table. This helps her see trends that may prove useful for her future classes.
Number of hours Studied Test Score Out of 100
0 62
1 66
2 70
3 74
4 78
6 86
9 98
FPGet in Line3.12
Explore10 min
80–90 min
Instructor note: Instruct stu-dents to work on #1–6 in groups. Go over answers.
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Discover20 min
Instructor note: Bring the class together to complete #7–10.
Let’s look at this situation and the data using a graph. 7. Create a graph using the coordinate system provided.
64
72
80
88
92
96
100
60
68
76
84
21 43 5 6Hours studied
Final Exam
7 8 9 10
Test
sco
re
Instructor note: Discuss the setup of the graph and the choices for scales on the axes to expose students to the idea of domain (rea-sonable values for the independent variable) and range (reasonable val-ues for the dependent variable).
It’s reasonable to have fractions and decimals for the hours studied but not negative values. The hours can continue to increase somewhat, but there is a physical limit. Also, there is a limit to what the test score can be.
Draw a right triangle from (1, 66) to (2, 70) on the line. Ask students for the rise and run of the triangle to find the slope of the line. m = 4 (4 additional points for each hour studied).
8. Where do the slope and the number 62 appear on the graph?The slope is the rise compared to the run when moving between points on the line. The point (0, 62) is where the line crosses the y-axis.
Let’s look more closely at how the equation of a line and its graph are related.
Explain that while we graphed the line by plotting points, we could use the slope to find more points on the line once a point on it is plotted.
A closer Looklearn
InterceptThe point at which a graph crosses an axis is called an intercept. If the graph crosses the x-axis, the point of intersection is called an x-intercept. If the graph crosses the y-axis, the point of inter-section is called a y-intercept.
For example, this graph has an x-intercept of (-2, 0) and a y-intercept of (0, 4).
look iT uP
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The formula T = 4h + 62 is in a very useful form because it clearly shows the slope and y-intercept of the graph. The equation is in slope-intercept form.
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Instructor note: Although it is conventional to use y = mx + b, it is often easier for students to think of linear equations as y = b + mx . This order is used when graphing with the slope and y-intercept and it’s how most contex-tual problems are described with an equation: a starting point and change from it.
look iT uP
slope-intercept formLinear graphs have equations that can be written in the form y = mx + b, called slope-intercept form. x represents the independent variable, and y is the dependent variable. When an equation is in slope-intercept form, the coefficient of x is the slope. The con-stant b represents the y-coordinate of the point where the line intersects the y-axis. The ordered pair (0, b) is the y-intercept.
dependent variable graphed on vertical axis
independent variable graphed on horizontal axis
y = 50x + 200
slope (change)
y-coordinate of y-intercept (starting value)
For example, imagine you have $200 in a savings account and you deposit $50 each month. This equation has m = 50 and b = 200. So the line has a slope of 50 and y-intercept of (0, 200). We can read these values only from an equation in slope-intercept form.
PracTiceWe have now learned that linearity can be seen from a table when there is a constant rate of change, on a graph when the points fall in a line, and in an equation of the form y = mx + b.
Instructor note: Remind students that every number has a sign. Don’t separate the sign from a number.
Instructor note: Instruct students to work in groups. Check answers and clear up issues. Signs and implied coefficients usually present a challenge.
Instructor note: After going over answers to the practice problems, work with the class on example 2.
5 min
10–15 min
exAmPle 1 For each linear equation, list the slope and y-intercept. Remember, the y-intercept is an ordered pair, not a number. a. y = -3x - 2 m = -3, y-intercept is (0, -2)
b. y =x2
- 8 m =12
, y-intercept is (0, -8)
c. y = 4 - x m = -1, y-intercept is (0, 4)
PRACTICe 1 For each linear equation, list the slope and y-intercept. Remember, the y-intercept is an ordered pair, not a number. a. y = -x + 1 m = -1, y-intercept is (0, 1)
b. y = -x m = -1, y-intercept is (0, 0).
c. y = 4 m = 0, y-intercept is (0, 4).
d. T = 4h + 62 m = 4, y-intercept is (0, 62).
Let’s graph a linear equation in two ways. The first method requires finding points that appear on the line. Organizing the points into a table is helpful. The second method uses the slope and y-intercept.
exAmPle 2 Complete this table for the following linear equation: y = -5x - 6.
x y
-1 -1
0 -6
1 -11
2 -16
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If you recognize the slope and y-intercept of a line, you can often graph it very quickly. To graph a line, we need to know a point to get a rough location on the grid and also how the line is tilted. The y-intercept gives us a location, and the slope tells the steepness.
Find the slope from the equation y = -5x - 6 and verify it by finding the rate of change in the table. m = -5
Find the y-intercept from the equation and verify it in the table. (0, -6)
Plot the y-intercept. Use the rise and run of the slope to move to another point on the line.
x–12–16–20 –8
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0–4 2 4 6 8 10 12 14 16 18 20x
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Instructor note: Remind stu-dents that although only 2 points are needed to draw a line, a third point provides a check.
We can choose any values for x. The numbers in the table are whole numbers and are small, which make calculations faster. Notice that we are picking x, plugging it into the equation, and finding y. The value of y depends on whatever we choose for x. We can connect the points with a line because the rate of change is constant, mak-ing the pattern linear.
Plot the points and graph the line.
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–20Instructor note: The slope is -5, which is easier to work with as
the fraction -51
. We can move down
5 and to the right 1 from the y-intercept. Or we can think of the
slope as 5
-1 and move up 5 and left
1. Do this again to get a third point.
Students should notice that this graph is the same as the first graph.
Instruct students to work on prac-tice 2. Check answers. Ask which method they prefer.
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b. Graph this linear equation by using the slope and y-intercept.
m =-34
, y-intercept is (0, 5)
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5–10 min
PRACTICe 2 Suppose y = - 34
x + 5.
a. Graph this linear equation by making a table of points. To avoid fractional y-values, choose x-values that are multiples of the denominator of the slope.
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x y
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Instructor note: Remind students that we have used the method of plotting points (as in part a) throughout the book so far. The method used in part b is a shortcut but it only works for lines.
leSSON 3.12 GeT IN lINe 347
When working with real data, we need to be able to interpret the slope and y-intercept, not just find them. Keep in mind that slope compares the rise and run or the dependent and independent variables. We can use this idea to write a ratio to interpret the slope.
exAmPle 3 If gas costs $3.65 per gallon with a $5.50 car wash, a formula for the cost (C) of g gallons of gas and the car wash is given by C = 3.65g + 5.50. This equation, or model of the situation, is also a function since there is only one cost associated with each value of g. The equation is linear. If it helps, think of the equation as y = 3.65x + 5.50. List the slope and y-intercept.
m = 3.65, y-intercept is (0, 5.50)
The values of the slope and y-intercept matter, but their meaning is also important.
Slope =change in ychange in x
=change in cost
change in gallons=
3.651
Interpret the slope in words: For each gallon of gas, we will pay an additional $3.65.
Interpret the y-intercept in words: The car wash, without any gas, would cost $5.50.
5 min
Instructor note: Explain the Sticky Note. Instruct students to work in groups for practice 3. Go over answers.
sTickyKeep in mind that slope-intercept form does not have to use the letters y and x. If you see an equation in this form, it is in slope-intercept form:
Dependent = Number ~ Independent + Number Variable Variable y = mx + bThe numbers, which represent the slope and y-intercept, can be positive, negative, or zero, and are not necessarily identical.
note
PRACTICe 3 For each problem, write an equation to model the situation. Interpret the slope and y-intercept in words. Also list which variable would be on the hori-zontal axis and which would be on the vertical axis.
a. Model the cost if you buy an iPod for $200 and download songs for $1.29 each.C = 1.29s + 200 would give the cost, C, of the iPod and s songs.
The slope gives the price per song of $1.29. The y-intercept would be (0, 200), indi-cating the cost of the iPod with 0 songs downloaded. The number of songs, s, would be on the horizontal axis. The cost, C, would be on the vertical axis.
5–10 min
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b. Model the weekly salary a salesman earns if the salary is a combination of his base salary and commissions on his sales. His base weekly salary is $500 and he makes $50 for each sale.S = 500 + 50n, where n is the number of sales he makes and S is his weekly salary. The slope gives the commission per sale of $50. The y-intercept would be (0, 500), indicating his salary if he made 0 sales. The number of sales, n, would be on the hori-zontal axis. S would be on the vertical axis.
In this cycle, we have defined linear equations as equations of the form ax + b = c, where a is not zero. But in this lesson, we have also referred to equations in y = mx + b form as linear. Let’s connect these ideas. 9. Suppose y = 2x + 4.
a. Graph the equation using the slope and y-intercept.
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b. Pick an ordered pair from the graph that appears to be on the line and confirm that the ordered pair is on the graph in part a.Answers will vary.
c. Pick another ordered pair from the graph that appears to be on the line and con-firm that the ordered pair is on the graph in part a.Answers will vary.
d. How are the points on the line related to the equation y = 2x + 4?The coordinates of the points on the line are the solutions of the equation.
e. How do you know for sure that the point (-100, -196) is on the graph of the line y = 2x + 4 even though you cannot see it in the graph in part a?Since the ordered pair is a solution to the equation, the point must lie on the line.
Connect10 min
Instructor note: Complete #9 as a class. This ensures students will make the necessary connec-tions between solving and graphing linear equations.
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Instructor note: Explain the contents of the Wrap-Up. Have stu-dents write an answer to the cycle question, “Why does it matter?” A prompt is provided to help stu-dents. Discuss homework.
Reflect5 min
WraP-uPWhat’s the point?Linear situations can be seen in tables, graphs, and equations.
What did you learn?How to find and interpret the slope and y-intercept from a linear equationHow to graph lines using tables and slope and y-intercept
Cycle 3 Question: When is it worth it?When is it worth drawing a graph to see and understand the slope? When are you able to get the information you need from the equation alone?
Answers will vary.
less
on
3.12 Homeworkskills MyMathLab
•Find and interpret the slope and y-intercept from a linear equation.•Graph lines using tables and slope and y-intercept.
1. List the slope and y-intercept: y = 3 - 2x.m = -2, y-intercept is (0, 3)
2. Use the slope and y-intercept from #1 to form the graph.
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concepts and Applications•Find and interpret the slope and y-intercept from a linear equation.
3. For each situation, write a linear model. Find and interpret the slope and y-intercept.
a. A student has $2,000 in debt from college loans, along with $500 payment for every month she has left to pay off her car. Model her total debt.D = 2,000 + 500m, where m = number of months and D = total debt
The slope is 500, meaning she has an additional $500 of debt for each month that passes in which she has not paid off her car.
The y-intercept is (0, 2000). This means that without the car debt, she would have $2,000 of debt.
b. A man has a starting weight of 205 pounds and is losing 10 pounds each month. Model his weight.W = 205 - 10m, where m = number of months and W = the man’s weight
The slope is -10, meaning he is losing 10 pounds per month.
The y-intercept is (0, 205). This means before he starts losing weight, when no months have passed, his weight is 205 pounds.
4. Imagine a company is offering newly hired employees three salary-pricing structures. One option computes raises by adding $1,000 and then a 3% increase. Another option computes raises by adding a 3% increase and then $1,000. A final option option com-putes raises by adding a 5% increase.
a. Write a formula for computing the new salary under each plan, where NS = New Salary and S = Old Salary.
Add $1,000; increase by 3% NS1 = 1.03(S + 1,000)
Increase by 3%; add $1,000 NS2 = 1.03S + 1,000
Increase by 5% NS3 = 1.05S
b. In each formula, what is the independent variable? S, the original salary in dollars
What are the dependent variables? NS1, NS2, NS3
c. Using the formulas you created, complete the table below. All amounts are in dollars.
Old Salary, S NS1 = 1.03(S + 1,000) NS2 = 1.03S + 1,000 NS3 = 1.05S
10,000 11,330 11,300 10,500
20,000 21,630 21,600 21,000
30,000 31,930 31,900 31,500
40,000 42,230 42,200 42,000
50,000 52,530 52,500 52,500
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d. Simplify the first equation and list the three equations below. What do you notice about their slopes and y-intercepts?
NS1 = 1.03S + 1,030
NS2 = 1.03S + 1,000
NS3 = 1.05S
The slopes are the same for Options 1 and 2. The first equation’s value of b is exactly $30 more than the value of the second equation. The third equation has a different slope and y-intercept than either of the first two.
e. Of Options 1 and 2, which will always be better and why?Simplified, we can see that the rate of change is the same, but Option 1 will always be $30 more than Option 2.
f. Since the three lines are very close to each other, a specific graph is not helpful in this case. Instead, use only the y-intercepts and slopes to make a rough sketch of all three lines on the same graph. Use the fact that lines with the same slope are parallel.
Old salary
New
sal
ary
Option 1Option 2Option 3
New Salary vs. Old Salary
g. Solve an equation to determine for which original salary amount the first two options produce equal new salaries.
1.03(S + 1,000) = 1.03S + 1,000; no solution
There are no salaries for which the first two plans produce equal new salaries.
h. Solve an equation to determine for which original salary amount the second and third options produce equal new salaries.
1.03S + 1,000 = 1.05S; S = $50,000
For an original salary of $50,000, the second and third plans both produce the same new salary.
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5. In the Explore, you looked at how the number of study hours is related to a student’s exam score.Imagine the professor’s data had instead been the following:
Number of hours Studied Test Score Out of 100
0 62
1 66
2 48
2 70
3 74
4 78
6 86
6 78
9 98
9 74
a. Does this data represent a function? Did the original data in the Explore represent a function? Explain.The original data represented a function, but this data does not since there are values of the independent variable that are matched with more than one value of the dependent variable.
b. What could the instructor do to get an equation to model the data? Draw a scatterplot and find a trendline.
6. a. Do all lines have a y-intercept? Why or why not?No, vertical lines do not have a y-intercept (unless the vertical line is the y-axis).
b. What kinds of lines are not functions?Vertical
7. Decide whether or not each equation is linear. If necessary, complete a table to help you decide. If the equation is linear, identify the slope and the y-intercept.
a. y = x - 4 linear slope = 1, y@intercept = (0, -4)
b. y = 4 - x linear slope = -1, y@intercept = (0, 4)
c. y =1x - 4 not linear
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d. y =1
(x - 4) not linear
e. y =14
x linear slope =14
, y@intercept = (0, 0)
f. y =(x + 1)
4 linear slope =
14
, y@intercept = a0, 14b
