“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%.
Showing posts with label Algebra. Show all posts
Showing posts with label Algebra. Show all posts
Saturday, December 31, 2011
Wednesday, November 30, 2011
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%.
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%.
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