“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%.
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%.
Monday, October 31, 2011
Introducing Exponents
In the second semester of Algebra 1, students encounter exponents and the rules of exponent multiplication and division. The power of exponents and the huge effect of a continual increase in the rate of increase (which is what an exponent greater than one provides) need introduction as well. This lesson uses the NROC video presentation from Algebra 1—An Open course, Unit 7, Lesson 1, Topic 1: Rules of Exponents followed by a concrete example wherein the number 2 is raised to the powers of zero through 64. The example comes from a classic math story in which grains of rice are doubled on each successive square of a chessboard resulting in the last square receiving 2^64 grains of rice. Students then work with the numbers from this example to explore just how big a value 2^64 really is.
Learning Objective(s)
• Understand exponents
• Be able to simplify basic fractions involving variables raised to exponents
• Be able to multiply variables with exponents
• Be able to explain how raising to a power is different than multiplying a number by that power.
Assessment Type
This is an introductory assignment that presents the basics of exponent use with constants and variables. It provides a formative assessment of student understanding of exponents and is designed for the 55minute high school algebra class.
Assignment Details
1. 15min--Play the video presentation from Algebra 1—An Open course, Unit 7, Lesson 1, Topic 1: Rules of Exponents which explains basic exponent concepts of exponent notation, multiplication and division. Students should take notes and then summarize key points as a whole class at the end. After this, check for understanding with a few example problems and assign any textbook work for homework. Be sure that students are NOT just multiplying by 2 or using 2x and x^2 interchangeably.
2. 10min--Tell a version of the Rice Grain Story, where someone asks a sultan (or raja or king) for payment in the form of rice on a chessboard. One grain on the first square, 2 grains on the second, 4 grains on the third, 8 on the 4th and so on. Write this as 2^0, 2^1, 2^2, and so on. Ask if this payment is more valuable than a pound of gold on each square and have students discuss. (Someone might have to look up how much a pound of gold is worth.)
Now, bring out one or more chessboards and a large rice sack to illustrate. Have student volunteers count out grains of rice to fit on the board until they can’t anymore. How many squares did that take?
Get someone to figure out how much rice would be on the last square. (If you have a computer projector, you can use this to do a quick aside on using a spreadsheet for calculations. Otherwise handheld calculators work.) 2^64 is 18446744073709551616. Yikes!
Redo problem with 3 grains of rice (3^0, 3^1, 3^2…) to make sure students don’t confuse the doubling for correct use of exponents. This also shows just how fast exponents increase in size…and how useful they are as 2^64 is easier to write out then the huge number it becomes.
3. 20min--Have students try to get a sense of how big 2^64 is. They can go about it several different ways depending on interest and what you have available. A small group of students could use a computer spreadsheet to see just how much numerically summing from 2^2 to 2^64 is. Meanwhile, another group can approach this as a storage problem. How much space would 2^64 grains of rice take up in a cubic warehouse? Have the students actually calculate this by filling a small box (shoebox for example) with rice. Have a measuring tape and half cup, quarter cup, measures available. Other students could try to compare the number of grains calculated to how much rice is grown yearly in the world. Finally, how much would that all weigh? Final numbers could be put up on a single poster displaying the results. Anyone finished begins on homework. To get a sense of what they’ll come up with, try this page.
4. 10min--Assign your cleanup people. Anyone not cleaning up is helping create the class results on one large poster and/or writing an answer to this Exit Slip Prompt: A student of mine once said, “5^2 is 10, 5^3 is 15, and 5^4 is 20. It’s so easy!” Could you kindly explain to this student where they went wrong and help them find a trick or slogan that will let them remember how powers work. Students who finish can share aloud. Summarize findings and dismiss the class!
Instructor Notes
• When doing example problems at the beginning of class, be sure to do a squared problem and a cubic problem and mention the relationship between squaring and the area of squares and cubing and the volume of cubes.
• To keep things interesting for successive class periods have them pretend to fill chessboards with 3^0, then 3^1, 3^2 of 4^0, 4^1, 4^2 and so on instead of powers of 2. Alternately, do not show the next class more than a glimpse at the previous class’s poster. Make it known that class posters will be voted on the next day (classes cannot vote for their own poster). The poster voted best will be kept on the wall.
• If there are not enough rice tasks to go around, a group could be assigned to summarize the morning’s exponent notes in a class poster as well.
• Show the effect of a negative exponent the next day (or the day after the posters have been voted on, as then you’ll have large scrap paper) as folding paper in half repeatedly. The How many times do students think the paper can be folded before it becomes too thick to fold?
• During summarizing the exit slip results, “Exponents are repeated multiplying of a number. Multiplication is repeated adding of a number,” is a slogan that helps many students.
Rubric
As this is a formative assignment, participation is what will be graded here. Any student who stayed on task and turned in a complete exit slip (or signed their name to the results poster or was a cleanup person) 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 correctly completed exit slip. (If students had time to discuss their answers at the end, you could require students to revise their answers and grade correctness more strictly. Don’t forget that some students were assigned cleanup or poster creation during this time, and those students should receive credit for those activities instead of the slip.)
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 exponents
• Be able to simplify basic fractions involving variables raised to exponents
• Be able to multiply variables with exponents
• Be able to explain how raising to a power is different than multiplying a number by that power.
Assessment Type
This is an introductory assignment that presents the basics of exponent use with constants and variables. It provides a formative assessment of student understanding of exponents and is designed for the 55minute high school algebra class.
Assignment Details
1. 15min--Play the video presentation from Algebra 1—An Open course, Unit 7, Lesson 1, Topic 1: Rules of Exponents which explains basic exponent concepts of exponent notation, multiplication and division. Students should take notes and then summarize key points as a whole class at the end. After this, check for understanding with a few example problems and assign any textbook work for homework. Be sure that students are NOT just multiplying by 2 or using 2x and x^2 interchangeably.
2. 10min--Tell a version of the Rice Grain Story, where someone asks a sultan (or raja or king) for payment in the form of rice on a chessboard. One grain on the first square, 2 grains on the second, 4 grains on the third, 8 on the 4th and so on. Write this as 2^0, 2^1, 2^2, and so on. Ask if this payment is more valuable than a pound of gold on each square and have students discuss. (Someone might have to look up how much a pound of gold is worth.)
Now, bring out one or more chessboards and a large rice sack to illustrate. Have student volunteers count out grains of rice to fit on the board until they can’t anymore. How many squares did that take?
Get someone to figure out how much rice would be on the last square. (If you have a computer projector, you can use this to do a quick aside on using a spreadsheet for calculations. Otherwise handheld calculators work.) 2^64 is 18446744073709551616. Yikes!
Redo problem with 3 grains of rice (3^0, 3^1, 3^2…) to make sure students don’t confuse the doubling for correct use of exponents. This also shows just how fast exponents increase in size…and how useful they are as 2^64 is easier to write out then the huge number it becomes.
3. 20min--Have students try to get a sense of how big 2^64 is. They can go about it several different ways depending on interest and what you have available. A small group of students could use a computer spreadsheet to see just how much numerically summing from 2^2 to 2^64 is. Meanwhile, another group can approach this as a storage problem. How much space would 2^64 grains of rice take up in a cubic warehouse? Have the students actually calculate this by filling a small box (shoebox for example) with rice. Have a measuring tape and half cup, quarter cup, measures available. Other students could try to compare the number of grains calculated to how much rice is grown yearly in the world. Finally, how much would that all weigh? Final numbers could be put up on a single poster displaying the results. Anyone finished begins on homework. To get a sense of what they’ll come up with, try this page.
4. 10min--Assign your cleanup people. Anyone not cleaning up is helping create the class results on one large poster and/or writing an answer to this Exit Slip Prompt: A student of mine once said, “5^2 is 10, 5^3 is 15, and 5^4 is 20. It’s so easy!” Could you kindly explain to this student where they went wrong and help them find a trick or slogan that will let them remember how powers work. Students who finish can share aloud. Summarize findings and dismiss the class!
Instructor Notes
• When doing example problems at the beginning of class, be sure to do a squared problem and a cubic problem and mention the relationship between squaring and the area of squares and cubing and the volume of cubes.
• To keep things interesting for successive class periods have them pretend to fill chessboards with 3^0, then 3^1, 3^2 of 4^0, 4^1, 4^2 and so on instead of powers of 2. Alternately, do not show the next class more than a glimpse at the previous class’s poster. Make it known that class posters will be voted on the next day (classes cannot vote for their own poster). The poster voted best will be kept on the wall.
• If there are not enough rice tasks to go around, a group could be assigned to summarize the morning’s exponent notes in a class poster as well.
• Show the effect of a negative exponent the next day (or the day after the posters have been voted on, as then you’ll have large scrap paper) as folding paper in half repeatedly. The How many times do students think the paper can be folded before it becomes too thick to fold?
• During summarizing the exit slip results, “Exponents are repeated multiplying of a number. Multiplication is repeated adding of a number,” is a slogan that helps many students.
Rubric
As this is a formative assignment, participation is what will be graded here. Any student who stayed on task and turned in a complete exit slip (or signed their name to the results poster or was a cleanup person) 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 correctly completed exit slip. (If students had time to discuss their answers at the end, you could require students to revise their answers and grade correctness more strictly. Don’t forget that some students were assigned cleanup or poster creation during this time, and those students should receive credit for those activities instead of the slip.)
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%.
Friday, September 30, 2011
Applying Functions: A Roller Coaster Project
After viewing “Snowboarding,” the online tutorial simulation from Unit 3 of NROC’s Algebra 1—An Open Course, that reviews the properties of functions and relations, students draw a graph representing the hills and loops of a roller coaster. The students use coordinates from their graph to discuss different types of relations (functions, linear functions, etc.). Finally, they create a scale model of a rollercoaster, thereby applying their learning to the use of a proportional function in a real world situation. This blog post builds on the “Design a Roller Coaster,” also from Unit 3.
Learning Objective(s)
• Identify linear functions, relations, and non-linear functions from graphs.
• Create a linear equation from two points on a graph.
• Use a proportional function in a real world endeavor.
Assessment Type
This capstone project allows students to showcase their understanding of functions and relations as well as apply them to a 3-D construction project.
Assignment Details
Applying the abstract concepts of functions and relations to real life can be difficult for a student. Projects and real life examples can help. The information below is an overall outline of how such a project can be introduced. Also provided is a link to a handout to help guide students through the project.
1. Warm Up: Have your students respond freely in a journal to the following prompt. “How do Roller Coasters work? What makes them fun? What (if anything) do they have to do with functions?” (Example answers: The work on gravity. Getting pulled around and going fast are fun! How fast a coaster goes is a function of how far it’s fallen. How pulled you feel when going around a corner is a function of how tight a turn is and how fast you’re going.)
2. Explain that today you are going to begin a project that will begin with using functions and relations to design a track and end with building scale models of roller coasters. First, you’re going to see a tutorial simulation that uses snowboarding tracks to show you how to represent a track or path as a graphed relation between height and distance. It will help you identify functions and non-functions using graphs and tables also.
3. Show the simulation, having students discuss and vote on correct answers. When finished ask: If shown a graph of another track, a roller coaster track, could you do what we did here?
4. Students work in pairs at computers or individually with graph provided as part of the “Design a Roller Coaster,” project from Unit 3 of Algebra 1—An open Course. They should answer questions one and two either from the website or from the handout I created to go with the project. Note that the students are being asked to do the same thing as in the snowboarding tutoring simulation they just viewed: Identify sections of the track as representing a linear function, non linear function, or not a function (it does not pass the vertical line test). They are NOT being asked to generate more advanced functions from the graph provided. You can have them write an equation for the portion of the graph that represents a linear function.
5. Introduce them to the project idea of building a scale model of a roller coaster. When one scales something, one uses a linear, proportional function, so this provides awesome real life use of the math they’ve been learning. Essentially, they will need to generate a scale factor using the ratio between the height their coaster would be in real life, and how tall the maximum height they plan for their model. This “factor”—the ratio between the true height and the height of their model--will determine all other heights in their model.
This project is based on the NROC Algebra 1--An open Course Rollercoaster Project, but a bit more fleshed out. To me, hands on building and real life applications are what make projects awesome, so I’ve added it in! See the handouts I’ve made to guide this project. WARNING: They are not complete, but I’d love your help to make them that way! I’ve posted what I have in a Shared Google Doc. Please feel free to edit and improve, and let me know what you’ve done! You’re welcome to edit the original if you feel confident or to send me written up ideas for future inclusions.
Instructor Notes
• Fun and innovation should be encouraged. If students would rather make a scale model of their house, the school, an airplane, or most anything, have them go for it! No kits, though.
• Many other functions apply to roller coasters than scaling functions for models. For example, if friction is ignored, the speed of a rollercoaster at any point along the track is determined by how far it has fallen from rest at the top of the coaster’s highest hill. As an extension project, have some students try to find that relation, determine whether it is a function, and if so can they use it to predict speed? (for the instructor: It’s a non-linear function sqrt(2gh) is what you’re looking for, where “g” is acceleration on earth from gravity and “h” is the height fallen.
• For students struggling with finding the equation of a linear portion of a graph invite them to review the NROC presentations from Unit 3 and perhaps redo the tutorial Snowboarding simulation from Unit 3 on functions and linear proportionality.
• In a classroom with limited space resources, students can create a scale illustration on large butcher paper of their planed coaster. Remember, they should still report the scale factor between their original graph and their poster as well as the scale factor between their poster and real life.
• This is a very scalable activity (pun intended). If you don't want to do the full project, you can still watch the snowboarding tutorial sim and discover the equation of a line on the NROC Algebra 1 Roller Coaster Project website. Then, instead of having students make coasters, print out pictures of coasters. Have students find out how tall the coaster is in real life and have them figure out: the proportionality between the image and the actual coaster, 2. The slope (or entire equation) of some linear portion of the coaster if it is projected onto an X,Y coordinate system.
Rubric
Please refer to the Google document handout.
Friday, August 12, 2011
Virtual Math Manipulatives for Algebra
When I last posted about the NROC text resources, I focused on the use of the written text itself, but did you notice the manipulatives? Take a look at Unit 4, Lesson 1, Topic 3 of NROC’s Algebra 1—An Open Course. There, you’ll see a manipulative that relates the slope “m” and constant “b” of a linear equation in slope intercept form to its graph. Below is a snapshot of the manipulative.
Like physical manipulatives, virtual manipulatives are awesome tools for students trying to really understand and internalize (I’d use the term “grok,” but I doubt that word is commonly understood, these days) how a mathematical concept works. Here is a way to use this tool in your classroom.
Learning Objective(s)
• Understand how graphing is used to represent solutions to a linear equation
• Recognize how changing coefficients and constants change the graphed solution.
Assessment Type
The formative assessment mentioned here should be employed as part of an introduction or reinforcement activity before students are thoroughly familiar with the concepts referred to.
Assignment Details
In a classroom with only one internet enabled computer, one can have one of NROC’s manipulatives running on through an overhead projector. Explain the basics of what they're seeing in the manipulative, and then ask them to predict what will happen when you toggle one of the variables.
For example, to use the manipulative mentioned, remind students of the y=mx+b form of linear equations, and possibly run them through how to graph one using an X,Y table. Hint that you’re going to be showing them a shortcut soon. Read the introduction to the manipulative aloud to the class and field any questions, but do not use any of the slider bars to change the manipulative. Then, ask “What will happen to the graph if we increase the value of b in the manipulative below? How about if we make B negative? ” Have them record their prediction and share their prediction with one partner. Then, “What will happen if we make ‘m’ greater? If ‘m’ is a fraction? Negative?”
If at all possible, though, these manipulatives should be given to students directly to explore on their own. You’ll need to give them basic direction on how to make the manipulative go, but once you have, allow them to have fun exploring the manipulative. Eventually, structure their investigation, asking them “What happens if…?” questions and “Why does…?” questions. I recommend having them write down their responses for you. (See the Example Questions below)
So, that’s a great way to use virtual manipulatives in your classroom. I hope you have fun with them! Yet, what do you do if you want a manipulative, but NROC doesn’t have it in its text?
I have two go-to sources, Wolfram Math Demonstrations and Geogebra, both of which have large libraries of manipulatives and a build-your-own option if you need something new. Both are absolutely free, although user accounts are required for Geogebra. Both can also be embedded in your webpage. In fact, Geogebra is what was used to make the demonstrations made in the NROC text.
See my hints and tips below for more information on getting started with these two great manipulative resources. If you build your own, give me a link to it in the comments section. I’d love to see it!
In summary, manipulatives are great because they instantly provide a playful math experience while allowing the student to explore and internalize “If I change one thing, what happens to another?” You can access manipulatives through NROC’s text, by visiting the Wolfram Math Demonstrations page, or making your own with Geogebra!
Example Questions
1. What do you think will happen to the graph if we increase the value of ‘b’ in the manipulative? How about if we make b negative? What really happened when ‘b’ changed?
2. What do you think will happen if we make ‘m’ greater? What if ‘m’ is a fraction? Negative? What really happened when ‘m’ changed?
3. Overall, describe what ‘m’ changes in a graph of y=mx+b? What about b?
4. *Extra Credit Challenge!* If you have two y=mx+b equations graphed on the same page and ‘m’ is the same in both, but 'b' is different, what is the special name for how these two lines’ relationship?
Rubric
4 Points for participating well in the class activity and discussion.
2 Points per question (6 total). The first point is for any honest attempt at a prediction given in a complete sentence. 2nd point is for noting the correct action of the graph when transformed. No penalty for skipping the Extra Credit, but take the opportunity to discuss the answer later as it introduces parallel lines.
Total: 10 points
Instructor Notes
Hints and Tips for Geogebra:
Download Geogebra and register with them to access their materials.
Once you register for Geogebra, go back and log in. Browse the wiki-library of resources, but be patient as things take a while to load. Geogebra's library is divided by language—you won’t see a list of manipulatives on the “Main” page. Look instead under the link that says “English.”
If you want information on how to create or embed a manipulative in your website, look in their "Introductory Materials" for orientation. Their format is do-it-yourself and share. Again, if you make a manipulative, please post me a link of your creation! I’d love to see it!
Hints and Tips for Wolfram Math:
You’ll need to download a CDF player (free) to use the demonstrations, but wow! It’s worth it.--CDF Player
To navigate, try looking within the topics list in the drop down menu linked to here.--
Wolfram Math Demonstrations Topics List. You can also do a keyword search.
If you get stuck or find yourself with questions on how to make your own demonstration--Wolfram FAQ.
Also, if you see a demo that’s almost what you want, but not quite, emailing them is very effective. I usually get a response within a week. Or, if you’re at all familiar with Mathematica and/or programming you can make a DIY manipulative here too! Have a great time!
Like physical manipulatives, virtual manipulatives are awesome tools for students trying to really understand and internalize (I’d use the term “grok,” but I doubt that word is commonly understood, these days) how a mathematical concept works. Here is a way to use this tool in your classroom.
Learning Objective(s)
• Understand how graphing is used to represent solutions to a linear equation
• Recognize how changing coefficients and constants change the graphed solution.
Assessment Type
The formative assessment mentioned here should be employed as part of an introduction or reinforcement activity before students are thoroughly familiar with the concepts referred to.
Assignment Details
In a classroom with only one internet enabled computer, one can have one of NROC’s manipulatives running on through an overhead projector. Explain the basics of what they're seeing in the manipulative, and then ask them to predict what will happen when you toggle one of the variables.
For example, to use the manipulative mentioned, remind students of the y=mx+b form of linear equations, and possibly run them through how to graph one using an X,Y table. Hint that you’re going to be showing them a shortcut soon. Read the introduction to the manipulative aloud to the class and field any questions, but do not use any of the slider bars to change the manipulative. Then, ask “What will happen to the graph if we increase the value of b in the manipulative below? How about if we make B negative? ” Have them record their prediction and share their prediction with one partner. Then, “What will happen if we make ‘m’ greater? If ‘m’ is a fraction? Negative?”
If at all possible, though, these manipulatives should be given to students directly to explore on their own. You’ll need to give them basic direction on how to make the manipulative go, but once you have, allow them to have fun exploring the manipulative. Eventually, structure their investigation, asking them “What happens if…?” questions and “Why does…?” questions. I recommend having them write down their responses for you. (See the Example Questions below)
So, that’s a great way to use virtual manipulatives in your classroom. I hope you have fun with them! Yet, what do you do if you want a manipulative, but NROC doesn’t have it in its text?
I have two go-to sources, Wolfram Math Demonstrations and Geogebra, both of which have large libraries of manipulatives and a build-your-own option if you need something new. Both are absolutely free, although user accounts are required for Geogebra. Both can also be embedded in your webpage. In fact, Geogebra is what was used to make the demonstrations made in the NROC text.
See my hints and tips below for more information on getting started with these two great manipulative resources. If you build your own, give me a link to it in the comments section. I’d love to see it!
In summary, manipulatives are great because they instantly provide a playful math experience while allowing the student to explore and internalize “If I change one thing, what happens to another?” You can access manipulatives through NROC’s text, by visiting the Wolfram Math Demonstrations page, or making your own with Geogebra!
Example Questions
1. What do you think will happen to the graph if we increase the value of ‘b’ in the manipulative? How about if we make b negative? What really happened when ‘b’ changed?
2. What do you think will happen if we make ‘m’ greater? What if ‘m’ is a fraction? Negative? What really happened when ‘m’ changed?
3. Overall, describe what ‘m’ changes in a graph of y=mx+b? What about b?
4. *Extra Credit Challenge!* If you have two y=mx+b equations graphed on the same page and ‘m’ is the same in both, but 'b' is different, what is the special name for how these two lines’ relationship?
Rubric
4 Points for participating well in the class activity and discussion.
2 Points per question (6 total). The first point is for any honest attempt at a prediction given in a complete sentence. 2nd point is for noting the correct action of the graph when transformed. No penalty for skipping the Extra Credit, but take the opportunity to discuss the answer later as it introduces parallel lines.
Total: 10 points
Instructor Notes
Hints and Tips for Geogebra:
Download Geogebra and register with them to access their materials.
Once you register for Geogebra, go back and log in. Browse the wiki-library of resources, but be patient as things take a while to load. Geogebra's library is divided by language—you won’t see a list of manipulatives on the “Main” page. Look instead under the link that says “English.”
If you want information on how to create or embed a manipulative in your website, look in their "Introductory Materials" for orientation. Their format is do-it-yourself and share. Again, if you make a manipulative, please post me a link of your creation! I’d love to see it!
Hints and Tips for Wolfram Math:
You’ll need to download a CDF player (free) to use the demonstrations, but wow! It’s worth it.--CDF Player
To navigate, try looking within the topics list in the drop down menu linked to here.--
Wolfram Math Demonstrations Topics List. You can also do a keyword search.
If you get stuck or find yourself with questions on how to make your own demonstration--Wolfram FAQ.
Also, if you see a demo that’s almost what you want, but not quite, emailing them is very effective. I usually get a response within a week. Or, if you’re at all familiar with Mathematica and/or programming you can make a DIY manipulative here too! Have a great time!
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