--- tags: mth201 --- # Information on Learning Targets and Checkpoints, MTH 201-04 Winter 2021 The purpose of this document is to give details on how Checkpoints are used to attain mastery on Learning Targets, what Checkpoint problems cover, how they are graded, what resources are approved for use on Checkpoints and how to use them, and let you know when Checkpoints will take place and what will be covered on each. **Complete transparency** is the goal; if you have questions or if there's something that needs to be added to this document, please let me know. Sample Checkpoints can be found in the *Learning Targets* area, where this document is located; as we take Checkpoints, old Checkpoints and their keys will also be posted in this area so you can use them for study and practice. ## How Checkpoints work **Checkpoints** are do-at-home tests that are given roughly every 10-14 days, that provide you with opportunities to **demonstrate skill on the Learning Targets** in the course. One of your primary responsibilities in the course is to **demonstrate as much skill as possible on as many Learning Targets as possible** through doing "acceptable" work on Checkpoints. Your skill on Learning Targets is indicated in the gradebook by a **rating**. There are three ratings available: - **No Rating**. This, or a blank entry, will appear in the gradebook if you have *not yet done acceptable work* on a Learning Target through a Checkpoint. *Everyone begins the semester with "No Rating" on all 16 Learning Targets.* - **Proficiency.** Once you complete **one** Checkpoint problem on a Learning Target that meets the standards for acceptable work, you will "level up" to *Proficiency*. - **Mastery.** Once you complete **a second** Checkpoint problem on a Learning Target that meets the standards for acceptable work, you will "level up" to *Mastery*. **Every Checkpoint is cumulative**, with new Learning Targets represented but also new versions of older ones. This will allow you to reattempt, on a limited basis, any Learning Target you do not "level up" on. However **Learning Targets only appear on four consecutive Checkpoints**; for each Learning Target you will have four (4) chances to do two (2) problems at an acceptable level of work in order to earn a Mastery rating. (This doesn't count Checkpoints 8 and 9 at the end of the semester which will contain all 16 Learning Targets for two final rounds of assessment.) :::info **Example**: Alice starts the semester, like everyone else, with "No Rating" on all her Learning Targets. Checkpoint 1 contains problems for Learning Targets 1 and 2; on this Checkpoint, Alice does acceptable quality work on the problem for Learning Target 1 but unfortunately her work on Learning Target 2 does not meet the grading standard. So following Checkpoint 1, her rating is *Proficient* on Learning Target 1 and *No Rating* on all the others. When Checkpoint 2 happens, it contains new versions of the problems for Learning Targets 1 and 2 and a new problem for Learning Target 3. Alice works all 3 problems; she does acceptable work on all three. This levels her up to *Mastery* on Learning Target 1, and to *Proficiency* on Learning Targets 2 and 3; she still has *No Rating* on the rest since she has not attempted a problem on those yet. Alice can continue to attempt and re-attempt Learning Target problems throughout the semester to demonstrate evidence of skill. However, once she has reached *Mastery* level on a Learning Target, *she does not have to attempt this any more during the semester* since she's reached the highest possible rating; for example she does not need to attempt any more problems for Learning Target 1 following Checkpoint 2 since she reached *Mastery* level after that Checkpoint. ::: ## Items covered on Checkpoint problems A list of all 16 Learning Targets can be found in [the Appendix to the syllabus](https://hackmd.io/Bnpy2q58StW8mIt2mqaSMA#Appendix). Below is a list of tasks that you may be asked to perform on Checkpoint problems for each of those Learning Targets. Please note this is just a representative sample; you might be asked something different, but always on the same level of difficulty as what's listed here, and aimed at the same Learning Target. You can see specific examples in the Sample Checkpoints posted on Blackboard. **General note:** Any time you see the word "function" below, on a Checkpoint this function may appear as a formula, or as a table, or as a graph; or as a combination of these. Be prepared to do your work using *any* representation of a function. | Learning Target | Tasks | | ------ | --------------- | | 1 | Given a function and an interval, compute the average rate of change of the function on that interval. Or, given the position function for a moving object and a time interval, compute the average velocity of the object on that interval. | | 2 | Given a function, find the limit of the function as the input approaches a point (possibly from just one side) or at infinity using algebra, table estimation, or a graph. | | 3 | Given a formula for a function (usually a [second-degree polynomial](https://themathpage.com/aPreCalc/quadratic-equation.htm)), write out the correct limit expression that would compute the derivative of that function at a point or in general; then work through the limit to find the derivative. ("Derivative" might be phrased as "instantaneous velocity".) | | 4 | Given a real-world context, give a notational expression for the rate of change; state the units of the derivative correctly; give a correct and simple interpretation of the meaning of the rate of change; and estimate the rate of change using forward, backward, and central difference approximations. | | 5 | Given graphical or numerical information about a function or one (or both) of its first two derivatives, draw correct conclusions about the missing information. (For example, given a graph of $f$, state where $f'$ is positive and negative and where $f''$ is positive and negative. Several variations on this idea are possible.) | | 6 | Given 4-5 simple functions (constant, power, polynomial, exponential, and sine/cosine functions), use basic rules to compute their derivatives and answer simple application problems (slope of a tangent line, rate of change, second derivative). | | 7 | Given products, quotients, and composite functions, use the Product, Quotient, and Chain Rules to compute their derivatives and answer simple applications problems (see above). **Note:** On this Learning Target, you will be asked to *state the rule you are using* and in the case of the Chain Rule you'll be asked to *state the inside and outside functions involved in the composition*. | | 8 | Given "advanced" functions (meaning logarithmic, trigonometric, and inverse trigonometric functions along with simpler functions that are combined with these), find their derivatives and answer simple application questions. **Note:** On this Learning Target, you will be asked to *list the rules you use in the correct order in which they are used*, and in the case of the Chain Rule you'll be asked to *state the inside and outside functions involved in the composition*. | | 9 | Given a function (generally as a formula), use Calculus and a correctly-formatted sign chart (see below) to find its critical points and its intervals of increase and decrease. | | 10 | Given a function (generally as a formula), use Calculus and a correctly-formatted sign chart (see below) to find its inflection points and its intervals of concavity. | | 11 | Given a function (generally as a formula) that is continuous on a given closed interval, use the Extreme Value Theorem method to find its absolute minimium and maximum values. | | 12 | Given a simple (WeBWorK-level) applied optimization problem, set it up, find the point where the target quantity is optimized, and give a mathematical explanation for why the quantity is optimized there. | | 13 | Given a function as a geometrically-regular graph (made up of lines and circle parts) or as a formula, find some combination of the area between the graph and the horizontal axis, net change on an interval, or displacement using geometry or by setting up and evaluating a Riemann sum. | | 14 | Given a function as a geometrically-regular graph, or as a formula that has a geometrically regular graph (for example $f(x) = \sqrt{1-x^2}$), compute one or more definite integrals of that function using geometry. Or, given one or more known definite integrals, use Properties of the Definite Integral to compute related integrals. | | 15 | Given a real-world rate of change situation, interpret the total change as a definite integral; state the units of an integral; and answer questions related to the rate of change and total change. | | 16 | Given 3-5 definite integrals, compute their *exact* values using the Fundamental Theorem of Calculus (i.e. by finding antiderivatives and evaluating/subtracting) | ## Overall grading criteria for Checkpoints ***Each* Checkpoint problem requires at least the following from a solution in order for that solution to be considered "acceptable"**: - **There can be no instances of significant errors.** A "significant" error is one that is *directly related to the Learning Target itself* and *causes the solution to fail to provide conclusive evidence of mastery*. Examples include (but are not limited to) the following: - A *significant computational error* that shows more work needs to be done on mastering the computation (for example: getting the subtraction reversed in the Quotient Rule; or computing the derivative of $\cos(x)$ as $+\sin(x)$ instead of $-\sin(x)$) - A *significant conceptual error* that demonstrates the need to understand the concept further (for example: Getting the units wrong on a derivative; interpreting a positive first derivative as concave up; etc.) - An *unclear explanation* that demonstrates the need to understand the concept further - *Significant omissions* including not doing a part of a multi-part problem (even if by accident); or leaving out an essential part of a solution, for example the argument at the end of an optimization problem that the critical point optimizes the quantity - A *highly disorganized presentation of a solultion* --- That is, the solution is so messy and incoherent that it is not easy for the reader to determine if the student has mastered the concept - A *copy error that oversimplifies the problem* -- For example, copying down $f(x) = e^{x^2}$ on a derivative question as $f(x) = x^2$ or $f(x) = e^x$. - **There can be no more than a single instance of a "simple" error.** A "simple" error is an error that is *not directly related to the Learning Target itself* and *doesn't get in the way of seeing that the student has mastered the concept.* Examples of simple errors include: - *Errors in arithmetic or algebra that are not central to the Learning Target and do not oversimplify the problem*. For example, working through a derivative and everything is correct except a minus sign was dropped in the final answer. - *Copy errors that do not oversimplify the problem.* For example, copying down $f(x) = e^{x^2}$ on a derivative question as $f(x) = e^{2x}$ is a mistake, but the derivative that results is roughly the same level of difficulty as the correct function, so I will read with your solution to make sure the answer and process are correct. :::warning Generally speaking, **you are not being graded on your answers but on your explanations, processes, and reasoning**. While correct answers are expected and required (except for simple errors), the evidence of your mastery of the Learning Targets does not come from the answers; it comes from your work that leads to the answer. So **make every effort not only to provide right answers but also full, correct, clearly expressed solutions** that back up your answers. Remember *you are doing work here to convince the reader that you have mastered the ideas*. Like in a courtroom, provide ample evidence, clearly expressed, that you have done so. ::: Finally please note that: 1. *Two simple errors in the same problem, no matter what the type, results in unacceptable work.* It is acceptable to make a simple error once, but not twice. 2. *Errors that are "simple" in one context may be significant in another.* For example, dropping a minus sign might be a simple error on Learning Target 6 but considered significant for Learning Target 13 where the minus sign is an important part of the concept. 3. *To avoid all forms of error, use the approved tools listed below to double-check all your work before submitting it*. For example, you can use [Wolfram|Alpha](http://www.wolframalpha.com) to double check the answers for derivative calculations; you just need to supply a complete and clear solution. In this way, your work should never really contain errors unless they are significant conceptual misunderstandings. ## Specific grading criteria for Learning Targets In addition to the overall standards for acceptability given above, here are standards for each individual Learning Target. | Target | Acceptable work means... | | ----- | ---------- | | 1 | All answers must be correct unless a mistake is the result of a simple error; the setup for each calculation must be shown and correct. | | 2 | All answers must be correct unless a mistake is the result of a simple error; the setup for each calculation must be shown and correct. Algebraic approaches must use correct algebra. | | 3 | **All limit setups must be 100% notationally correct**, including: the limit must be present and on the correct side of the equals sign; the limit must have the correct notation underneath it; and the difference quotient must be correctly stated. *Any* omissions or errors on this element will result in unacceptable work. The computation of the limit must use correct algebra throughout and result in a correct answer. | | 4 | Every instance of this Learning Target will present you with a derivative and ask for its units; you **must** give correct units here. Giving them incorrectly here and then correcting yourself later on is considered unacceptable work. The explanation of the derivative must use simple language and no technical jargon; and it must explain the derivative in terms of rates of change. The answers on approximation questions must be correct and the setup must also be correct. | | 5 | The conclusions drawn from the given information must be correct *and* backed up by a *clearly expressed* explanation that is *also* correct. That is, correct answers with no explanation, incorrect explanations, or unclear or incoherent explanations will not be considered acceptable. | | 6 | All answers must be correct; if the work requires more than one step, each step must be shown. | | 7 | All answers must be correct and accompanied by complete, correct solutions. You must also state which rule you are using and in the case of the Chain Rule, correctly state the inside and outside functions involved in the composition. | | 8 | All answers must be correct and accompanied by complete, correct solutions. You must also state which rule you are using *in the correct order in which they are used* and in the case of the Chain Rule, correctly state the inside and outside functions involved in the composition. | | 9 | First, your first derivative computation and determination of the critical points must be correct except for one simple error allowed. Second, there must be a correctly setup and labelled first derivative sign chart used that includes: a clear list of all the test points used and their results for the sign of the derivative; a number line with the critical points clearly labelled; a clear indication of the sign of the first derivative on each interval created; and a clear and correct indication of the behavior of the origial function on each interval. Finally, there must be a clear statement of the intervals on which the function is increasing and decreasing. | | 10 | First, your first and second derivative computations must be correct except for one simple error allowed. Second, there must be a correctly setup and labelled second derivative sign chart used that includes: a clear list of all the test points used and their results for the sign of the derivative; a number line with the critical poitns of $f'$ clearly labelled; a clear indication of the sign of the second derivative on each interval created; and a clear and correct indication of the concavity of the origial function on each interval. Finally, there must be a clear statement of the intervals on which the function is concave up and concave down and a clear statment of the inflection points. | | 11 | The first derivative computation and determination of the critical points must be correct except for one simple error allowed. No extraneous points (for example critical numbers outside the interval, or numbers that are neither critical numbers nor interval endpoints) should be tested. A clear indication of the points being tested and the results of the test must be given, and a clear statement of the absolute minimum and maximum must be made. | | 12 | If the problem involves a diagram, it must be clearly drawn and correctly labelled with the variables being used. It must be clear from either the diagram or a separate declaration what each variable stands for in the problem. You must clearly state what quantity is being optimized, then give a clear statement of a formula for that quantity that includes intermediate steps if needed. If you use a constraint in the problem to arrive at the target formula, it must be clearly stated and correctly derived. You must use correct calculus to find the critical value(s) of the target formula. Then **you must give a correct and clear mathematical argument for why your answer actually optimizes the target quantity.** This can be done using the First Derivative Test, Second Derivative Test, or (in some cases) the Extreme Value Theorem. | | 13 | All answers must be correct unless a mistake is the result of a simple error; the setup for each calculation must be shown and correct. | | 14 | All answers must be correct unless a mistake is the result of a simple error; the setup for each calculation must be shown and correct. | | 15 | All explanations and conclusions must be correct, clearly expressed, and coherent. The units of the integral must be correct. | | 16 | **UPDATED 2021-04-12** Each integral calculation must do *all* of the following: (1) clearly state a correct antiderivative for the integrand; (2) show the "Fundamental Theorem step"; (3) give an *exact* value of the result that is *fully simplified*; and (4) give a decimal form of the answer that agrees with the exact value to four decimal places. | ## Approved resources for Checkpoints You are approved to use the following resources on all Checkpoints: - The _Active Calculus_ textbook and any external site that it linked inside the textbook. - Any video or document posted to the class Blackboard site or posted by the professor to Campuswire. These include the lecture videos for Daily Preps; slides from class meetings; Jamboard PDFs from class meetings; and any tool, document, or video that is linked in a Daily Prep assignment. - Class Zoom recordings. - The websites [Wolfram|Alpha](http://www.wolframalpha.com) and [Desmos](http://www.desmos.com). **Any resource NOT included on this list is to be considered off-limits and not approved for use on Checkpoints. Evidence of using unapproved resources will be considered academic dishonesty.** This includes: - Solution websites such as Chegg - Other students in the class, or past students from other MTH 201 classes - Other textbooks or videos not included above Please see the [Collaboration and academic honesty policy in the syllabus](https://hackmd.io/Bnpy2q58StW8mIt2mqaSMA#Collaboration-and-academic-honesty) for more. If there is a resource that is not on the approved list that you'd like to use, please ask me (Talbert) for permission. Additional resources may be added to this list later. ## Schedule for Checkpoints and Learning Targets Below is a duplicate of [the schedule found at the end of the course syllabus](https://hackmd.io/Bnpy2q58StW8mIt2mqaSMA#Schedule-of-Checkpoints), and as stated in the syllabus, it is subject to change depending on how things go in the course. All changes to this schedule will be announced well in advance. | Checkpoint | Learning Targets | Date assigned | Date due | | --------- | ---------------- | ------------- | -------- | | 1 | 1, 2 | Feb 1 | Feb 3 | | 2 | 1, 2, 3 | Feb 8 | Feb 10 | | 3 | 1, 2, 3, 4, 5, 6 | Feb 22 | Feb 24 | | 4 | 1, 2, 3, 4, 5, 6, 7, 8 | Mar 8 | Mar 11 (Thursday) | | 5 | 3, 4, 5, 6, 7, 8, 9, 10, 11 | Mar 22 | Mar 24 | | 6 | 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | Apr 5 | Apr 7 | | 7 | 7, 8, 9, 10, 11, 12, 13, 14, 15, **16** | ~~Apr 12~~ Apr 13 | ~~Apr 13~~ Apr 14 | | 8 | 1--16 | Apr 19 | Apr 21 | | Mini-checkpoint | 12, 13, 14, 15, 16 | ~~Apr 23 (Friday)~~ Apr 22 (Thursday) | Apr 25 (Sunday) | | 9 | 1--16 | Apr 27 (Tuesday) | Apr 29 (Thursday) | [![hackmd-github-sync-badge](https://hackmd.io/A1vNFWQOR5m5qUJtR1X_Aw/badge)](https://hackmd.io/A1vNFWQOR5m5qUJtR1X_Aw)