“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.
• 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.
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.
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.
• 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.
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.
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%.
Basic Ground Rules:
Homework will be assigned weekly or so, and due at lecture the following Monday (so everyone gets a crack at recitation). Since the first two weeks have lecture only on Wednesday, the first homework will be due on Wednesday September 6. The homework will often contain problems that are intended for you to present in recitation, so we can all practice civil mathematical discourse.
There will be two midterm exams, on Wednesday October 4 and on Wednesday November 8. We'll talk about how these will work as the time comes closer. The final exam is scheduled for Thursday December 14 from noon to 2 in DRL A2..
Homework and class notes:
- Week 1 - real numbers (long version)
- Homework 1
(due September 6).
For this week, only discussion problems 1-4 are due to be discussed, and only problems 1 and 5 of the hand-in section need to be handed in on September 6. We'll get to the rest later. Also, there was a typo in the very last problem (it said "for all b" when it should have said "for all n"), but it's fixed now.
- Solutions to (problems 1 and 5 of) Homework 1.
- Week 2 - other axioms (Updated, 9/18).
- Homework 2 (due September 18)
- Solutions to Homework 2.
- Homework 3 (due September 25)
- Week 3 - derivatives
- Week 4 - an integral, and applications.
- Some practice problems for the first midterm. (typo in #3 corrected)
- Solutions to practice problems for exam 1.
- Solutions to homework 3
- First midterm.
- Solutions to first midterm
- Homework 5 (due October 18 technically, homework 4 was the last two problems on the exam).
- In "to be handed in" problem #4 in this week's homework (p. 149, problems 4 and 6), you don't have to do problem 4 -- it duplicates part of problem 6 and it's wrong(!). Thanks, Nathan A for noticing!
- Here is a paper that gives a nice proof of the irrationality of pi.
- Solutions to homework 5
- Homework 6 (due October 25) (modified as of 1pm, 10/18).
- Notes part 5 Uniform continuity and the integral.
- Solutions to Homework 6
- Homework 7 (due November 1)
- Solutions to Homework 7
- Some practice problems for the second midterm.
- Can't get enough? Here are some more practice problems.
- Solutions to (the first set of) practice problems.
- Solutions to most of the second set. (There was a typo in 4(e) but it's fixed here -- it said greater than or equal to when it should have said just greater than.
- The second exam.
- Solutions to second exam
- Notes part 6 Series and uniform convergence.
- Homework 9 (due November 20)
- Solutions to Homework 9
- Homework 10 (due November 29)
- Solutions to Homework 10
- Homework 11 (due December 11)
- Solutions to Homework 11 (well, most of them)
- Notes part 7 Metric spaces.