1 | Page Information and Sample Lesson for MLCS Kathleen Almy, Heather Foes Rock Valley College Emails: [email protected][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 21 st 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.
Transcript
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Information and Sample Lesson for MLCS
Kathleen Almy, Heather Foes Rock Valley College Emails: [email protected]
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.
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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MLCS Content by Unit
Each unit begins with an open ended problem and a thematic question that will appear in every unit. For example, every lesson in unit 3 ties back to the question, “when is it worth it?”
Every unit contains skills, concepts, and applications from all four strands: numeracy, proportional reasoning, algebraic reasoning, and functions. Student and mathematical success components as well as geometry and statistics appear in every unit. Additionally, every unit has at least two articles that students read and use in an activity.
The following lesson comprises just a sample and does not encompass all facets of the text. The text has a wide variety of lessons in terms of length, complexity, focus (skills, concepts, or applications), and content area (numeracy, proportional reasoning, algebraic reasoning, functions, geometry, statistics). Additionally, there are open-ended problems that can be used as projects.
Lessons are still being edited and professionally produced by Pearson Education so this will not be the final product. Additionally, specific lessons can be chosen to suit a school’s needs to create a custom workbook for a course.
An instructor’s appendix is also available for the text to give further information, materials needs, timing, objectives addressed, and content area(s) addressed.
Unit 1 Venn diagrams Cartesian coordinate system Ratios & rates in common contexts Fraction review Pie/bar graphs, scale a fraction to percent Scatterplots, independent vs dependent variables Convert units by multiplying/dividing Increase a number by a percent, generalize calculations, definition of function Variables vs. constants, equations vs. expressions Read and interpret points on graph Scale fractions, proportionality Inductive vs. deductive reasoning, counterexamples, patterns Percent of increase/decrease Linear vs. exponential growth Shapes of graphs including constant, normal, linear, exponential (scatterplots) Sierpinski triangle (Area, perimeter, similarity) Volume concepts
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Unit 2 Integer concepts, notation, operations Means (conceptually and computationally) Whole number exponent properties Like terms, polynomial terminology and addition/subtraction Order of operations and formulas Distributive, commutative, associative properties, multiplying polynomials Writing and simplifying expressions Pythagorean Theorem Slope Distance formula Operations & relationships Functions with order of operations Role of rounding in error
Unit 3 Weighted means Correlation, median, mode Standard deviation Numerically and algebraically solving 1 step equations Volume/surface area of rectangular prisms & cylinders, Pareto charts Solving linear equations physically (manipulatives or pictorially) and in written form Linear equation applications (model, then solve) Modeling linear situations with tables, graphs, equations Slope and y-intercept from tables, graphs, and equations Variation Rational function modeling Solving non-linear equations Compounded error (volume of a cylinder)
Unit 4 Dimensional analysis, scientific notation Solving proportions algebraically (without scaling) Probability concepts, area of circle Writing equations of lines using slope and a point or two points Exponent properties with negative exponents Literal equations, GCF, concept of factoring Systems of 2 equations, solving by graphing and substitution Quadratic function modeling Modeling an exponential function Residuals, z-scores, compound inequalities pH, order of magnitude
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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.
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Implementation Ideas:
Lessons from the pilot
• Students want to mimic. • Teach students how to study. • If students will work, they can succeed. • Mastery learning in online systems ≠ learning. • Context improves connections and understanding. • We cannot help them all, but we can accelerate the process for many.
Areas to focus on when developing your course
• Advising • Pace • Materials • Continual assessment • Training sessions and materials • Adjunct support
These areas are identical to areas to be addressed when redesigning any developmental math course.
Getting Started
1. Choose materials that support this course and all faculty who will teach it.
2. Create a faculty collaboratory for pilot.
• Sit in on other MLCS classes is possible • Meet with other MLCS instructors regularly • Test with common instruments
3. Train faculty new to this course so they are successful.
4. Assess and improve the course so that it works for your students.
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2.2 It’s All Relative
Explore
Facts about atoms and ions:
a. Atoms contain protons, neutrons, and electrons.
b. Protons have a positive charge, electrons have a negative charge, and neutrons have no charge.
c. An atom, by definition, is neutral and has the same number of protons and electrons.
d. An atom can gain or lose electrons and become a charged particle called an ion.
Positive Ion: has more protons than electrons
Negative Ion: has more electrons than protons
Consider the examples in the following chart to answer the questions that follow.
1. Use the information given above to determine the sign of the numbers in the proton and electron columns. Indicate this in some way on the chart.
Protons are positive, electrons are negative.
2. Considering the signs of the numbers in the proton and electron columns, where do the numbers in the charge column come from?
Charge = number of protons + number of electrons
3. What do you notice about the number of protons for the sodium atom and sodium ion? Does this appear to be true for chlorine and aluminum?
The atom and ion versions of an element have the same number of protons.
4. The abbreviation for an ion includes either a positive or negative exponent. The abbreviation for an atom does not have an exponent. Use the examples in the table to determine the significance of the exponent on the abbreviation for an ion.
The exponent is the charge. If the charge is +1 or -1, only a plus or minus sign is listed. The 1 is implied. If it is an atom and the charge is 0, no exponent is written.
Instructor note: As a class, go over the ground rules for atoms and ions and do #1 with them.
Instruct students to work in groups to answer #2 – 4. Chemistry knowledge is not necessary. We are working with a set of ground rules in a new situation.
After 10 minutes, go over answers. Instruct students to complete #5.
10 - 15 min
Number of Protons
Number of Electrons
Charge
Sodium atom 11 11 0 Sodium ion 11 10 +1 Chlorine atom 17 17 0 Chloride ion 17 18 -1 Aluminum atom Al 13 13 0 Aluminum ion Al +3 13 10 +3
Instructor note: Have students put a ‘+’ above the protons column and a ‘-‘ above the electrons column.
This lesson gives a glimpse at instructor materials and the resources included. The first two problems in the homework are a check for students to complete after they have completed the accompanying MyMathLab assignment.
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5. Use this information and the following fact to complete the chart.
An atom can only become a charged ion by gaining or losing electrons, not protons. An atom and an ion of the same element will always have the same number of protons.
Discover
Negative numbers may seem abstract since we can’t see -3 apples. However, there are naturally occurring situations in which we may have less than zero, such as debt, and negative numbers satisfy the need to describe such situations. All numbers were created to deal with issues that occur in life, whether we are dealing with debt or counting a set of objects. 6. Describe the kind of numbers you would use to address each situation.
a. How many cars are in the parking lot?
Whole numbers: 0, 1, 2, 3, … The counting numbers (1, 2, 3, …) would not be sufficient since there could be zero cars in the parking lot.
b. What is the depth of a submarine (relative to sea level) if it is descending at a rate of 200/feet per hour?
Negative numbers
c. I purchased a candy bar and plan to share it with my children and husband. How much will each person get? Fractions
d. Jake uses a spreadsheet to keep track of his finances. He lists all purchases and bills for the month as well as any
income. How much money does he have left over after paying his bills? Decimals
In mathematics, we discover sets of numbers that will help solve a problem or satisfy a need. Next, we develop notation to describe them. Then we learn how they behave and what rules they must follow.
10-15 min
# Protons #Electrons Charge Hydrogen atom 1 1 0 Hydrogen ion 1 0 +1 Oxygen atom 8 8 0 Oxygen ion 8 10 -2 Calcium atom 20 20 0 Calcium ion 20 18 +2
Instructor note: Go over answers. Discuss the Discover introduction. Complete #6 as a class.
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7. a. What is the opposite of -8? 8
b. Find 11− . 11
c. What is the sign of 24? The size? Sign is positive, size or absolute value is 24
d. Read and simplify: - (- 6) The opposite of -6; 6
e. Read and simplify: −|−6| The opposite of the absolute value of -6; -6
A negative number is a number less than zero. We will indicate a negative number by writing a negative sign before it. Every number, other than zero, has a sign. Think of each number as made of two parts: its sign and its size.
- 5
Sign Size
“Size of a number” is the informal way of saying how far the number is from zero. Two numbers with the same size but opposite sign are called opposites. For example, 17 and -17 are opposites. Notice 17 and -17 are both 17 units from zero, but on opposite sides of zero. -17 can be read as negative 17 or the opposite of 17. So the symbol for opposite is the negative sign.
The set of whole numbers, together with their opposites, form the integers. When we draw a number line, we often label it with integers.
In mathematics, we refer to the size of a number as its absolute value. To indicate absolute value, we write vertical bars around the number.
For example, the absolute value of -5 is 5 since -5 is five units from zero. Absolute value of -5 is written as |−5| = 5.
Instructor note: Discuss the Look It Up. Instruct students to work #7 to practice these ideas. Go over answers.
8. When a situation necessitates a negative, there are many ways to express it. On a budget sheet, you may see negative quantities in red or parentheses. Mathematically, we write a negative number by placing a negative sign in front of it, like - 4. This is similar to a subtraction symbol. How do we indicate a number is positive?
It will have no sign in front of it. It could have a + sign but that is uncommon outside of math books.
9. Is zero positive or negative?
Neither; it has no sign.
10. The following examples do not use negative or positive numbers but they each have a positive or negative meaning. Determine if the meaning is positive or negative. If it is negative, write it using a negative number.
a. 100 feet below sea level
Negative, -100
b. $500 in the black
Positive
11. Below are negative numbers from real-life situations. For each one, interpret what it means. Notice sometimes negatives are used for loss and sometimes they are not.
a. -30 ◦F
30 degrees below zero
b. Poll has a +/- 3% margin of error
For poll results, a range of likely values is found by taking the percent in the poll and adding and subtracting the margin of error.
Connect
The periodic table of elements lists a great deal of helpful information about each element. Use the graphic below and the rules that follow to answer the questions.
Atomic number 9
F
Fluorine
18.9984032
Atomic mass So Fluorine has a mass of 19. It has 9 protons and 10 neutrons. Since fluorine is a neutral atom, it also has 9 electrons.
Rules:
a. The atomic number is the number of protons.
b. Round the atomic mass to the nearest whole number to get the mass of one atom of the element.
c. The mass rounded to a whole number is equal to the number of protons plus the number of neutrons.
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12. For each element, find the number of protons, neutrons, and electrons in one atom. a. Mass = 39, P= 19, N = 20, E = 19
19
K
Potassium
39.0983
b. Mass = 207, P= 82, N = 125, E = 82
82
Pb
Lead
207.2
Reflect
Lesson wrap-up:
What’s the point?
Negative numbers are common. Often they describe a quantity less than zero, such as debt. But, they also describe being below a reference point (like sea level) just as positive numbers describe being above it. They provide a way of describing relative positions.
What did you learn? How to interpret signed number situations
How to find the opposite and absolute value of a number
Cycle 2 Question: Why does it matter?
Why does the sign of a number matter? Answers will vary.
5 min
Instructor Note: Explain the contents of the wrap-up box. Have students write their answer to the cycle question, “why does it matter?” A prompt is provided to help guide their answer. Discuss homework.
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2.2 Homework
Skills • Interpret signed number situations. • Find the opposite and absolute value of a number.
1. Simplify: - |-15|
-15
2. During a quiz show, a contestant loses 2500 points. Indicate their loss with a number.
- 2500
Concepts • Interpret signed number situations.
3. The following situations do not use negative or positive numbers but they each have a positive or negative
meaning. Determine if the meaning is positive or negative. If it is negative, write it using a negative number. a. $10,000 in debt
Negative, -$10,000
b. 400 B.C.
Negative, -400
4. Below are negative numbers from real-life situations. For each one, interpret what it means.
a. -20 rushing yards in a football game
Loss of 20 yards
b. Stock market performance for the day: -120.62 points
Loss of 120.62 points for the day
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5. Graph the following numbers on the number line provided. Make a dot at the appropriate location and write the number above it.
a. 4.5 b. -2 c. 2/3
d. -6.7 e. -16/5
d e b c a
6. As we go right on a number line, the numbers increase. If two numbers are graphed on a number line, the larger one will be on the right. For each pair of numbers, graph them and then circle the larger one. In the circle, between the numbers, write > or <.
a. 6 2/3 ⃝ 6.7
<
b. -3 ⃝ -4
>
c. -5 ⃝ -5.2
>
d. -5 ⃝ -4.8
<
7. One student says -5 is bigger than -4 and uses money as the analogy: “If I owe $5, I have a bigger debt than owing $4.” What is wrong with their argument?
The bigger the debt, the more the person is “in the hole.” The less debt, the more money a person has. Assuming all other aspects of their financial situations are the same, the person with the $4 debt has a smaller debt and therefore more money than the person with the $5 debt.
8. Looking forward, looking back
a. What is the opposite of 7? -7
b. What is the opposite of your result from part a? 7
c. So the opposite of the opposite of 7 is _7____. That is, - (- 7) = _7____.
d. Generalize this. For any number a, - ( - a) = __a____.
Art note: Number lines for #6 should go from -7 to 7. AIE should show both points for each part.
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4.15 A Little Less
Connect
Water bottle manufacturers have used various techniques over the past few years to reduce the amount of plastic used in each bottle in an effort to reduce the overall amount of plastic that will eventually become refuse. Some companies are using smaller water bottle caps in terms of the height of the cap. A company claims that they have reduced the amount of plastic used with this change by 30%. Directions: Answer the questions below to analyze the change in the amount of plastic produced by using smaller
water bottle caps. 1. First, let’s find the volume of an existing plastic cap that has the new smaller height. Since this is difficult to do
directly, we can use the following formula: weightVdensity
=
If the weight of the cap is 0.95 g and the density of the plastic used in the cap is 0.925 g/cm3, find the volume of the cap.
2. The company claims this cap has 30% less plastic than their previous cap. Find the volume of the original, larger
cap. 3. How much plastic is saved with the new, smaller caps? 4. If there approximately 50 billion water bottles produced each year, find the amount of plastic that would be
saved each year with the new shorter caps. 5. The amount of plastic saved is such a large number that it is difficult to comprehend. Let’s convert it to
quantities that are more tangible. Convert the volume of plastic to cubic yards since many large quantities, such as concrete and mulch, are sold in this unit.
This lesson has no new content developed but allows students to apply and connect several ideas from the unit.
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6. Since that result is still fairly large, let’s go one more step to make sense of it. An Olympic swimming pool holds 88,000 ft3. Find the number of Olympic swimming pools that could be filled with the plastic saved.
7. Is this measure worth it?
Reflect
Lesson wrap-up:
What’s the point?
A seemingly small quantity can add up to a lot when used repeatedly. What did you learn? How to use dimensional analysis to analyze a problem
Cycle 4 Question: How big is big? How did such a small amount of plastic saved per bottle result in such a large overall savings?
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4.15 Homework
Concepts • Use dimensional analysis to analyze a problem.
We typically think of bottled water as cheap and gasoline as expensive. Is that true? Below are a few commonly used liquids and their prices. a. Find the cost of filling a 15 gallon gas tank with each liquid. Use dimensional analysis and the conversion:
1 Liter = 0.26 gallons. Bottled water: $2 for one liter
Gasoline: $3.85 for one gallon Correction fluid: $1.40 for 22 mL Printer ink: $30 for 42 mL Milk: $3 for one gallon
b. Order the cost of filling the tank from least to greatest.
c. What is your reaction to these findings?
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4.8 Chain, Chain, Chain
Explore
Alkanes are a type of molecule with only hydrogen and carbons atoms, joined with single bonds. Single bonds are shown by a straight line connecting a C with an H. The first few straight-chain alkanes are drawn in the table below, starting with methane, the smallest alkane. 1. Write the formula for the first four alkanes by counting the number of carbon atoms and the number of
hydrogen atoms in the structural formula. The first two have been done as an example. Draw pictures for the rest of the alkanes in the table and write their formulas.
Name Structural Formula Formula methane
ethane
propane
butane
pentane
hexane
heptane
octane
nth alkane CnH______
This lesson allows students to learn algebraically how to write a linear model. So far in the course to this point they will have used inductive reasoning and intuitive approaches to writing linear models.
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2. Since the number of hydrogen atoms increases by _ every time the number of carbon atoms increases by 1, there is a __________ relationship between the number of hydrogen atoms and the number of carbon atoms.
3. To represent the relationship between the number of carbon atoms and the number of hydrogen atoms as a
linear equation, write the information in the table in the form of ordered pairs.
4. Find the slope of the line representing this data and interpret it in words.
Discover Since this data is linear, we can write a linear model that gives the number of hydrogen atoms for a given number of carbon atoms. To date, we have used inductive reasoning to find a relationship between two variables. While this does not require a lot of math, it can be time consuming and frustrating. Let’s develop a procedure to write a linear model from data. 5. a. What is the form of an equation for a linear model?
b. Rewrite the formula for a linear model in terms of H and C. Use the table to determine which variable
will be independent and which will be dependent.
H = the number of H atoms and C = the number of C atoms c. Substitute the value of the slope for m.
d. To complete the model, we need to find a value for b. One way to do this is to substitute an ordered pair into the equation and solve for b. We can do this using the first ordered pair (1, 4). After substituting in C = 1 and H = 4, we have:
e. Solve this equation to find the value of b: f. Write the completed linear model:
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6. Explain the meaning of the y-intercept in the model, if possible. 7. On the following grid, create a graph that shows the ordered pairs and the linear model.
8. Use the model for straight-chain alkanes to answer the following questions. a. If an alkane has 10 carbon atoms, how many hydrogen atoms does it have?
b. If an alkane has 14 hydrogen atoms, how many carbon atoms does it have? c. Is it possible to have a straight-chain alkane with an odd number of hydrogen atoms?
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How it works
To write the equation of a line: 1. Find the slope of the line. If it is not provided, use the table, graph, or points provided to find the slope of
the line connecting them.
2. Write the slope-intercept form of a line: y = mx + b. You may need to change the variables x and y to variables that are meaningful for the problem.
3. Substitute the slope for m and an ordered pair on the line for x and y. Solve the resulting equation for b. NOTE: If you can see the y-intercept from the table or graph, you can skip this step. 4. Write the slope-intercept form with the values of m and b included.
For example, assume a line passes through (-2, 6) and (3, 11). Since the slope is not given, we need to find it.
Since we do not have the y-intercept, we will pick a point on the line and substitute it into the equation to find b. (3, 11) is easier to work with than (-2, 6) since is has no negative numbers in it. Plug in m = 1 and the point (x, y) = (3, 11) into y = mx + b and solve for b:
Writing the linear equation, we get: y = 1x + 8 or y = x +8.
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9. Find the linear equation for each situation.
a. m = 4, line passes through (0, -8)
b. m = 4, line passes through (1, -8)
c. Line passes through (2, 6) and (4, 10)
d. Horizontal line passing through (-6, 7).
d. Vertical line passing through (1, -2).
Draw a graph with a vertical line passing through (1, -2).
Complete this table with points from the line:
x y
Use inductive reasoning and the table to write an equation.
Notice, horizontal lines have the form y = b where b is a number.
Vertical lines have the form x = a where a is a number. Equations for vertical lines cannot be written in y = mx + b form.
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Connect
10. Carbon chains also exist in a cyclic form with double bonds as shown in the table. Write the formulas indicating the number of carbon and hydrogen atoms in each structure. Then complete the last column in the table by writing an ordered pair to indicate the number of carbon atoms and the number of hydrogen atoms in each structure.
Name Structural Formula Formula (C, H) cyclopropene
C3H4 (3, 4)
cyclobutene
cyclopentene
cyclohexene
nth alkene CnH_______ (n, ) 11. Find a linear model that gives the number of hydrogen atoms for a given number of carbon atoms.
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Reflect
4.8 Homework
Skills • Write the equation of a line using a point and slope or two points. • Model linear situations in context.
Skill Check
1. Write the equation of the line passing through (6, -3) and (-5, 2).
2. A student pays $1175 to take 4 credit hours at a local college. Another student pays $2175 to take 8 credit
hours at the same college. Write a linear model for the cost of attending the college based on the number of credit hours a student takes.
Lesson wrap-up: What’s the point?
Sometimes a linear relationship is apparent from the description of the situation. When it is not, we can use the linear equation form y = mx + b and information about the situation to write the equation. What did you learn? How to write the equation of a line using a point and slope or two points How to model linear situations in context
Cycle 4 Question: How big is big? At what point should you use y = mx + b to find a linear model? When does a problem become too big or messy for intuitive approaches?
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Concepts Use the model for cyclic alkenes from the lesson to answer the following questions. 3. a. If a cyclic alkene has 12 carbon atoms, how many hydrogen atoms does it have? b. If a cyclic alkene has 12 hydrogen atoms, how many carbon atoms does it have? c. Is it possible to have a cyclic alkene with an odd number of hydrogen atoms?
d. How many carbons are necessary for this model to make sense? 4. On the following grid, create a graph that shows the ordered pairs and the linear model. Use your answer from
