Teaching Math Blog

Factoring and Checking Answers

2011-12-31T17:44:00.000-08:00

“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

2011-11-30T16:53:00.001-08:00

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.

Note: If you do not have direct NROC access, recordings from the course are available at http://new.hippocampus.org/. Look under course “Presentations,” subcategory, “NROC Collection,” for the title Algebra 1—An Open Course (2011). Once you’ve selected this course, from the second frame under “Topics,” select “Polynomials” and “Adding and Subtracting Polynomials" respectively.

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%.

Introducing Exponents

2011-10-31T13:31:00.000-07:00

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%.

Applying Functions: A Roller Coaster Project

2011-09-30T16:59:00.000-07:00

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.

Virtual Math Manipulatives for Algebra

2011-08-12T14:04:00.000-07:00

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!

Strengthening Language Skills While Reviewing Graphing Inequalities

2011-07-30T21:34:00.000-07:00

While working to figure out from context which words have been removed from a paragraph explaining systems of inequalities, students will both reinforce their understanding of graphing systems of inequalities and their ability to use and understand academic English. Students first fill in the teacher created cloze dealing with systems of inequalities. Then, they make and share one of their own. A systems of inequalities cloze worksheet and key (based on a paragraph from NROC's Algebra 1--An Open Course, Unit 6, Lesson 3, Topic 1: Graphing Systems of Inequalities) are provided.

Learning Objective(s)
• Understand how graphing is used to represent solutions to systems of inequalities
• Recognize and use proper English grammar and syntax when communicating about algebra

Assessment Type
This formative assessment should be employed as part of a reinforcement activity or review after students are familiar with the terms and concepts referred to.
Assignment Details
A truly useful but often overlooked item in the NROC Algebra 1 course is the topic text. Written in a conversational style, the text re-explains the same concepts addressed in each lesson’s recorded presentation, providing more details and example problems. Unlike a traditional offline textbook however, it can be copy/pasted into presentations and lecture notes. So long as proper credit is given to its authors, you’re good to go. Here the topic text is used to make a “cloze” activity.

Used since the 1950’s, a fill-in-the-blank cloze exercise is designed to strengthen grammar and syntax skills. You will see the cloze worksheet (and key) on systems of inequalities were made by taking the NROC text and then removing every seventh word. Thus a puzzle rather than a fill-in-the-blank quiz is created—one in which a non-math preposition or verb is as likely to be omitted as a math term. In this assignment, as students first solve, then create, and finally trade-and-solve their clozes, they will use and discuss many aspects of the English language. This cloze activity could be used in many contexts, but this lesson assumes a 55 minute high school class session. Have on hand a class set of the systems of inequalities cloze worksheet and one copy of the key.

15min – As students enter, hand out the systems of inequalities cloze and have them begin work figuring out the missing words. After 10 minutes, have them compare answers with a partner. During this time, circulate, providing insight to any stuck students. Explicitly use grammar terms. If needed, you can brush up on prepositions and adverbs here or many other places online.

5min – Review answers as a whole class. Some suggested answers will be different but still technically correct based on the context. Discuss variance of answers, but accept all that work. Check the results with your key and discuss why the author(s) might have chosen one word over another. Explain that students will now have the make their own cloze puzzles and will then trade and complete them.

15min – If you have internet enabled computers available, have students go to the topic text portion of several of the previous lessons on the NROC Algebra 1--An Open Course. If you don’t have computers, this same activity can be done using a textbook’s text. Instead of “copy/pasting” as describe below, students will re-write and create a cloze paragraph by hand using their class textbook. They should note the textbook and page number as the text source and cloze key.

For those of you using the NROC text, this image shows where to click to find the topic text:

Since the example cloze deals with Unit 6, Lesson 3, Topic 1: Graphing Systems of Inequalities, text from Unit 6, lessons 1, 2, and 3 all make sense as text sources. This will provide the wanted review of the unit. Students should choose and copy/paste one paragraph from the lesson of their choice into a word processor and save it with a descriptive title (I recommend “NROCAlgebraUnitLessonTopic_Cloze”).

At the top of their document they should put (write the following on the board for them):
“Cloze and Key Created by: (First and Last Names)
Text Taken from: NROC Algebra 1—An Open Course Unit _(#)_ Lesson (#), Topic number: Topic Name
Date:____________”
This gives credit to the original authors of the text as well as those creating the cloze.

Beneath this, the students should paste the paragraph’s text, making sure their selection will fit on the page twice and removing unwanted formatting (extra spaces and unwanted underlines from hyperlinks, for example). Then, they should bold and underline every seventh word. Now they have their key! They should copy/paste their edited paragraph below the first one, leaving about 5 spaces in between.

At the top of the second paragraph, they should write:
“Cloze Created by: (First and Last Names)
Cloze Solved by:_________ Date:________”

Now, they should delete every bolded/underlined word from the second paragraph, replacing what they’ve removed with a blank line. Finally, they should save their work and print out a copy.

10min--Once they have their copy printed, they should put their names in the appropriate places on the printout. Then, they cut the worksheet from the key and trade with another finished group who has done a different section of text. Each group races the other to complete the traded clozes. Afterwards, they should check their answers against the key. Groups that finish early can be given time to work on review homework or invited to make a cloze from scratch on any funny but classroom appropriate topic they want to share with the class and solve for fun.

5min--For wrap up, have students volunteer with which sentences were hardest to figure out and why. Summarize any key concept and/or syntax difficulties or vocabulary difficulties that were run into.

Instructor Notes
•Your students have just made you a large number of worksheets that can be used for future years. Save them! Either copy them off the computers or have them print out an extra hard copy for you.
•Encourage students to use and share hotkeys. In Microsoft word, for example, “CRTL+C” is copy, and “CRTL+V” is paste. “Shift+hypen” makes an underline.
•Remember, ELL students will have much more difficulty with a cloze. Offer them a bit more time and scaffolding for their work or pair that with a kind stronger student. Just be sure that the stronger student can explain why certain words are being chosen and does not just complete the worksheet solo.
• Since they’re making and solving puzzles, get them revved up. Pull out a timer when they solve each other’s clozes! They are racing each other within the class but also TOTAL time against other classes. Once you explain what they are to do after they’ve printed their work, start the timer. After the first group finishes, hand over timer and work collection duty to that group so you can help others.

Rubric
Grade on effort and completeness, not on correct answers. Grade by looking at all the names on the completed clozes. Each student’s name should appear twice, once as a creator and once as a solver.

3pts – Student’s name appears on a correctly completed cloze key.
2pts - Key formatting is correct and answers provided in key.
5pts-- Student’s name appears on a correctly completed cloze.
(Partial points for an incomplete cloze)

10pts Total

How to Record Your Own Step-by-Step Algebra Problems

2011-07-14T11:41:00.000-07:00

The NROC Algebra 1 course includes many step-by-step worked example problems recorded by Salman Khan of Khan Academy. What do you do if you find yourself in need of an additional or alternate example? This post will help any teacher who wishes to create additional example problems for their students.

Learning Objective(s)
• For the Teacher: Learn how to create and distribute a recording of your own.

Assessment Type
This post is aimed at teachers, not students. It uses the worked examples relevant to NROC's Algebra 1--An Open Course, Unit 6 Topic 1: Solving Systems of Linear Equations by Graphing to show how one can add to the provided worked examples and supplement with examples made by the teacher.

Your finished example will look like the one linked to here:
Unable to display content. Adobe Flash is required.

Or, you can have students follow a link to reach a full size recording:
SolvingSystemsofEquationsbyGraphing3

Assignment Details
The worked examples in “Unit 6 Topic 1: Solving Systems of Linear Equations by Graphing” jump right in with some deep word problems. An example that goes straight to graphing the intersecting equations might help. Let's record that example, showing step by step solving and reviewing graphing. Of course, you can apply the same techniques to any problem you choose.

Set aside a few hours for this so you won’t feel rushed, and then have fun with it. After you’re familiar with the tools, making recordings is fast and relatively easy.

First, make sure you have an internet connected computer equipped with a microphone headset and drawing tablet. I’ve had good luck with using a Logitech USB headset and a Wacom drawing tablet. This will improve your sound quality and allow you to write out your equations neatly as you present. If you cannot get a tablet, I suggest you type out your work on separate PowerPoint slides. If you only have an area or laptop mic, give it a try and see if you find the audio acceptable.

Next you’ll need recording software. Download the free version of Jing if you’re only going to do simple recordings lasting five minutes or less. If you’re looking for something more advanced, Camtasia Studio provides many more editing and presentation options. It costs a bit, but it does have a free trial period and frees you from the pesky 5 minute limit. Both programs have the ability to record a selected area of your computer screen and the audio to go with it.

Now, set up your graphics. Since we’ll be talking about line graphing, I’m going to want a coordinate axis set up in advance. I used screencapture (fn+f11 on my PC) to pull an image of a coordinate axis from one of the tests I use. I also plan to work with three slides, so I’ll paste the image I want into PowerPoint. I plan to have an introduction slide, a working the problem slide, and a summary or “right answer” slide. Since PowerPoint does not have a straight line tool available in presentation mode, when my problem involves graphing I like the neatness of a final answer screen. Now, my graphics are ready to go as Jing will allow me to play the PowerPoint while recording.

Work your problem. Before recording, take a moment to work your problem from beginning to end while taking notes on the problem’s key points. I also recommend talking through your problem aloud while you solve it, as you’ll be less likely to stumble later. Remember, you’ll need to keep things brief to stay under 5 minutes.

Secure the space. Make sure your phone is off, and put a sign on your office door saying, “If I’m talking I’m recording. Please leave me a note, but do not disturb!” You might want to invest in a small dry erase board or some Post-It notes for your door if you’re likely to do this often.

Record! Check your audio at the beginning of each recording session and have your practice notes on hand. Expect to make errors the first time you do it, but try to reach the end of the problem even if you mess up early. This will help you ensure you have your timing right. Keep your voice dynamic! You’ll improve with practice, so keep at it! Check your recording for good sound before publishing.

Publish your recording. Jing provides you with a file that you can save for viewing in your class or upload into a server of your choice. Screencast.com provides free accounts with 2GB storage and works well with Jing for this. Their sharing tutorial is linked to here. Essentially, once you upload to screencast.com, you will have a URL that you can paste into a website or email to a student.

There you have it! You’ve created a recording that explores additional concepts for your students that they can access at home and any time that they’d like. Perfect!

Instructor Notes
• If you need multiple slides, use Microsoft PowerPoint, or a similar presentation tool. Most will allow you to do markup while presenting. If you just need one screen to markup, Microsoft Paint or any basic image editor will suffice.
• I allow myself one caught error per recording. Any more than that and I believe the recording needs to be redone.
• If you want to capture an image from your screen to use in a slide presentation, on a PC hit the “Fn+F11" hotkey combo or the "print screen” button. This takes a snapshot of your entire screen. You can paste that snapshot into a PowerPoint slide or paint program. You can even use this to grab an image from the NROC presentation or worked examples if you need to further explain an example used there.
• If you want your students to demonstrate that they’ve viewed a recording, assign them to take notes on it and email or upload to you a scan or digital photograph of it.
• If you’re using an online learning management system such as Angel and have published an example recording via a link, the system can track which users have viewed that link. Assuming no two students work together, you can use this to track which students have viewed a recording as well.
• Keep things short, breaking things up into multiple videos as needed. This helps you as well if you’re using the free Jing software. It’s sure frustrating to have made a great recording only to stumble in the last minute.

Rubric
If you complete a recording and get it to your students, let me know about it here and give yourself an A!

Tutor Sim: Feedback While Reviewing Functions

2011-06-30T09:25:00.000-07:00

The NROC tutor simulations can be used for individual review or whole class game play. As the student(s) answer a series of interactive multiple choice questions, the simulation first provides hints and then, at the end, suggests topics to review. With a little introduction and wrap up, the sims can be powerful review tools. This lesson plan uses the simulation from Unit Three - Tutor Sim: Snowboarding, which reviews the fundamentals of functions and their graphs. This lesson could be used with any NROC simulation. It describes the individual and whole class use of the tutor simulations.

Learning Objective(s)

Review concepts before an exam.
Identify areas for future study.
Review slopes, proportionality, functions and their graphs.

Assessment Type

This assessment can be one of the last formative assessments done while preparing for a summative assessment. If done individually, it can help a student self-assess their learning needs. If done with a whole class, it can be used at the beginning of a class session to help the teacher get a sense of what the class still struggles with.

Assignment Details

Before beginning this with your class, take the time to run yourself through Unit Three's "Tutor Sim: Snowboarding." I recommend you get some answers wrong on purpose, so you can see how the simulation reacts. Upon completion, you'll also see the feedback given by topic at the end.

The two timed lesson plans below assume a 55 minute class session in a high school classroom.

If students can work individually with internet enabled computers:

10 min– Warm Up. I have a warm up that I call, "Easy, Medium, Hard." Basically, as students enter the room I invite them to choose one easy, one medium, and one hard question from their homework and to write those three problems on the board for everyone to solve. After a few minutes working on the problems we share and discuss answers. If you have a trusted student or TA, it might help you to have them go around to each computer in your classroom and make sure the web page for the tutor sim is loaded and ready to go.
15 min—Have students get out their note paper and introduce them to the NROC Tutor Sim: Snowboarding then have them work through it at their own pace on their computer. I recommend that you require them to summarize each question (draw any graph given, write down key data and what you're asked to find), and to show both their work and answers (not just the letter of the answer). They need to work through the sim honestly (not just guess and check) because at the end of the tutoring session the sim will use their wrong answers to identify areas is which they are having difficulty.
15 min-- Having completed the tutor sim, students should copy down the suggested review sections as headings on a new sheet of paper, skipping 10 lines between each heading. They can then either fill those blank lines in with three example problems worked from their textbooks or with notes from going to and reviewing the presentations and problems from the NROC lessons dealing with the topic they had trouble with. If a student has no trouble, allow them to begin on their practice test or homework right away.
10 min—Gather the class together for a discussion of what topics are still giving them the most trouble. Provide some review on topics with which many students are struggling.
5 min—If you've not already done so, pass out any practice test or take home review sheet you have for students to complete. I often also offer 5% extra credit for 10 additional worked problems chosen by the student specifically to help them fill gaps that they realized they had today.

If one internet enabled computer and projector setup is available:

10 min– Warm Up. Do the "Easy, Medium, Hard" warm up described above. If you have a trusted student or TA, it might help you to have them get the projector setup and working while you help students with the warm up.
15 min—Have students in teams of four with one volunteer student running the sim for the whole class. As each question is presented, the teams should work together to solve the problem for a set amount of time. When time is called, the teams should show their answer to the student at front. Whichever answer appears the most, that is the answer picked. If your students like to be competitive, you can keep score. Require that the person writing down the answer switches with each question and remind students that all team members need to understand and agree on the answer.
5 min – Check in with the class about the topics covered with the sim and take a moment to summarize concepts and field questions. Now is a good time to pass out a practice exam or to introduce a set of review questions.
20 min—Allow the students to work in small groups on the review questions. If you've noticed a subset of students struggling on a particular topic, you can pull that group aside to work with you or on one of the NROC lessons. If another group finishes early, you can encourage them to play a math game from the section.
5 min—Check in with students before they go. Ask them to reiterate and summarize some of the key concepts on their test. They can write on a slip of paper the topic they plan to study most and one strategy (flash cards, choose extra problems from the book, work in a study note, review online) they can use to study their chosen topic.

Instructor Notes

Warning: The simulation does not have a back button. If needed, one can reload the web page and begin again.
Students will want to test the sim--meaning give wrong answers on purpose--to see what the sim does. Such curiosity is great! I would encourage students to do so AFTER going through the sim once earnestly.
If a student answers all sim questions correctly and has no topic that needed more work, they should show you the completion screen. You can then initial their paper so you know that they did not just skip that part of the assignment and can give them full credit.
One could make an assignment asking students to design their own tutor sim. Students could present the answer options as a flow chart and even look into creating a basic simulation using linked web pages.

Rubric

If students worked individually, have them turn in their notes and problems from the tutor sim along with their completed practice test.

10 Point Scale: Tutor Sim Notes

2pts – Problems from the tutor sim are numbered and organized clearly.

2pts – All problems are written out with key data and answers shown.

3pts – Follow-up topics are identified OR student showed you they'd made no errors on the tutor sim and had you sign off on their paper before leaving that day.

3pts – All follow-up notes or problems (three per topic) are written out and solved correctly OR your signature shows they were not required to do the follow questions.

Total= 10pts

10 Point Scale: Practice Test

5pts – Practice test is complete, readable, and received. Work is clear and understandable. Give an approximate percentage of 5 points based on what percentage of problems meet this criteria.

5pts – Answers are correct. Give an approximate percentage of 5 points based on what percentage of problems meet this criteria.

Total= 10pts

Algebra Equations to Budget a School Party

2011-06-10T11:48:00.001-07:00

Having learned the basics of making and solving equations, students use their knowledge to budget for an imaginary school event. Students can follow up with budgeting for a real event or fundraiser.

Learning Objective(s)

Translate real life situations into word problems.
Translate word problems into algebraic expressions and equations.
Apply algebra principles to everyday problem solving.

Assessment Type

This middle through high school appropriate summative project from NROC's Algebra 1--An Open Course, Unit 2 Team Project: Students Rule can act as a capstone assessment in which students showcase their skills by presenting results from their algebraic comparison of different pricing plans for a school event. Alternately, the problems from the project can be used as a standalone single class assignment.

Assignment Details

Before beginning this with your class, take the time to read through "Team Project: Students Rule" completely. You’ll find three interesting class party budgeting problems followed by a larger project based on presenting findings. Decide if you want to do this as a small, single class exercise (just doing and discussing the three word problems), or do you want to follow through with a full project. If you’re just doing the problems, you can paste them into a separate document from the overall project and print them as a worksheet that omits mention of the larger project. Otherwise, you’ll want to print the whole project prompt, one copy for each group.

The timed lesson plan below assumes you’re doing a standalone, 55 minute class session using and discussing the problems provided. Suggestions for fully implementing the project (with presentation and follow up investigations) follow.

10 min– Have your students write to this prompt, “If you had $200 to throw you and ten of your friends a party, what would you do? How about if you had $2000 and threw a party for the school?” Have fun talking about options, ideas, and possibilities. While you do so, find ways to turn the ideas into equations and record these on the board.
10 min—Today’s goal is to use algebra equations to answer real life questions about budgeting. Break students into small groups and hand out the instructions and three questions from the project. Students will record their answers individually even though they are working in a group. Expect to help them pretty heavily with the first problem, but do give them as much of an opportunity as possible to solve it on their own. I recommend giving them five minutes to read the problem to each other in their groups, then check in with them to see if they’ve come up with equations to represent the budget problem. If not, help them break it down into key points. Once equations are built that make sense, give them five more minutes to solve the equations they’ve come up with, and then again compare answers and discuss results.
20 min (10 min per problem) – Students should now complete problems 2 and 3 in small groups while you circulate and help if needed. If a group finishes early, either check their answer yourself or have them check their answer against the results of another group when it finishes. If a group is completely stuck, but you’re already helping a group, you can invite the stuck group to send out one of its members as a spy to see what a more successful group has come up with. Groups that finish early (and correctly) should be invited to graph their cost equations from one of the problems on a white board or poster for others to see. Require correct axis labeling (recommend that the cost goes on the y axis, the variable appropriate for each problem on the x axis) and clarity as to which equation goes with which line.
10 min—When all groups are finished, check that all have reached the same conclusions as to which vendors would be best to choose based on the budget numbers given. Discuss any discrepancies. Extend the discussion by referring to the graphs made by the groups that finished early. Have the students explain to you under what circumstances it would make sense to choose another vendor and justify it based on the graph. Ask them if to identify the point of intersection of the two equations from their problem. Can they tell you the meaning of that point? (Its where the cost and benefit would be the same regardless of which vendor was chosen.) This leads nicely into a discussion of systems of equations.
5 min—Working individually students should summarize the results of the day in their own words. They can add this to the end of the page on which they solved their problems. I sometimes call these “Dear Me,” notes, as the goal here is for the students to take note of any realizations that they’ve had during class.

Instructor Notes

If you’re working with a middle school class or lower level high school class, having them create presentations as described in the NROC project would be great. If your class has access to a computer lab completing the presentations should be do-able in about two more class sessions. I would ungroup the students for this part of the assignment, however, so each could learn the editing skills needed to make the presentation. A rubric is provided as part of the project.
To extend this, have the students create a budget and use their algebra to compare options for a real life situation. Competing cell phone contracts can be good for this. Or, of course, they could really research and plan a school party.
If you have a class interested in service learning, this would be a great introductory project for a charity fundraiser. Student groups could discuss and research a local need and create competing proposals for a fundraising event to meet that need. When all projects are presented, the class could vote on one project to actually attempt to bring to fruition.
If you know you have a student who has really been struggling with group work or making algebra equations in general, you can have that student go online and use the NROC tutor simulation “Building a Swimming Pool” which is also part of Unit 2 and covers turning a real life situation into an equation.

Rubric

A rubric for the project itself is provided in the NROC assignment description.

Alternately, if using a 10 point scale for grading only the word problems and group work in class:

5pts -- Time on task. Students stayed engaged and on topic working to solve the problems. All students in the group supported one another and invited contributions from each other.

3pts – All three problems are written out and solved correctly.

2pts – A separate summary of what was learned in class that day is provided.

3pts – All three problems are written out and solved correctly.

Total= 10pts

Constructing the Idea of Absolute Value

2011-05-27T14:07:00.000-07:00

Using the NROC absolute value game, “Absolutely,” students construct the meaning of the absolute value symbol. They then collaborate to create a set of class notes and hypothesize about how the symbol effects solving for X in an equation.

Learning Objective(s)
• Find the absolute value of numbers.

Assessment Type
This 55minute high school level lesson introduces absolute value using “Absolutely,” a game from NROC's Algebra 1--An Open Course, Unit 2, Lesson 2, Topic 1: Absolute Value. The students act as collaborative investigators to solve the mystery of absolute value. The class activity is followed up with a written assignment to assess understanding.

Assignment Details
10min— Warm up your class with a reminder about negative numbers.

Give them a few problems to solve as bell work, or see if you can use Socratic Questioning to lure them into explaining negatives to you.

10min— Introduce the new symbol | |, but not what it means. Tell the students it’s up to them to figure out how it works.

As a framing device, the lesson can be built up as an archeological investigation in which the symbol, |x|, has been found as part of a machine left by a vanished civilization. The students must discover what this key symbol means!

If using the framing device, you can claim that there’s a number code that needs to be figured out and tested to unlock a tomb, but you’ll only have one chance to get it right! Put on the board a few absolute value problems, the answers to which will be the code.

15min— Now that students know their goal is to find them meaning of |x|, have most of your students work in pairs at computers playing the *Absolutely* game and noting what they find.

They are your “Field Researchers.”
“Absolutely” has students order integers (with or without absolute value symbols) from least to greatest by flipping around pairs of numbers mounted on gears. It provides deeper understanding than a standard introductory lecture on absolute value because of the higher level of cognitive engagement and the immediacy of feedback in the game situation.

15min continued— Meanwhile, have a few students collecting information to make a set of class notes on the white board.

They are the “Senior Investigators” creating “Research Notes.”
Clipboards and white lab coats can be fun to use.
Encourage revision and neatening of the notes. This is an opportunity to teach good note organization skills.

5min— When ready, the note takers should present the class notes to the people who were at the computers. Everyone copies the notes into their own notebooks.

Present this as a consultation between the field teams and the senior researchers to ensure consistency before attempting the key code.
Encourage more revision and feedback from the field teams.

5min—Name the symbol.

The class should propose a name for the symbol they now understand so well.
Reveal the symbol’s established name is absolute value and discuss its use.

5min— As a class they should now discuss the answers to the key code problems. It’s up to you whether you give them some sort of prize when they get it right, but use their answers as a wrap up for the class being sure to summarize the absolute value findings.

If time: Invite students to hypothesis about how |x| + 7 = 10 will be solved.

5min— Homework – Students must write a paragraph explaining what the |x| means and including a few examples of their own making.

Prompt: Your research has gained national attention and you need to write a one paragraph summary (at least 7 sentences) for newspaper publication explaining your techniques and findings. An original example must be included.
Extra Credit Extension: |x| has been found in expressions and equations in nearby tombs. In a second paragraph, develop a hypothesis about how |x| + 7 = 10 will be solved OR how a negative number placed in |2x| + x would be evaluated.

Instructor Notes

For the negative number review, the distance you walked to school can be a good example. Have +1 mile represent the distance to school and -1 mile represent the distance home. Overall, if you’re trying to figure out where you are at the end of the day, you’d add the positive mile and negative mile together to get 0 miles away from home. For tomorrow’s warm up or as part of class wrap up, you can introduce the idea that if you wanted to find the total distance walked (not just where you ended up), you’d do 1 + |-1| since direction wouldn’t matter.

Keep things fun! Props like clip boards or a few lab coats can help your researchers get into their roles.

If short on time, omit the “Key Code” questions or move them to homework.

To encourage time on task, you can have each senior investigator working on separate boards with their teams of field researchers. Class notes are then moved to a single white board as part of a peer review process. The most successful research group will receive National Science Foundation funding (or be first dismissed from class)!

Choose students who are not shy and who might have trouble with sitting still as your senior researchers. Advanced students also often like the Senior Researcher roll.

You can also have two folks working at the boards as “Research Recorders” that your senior researchers report to if you need to have roles for more students.

Pair a student who is likely to struggle with a stronger, helpful student.

Rubric

If notes are kept in a binder by each student, just mark them with a plus, check, or minus in a brightly colored pen as you circle the room. Later, a TA or you can count up the total number of pluses, checks, or minuses to calculate an overall grade for the notebook. Commonly, plus is 100%, check is 75%, minus is 50%, and 0 (earned if nothing is there or the students significantly disrupted) is 0%.

Alternately, if using a 10 point scale and collecting the notes and the homework paragraph the next day as a single assignment the following rubric can be used:

Notes and Paragraph Rubric

5pts – Class notes are neat, complete and clear. Points can be lost from this section if student was disruptive the previous day.
3pts – The paragraph explaining absolute value uses complete sentences that successfully communicate the meaning of |x|.
2pts – An original example is included and is correct.

Total= 10pts

Writing Real Life Algebra Equations

2011-05-15T15:42:00.000-07:00

After viewing the creation of an equation describing a real life situation, students create and illustrate their own algebra equation.

Learning Objective(s)

Translate real life situations into word problems.
Translate word problems into algebraic expressions and equations.

Assessment Type

This introductory lesson uses the presentation from NROC's Algebra 1--An Open Course, Unit 2 Topic 1: Solving Equations. It includes a low stakes formative assessment in which high school students create and illustrate a problem of their choice.

Assignment Details

Let your class warm to the topic of making real life algebra equations. You can ask them to give examples of when they use math in real life or give them an example of your own. Have they ever had to write their problems down to figure them out? Tell them their objective for the day is to create a real life algebra problem of their own.
View Unit 2, Topic 1: Solving Equations (4min). In this presentation, an equation is made to solve for the number of batches of cookies a person can make if they only have so much flour. In the presentation, "equation" is defined and properties of equality are also discussed. If you wish, you can have your students take notes.
Lead the class through making a few example cookie batch equations—have fun with letting them choose ingredients and see how much they can do on their own (Socratic questioning techniques can help with this). Be sure to use correct algebra terminology as you go.
When comfortable, break students into small groups (I suggest heterogeneous grouping) encourage them to discuss their favorite cookies amongst themselves to create a fun, excited atmosphere and lower apprehension about errors.
Have students take about 30 min total to make and illustrate an equation for a cooking situation of their own. They should choose what they are going to make and what variable to solve for. (How many batches, X, can they make if they have 12 cups sugar to use up and the recipe calls for 4 cups of sugar in each choco-caramel cookie batch?)
Have each group show you and each other what they've come up with after about 5-10min. This allows you to check their progress and give feedback as needed.
Circulate the room checking on whether students have successfully made a real life equation. Once you approve a group's equation, give them a large piece of paper and marker to make their idea and equation into a large poster for your classroom.
Students do not need to present a solution on their poster, unless they are done early and you need them to keep working on something! I prefer for the students to make the equation illustration with a large blank solution space. Then, the student-generated problems can be used as class warm ups in the future or student groups can trade and solve later.

Instructor Notes

Fun and innovation should be encouraged. If students would rather make up an equation about car parts and cars, have them go for it!
For a student who is feeling really stumped, the Unit 2 Tutor Sim, "Building a Swimming Pool,” can help with moving into representing word problems with symbols.
For students needing a challenge: Can they make up a real life problem that involves more than one step, a problem involving addition and multiplication for example?
For older students, a more grown up topic, could be used to make equations. Budgeting for a shopping trip or a vacation for example.
This is a very scalable activity. If you don't have much class time, individuals can just set up their equation problems in class, then illustrate at home.

Rubric

As this is a formative assessment, emphasis should be on participation and an honest attempt at completing the task assigned.

If in a rush, you can use a basic, "Plus, check, minus," scale to represent 100%, 85%, 75%. Odds are low a group will earn less than that.

Alternately, if using a 10 point scale:

5pts -- Time on task. Students stayed engaged and on topic working to create their problem and present it. All students in the group supported one another and invited contributions from each other.
3pts -- Students created a neat and clear equation representing their problem and incorporating feedback you gave them. A word problem is written out, variables are defined, and the equation is shown.
2pts -- The equation is completely correct.

Total= 10pts

Hippocampus Correlation
Algebra 1--An Open Course, Unit 2 Topic 1: Solving Equations

Musings on Math and the Art of Problem Solving

2009-09-12T06:46:00.000-07:00

For math, as for most topics, there are those students who love it and can't get enough, and there are those who find math as mysterious as a foreign language written in an entirely new alphabet, who are constantly frustrated and annoyed at having to learn something they find so perplexing.

In an ideal world, we could present students with math concepts and ideas from an early age, in a fun and engaging way that gives them the gift of numerical competency and a more comprehensive understanding not just of the how-to's of math, but of the more conceptual whys of math.

Like artists who learn to see colors and shapes in artistic ways, children and young adults can be taught to notice math in everyday activities--apportioning a pizza or pie, building a birdhouse, mixing up a recipe, balancing a checkbook, understanding compound interest on a savings account--exposing it not as a foreign language, but as a vital subject that permeates our daily activities in ways too numerous to, well, count.

(Note: a good resource for the youngest students is "Kitchen Table Math," a book by Chris Wright, Ph.D., and available through the Art of Problem Solving web site.)

It is the conceptual understanding of the whys of math that helps students to see problems in a larger context and, in later academic work, to be able to solve multi-step problems using many strategies that work in concert. Rote memorization of formulae or "to-do" steps may help a student get some answers correct on a specific quiz, but do less for their larger problem-solving abilities. It is also this ability to strategize and logically solve math problems that can spill over into solving problems that crop up in engineering, chemistry, biology, computer science, and other topics that American students seem to find too difficult to bother with.

Gifted math teachers understand that it's the relevance to a student's everyday life experiences that can trigger the sometimes elusive "ah-ha!" moment, which indicates they finally understand how to solve a problem.

In an interview I did on my radio show last week with Nigel Nisbet, a math expert with the Los Angeles Unified School District who is currently working on a math project for HippoCampus, there is a discussion of how students are not necessarily mastering the basic math skills (addition, subtraction, multiplication, division, etc.) taught in the early grades. This lack of mastery tends to first be revealed when students start struggling with algebra. Nigel argues that those standardized tests given in the elementary grades really can help parents, teachers, and schools see how well students are mastering those basic topics. Catching this early leaves enough time left to remediate if a deficit is found. (Listen to the interview here.)

My take from the interview is that if a student manages to bump along with a half-baked understanding of math until they hit algebra, it's going to be much more difficult to go back and fill in the missing information, and by then it's much more likely that the student will be developing an aversion to math that could haunt them throughout high school (and possibly into college).

For the math-loving student who either isn't challenged at school, or is challenged and is looking for like minds with whom to collaborate to learn new strategies, I have a recommendation. Richard Rusczyk, whose own gifts with math are surpassed only by his gift for teaching math to others, has a whole online following that includes many of the brightest young math minds in the country. Richard and a core of other dedicated yet unconventional teachers have taken matters into their own hands, creating an extracurricular community of math-lovers who relish digging into a juicy problem and finding creative and clever ways to break it down and solve it. His company and online community are aptly named, "The Art of Problem Solving." I've done several interviews with Richard on the Countdown to College Radio Show, three of which can be found here.

Note: Guest blogger Beth Pickett is the producer and host of Countdown to College Radio Show, and as such as interviewed numerous teachers and math experts on the current state of math in the U.S. today, and the opinions she has expressed here are those she has formed after listening to those who are on the front lines of teaching math. She is also the primary blogger for the HippoCampus Countdown to College blog.

Elementary Math Education

2009-06-14T20:25:00.000-07:00

So far in this blog I have really focused on high school education. I want to talk this week about elementary school math education, especially on how the Internet can help younger students grasp early mathematical skills.

I think the Internet is a great place for kids to practice their skills through the use of games. I searched to find some interesting math sites for younger students. It is important that students are monitored and that the sites are reliable, since we are dealing with a younger audience.

First, I found the website King's List of Math Activities. This is a great list of websites that deal with math activities, many for the younger audience. For example, there is an addition game with frogs from the BBC website that would be very fun for students to try.

APlusMath is another great website. It has a link to a game room that allows students to choose from a list of math games. There are even flashcards and math word puzzle games available.

CoolMath is a site that I have talked about before. This site has many fun math games from younger students up through Algebra.

The Classrooms website also has a great list of online activities for students, especially the younger elementary level math students.

I think it's important that we make math education as fun as possible for students from the very beginning. We all know that math can be a difficult and scary subject, so why not start the process on a good note. Students learn when they are playing and having fun. The Internet can be a great place to start that process.

Online vs. Brick and Mortar Schooling

2009-06-01T07:50:00.000-07:00

Lately I have had a lot of questions about why students and parents would choose online over traditional brick and mortar schooling. I am no expert, but I have worked in both arenas and have my own opinions about the growing trend of online schooling. It is no secret that online schools are popping up in every state in great numbers. I project manage math curriculum for a company that writes online courses. I've also taught in both a traditional high school and a brick and mortar high school.

So why would a student choose an online program? I think its important to understand that there are different types of online schools. Some students choose a full-time program where they take all their classes online. Some students are involved in supplementary programs where they might take one or two classes that are not offered in their traditional high school. This is often the case in smaller, rural schools that can't staff teachers for every subject. There are also hybrid schools that are being developed where students take a portion of their courses online and a portion of their courses in a classroom. I believe it is this hybrid classroom that will become the most popular. This combination of convenience mixed in with the social aspect will be very appealing to students and parents.

From my experience in the online schools in Colorado, we have a reverse bell curve. We have a lot of very high achieving students who like working on their own, like to be able to set their own schedule, and like working at their own pace. They are able to take classes that may be at a higher grade level then what they are used to and they feel comfortable working in a home environment. We also have the low achieving students, who may need more time to grasp subjects, and feel that working at home is the best alternative. We don't have as many of the students that fall in the middle of the bell curve. It is very important to note that this is the not the case in all states. This just seems to be the demographic in the state of Colorado.

Parents have told me in the past that they chose online education because if fit the personality of their children. That is an important aspect in this whole process. Online education is not for everyone, but the students that thrive, really enjoy the opportunity to work from home, in a safe and encouraging environment.

I hope that answers some of the questions. Please let me know your thoughts.

HippoCampus

2009-05-18T22:30:00.001-07:00

Throughout the time I have been writing this blog, I have referred back to the HippoCampus website. I want to do this week's blog on what is available on this website. I feel like this website has a lot to offer students, especially in the discipline of mathematics.

Entire classes are available on the website. A very easy to follow menu is presented for each class that divides up the course based on the concepts. A student clicks on a link and is taken to an interactive lesson.

For example the following is the lesson on number properties in Algebra 1.
Another example is a lesson on limits from the Calculus class. The Calculus class is even subdivided into AP and non AP, AB and BC, General and Introductory. Each level is catered to the level of the learner.

I have used these classes in my own course writing, directing students to these pages for more information, more practice, and remedial work. Students have been happy with the lessons and feel like this is a great addition to their curriculum. Our overall goal should always be to give our students as much information as we possibly can, directing them to helpful websites, good examples, and the discovery of learning.

The HippoCampus website has Algebra, Algebra (Spanish), Calculus, Calculus (Spanish), and Statistics.

Rubrics in Mathematics

2009-05-05T08:06:00.000-07:00

As I write more for the online curriculum world, I find it is extremely important to include rubrics with any assignments that are created. If you think about the environment that online students work in, they are home, assignments are done on their own, and it may take teachers a while to return phone calls and emails, students really need to know how they are being graded. In a brick and mortar environment, I hardly ever used rubrics. Students did homework from the book, they completed short answer assignments, and occasionally they would complete a project that required a rubric. The difference was that I was always there to answer questions and clarify any grading issues. That is not the case in the online environment.

I don't know about you, but I have a very hard time writing good rubrics. I either don't explain enough or I go into so much detail that my students get lost in all the wording. I found some helpful websites that can be used to write rubrics. I wanted to share these with you so that if you are rubric challenged like myself, you know there is a little help out there.

Rubrician.com gives you a list of websites that offer help and templates for rubrics. I found that this site is a great place to start if you are writing a rubric.

Rubrics4teachers.com has a great list of sample rubrics for all math levels and math classes. This is a great place to get ideas.

Mathforum. com also has a teacher to teacher post that has a list of sample rubrics. This is another great place to get some ideas.

Finally, I found this great website from missioncollege.org that gives you steps to creating a rubric, examples of different kinds of rubrics, and shows examples for future use.

Do you have any good websites that can help us with writing rubrics?

Sampling Methods

2009-04-26T18:39:00.000-07:00

I recently wrote a statistics unit in an Algebra 2 class. Looking at the state standards I noticed that they include sampling methods in statistics, which is something I don't ever remember being taught when I was in high school. The statistics chapter was added since many of our high school students will not go on to take an AP or non-AP Statistics class, but the knowledge is important. The sampling methods was a subject I had to do some research on in order to include in the unit, but I thought it had some amazing results.

It occurred to me that students talk about measures of central tendency, variance, standard deviation, etc, but until college, I was never taught how the information I used for these calculations showed up on my plate. I was glad to see that sampling methods made their way into our content standards.

The following are some really helpful websites that I found that can teach students the difference between the sampling methods and why these differences are so important.

The first is a statistics glossary page that has links to different kinds of sampling methods such as random sampling, cluster sampling, etc.

The following is a slightly weird site. The Australian Bureau of Statistics has a great website outlining all the different types of statistics methods.

The Star Trak website offers an AP Stats tutorial on sampling methods.

The stat.yale website has a great one page site that goes through all the sampling methods. It is concise and very informative.

Any other helpful websites that you found?

ACT/SAT Practice Questions

2009-04-19T19:22:00.001-07:00

With the ACT and SAT exams coming up, I wanted to give students and teachers some websites that have good review and practice questions. Most students are quite nervous about the math portions of these exams, and to be honest, they are not easy. We want to make sure we give our students the best chance on these exams.

The following are some great websites to share with your 11th and 12th graders:

The website actstudent.org has a long list of sample math questions from previous ACT's. I really like this site because it gives students the answers and also shows them why each of the incorrect answers will not work.

Kaplan test prep also gives students some sample questions with answers. The reason I like this website is that each of the problems is worked out step by step.

The website act-math-practice breaks down the different math subjects which should allow students an opportunity to find out which areas are weakest for them.

I also found some good SAT math questions. Check out the following websites:

The college board website has great SAT math review questions.

The test preview website allows students an opportunity to check their math skills before attempting the test. Scroll down to the the bottom for a breakdown of concepts.

As always, please share your finds with the rest of us.

Math Videos

2009-04-12T22:29:00.000-07:00

If you have ever taught online or if you have ever wished you could send a recording of yourself home with students, there may be an answer. Many online programs use videos to help students understand material. Many are paid sites but there are a few good ones out there that are free to students. That is the focus of the blog for this week. We are always searching for new ways to help students, why not give them as much information and help as we can.

The following is a list of websites that provide students with free videos and helpful tutorials in mathematics:

1. The mathplayground has an extensive list of topics with short tutorials that explain a list of concepts to the students.

2. Youtube and Teachertube are also great places to get videos. I recommend that teachers track these videos down for students to check for validity and for appropriateness.

3. The University of Colorado at Denver has a website with streaming videos about mathematics.

Can anyone share any more finds?

Math Anxiety

2009-04-05T09:42:00.000-07:00

I was talking with my husband yesterday about last week's blog, math strategies. He commented that he has always had math anxiety, even though he is a successful engineer. My first thought was, didn't you have to take a lot of math classes? The answer was yes, and he hated every single one. Now that is sad. For those of us that teach math, we might not be able to understand that kind of anxiety. The numbers, formulas, and concepts come very easily to us. For others, the thought of going into a math class or working on a set of math problems sends them into a panic. I never particularly liked reading Shakespeare, but I didn't panic when I opened up one of his books. Only math seems to produce this kind of behavior in people.

I took some time to look for options that we can give students when they are feeling this kind of anxiety. We are coming up on ACT/SAT/AP exams, and thought this may be a good time to produce a list of websites that students can visit and get help with some of the anxiety they are feeling. Though this is not a cure, it is another place for students to turn without having us keep telling them to relax. My husband never did.

The first place I found is from math.com. This website offers a list of common math anxiety symptoms that students may have, and offers students options to cope with these anxieties. I thought the best one was "ASK FOR HELP". I'm sure as teachers we often wonder why students don't just ask for help. After talking with my husband, it seems that he was embarrased that he would look stupid. He would rather suffer in silence.

The next was a help for math anxiety website that I came across. The reason I liked this website is that they have a link to why math is different then any other subject and offer exercises for students to complete to help them with their math anxiety.

I also found this top ten list from mathpower.com that gives students a short list of things they can do to overcome math anxiety.

The next helpful website I found is from math.about.com. This one page site offers the reasons behind math anxiety, the misconceptions about math anxiety, and some helpful solutions.

The final website I found was from collegeboard.com. This is just a short one page list of helpful solutions to math anxiety.

Please share other ideas that you have found to help students cope with math anxiety. If we can stop it in grades K-12, students will enjoy math more and be more successful in college.

Math Test Taking Strategies

2009-03-29T19:30:00.000-07:00

I know that one of the most stressful times for students is test taking days. When you say “math test” you can almost see the terror in the eyes of most of the students. I have spent some time thinking about why that might be. Students read a book in an English class and feel comfortable answering some question. On the other hand, they learn a concept such as graphing lines, and all of a sudden feel like there is no way that they can answer any questions. This fear and anxiety triples as students think about standardized tests such as the AP Exam, SAT, and ACT.
After really thinking about this in my years of teaching I decided that math is a pretty black and white subject. Students either get the question right or wrong and that scares them. They can interpret a book, but either you graphed the line correctly, or you didn’t.
There are a couple of test taking strategies that I have often shared with students.
1. Go through old exams, quizzes and homework.
2. Identify your weak areas and work on those first.
3. Eat a good breakfast the day of the test.
4. RELAX!!!!!!! It is one of many tests you will be taking throughout your life.
5. Glance over the test when you get it and come up with a quick plan.
6. Manage your time. Don’t spend too much time on any one question.
7. Read the questions very carefully. If you don’t understand the directions, ASK!!!
8. If you don’t know a question, then skip it. Do not dwell on it.
9. Did I mention relax!!!!

I also found some good websites to visit with test taking strategies. The following are general test taking strategies for math tests:
Website 1
Website 2
Website 3
Website 4
The following website is geared towards the AP Exam:
AP Test Taking
The following are geared towards the ACT and SAT Exams:
Website 5
Website 6
Website 7
Website 8

See you all next week.

Calculus Again

2009-03-22T15:45:00.000-07:00

I will be starting a Calculus project in the next couple of weeks and I was thinking what it was like for me to take Calculus back in high school. Internet was just starting and it was not readily available in schools. I had a very old and clunky TI-82 Calculator. I think it may have been the first graphing calculator they made. I think of what the students have available to them now, and I marvel. I know some teachers think that students rely on technology too much, but I think we need to provide students with different options. Of course students should be able to take a derivative, but does it make sense at some point for students to struggle though large derivatives if we are testing another skill?

I looked through the internet to find some helpful sites for students and for teachers.

The first is a complete Calculus class on Hippocampus.org. There is also a Spanish version of the course which is wonderful. Math is supposed to be the universal language so it is wonderful that language is not a barrier in this case.

I also found a wonderful website with an extensive list of graphics that can be used by students and teachers.

Cow.math offers a list of small calculus modules. These modules can be used by students as extra practice or students can use them for remediation purposes.

I also looked for websites to help students with their TI-83 calculators. The website prenhall.com has an incredible table of contents for every TI-83 calculator that has been made.

For those students that are interested in the history behind Calculus, I found a great website that outlines how we came to be studying this subject.

As always, please share your finds with the rest of us and see you next week.

Graphing Linear Functions

2009-03-15T22:09:00.000-07:00

Just recently I was working with a student on graphing linear functions during a tutoring session. She asked if there was anything she could work on to continue with her practice. This prompted me to get on the computer and do a search for some lessons and interactives that might help her.

I started my search with Hippocampus.org. The full Algebra 1 class has an entire chapter devoted to graphing linear functions. This is a wonderful place for students to start if they are having trouble with the concept or need a refresher.

I also came across some great interactives that can really help students. The first is an online graphing utility that I have used in the past called GCalc. This will allow students the ability to graph any function, including linear functions.

The second is a Shodor Interactive that will also allow students to graph linear functions. You may ask why give students an opportunity to graph functions with technology instead of with paper and pencil methods. I am a huge proponent of doing both. Students need to know the mechanics of graphing functions on their own, but we should also expose students to the technology that is available to them.

I also found two sliding interactives. Students are able to move a slide that changes the slope and y-intercept of an equation and they can see how these changes affect the graph of a linear function. One of these interactives can be found at mathsnet and the other at id.mind.

Finally, I found a site that offers free graph paper. I think that is also important when students are practicing these skills.

Geometric Proofs

2009-03-09T22:38:00.000-07:00

I believe hands down the toughest subject for students is proofs. There are a few reasons why that is the case. Number one, proofs tend to be taught very early in a geometry course. There is no way around it, but it causes students some problems. You have to teach proofs in order to work with theorems in a Geometry course, yet you teach proofs before students know a lot of definitions, postulates and theorems. The subject seems to come out of nowhere for students and causes some stress.

I want to give everyone a few websites that students can refer too while working with proofs. Hopefully this extra information will help students transition their thinking from Algebra to Geometry and especially geometric proofs.

Sparknotes have a great set of pages that can really help students. The website includes a table of contents that is very thorough.

Proofs for Dummies is another good website that seems to be affiliated with the popular books.

Another great website is How to Write Proofs. Again, there is a comprehensive table of contents that includes direct proof, proof by contradiction, counterexamples, just to name a few.

The Library of Math has a huge collection of proofs that students can use as a reference. Sometimes students need a few examples to see how proofs are constructed.

If anyone has any other websites to share I would love to see them. I have written and taught Geometry and this is always a difficult subject for me to explain to students well. Share your experiences in the classroom as well. I know I will learn a lot.

Matrix Multiplication and Solving Systems of Equations

2009-03-03T00:14:00.000-08:00

Last week we talked about the basic operations of matrices and how to find a matrix inverse. This week we will continue to our discussion of matrices by talking about matrix multiplication and solving systems of equations using matrices.

I recommend starting the discussion by talking about when matrix multiplication can occur. Remember that if two matrices can be multiplied, their inner dimensions are the same. For example, if matrix A has dimensions 2 x 4 and matrix B has dimensions 4 x 5 then the matrices can be multiplied since the inner dimensions of 4 are the same. The product matrix will have dimensions 2 x 5. It is important to start with this fact so that students do not spend time working on problems that can’t be completed.

The following are some helpful websites that I found that can help students and teachers work through the concept of matrix multiplication.

The Math Warehouse website has a great step by step explanation of matrix multiplication. There are great tutorials as well that take students through the entire process.

Most graphing calculators will do matrix multiplication for students but we can’t expect all students to be able to own or use a graphing calculator. There are some great online calculators such as the Easy Calculations Calculator and the Analyze Math Applet. Students can use these sites to check their work.

Solving systems of equations is also a concept that utilizes matrices, especially when working with three equations and three unknowns. Matrices are the easiest way to solve these types of problems. Spark Notes and Cliff Notes both have great summaries of how matrices can be used to solve systems of equations. I also found a great tutorial on how to use graphing calculators to solve systems of equations using matrices.

As always, please share your finds with the rest of the readers. Next week we will talk more closely about proofs.

Matrices

2009-02-24T07:15:00.000-08:00

I was recently working on writing an Algebra 2 class when I came across the section on matrices. I thought that this would make a great topic for our weekly blog. Matrices are so useful and interesting for students that I thought it was a good idea to share what I have found so far online. There are a lot of great interactives to help students with matrices. Also, not every student will have access to a graphing calculator. There are some great tools to help students find inverses of matrices and to solve matrix equations.

Basic Operations with Matrices
Basic operations include adding and subtracting matrices and scalar multiplication of matrices. The following are some great websites to help students understand these concepts.
I would have students start with this short lesson on adding and subtracting matrices that includes practice at the end of the page. Purplemath.com also has a great lesson on adding and subtracting matrices. There is also a lesson at purplemath.com that talks about scalar multiplication. I also recommend looking at the tutorial on scalar multiplication. It is a great step-by-step lesson that can help students through this concept.

Inverse of a Matrix
I recommend starting with a few lessons on how to find the inverse of a matrix. Once again purple math has a very good lesson on finding the inverse of a square matrix. I also recommend this lesson finding the inverse of a square matrix. I found this interactive from easycalculation.com to find the inverse of a 3 x 3 matrix. Analyzemath.com also has an interactive that helps students find the inverse of a 3 x 3 matrix.
For those students that have a graphing calculator I recommend this graphing calculator instruction site to help them navigate their graphing calculator when it comes to matrices.

Next week we will focus on matrix multiplication and solving systems of equations using matrices.

Taxicab Geometry

2009-02-16T20:57:00.000-08:00

In an effort to continue to talk about non-traditional subjects in math, I want to do this week’s blog about taxicab geometry. I first became familiar with taxicab geometry while taking a graduate course. I then taught the subject in an online classroom. The first thing I noticed, was the excitement of the students at seeing something that was not traditional. I wanted to share some websites that I found that can help you get started in adding taxicab geometry into your classroom.

You can start with the Think Quest website called Enough of Euclid, Taxicab Geometry. This is a great website to start your discovery. Make sure to click on the application link at the bottom of the page.

The Taxicab Geometry website is also a great starting tool. It goes through the basics of taxicab geometry including the definitions and examples of taxicab distance and taxicab circles.

The taxicab geometry overview website is another great tool to get you started with incorporating taxicab geometry into your classroom.

Taxicab geometry and trigonometry website links taxicab geometry to trigonometry which is a great cross subject link.

Finally, I found some very good projects that you can incorporate into your classroom. The Taxicab Treasure Hunt is a great interactive website that students can go through on their own. The taxicab project would be a great project to use in a brick and mortar classroom.

As always, please share your finds with us and see you next week.

Non-Euclidean Geometry

2009-02-08T20:31:00.000-08:00

I wanted to talk about something a little different this week. I have noticed that many of the state standards now ask teachers to show students the differences between Euclidean and Non-Euclidean Geometry. Non-Euclidean Geometry is an interesting way to help students see the many aspects of math. When I have taught these concepts I find that students are very interested in Non-Euclidean Geometry and have a renewed sense of excitement as the subject is being taught.

I have compiled a list of useful websites to help you think about how to add Non-Euclidean Geometry into your lessons. Enjoy and please add to the list if you find any more interesting sites.

I found a very useful website describing Non-Euclidean Geometry called What is Non-Euclidean Geometry? This may be a great place for students to start their explorations.

Spherical Geometry is a great way to introduce students to Non-Euclidean Geometry. The Geometry of a Sphere website has an extensive table of contents that goes through many topics related to Spherical Geometry. I also found a Spherical Geometry Demo that is a great visual and interactive representation of Spherical Geometry. For those of you looking for an in-class activity check out the Earth Geometry Lesson. For students that are really interested in the topic teachers can discuss the Spherical Pythagorean Theorem.

You may also branch out to Hyperbolic Geometry and I recommend a great applet to start off the discussion.

Enjoy these great websites and please continue to share your finds.

Geometry

2009-02-04T08:09:00.000-08:00

All of our students will need to take a Geometry class. Whether online or in a classroom, Geometry is very visual and very interactive. This may be difficult for some students who are more comfortable with concrete math subjects such as Algebra. We can use the tools that are available online to help our students work through some of the difficulties they may face in Geometry. These interactives and lessons also allow us to teach concepts online that may be difficult to do without manipulatives. For example, constructions still need to be taught online and without interactives it would be hard to do.

The following is a list of online sources that can help you with Geometry. Please continue to add to this list. Your contributions are greatly appreciated.

Geometry Lesson Plans:
There are a lot of great lesson plans available online. The Cut-The-Knot website had great lesson plans, interactives and articles about Geometry.
The Math Forum website had great lesson plans.
The Teach-Nology website also has a great list of Geometry lesson plans.

Geometry Interactives:
The Matsnet website had interactives on points, lines and planes.
The NLVMU website had online manipulatives of all kinds.
The Shodor Interactive website has interactives based on Geometry. You just need to scroll down to see the Geometry list.

Applications of Geometry:
Hippocampus.org has a list of all the ways that Geometry is used in other subjects.

Please continue to add your ideas with the rest of us.

Statistics

2009-01-26T18:15:00.000-08:00

This week’s topic is a completion of last week’s discussion on Probability. If we talk about Probability, then we need to add a discussion about Statistics. The two topics are related and overlap frequently.

It is difficult at times to run statistical experiments in both the online and the brick and mortar setting. It is also difficult to convey to students the nature of Statistics and its importance in real world applications. There is a wealth of information and activities online that can make the study of Statistics more applicable and more interesting for students. I recommend that you work from the following list of useful websites and build on the library. Please share your finding with the rest of the readers.

1. Online Textbook
Hippocampus.org has a full Statistics class online. This course includes content, tutorials, examples, and interactive activities. Students can use this tool in conjunction with their regular class. This is a great tool for remediation, extra practice, or as an entire course.

2. Interactives
Shodor.org has an entire list of interactives that students can use for all their math classes. Scroll down the list until you come to Statistics.

3. Graphical Displays of Data
One of the most important concepts in Statistics is graphical displays of data. This includes bar graphs, frequency diagram, histograms, etc. Cliffnotes.com does a great job going through all the different ways that statistical data can be represented. There is a short but thorough lesson on each type of tool.

4. Measures of Central Tendency
The study of measures of central tendency is also very important in Statistics. The mean, median, and mode make up the basics in the collection and analysis of data. The following three websites are very helpful in the study of mean, median, and mode.
Website 1: Connexions
Website 2: Tutorial from Virginia Tech
Website 3:

Please continue to share your own finding with the rest of us. Next week we will discuss Geometry.

Probability

2009-01-19T09:42:00.000-08:00

This week I want to discuss probability and how the interactives and lessons on the internet can help in your online and brick and mortar classrooms. Probability is an exciting and hands on way to teach mathematics to students. Often, mathematics is a subject that is taught for the sake of concepts in the future and students do not have a lot of buy-in to the lessons being taught. Probability makes mathematics very concrete and at the same time interactive which allows students an opportunity to experiment and get involved. I have taken some time to find some interesting and useful online sites that can help students. As always, please feel free to share other books, sites, and resources that you have found or used in the past.

Probability Lessons

The lesson from Math Goodies is a very concise and quick overview of probability which also includes interactive examples. This is a great site for students who need extra remediation or are interested in more practice.

About.com has two very good lessons on probability. The first is a comprehensive table of contents that goes through probability from the very basic all the way to much more complicated. The next is a lesson that is entirely geared for high school and middle school students.

Interactives

The internet also provides a wealth of interactives that allow students to run experiments easily and frequently. Shodor.org has an interactive spinner that allows you to change the area of each of the regions, therefore changing the probability. Illuminations from NCTM has an adjustable spinner as well. They also have directions and lesson plans for a probability game called Sticks and Stones. This is a great idea for a brick and mortar classroom.

Links to Other Disciplines

As always, it is important to link probability to other educational disciplines so that students understand that these concepts are all related. Hippocampus.org links probability to Calculus and to Evolution. These short videos can be shown in a classroom to reiterate to students that math does not happen in a vacuum.

I hope that these sites are helpful. Please continue to share your findings.

Graphing Trigonometric Functions

2009-01-11T07:37:00.000-08:00

This week’s topic is graphing trigonometric functions. When it comes to graphing, the internet provides a wealth of interactives and tutorials that can be utilized by students struggling with graphing. I want to start out by saying that I do not advocate students skipping paper and pencil methods of graphing, but I’m sure we have all noticed that some students have a very difficult time visualizing graphs. The websites I will provide this week should be used as supplemental materials.

When students learn the affects of amplitude, period, and phase shift on trigonometric functions they often have a hard time conceptualizing how changing one of these parameters affects the graph. I have watched students plug in numbers into an equation instead of analyzing the equation for changes. I want to provide you with interactives and graphing utilities that can help students easy change amplitude, period and phase shift to practice how these changes affect the graphs of trigonometric functions.

1. The GCalc graphing utility can easily graph trigonometric functions if students do not have graphing calculators. I recommend using this utility to have students quickly check their work and to analyze graphs for changes in amplitude, period, and phase shift.

2. The website from Illuminations at NCTM allows students to instantly change the amplitude, period, and phase shift to see the changes to sine, cosine, tangent, cosecant, secant, and cotangent functions.

3. The website from Analyze Math allows students to work on separate interactives for all the trigonometric functions. Scroll down to Trigonometric Functions and then choose an interactive.

4. The website at Interactive Mathematics provides students with a lesson on graphing sine, cosine, and tangent functions while providing an interactive for each of the lessons.

I hope that these websites and interactives will be helpful in your classroom and for your students.

Factoring

2009-01-08T21:36:00.000-08:00

Like fractions, factoring is a skill that students need to master in order to build other mathematical concepts. It is also one of the skills that is hard for some students to grasp. This week’s blog is intended to give you some possible lessons and internet sites that can help students with factoring and give you some ideas and resources in your own classrooms.

The website Hippocampus.org has free online lessons that cover factoring.

1. Factoring Trinomials

2. Factoring Using Tiles

3. Factoring Difference of Squares

There are also tutorials available online that can help students work through all the different aspects of factoring. For example, the West Texas A&M University website Virtual Math Lab has a tutorial that includes helpful videos and example of factoring from Greatest Common Factors to factoring trinomials. At the end of the tutorial is another helpful list of websites that can be accessed for extra help..

Another helpful website comes from the Oswego City School District. This link takes students to a table of contents that covers factoring form the simplest to more difficult examples.

I believe the greatest problem that students have with factoring is lack of practice and confidence in their abilities. The websites above give students an extra set of lessons and practice problems that can help them become more comfortable with factoring.

Multiplying and Dividing Fractions

2008-12-28T11:14:00.000-08:00

Last week we talked about adding and subtracting fractions, which most of us know is a very difficult subject for students. This week I want to add to this discussion and give some helpful websites to help students multiply and divide fractions. I think it is vitally important that students be proficient in working with fractions on all levels and with all types of computations. I have found in the past that students will know how to do a problem conceptually, but will struggle with simple computations, mostly involving fractions. Students need to be comfortable with these common computations so that they can start thinking of mathematics on a more difficult and conceptual level in higher classes.

The following is a list of websites that include games, interactives, practice items, and lessons that help students with multiplying and dividing fractions. Please feel free to add to this list as you explore this subject further. Also, please share those findings with the rest of us.

Lesson on Multiplying and Dividing Fractions

This website has two lessons. Lesson 1 involves multiplying fractions while lesson 2 works through dividing fractions. Both lessons have an extensive list of practice problems with answers that students can use to check for understanding.

Math Workout: Multiplying and Dividing Fractions

These two websites are from the same source. These websites allow students to practice and work on dividing and multiplying fractions.

Multiplying and Dividing Fractions by Using Circles

These sites allow students to see the multiplication and division of fractions visually by using circles.

Dividing Fractions

This site allows students to enter their own fractions and then shows a detailed explanation of how the student’s problem was completed.

Math Playground: Multiplying Fractions

This website allows students to practice their skills on computer generated problems.

These are just a few of the activities that are available to students online. I hope that this is helpful in getting you started with your search. Next week we will move into another difficult subject, factoring.

Adding and Subtracting Fractions

2008-12-21T19:24:00.000-08:00

Working with fractions is one of the most difficult concepts for students. This week I am going to go through some interactives and web tools that will help students add and subtract fractions with both common denominators and without common denominators. In my experience students have problems with adding and subtracting fractions in all the levels of mathematics, from Math Foundations to AP Calculus. It seems to be a concept that students don’t grasp early on, and then struggle with throughout their math education. The following is a list of interactives and activities that can really help students work through their struggles with adding and subtracting fractions. I believe adding a visual element to this concept will help students tremendously. This list is just a few of the interactives that I have found online. Please feel free to look even further and share your findings with all of us.

Fractions Tutorial

This is a tutorial that goes through many concepts dealing with fractions, including adding and subtracting fractions.

Adding and Subtracting Fractions Interactive

This interactive shows a visual representation of a fraction while working through addition/subtraction problems with common and unlike denominators.

Create Your Own Game

This interactive allows students to create their own game depending on what skill they want to practice.

Catch the Fish Game

This game asks students to select the correct answer to an addition/subtraction problem while a fish scoops the correct answer up.

Math Splat Game

This game asks students to select the correct answer as they are riding in a boat. The answers are held up by bugs flying by the window of the boat.

Fractions Quizzes

This is a list of fraction quizzes that students can complete to practice their skills.

I hope that this is helpful for students who are struggling with adding and subtracting fractions. Next week we will work through multiplication and division of fractions.

Adding and Subtracting Integers

2008-12-14T21:26:00.000-08:00

One of the most difficult concepts that students struggle with in mathematics is adding and subtracting integers. This happens for students from math foundations all the way to AP mathematics. Like fractions, this is a subject that students do not always grasp at the beginning of their exploration of mathematics, and therefore struggle with throughout their classes.

I wanted to offer some online resources that I have found to help students work through some of the difficulty of adding and subtracting integers. The first is a great overview of how integers fit into the scheme of all numbers. This tutorial can be found on HippoCampus.org and should be the first place a student starts in their exploration of adding and subtracting integers.

Next, there are some great free videos that can help students. The first two videos talk about adding and subtracting integers using tiles while the next two videos talk about using a number line.

Adding Integers Using Tiles

Adding Integers Using a Number Line

Subtracting Integers Using a Number Line

Students can also practice their skills online. The following is a list of interactives that can help students:

Algebra Tiles Interactive

Number Line Interactive

Adding/Subtracting Problems

Adding/Subtracting Extra Problems

Adding and Subtracting Card Game

I hope these are helpful for your students working through adding and subtracting integers.

History of Mathematics

2008-12-06T13:02:00.000-08:00

As we move forward in our discussion of how we can use the internet to help us teach mathematics, this week I wanted to concentrate on how we can link mathematics to history. Our students learn math in a bubble, and they usually have no link to mathematical history or to famous mathematicians. This is a shame because there are a lot of famous problems and historical topics that can be linked to mathematics.

I think the best way for students to appreciate the history behind the study of mathematics is for them to either write a paper or do some research on the topic. In my own investigation of the history of mathematics I came across some good resources that I want to share with you.

1. You may ask students to do research in terms of important history topics. These topics can include the core classes we think of now, such as Algebra, Geometry, or can branch out to analysis, astronomy, and physics. They can also include a broad overview of the history of mathematics.

2. Students can also base their research on famous mathematicians even narrowing down their research to searches such as famous woman mathematicians.

3. Finally, students can do research on famous unsolved problems. Though some of these may be hard for students to understand, the discovery portion of the task and explaining a difficult problem to others is extremely beneficial.

I highly recommend adding this research component to any class. I don’t think we can afford to continue to teach mathematics without giving our students some background information.

Tessellations

2008-11-28T13:36:00.000-08:00

In an effort to continue giving you good ideas about how to use the internet to help with teaching mathematics, I wanted to use this week's blog to talk about tessellations. Tessellations are an interesting, hands-on, and informative way to teach mathematical concepts. Tessellations are pictures or tiles which are organized on a surface in a symmetrical way without leaving gaps or overlapping. They can be created and analyzed by students either online or through paper and pencil methods. I wanted to give you some ideas about how you can use these tessellations in your math class.

Of course we can use tessellations to enhance our knowledge of mathematics. For example, tessellations can be used to discuss symmetry. We can also look at the symmetry of animals which links tessellations to science. Tessellations can also be linked to geometry.

Oftentimes we have a hard time linking mathematics to other subjects. Tessellations can be used to make that link. For example, tessellations can be used to link to history. Students are often asking why they need to learn mathematics. A link to another class can help students answer that question. One idea is to use Webquests to link tessellations to history, art, and mathematics.

The internet also provides ideas and applications to teach tessellations. For example, Shodor Interactive allows students to quickly tessellate different geometric figures and allows students an opportunity to analyze the angles associated with the final tessellation. Students can also create tessellations using Paintbrush or Microsoft Word.

I hope that these activities and websites give you some good ideas on how to teach tessellations. Be creative and your students will follow.

Have a great week.

Helping Students Graph

2008-11-21T09:47:00.000-08:00

Welcome to the math blog. My name is Natasza Krajcovic and I will be contributing to this blog on a weekly basis. I have been teaching math at the high school level for six years, both online and in a traditional brick and mortar setting. I have also been writing curriculum for two years. This curriculum is used in online schools all over the United States. I hope that I can provide you with helpful information that you can transfer to your own classrooms, whether they are brick and mortar or online.

We have all witnessed students in our classes struggling with graphing, whether it is linear equations, inequalities, even up to trigonometric functions. Some may not be conceptually able to see graphs and others struggle with the mechanics of creating these graphs. Though I am not advocating getting rid of paper and pencil methods of graphing, I do want to give you some online options that can help students.

The Algebra 1 course on HippoCampus Algebra offers great tutorials to help students get through subjects such as the coordinate plane, graphing slope, graphing linear functions, and countless other subjects. Students can click on a link and hear a tutorial, see the voice script, and watch the tutorial as it progresses. There are also practice questions students can work through to check their understanding.

Some of our students may not have graphing calculators. If students need a graphing utility they can use GCalc Online Graphing. This free online option will graph anything from straight lines to sine graphs. This is a valid option if students do not have graphing calculators.

Finally, if students need to make scatter plots they can use Shodor Interactive Scatter Plot. This free utility allows students to graph points and changes the scale as they add points. They can even name each scatter plot to fit the problem they are working on.

Students can use these free websites to boost their understanding. They can use the graphing utilities to check work or to help visualize their graphs. What online sources are you using to teach math?