Differentiated Instruction while Studying Rational Equations

Rational equations cause some students great difficulty because they bring together multiple, difficult topics: fractions, quadratic equations, and rates. Students who’ve previously mastered these topics can succeed rapidly while those who’ve struggled before have difficulty mastering the basics.  This lesson uses presentations and simulations from NROC’s Algebra 1--An Open Course’s Unit 11, Lesson 2: Rational Expressions and Equations (also available on HippoCampus.org) to allow students to review material or move on to new material within one class assignment.

Learning Objective(s)

•    Understand how to solve rational equations.
•    Learn to apply rational equation solving techniques to real world problems.

Assessment Type

This differentiated assignment can be completed in any context (remote or classroom) where students have access to the internet. The assignment varies based on assessed student mastery, but all students will turn in a set of notes and practice problems.

Assignment Details

Before this class session, identify which students need additional review of rational equation solving techniques and which students are ready to go on. You can do this with an in-class quiz, based on completion of problems in a class warm up, or even after grading a chapter test on the material.

Students who are having trouble simplifying and solving expressions are not ready to tackle the word problems in the applications section. It would be almost impossible for them to identify whether they are making solving errors, simplification errors, or setup errors. They need to master the basics first or they will only be frustrated by applications.

 Have them view, take notes on, and work practice problems based on Solving Rational Equations.  You can use the practice problems associated with this topic. Have your students complete at least five of those after having viewed the video. Otherwise, assign related problems from the textbook.

Meanwhile students who have achieved mastery would be bored by further review. Have them move on to the NROC lesson, Applying Rational Equations, viewing and taking notes on it before working through the problems presented in, “Conserving Water,” the section’s applications-based tutorial simulation.

For homework, all students should complete anything not done in class. In addition, assign all students completion of five additional practice problems from the section or sections of their choice. For full credit, they must record the lesson’s name (and page number if using a textbook), problem number, and enough information from the problem for someone to answer the question asked without referring to the textbook or computer. They can do this on a separate page with its own title of, “Rational Expressions Homework--Problems of Choice,” or they can separate this from their class work with a horizontal line and the new title as a subheading.

Instructor Notes

•    Students who complete the remedial assignment should be given the opportunity to complete the applications assignment as extra credit. Likewise, students who’ve gone on to applications should be allowed to complete the other assignment for an equal amount of extra credit. This way, no one feels like they were denied an opportunity to earn credit for learning.
•    You can also allow those who are really struggling with rational expression basics to earn credit for revisiting the previous lessons’ material in a similar manner (taking notes and completing practice problems).
•    Do not allow self selection of which in-class assignment to do, as many students will group themselves by friends rather than ability.
•    If you have students who find they cannot complete the assigned practice problems redirect them into taking notes on the worked example problems from the same section (like Solving Rational Equations) or a related previous section’s video (like Simplifying Rational Expressions) instead. Ultimately, these students will earn the day’s credit for turning in a set of notes from the video and a set of worked practice or copied example problems. Re-assign the original practice problems to them as homework instead of allowing them the self-selected problems.
•    Anything not completed during class time can be finished as homework. Email the required links to your students or make them available on the class website.

Rubric

As there are three clear parts to the work assigned here, I would grade this on 15 point scale with 5 points per section.

5pts—Completion of the notes from the correct section and overall reasonable neatness of layout.
5pts — For the review group: Correct completion of at least 5 practice problems (or notes on example problems) from the assigned section. 
          For the applications group: Clear recording and correct completion of all problems from the tutorial sim.
5pts—Correct completion of 5 practice problems of the student’s choice, labeled as requested with work shown.

Note: For all problems (including those from the tutorial sim) work must be shown, answers written out, and enough information recorded to make it clear what was asked in the question.

Comments Invited

How do you like to differentiate instruction and assignments in your classroom? What problems have you run into and what successes have you had?

Algebra Factoring Simulations and Games—Now On HippoCampus.org!

Algebra tutor simulations and puzzles (including two on factoring!) from NROC’s Algebra 1--An Open Course are now available to everyone on HippoCampus.org! A student can be assigned to complete a tutorial or reach a certain score on a game and send you a screenshot that shows completion.

The puzzle and sim linked to here deal with factoring trinomials. If you’d like to browse more interactive algebra resources, go to the “Algebra and Geometry” section of HippoCampus.org and scroll down past the “Test Prep” in the leftmost frame until you see “Simulations.” Click on “Algebra 1—An Open Course (2011),” and a list will appear.  Click on the “Tutor Sim” or “Puzzle” you wish to view.

Learning Objective(s)

•    Understand how to factor trinomials.
•    Know how to take and submit a screenshot to a teacher.

Assessment Type

This assignment is designed for completion by the students at home. Completing the tutor simulation takes 10 minutes or less; the time spent on the factoring game can be about the same. The resources linked to here deal with factoring trinomials and are best used as part of a review (perhaps in conjunction with or just before assigning a more formal practice test) a bit before a test on the material.

Assignment Details

Students will access the tutorial and games at home via the internet and use them to review factoring trinomials. To show completion, they will send in a screenshot.

For students to reach these online tools, they will need access to an internet enables computer with working speakers. They’ll want to have the browser window maximized and also “maximize” their view of the tutorial or simulation by clicking on the “+ Maximize” button circled in red below. 

On a side note, if you wish to link to a puzzle, simulation, or any presentation on Hippocampus.org click on the little symbol that looks like a link to get the correct URL. It is third to the right after the maximize button circled above, next to the "+". Copy/pasting directly from your browser will not work.


The Tutor Simulation--Factoring: Perfecting the Long Kick in Soccer
The simulation linked to above uses an example where the student factors a quadratic equation with a lead coefficient other than 1. Assign the student to work until they reach the end of the tutorial simulation and receive feedback on their performance. They'll need to take a screenshot of the feedback to show completion. An end-of-sim screenshot example is here.

The Puzzle--Factoring: Math Factors

The puzzle linked to here has students select correct factors for quadratics from multiple possibilities. The format encourages guessing. If you want students to fully solve each problem instead, you may want to require that they turn in a written copy of their work as well as a high score screenshot.

Take a moment to play this game and decide what level or levels you wish your students to play. Assign the student to play until they reach a certain score (100, for example) on a certain level. Once they reach that score, they should use screen capture, and email (or drop box) an image showing the problems still on their screen and their score. You’ll want to see the problems on the screen, not just the score, to ensure they are working at the assigned level. An example screenshot is below with the score circled in red.

Instructor Notes

•    You will need to check that students know how to take a screenshot on their computers. Demonstrating this for them can help, but there are also a lot of online tutorials that they can Google. Here are two I found:    How to take a screenshot on a PC.    How to take a screenshot an a Mac.

•    They'll want to paste their screenshots into an image editor, such as Microsoft Paint, so they can crop them and save them in the correct format. They should take care when cropping their images to exclude any inappropriate or private information in the background of the screenshots they submit.


•    Be sure students understand which file formats are acceptable and how you wish files to be named “LastNameFirstNameFactoringGame.jpg,” for example.

•    If you have a class email list, it may be easiest to send students an email with links to the simulation or game and an explanation of how you wish them to submit their work (email or drop box and what file format) as well as links to the screenshot tutorials.

Rubric

Most students will either complete or not complete this assignment. Very few will submit an item that receives partial credit. If you receive a note like, “I earned 100%!” without screenshot proof, it is up to you as to whether you wish to credit it or not. In general, I recommend you assign total credit as you would for a normal homework assignment that takes about 10-15min to complete.

I hope your students have fun, and let me know what they think!

CAHSEE and Others -- High School Exit Exam Math Preparation

Many states require exit exams in math for high school students. The plan here focuses on preparation for the California High School Exit Exam (CAHSEE), but it can help students prepare for any summative high school math exam. California state materials and NROC resources, including resources from Algebra 1—An Open Course and from the Khan Academy Collection on Hippocampus.org, are used.

Learning Objective(s)
•    Review and practice math standards for a high school exit exam.

Assessment Type
This is for any high school math class or study group preparing for a general knowledge math exit exam. Before beginning the review, the instructor should look at the topics covered by the exam and plan accordingly.

Assignment Details
You have many preparation options for a cumulative math exam. The key things are: Begin early. Identify gaps. Fill gaps. Practice. Retest and recap. Many links identify information by topic, so if you’re working with a similar math exit exam from another state, these should still be helpful to you!

1.    Begin Early. Plan on one week of work per review topic. Shorten this if needed or if you’re only doing test preparation and not also working on other class material. Be sure to pepper later weeks with warm up questions from previously reviewed topics.

2.    Identify Gaps.  Have your students take a diagnostic test. They can do this at home or in class, but I do recommend grading in class so students are not tempted to cheat. Paper can be saved if you have access to a computer lab where each student can work separately. For the CAHSEE, the practice test linked here is a good one. Results can be checked with the key posted in the Appendix or use the CAHSEE Practice Test Key and Histogram I made that divides problems by topic. Even if preparing for another state’s test, look at the topics covered by the CAHSEE practice test. Based on topics covered, you might be able to use this practice test and histogram as a diagnostic tool.

3.    Fill Gaps. If your student has missed only a few problems, it might be best for him/her to only review what went wrong on those problems. For the CAHSEE, the link here leads to videos that explain the solution to each practice test problem.

For in-depth Review by Topic follow the link here. Topics covered are as follow: 

Number Sense- Fractions, percents, roots, decimals, proportions, scientific notation
Probability and Statistics- Basic probability, data analysis (graphs, mean, median, mode)
Algebra and Functions- Solve a linear equation, generalize patterns, make a symbolic expression from a written one, know slope, evaluate an expression, identify functions
Measurement and Geometry- Convert between units of measure, scale, translate objects, perimeter, area, volume, Pythagorean Theorem.
Math Reasoning- Recognize and generalize patterns, organize information, use logic in conjectures.
Algebra 1- Simplify polynomials and expressions, absolute value, roots, match graphs to linear equations, find slopes, intercepts, parallel lines, solve linear inequalities, rate, work and mixture problems, systems of equations.

4.    Practice.  Practice should occur both in class and at home. If you have access, you can pull problems from many practice problems in NROC’s Algebra 1--An Open Course. These are also listed by topic. Otherwise, pull questions from textbooks or the 2008 CAHSEE Released Test Questions. (Note: These are listed by topic and answer keys appear after each topic. They repeat the example and practice questions provided after the worked example questions linked to above).

5.    Review and Retest. Create or find a new practice test for your students. If you’re doing the CAHSEE, you can select the problems you want from those linked above to make a new test. Don’t worry about repeating old questions as long as you mix up the order. Use questions you don’t select for review games. In fact, you may wish to do a brief review game at the end of each week to keep topics fresh.

Instructor Notes
•    Look over the practice test results. Recruit high-scoring students as in-class tutors and/or helpers. Excuse them from review homework as a reward.
•    Tell your students to circle or mark any problem that they were unsure of, or on which they guessed the answer. Even if they guessed right, they should review that problem.
•    Out of time? Have students take the CAHSEE practice test and self grade with the histogram. At home, they should view the recorded CAHSEE practice test solutions taking notes for credit. 

Rubric
The first practice test should be participation credit, not percent correct, since you’re using it as a diagnostic rather than a cumulative tool. For in-class and weekly work, one can assign completion of practice problems and/or notes on recordings for homework.

The second practice test can be graded on percent correct and count as much as a regular end-of-chapter test in order to encourage students to take it seriously. Alternately, a score of more than a certain amount might be traded for an extension on a homework assignment or replace one low score on a previous test. A final option is to have a high score on this test count as an extra credit coupon (say, 5%) toward their final exam.

Factoring and Checking Answers

“Am I right?”
This is a question the student should be able to direct at him or herself, not the teacher. This lesson on factoring reinforces students’ ability to check their work while reviewing both factoring and polynomial multiplication. Students then use the practice problems from NROC’s Algebra 1—An Open course, Unit 9 - Factoring, Lesson 1, Topics 1, 2, and 3 (these cover factoring out greatest common factors and factoring simple or advanced trinomials by grouping) to develop skills in checking their own work.

Learning Objective(s)
•    Understand how to factor out the greatest common factor and to factor trinomials.
•    Practice checking that one has factored correctly. 
•    Practice solving and checking for correctness on all problems.

Assessment Type
This lesson is designed for the 55-minute high school algebra class, but can easily be modified to fit a variety of contexts.  It can be used when students are ready to practice trinomial factoring, well after the initial introduction of polynomial factoring. Alternately, it can be used as a general review of how to check one’s work and as standardized test practice if one selects a wider variety of practice problems from multiple topics.

Assignment Details
15min: Review factoring with warm up problems of your choice or use the factoring game, Puzzle: Match Factors, provided as part of this unit (Unit 9). If using the game, note that the different levels provide problems from different types of factoring.

10min: Brainstorm on the board all the different ways that one can check work on different types of algebra problems. Stress that the ability to self-check is important for taking final exams and standardized tests, not to mention in real life where one has neither a teacher nor a textbook to provide correct answers. For the algebra problems, provide examples of different types of problems if needed and/or have students pull examples from past homework and tests. Focus on the two ways that you can check factoring problems, as this is the newest concept.  Students should record the results of the brainstorm as notes.  (Examples of how answers can be checked are listed in Instructor Notes below.)

5min: Have students get out notebook paper for an in-class assignment that they will turn in at the end of the day. Pull up the practice problems from Unit 9, Lesson 1, Topic 1: Factoring and the Distributive Property, under the link titled “Practice” which covers factoring monomials. Show the students how to first solve the problem and record checking their work. The class assignment is to complete all practice problems from all three topics with both steps to the solution and work-checking shown. It’s up to you whether you want to require them to show their check two ways (both by evaluating and by multiplying the factors back together) or just one or the other. Also select and do an example problem from Topic 2: Factoring Trinomials by Grouping 1 to ensure students don’t get stuck here. A Topic 3 example may also be needed. Do note that NROC teaches factoring by grouping, not by “un-foiling” so be sure that your students understand this method before unleashing them on these problems.

20min: Students work to complete these problems on internet-enabled computers (working in groups as necessary), recording both the steps in their work and their answer checks on their paper as shown. Students who finish early can begin their homework or play an NROC math game of their choice.

5min: Check in with the class about the progress made. Which problems were the hardest to check and why? Anything that the brainstorm missed? Make sure to collect work from the day and that any homework assignment is recorded.

Instructor Notes
•    To check a “Solve for X” problem: Put in the variables that you solved for. Does the left side equal the right side when evaluated for the values found? (Common pitfalls: Arithmetic errors. Copying errors.)
•    To check a simplification problem: Take the expression and evaluate numbers in both the original un-simplified and simplified forms. For example, X + X + X + Y simplifies to 3x+y m Put in x=2, y=4 and see if 2 + 2+ 2+ 4 = 3*2+4.  Does it? Then you’re probably right. (Common pitfalls:  Arithmetic errors. Copying errors. False positives can occur, especially if students use the same number for two different variables, use 0 or 1, or use a number that is also a coefficient in the problem.)
•    To check a graphing problem: Use a graphing calculator if allowed. After graphing, choose two clear, whole number (x, y) points from your graph. Use the X coordinate and Y coordinate values in the equation that you made the graph with. The equation should be true when evaluated for each point (the left side should = the right).
•    Factoring: Check by multiplying back together. Can also check in the same manner as simplifying (evaluate with a number of your choice, factored and un-factored results should match).
•    Multiplying: Check by factoring. Can also check in the same manner as simplifying (evaluate with a number of your choice, multiplied and un-multiplied results should match).
•    Systems of equations: Check results in all equations. Graph with a graphing calculator, lines should intersect at the (x,y) point that matches the values solved for.
•    Solving by factoring: Evaluate solutions in original, un-factored equation.
•    For the brainstorm notes: Recording the results on a large class poster can be a great alternate assignment for a couple of strong students during the online practice portion of the class. This will also make it easy for any absent students to catch up on notes from today’s lesson when they return to class.
•    This assignment can be repeated with the “Review” problems instead of the practice problems as an introductory activity tomorrow or as a review before an exam that focuses on factoring.

Rubric
As this is an introductory assignment participation should be the focus of grading. Any student who stayed on task and turned in a complete exit slip (or provided class notes) should receive full participation credit for the day. If kept in an organized notebook, notes can be graded on a later day.
2pts--Arrived on time, stayed on task, and participated with class.
3pts—Work is neat and organized according to expectations.
5pts—Student completed the expected amount of completed practice problems showing the self-checking of answers.
Total= 10pts
You can also grade an activity like this with a rough “Plus, check, minus, zero,” format where a plus is worth 100% credit, a check is 75%, minus is 50%, and zero, 0%.

Polynomials--Introducing and Exploring

After monomials are introduced, second semester algebra 1 students work with operations on polynomials. Before they get into more advanced operations, it is critical for students to understand how to add and subtract these strings of numbers and letters.  NROC’s Algebra 1—An Open course, Unit 8, Lesson 1, Topic 1: Polynomials, contains a recorded introduction to basic terms and properties of polynomials. This lesson plan uses that recording as a jumping off point and asks students to predict the rules of polynomial addition and subtraction. They share their results and check their predictions using the presentations and problems from Unit 8, Lesson 1, Topic 2: Adding and Subtracting Polynomials.

Learning Objective(s)
•    Understand polynomial terms and how to evaluate polynomials
•    Practice making logical predictions about mathematical concepts 
•    Be able to identify like terms and add or subtract polynomials
•    Be able to explain why only like terms can be added and subtracted

Assessment Type
This is an introductory assignment that presents the basic polynomial terms, addition and multiplication. It provides a formative assessment of student understanding and is designed for the 55-minute high school algebra class. It encourages students to look ahead and think logically about math. It also assumes that enough internet enabled computers are available for students to work online in small groups.

Assignment Details
1.     5min: Play the presentation from Algebra 1— An Open course, Unit 8, Lesson 1, Topic 1: Polynomials while students take notes. The presentation uses the example of someone calculating printing costs for their magazine to introduce key polynomial vocabulary. Terms are discussed and a monomial, binomial, and trinomial are created. One volunteer should write a class set of notes on the board and all should check to see that their notes match reasonably well.  

2.    10min: Make up, simplify, and evaluate some polynomials for easy whole numbers. For example, use the three polynomials created in the video. I find turning the variables into concrete examples helps students to understand their use and meaning. Be sure to include a squared variable and also a polynomial that has terms containing both x and y at the same time. 

3.    15 min: Predict addition and subtraction rules for polynomials. Move students into heterogeneous groups. Take two of the more complex polynomials you’ve been using (or have a prepared graphic organizer containing some polynomials to use—if you make one, please send me a copy and I’ll post it here for others!). Students should predict the rules for adding together polynomials. They need to check their predicted rules by evaluating their results with numbers.  If they are correct, their simplified (added together) results should equal the same amount as their un-simplified results when both are evaluated.  They should then go on to working on polynomial subtraction. Four questions to explore and explain: How do you simplify monomial terms within a polynomial? How do you add together polynomials? How do you subtract polynomials? Within a term, does order matter when you evaluate? Explain your rules and use evaluation of results to show that your rules work.

4.    15min: Check results. Once students think they have a set of rules, they should check their results. They can do this by viewing and taking notes on NROC’s presentation on adding and subtracting polynomials. They should then complete as many as they can of the practice problems provided in the “Practice” section of that same lesson.

5.    10min: Summary and Exit Slip. The first group finished should write up a summary of their notes on the board. All students should write down any homework assignment before they exit. And all students should make sure their notes are as complete as the board notes.  Student should, on a half sheet of paper, write their answer to a basic check-in type question. This is their “exit slip,” from the class (be sure they know it is worth participation credit) and they turn it in as they leave.  Exit Slip Prompt: “My friend Jesse does not understand why (x+xy) + (x + xy) doesn’t equal 2x2y2. Please explain to Jesse what’s going wrong and how you could tell the answer wasn’t right.”


Instructor Notes
•    Heterogeneous grouping is recommended to help prevent any one group from just being stuck while another group is already finished. If you do not have enough computers and must run this as a single class exercise, you can do so. Just be sure to facilitate class discussion and allow multiple suggestions for addition and subtraction rules. Don’t jump to the right answer. At a preset time or whenever the time seems right, move from the discussion to the next recorded presentation. Then, use the practice problems as a whole-class quiz game. 

•    In part 4, I provide four questions. I recommend using the first question as a class example for how to evaluate numbers in their simplified and unsimplified results to check if a rule works. For example, X + X + X + Y might simplify to 3xy. How do we know it doesn’t? Put in x=2, y=4 and see if 2 + 2+ 2+ 4 = 3*2*4. It doesn’t? Then 3xy is wrong. Is 3x+y right? Check it the same way.

•    Warn students about false positives that can occur for their rules. Common sources for these false positives are reusing the same number for two different variables or using a number that is also a coefficient in the problem.

•    The NROC subtraction of polynomials presentation does not include an example in which a negative term gets subtracted. You’ll need to make sure this is covered later or students may not realize that addition results from the subtraction of a negative term.

•    A great question to include in a warm up for tomorrow: “My friend Jesse does not understand why (x + xy) - (x - xy) doesn’t equal 0. Please explain to Jesse what’s going wrong and what the answer should be.”

Rubric
As this is an introductory assignment, participation should be the focus of grading. Any student who stayed on task and turned in a complete exit slip (or provided class notes) should receive full participation credit for the day. If kept in an organized notebook, notes from the morning video can be graded on a later day.

5pts--Arrived on time, stayed on task, and participated with class.
5pts--Turned in fully completed exit slip.  (If students complete the exit slip with no feedback from you, take the time to separate them into piles as you check them off as complete. Make one pile for fully complete and correct, one for sort of correct and sort of complete, one for complete but incorrect, and a final one for incomplete. This can help you get a sense of how well the concept is being understood and can help you set up truly heterogeneous groups. You’ll just pick one name from each pile until you’re out of names and note these as the groups for next time.)
Total= 10pts

You can also grade an activity like this with a rough “Plus, check, minus, zero,” format where a plus is worth 100% credit, a check is 75%, minus is 50%, and zero, 0%.