Introduction to Baseline Proficiencies

These proficiencies address topics in the syllabus for Math 130 - 131. You must pass all the proficiencies by the end of Math 131. When you have passed a proficiency, you have passed it forever (i.e. if you make a mistake later in the semester on a topic for which you have passed the proficiency, you do not lose your “passed” status on that proficiency).

Although you do not officially have to satisfy the proficiencies until the end of the spring semester, it is critically important that you take them seriously from the beginning of MAT 130. This list of proficiencies includes information about when in the semester you should get extra help if you haven’t yet passed each proficiency. It is YOUR responsibility to seek out extra help and to do what you need to do so that you can pass them by the end of the course. You can ask your instructor, your TA, or the Office of Academic Advising for information about getting help with a baseline proficiency.

Ways to Satisfy Baseline Proficiencies

Each description of a baseline proficiency also includes a list of possible ways to satisfy that proficiency. These lists might be modified a little as the year progresses. In fact, if you can think of a way to satisfy a proficiency that is not on the list, please feel free to suggest it to your instructor. If it’s a good suggestion, he or she will bring it to the other Math 130/131 instructors for approval. Here are descriptions the most common ways of passing proficiencies.

Test/Quiz Questions: You will always have more than one opportunity to use test or quiz questions to satisfy a baseline proficiency. Generally, there will be several opportunities incorporated in your regular, in-class tests and quizzes. The Math 130/131 instructors will also arrange extra sessions for students wishing to try additional test questions.

Test/Quiz Revision: If you make significant progress toward satisfying a baseline proficiency on a test/quiz, but aren’t quite there yet, your instructor may give you the option of revising your answers. You will redo the questions, and then explain both your mistakes and your new solutions. These explanations may be oral or written (your instructor may only accept one of these for a particular problem). Your instructor may also ask you to solve problems similar to the ones on the test. You will have a brief interview with your instructor so that s/he can ask you a few questions to be sure that you really do understand your revisions.

Instructor Interview: If you believe you understand the baseline proficiency and feel you can explain your understanding better orally than in writing, you may arrange for an instructor interview. Sometimes such interviews can take place during the normal part of class, if your instructor deems it appropriate.

Detailed Description of Baseline Proficiencies

Topic I: Introduction to Problem Solving, Reasoning, and Communication

1. Problem Solving:

To satisfy this proficiency, you need to solve a non-trivial problem that is new to you, which involves making a table, finding a pattern, and doing at least one of the following things:

a. Extending the pattern to a large number, given by your instructor

b. Finding an equation for the pattern

c. Giving a “convince a skeptic” argument that the pattern continues

This is only one type of problem that we will be working on in class, but it is a very important type of problem, one that is especially important not only for developing problem solving skills, but also for developing algebraic thinking skills. Your instructor will help you get a sense of what constitutes a “non-trivial problem.” For some specific problems, you might not be able to use part c to satisfy the proficiency.

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview. There will be many test/quiz opportunities throughout both semesters to pass this proficiency.

You may also satisfy this proficiency with an out-of-class project, accompanied by an instructor interview, at which you will be asked to explain your solution and to extend it to a similar problem. This project must be an individual, not a group, project.

When to get extra help: If you haven’t satisfied this proficiency by Thanksgiving, you should start doing some extra work to try to finish it before the end of the semester. If you are still having trouble, Topic VII could help a lot, however, you should meet with your instructor as soon as you start study algebraic thinking, to make a plan to help you satisfy it by the end of the course.

2. Mathematical Writing

Ordinarily, you will use a regular assignment to satisfy this proficiency as a normal part of the course. To satisfy it, you must write a description of your solution to a non-trivial problem that meets the criteria outlined below. On your writing assignments, your instructor will indicate which criteria you need to work on in future assignments. If your assignment is very close to satisfying the baseline proficiency, your instructor may allow you to revise it so that it will meet all the criteria.

Please note that you should always do your writing for the course by yourself. You may use write-ups of problems that you worked on with others to satisfy this baseline proficiency, however, your writing must be your own. If your instructor suspects that the writing is not your own, he or she will take you to the Scholastic Review Board.

Your writing must meet all of the following criteria:

a. Someone who hasn’t seen the problem before can follow it. You need to include a clear description of the problem and of any material you incorporate from class. You need to explain concepts or vocabulary that are beyond what a typical student, who is at the level of a Math 130/131 student but is not in the class, would know.

b. Your paper includes written representations of the problem, such as pictures, graphs, and tables, that are appropriate to the problem, clearly labeled and presented, and that help advance your argument.

c. The writing flows. It moves from one idea to another in a logical fashion. Your ideas are clearly explained.

d. The writing is convincing. You don’t necessarily have to have solved the problem completely to satisfy this proficiency, but if there is a piece missing or a part you don't understand, you must tell the reader, rather than hoping s/he won’t notice. The reader can tell that you know what you are talking about.

e. With only a small number of minor exceptions, the paper is written using complete sentences and correct mathematical terminology, grammar, punctuation, and spelling.

Ways to satisfy the proficiency: Out-of-class writing as described above. You can also do an extra assignment if necessary.

When to get extra help: You will be working on writing in many of your other classes, and if you improve your writing in general, it will improve your mathematical writing. If you haven’t satisfied this proficiency by the start of the second semester, you should get help from the writing center or a learning specialist.

Topic II: Number Theory

3. Number Theory

To satisfy this proficiency, you must show that you understand main concepts of number theory: multiples, factors, prime number factorization, and divisibility rules. If you use test or quiz questions to satisfy this proficiency, you will have to correctly solve at least four of five designated problems (i.e. over 80% of five or more problems). Sample problems:

Are the following numbers prime or composite? 251, 252, 253, 255, 257.

Factor 7168 into primes.

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

When to get extra help: If you haven’t satisfied this proficiency while taking Topic 2, you should get extra help to prepare for a test covering Topic 2. If you don’t pass the proficiency while taking the test, then you should continue to work this proficiency outside of class and pass it as soon as possible.

Topic III: Rational Numbers I

4. Modeling Fractions

To satisfy this proficiency, you must demonstrate understanding of at least two models of fractions from different categories. Categories are:

1. Area Models: including fraction factory pieces, pies and pizzas, pattern blocks

2. Length Models: including fraction strips, rulers, cuisenaire rods

3. Group Models: including tiles, drawings of dots, etc.

In order to demonstrate understanding, you must solve a non-trivial problem, with a “convince a skeptic” explanation that includes either a demonstration with manipulatives or a drawing.

Examples of non-trivial problems (all require you to draw a picture or use manipulatives to convince a skeptic that your answer is correct):

What is 2/3 of 3/4?

If the purple Cuisinaire Rod represents 2/5, what rod represents 1/4?

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

When to get extra help: If you haven’t satisfied this proficiency during the first few weeks of Topic 3, you should get help as soon as possible. If you don’t understand models of fractions, it will be hard for you to do well in the rest of Topic 3.

5. Comparing Fractions

To satisfy this proficiency, you must show that you understand the size of fractions. If you use test or quiz questions to satisfy this proficiency, you will have to correctly solve at least four of five designated problems (i.e. over 80% of five or more problems). The designated problems will be a wide assortment of types of problems, such as the following:

Placing fractions on a number line.

Ordering lists of fractions.

Finding fractions between two other fractions.

Identifying fractions that satisfy a specific criterion, e.g. less than 1/2.

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

When to get extra help: If you haven’t satisfied this proficiency near the end of Topic 3, you should get extra help to prepare for a test covering Topic 3. If you don’t pass the proficiency while taking the Topic 3 test, then you should continue to work this proficiency outside of class and pass it as soon as possible.

Topic IV: Rational Numbers II

6. Operations with Fractions

Operations with fractions: addition and subtraction

To satisfy this proficiency, you must solve one addition and one subtraction problem involving fractions and give a “convince a skeptic” argument to explain why your solution is correct. If you use algorithms, you must justify why they work. You may also use manipulatives or drawings. Problems use common fractions; they are not overly complicated, but addition and subtraction problems include fractions with different denominators. Sample problems: 3/4 + 2/3, 5/6 + 13/9, 9/4 7/8.

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

When to get extra help: If you haven’t satisfied this proficiency near the end of Topic 4, you should get extra help to prepare for a test covering Topic 4. If you don’t pass the proficiency while taking the Topic 4 test, then you should continue to work this proficiency outside of class and pass it as soon as possible.

Operations with fractions: multiplication

To satisfy this proficiency, you must solve one multiplication problem involving fractions and give a “convince a skeptic” argument to explain why your solution is correct. If you use algorithms, you must justify why they work. You may also use manipulatives or drawings. Problems use common fractions; they are not overly complicated. Sample problems: 2/3 x ; 3/2 x 1/3.

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

Operations with fractions: division

To satisfy this proficiency, you must solve one division problem involving fractions and give a “convince a skeptic” argument to explain why your solution is correct. If you use algorithms, you must justify why they work. You may also use manipulatives or drawings. Problems use common fractions; they are not overly complicated. Sample problems: 2/3 ; 3/2 1/4.

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

7. Comparing Rational Numbers

To satisfy this proficiency, you must show that you understand the size of rational numbers. If you use test or quiz questions to satisfy this proficiency, you will have to correctly solve at least four of five designated problems (i.e. over 80% of five or more problems). The designated problems will be a wide assortment of types of problems, such as the following:

Placing rational numbers on a number line.

Ordering lists of rational numbers and fractions.

Finding rational numbers between two other decimals.

Identifying rational numbrs that satisfy a specific criterion, e.g. less than 0.5 or more than .

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

When to get extra help: If you haven’t satisfied this proficiency near the end of Topic 4, you should get extra help to prepare for a test covering Topic 4. If you don’t pass the proficiency while taking the Topic 4 test, then you should continue to work this proficiency outside of class and pass it as soon as possible.

8. Estimation with Rational Numbers

To satisfy this proficiency, you must demonstrate abilities in estimating the solution of addition, subtraction, multiplication, and division problems, using both fractions and decimals. If you use test or quiz questions to satisfy this proficiency, you will have to correctly solve at least four of five designated problems (i.e. over 80% of five or more problems), without using a calculator. Sample problems:

Which of the following is closest to 411/101 123/61 (circle the answer):

a) b) 1 c) 1 d) 2 e) 2 f) 3

Which of the following is closest to 21.11 x 0.103 (circle the answer):

a) 0.22 b) 2.2 c) 22.0 d) 220 e) 2200

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

When to get extra help: If you haven’t satisfied this proficiency near the end of Topic 4, you should get extra help to prepare for a test covering Topic 4. If you don’t pass the proficiency while taking the Topic 4 test, then you should continue to work this proficiency outside of class and pass it as soon as possible.

9. Using Zero

To satisfy this proficiency, you must demonstrate understanding of how to correctly use zero in multiplication and division problems. You must include “convince a skeptic” explanations with your answers; it is not enough to simply memorize. If you use test or quiz questions to satisfy this proficiency, you will have to correctly solve designated problems of the type a 0, 0 b, and c x 0, where a, b and c are not zeros, without using a calculator.

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

When to get extra help: If you haven’t satisfied all parts of this proficiency near the end of Topic 4, you should get extra help to prepare for a test covering Topic 4. If you don’t pass the proficiency while taking the Topic 4 test, then you should continue to work this proficiency outside of class and pass it as soon as possible.

Topic V: Percents.

10. Percent Problems

To satisfy this proficiency, you must solve at least four of five designated percent problems and give a “convince a skeptic” argument to explain why your solution is correct. If you use algorithms, you must justify why they work. You may also use manipulatives or drawings. For example, you could be asked to fill the blanks of the following problems:

20 is ______% of 80 _____ is 10% of 12 9 is 45% of _____

and explain why your answers are correct.

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

When to get extra help: If you haven’t satisfied this proficiency near the end of Topic 5, you should get extra help to prepare for a test covering Topic 5. If you don’t pass the proficiency while taking the Topic 5 test, then you should continue to work this proficiency outside of class and pass it as soon as possible.

Topic VI: Probability

11. Basic Probability

To satisfy this proficiency, you must solve at least two problems, one that refers to finding a simple sample space (e.g. die or spinner) and one that is a little more complicated (two dice, two spins on a spinner). Sample questions:

What is the probability of getting an odd number on a number cube with numbers 1 through 8?

What is the probability of getting a five two times in row when rolling a dice numbered 1 through 6?

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview. Possibly might be included in a project near the end of the course.

When to get extra help: If you haven’t satisfied this proficiency near the end of Topic 6, you should get extra help to prepare for a test covering Topic 6. If you don’t pass the proficiency while taking the Topic 6 test, then you should continue to work this proficiency outside of class and pass it as soon as possible.

Topic VII: Algebraic Thinking

12. Representing Relationships between Two Functions as Tables, Graphs, and Formulae, and Connecting Representations

To satisfy this proficiency, you must demonstrate abilities to represent relationships between two quantities (one depends on the other) as graphs, tables, and formulae. You have to demonstrate abilities to connect different representations, for example, matching tables to graphs. You must solve at least two of three designated problems and give a “convince a skeptic” argument to explain why your solution is correct. Sample problems:

1. Represent the relationships between the cost of gas and the miles driven as a table, as a formula, and as a graph, if it is known that the car travels 20 miles per gallon and gasoline costs $1.50 per gallon.

2. Match each of the graphs with the correct equation from the list below. Graphs have different scales, and there is one extra equation. Write the letters corresponding to the equations in the appropriate spaces on your answer sheet.

A) B) C)

D) E) F)

3. Sketch a graph that describes the following situation using time as an independent variable (the horizontal axis) and speed as a dependent variable (the vertical axis):

During your commute home from work you start from your parking space and speed up to 30 miles per hour. You pass by a school and slow down to 20 miles per hour temporary and then accelerate back to 30 miles per hour. You stop for a light and then enter a highway speeding up to 50 miles per hour. After several minutes you get off highway and drive at a speed 35 miles per hour until you get home.

Ways to satisfy the proficiency: Test/quiz questions, test/quiz revision, or instructor interview.

When to get extra help: If you haven’t satisfied this proficiency near the end of Topic 7, you should get extra help to prepare for a test covering Topic 7. It is your last topic.

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