Made for Math Monday * Foldable for introducing matrices

I don't think this is a very difficult topic for most students, but I do occasionally have a couple who struggle and could use a little more time working on the basics of matrices, so I made a foldable to keep most of the class busy while those students got the time they needed to process and absorb the material.

First, take a 8.5 x 11 sheet of copy paper and place it in the landscape position (wider than tall).  Fold in about 1-1.5 inches on both sides.  Next, while the sides are folded in, fold the entire paper in half both ways (lengthwise and width-wise)  I found the template for this foldable a while ago and just hadn't before used it.  Now that I have use it, I'm trying to figure out where else I can use it! (funny thing is I didn't even realize it was for matrices until I wrote the blog posting).

Here are 2 versions of the front of the foldable I did.  I like that the foldable has 4 pages within, each one having a little pocket to hold something

The RED CLOCK thing I did was a mnemonic I did to try and help the students remember that in defining the dimension of a matrix that it does ROWS and then COLUMNS


I told the students to make the time a number that meant something to them.  We just started a new trimester and I only know about 25-30% of the students from last trimester.  This gives me a way of connecting with a new student and the students a way of personalizing the foldable.









Next, here are pages 1-2.  I would like to point out that I had students label each part as one would label the elements of a matrix (a (sub) 11, a (sub) 21, etc.

 As we did the foldable I didn't give any instruction on how to perform any of the operations.  We put the foldables aside and did examples of adding, subtracting and multiplying by a scalar.

For homework they were to write instructions on the pages a (sub) 12 [adding matrices] a (sub) 13 [subtracting matrices] and a (sub) 14 [multiplication by a scalar] as well as make up 4 addition, 4 subtraction and 4 multiplication by a scalar problems - on half-index cards, which fit wonderfully in the pockets.  Students will quiz their partners as I check the assignment on Monday.

I'm just not a very good test taker

I must hear these words at least 3-4 times I give a test. (I'm just not a good test taker).

I've come to realize that this doesn't actually mean what the student is trying to say.

After hearing this from a number of students, I peeked into their other grades and found that many of these same students were doing quite well in other subjects but struggled in math.  Upon asking them about their test-taking skills in these other areas the overwhelming response was that those other tests are easier.  Yet I have students who will still say that they are poor test takers even on assessments with very high class averages.

I'm convinced the "I'm not a good test-taker" comment is more a cry for help from students who are struggling but who lack the skills necessary to improve their scores alone.

Essential skills for studying in math

• you must know what material will be assessed
• you should know what kinds of questions your teacher will likely ask (multiple choice, constructed response, etc)
• you should identify your own weaknesses and focus your study there
• you should NOT wait until just before the test

What better students will frequently do in addition:

• make up practice questions
• even if you just take a problem worked out in class and just change 1-2 numbers and work that out
• find another student in that class and discuss the examples
• better yet, challenge each other with the additional examples you've made
• Seek out assistance _online, IRL, etc...

Any other strategies you can suggest?  

Domain and Range Foldable - Made for Math Monday

I did this foldable about a month ago and keep meaning to add it to my blog.

I had a discussion last year about what the AP Calc teacher finds that he wishes the incoming students understood better.  Among the topics he suggested was domain and range.  Traditionally we do inequality form in Algebra 1 - Precalc, but in Calculus he needs the students to understand interval notation.  Now I was better versed with Set Builder notation, but he said he preferred interval notation.  (which I now admit that at the time I knew nothing about;-)

Last year I discovered that the students I had (and I started teaching the class late due to a teacher leaving) found domain and range confusing no matter the notation I asked them to use.  This year I decided to do better and I wanted to get right into interval notation.  So, I came up with a simple 3-window foldable.

First window is the definitions of Domain and Range.  DIX ROY and the definitions of discrete and continuous in relation to domain and range.

Second window is primarily for (what they have seen before) Inequality Notation.  It also includes when an open dot is used versus a closed dot and an example is given with the domain and range given using Inequality Notation

The Third window is for Interval Notation.  Here we defined when parentheses are used as opposed to brackets.  Another example is given with the domain and range in interval notation this time.  (I think this window is a little sparse and would like any suggestions anyone has)

We did it by hand this first time, but I have now made a handout version to clean things up and make student copies more uniform (which should eliminate some of the errors I saw).

This was a success I would say, because I saw many students using their foldable over the next week+ until they didn't need it anymore.

Below you can see a student's hand-made foldable.  I will add my new typed version for Made 4 Math Monday this next week Foldable inside

Mind-reading and expected value.

I just finished looking at Dan Meyer's blog and saw that part of the most recent posting discussed uses for dice in class.   I have a trick I use with my students, let me tell you about it.

I give out 5-6 sets of three dice.  I have the students roll them and then add up all the numbers which cannot be seen (bottom, middles and middles).  Once they have the sum, they sit back with the dice still stacked and I "read their minds" to get the sum.  I have math on my side, and as long as they add correctly I can get the sum.

Last year I tied this activity into expected value in probability and it was a hit.   I don't want to give too much away, as my students have been following my blog as part of their summer enrichment.  Suffice it to say I can "read" the answer from their thoughts from across the classroom pretty well as there is only 1 piece of information I need.

It's a great mini lesson that is always great for engagement and I like that it's the kind of thing that the students take home and show their families.

ACT Readiness -pt 7- Graphical Representations

The next Standard relates to Graphical Representation.  These problems will either require a student to read information out of a graph, ie slope; or they will require a student to describe what happens to the graph, given a set of numbers or equation.

Upon first glance, I notice that many of the skills listed include an asterisk.  These standards are assessed on the ACT and Plan tests, but not the Explore test. Standards with a dagger are only assessed on the ACT.

The two examples here are higher leveled questions.  The first one requires that you understand how points change when a shape is translated in the coordinate plane.  The second one is interesting as the axes are not labeled with regard to scale, but the instructions tell you the scale is the same which is critically important to answer this question.  Do you understand how the answers were reached?

My students, ask yourself, how good do you understand the skills listed in the matrix?  Do you feel you could do each of these skills if asked?   Let's play taboo again:  Please list 5 more words to be restricted when trying to describe slope, ie rise/run.  Do the same with Parallel, Perpendicular,Vertex and Center (of a circle).  Email these to me, same email address as before.

ACT Readiness -pt 6- Expressions, Equations and Inequalities

If I ask my students, "What is Algebra all about?"  I can guarantee that at least 1/3rd will say solving equations.  I'm not sure this is necessarily bad, after all we do spend much of Algebra dealing with equations.  I hope that one day most of my students will answer that Algebra is all about the processes by which rationally problems are observed, analyzed and then solved.  (or something like that)

Here I present, Expressions, Equations and Inequalities.  Of the three, I would say that the last is most confusing and yet also the most "real-life", and yet the hardest to use in the classroom, without being trivial. Its also one of the hardest things for most students to use.

24-27 is the "meat and potatoes" or "curry and rice" of most Algebra classes.  I spend more time helping students hone these skills than just about anything else.  I know that most years my students have had to practice multiplying binomials (20-23), especially a binomial squared.  Factoring was also a big issue in recent students as well.

What steps do you need to use to solve the previous problem?  Do the decimals with different place values make the problem more difficult, or make it look more difficult?  Below are some more advanced examples:

The score range for each of these problems is the same.  The first problem requires greater literacy skills and the second requires greater computational skills.  We all struggle in different areas, which do you find more difficult?

The next range includes :"write expressions, equations and inequalities for common algebra settings".  We should be better at this.  We will be better at this.

If you are one of my students then the practice provided for this standard includes:  translating expressions and Equations Basketball

ACT Readiness- pt 5- Numbers and their Properties

The next set of standards will be Numbers and their Properties.  I hate to think that this is an area of weakness, but after years of seeing students counting on their fingers and the difficulty which my students have with factors (both prime factorization and factoring standing out as examples) I can't say that it isn't a possible source for student struggling.

Why does this first problem start with y not equal to 0?  Do you agree with the asterisk'ed answer?

How could you "work out" this second problem?  Would a list or table simplify calculating the answer?  How would you set up one of these to do this?

I look at the first column and seeing fractions, even equivalent ones leaves me wondering.  Also, I look at one of the higher columns and I see rules of exponents and complex numbers, and I know these are skills my students can accomplish.  Makes me feel like more and more that I need to know exactly where our students have struggled in the past.

I have a couple of ideas for my lovely students to do on this one.  First off, Beginner , That's COOL! and (if you use IoS, then FactorMan).

One more thing...on paper please list out the numbers 10 - 25.  For each number list out possible ways to multiply two numbers to get that number as an answer and then find the sum of those factors. (you need only do the positive factors and stick with whole numbers)

For example 24:
1*24: (sum) 25
2*12: (sum) 14
3*8: (sum)  11
4*6:  (sum) 10

Please keep the work from these "challenges" until after school starts.  Details will be given as to why before the end of the first week of school...

ACT Readiness- pt 4- Standards Functions

 The next standard is functions. All questions in this standard are applicable ONLY to the ACT, not the Plan or Explore tests.  There are generally not a lot of trigonometric questions on the ACT, but here is where most of them will be found.

There is only 1 example for this standard.  It is, predictably, a trigonometric example.




 I know the temptation is to get a numeric answer, but let me remind you that ACT questions are supposed to be written in a way that they can be solved WITHOUT a calculator (and I don't know the sin,cos or tan of 70 degrees off the top of my head).

For my students, let's first review the Unit Circle and a Real life Trigonometric Function.  Do a web-search for data relating to another real life trigonometric function and copy data which could be graphed to be trigonometric, please send it to me.

ACT Readiness - Pt 3 - Measurement and Plane Figures

This one will be a two-fer as these standards are generally covered in Geometry (rather than Algebra).

 Area and Perimeter of common shapes are usually pretty straightforward for students, unless they confuse the two.  Additionally, at times a formula may be given and being able to identify what each letter in the formula represents may be tricky.

Lots of triangle material here.  Pythagorean Theorem is listed in two of the score ranges, and after the session the Global Math Department had on it a few months ago, even I learned something. I've got to come up with a way to share some of those insights with my students.

For my students, tackle this Activity.  There are 5 parts, try each one and don't give up until you understand the theorem and can apply it to solve problems.

ACT Readiness Standards pt 2 -Probability, Stats and Data

This is the second posting in a series primarily aimed at my students regarding the ACT College and Career Readiness Standards.  My goal is encouraging self-assessment so that my students know where their weaknesses might lie, and hopefully getting them thinking about strengthening their skills before the start of school, or the official ACT assessment this upcoming school year.











The second list of topics in this series is Probability, Statistics and Data Analysis.

In our school, students first see probability and statistics in Algebra 2 (though there are many related topics which should be addressed in middle school mathematics).  A review of probability comes up in Algebra 2; for students wishing to learn more about these topics Statistics is available as an elective in their Senior year.

For the first example, students need to know which one word in the question?  Without knowing this math term, this is an impossible task, with this knowledge the question should be straightforward.

While this is a "higher leveled" question, I believe that it is also one which we should be able to figure out quickly.  Notice for a moment the wrong answers.  A.  We're looking at one of the 12 jelly beans, B. We're looking at 1 of 5 green jelly beans.  Can you guess how they arrived at the other two choices?

As mentioned in the first post in this series, our school's average Math ACT score is in the second range.  With regard to these topics, this means that our students (on average) cannot find a missing data value or compute simple probabilities (20-23), manipulate data from tables or graphs (24-27) or compute weighted averages.

As I will be teaching both Algebra 2 and Precalculus, I will be working more spiraling of probability & statistics topics into both classes.

If you are one of my students, then I challenge you to review basic probability.  I've got a special challenge for you this time.  Read through Bear in the Moonlight and then follow it up with James Tanton is the man!  Once you've done that, email me to let me know you did it.  (mr**&%hills@pre##$%#calc@gmail.com) remove spaces and special characters)

ACT college and career readiness standards Mathematics Part 1

I have found that I am rarely surprised by the ACT scores, in mathematics, that my students achieve. "A" students generally score better than most other students, but even these students do not score to their potential on the ACT.  I'm sure that nerves and distractions play a part, but I also believe that many students over-estimate their comprehension and ability with using some of the mathematical skills which they've "learned".  I'm hoping through a series of blog postings that I can encourage some self-assessment among my students

ACT has prepared a document in which they give score ranges and the skills which they associate with students who can accomplish those skills.  I plan on making a series of blog postings to share with my students, especially the Juniors, to help them better understand what they need to get the scores they believe they deserve.

If you look at the matrix, you will find that the skills required for higher scores are far more involved.

The average math ACT score in our school is in the second range. This implies that many of our students, including some of the honors students, can solve "routine" one and two step problems, but once rate, taxes or averages are required the problems become too difficult.  (20-23)  Solving problems which require planning or converting measurements (24-27) should be well within the skill set of a high school math student.  (especially given that ACT problems are written so that one does not require a calculator to solve them).

 The answer with the asterisk is the correct answer.  Take a moment and work out each one and be sure you understand why the asterisk'ed answer is correct.

If you are one of my students, reading this blog posting as part of our Summer Review Challenge, then I would like you to take a piece of notebook paper and write (or find) 2 examples which would fit into each of the lowest 3 ranges of scores.  (so that is how many problems altogether?)  Take a picture of them, or scan them and please email them to the email address I send in the previous remind101 messages.

Productive Struggle

For my few followers, please excuse my absence.  My dad passed within the last 2 weeks of school letting out; and I had to travel across the country and have been dealing with a 'caregiver' committing mass fraud, so much so that my mom nearly was kicked from her nursing facility.

Not what I wanted to post about.....

Letter to my students next year....

Hello....

        We are here, together, for the purpose of learning math.  But there is a problem with that statement.  Anyone care to point out what it is?  (all options should be discussed)

        I don't seem to be doing any of the learning, at least not observably so.   To me this is a problem.  How can I expect you to be learners, when it doesn't appear that I am one myself?  I promise to put myself in the same position, that of learner, right beside you.  I will ask questions I find interesting, and would love suggestions from the class, without knowing the answers.  Please do not expect that I will always just explain things, for I don't know everything.    My goal is for us to collectively struggle at times.

       Learning about math is a struggle.  No one ALWAYS finds it easy, not really understanding 100% of it.  I learn something new about the content which I teach, every single year.  There are things which I will show you, primarily because they important or in a couple cases just cool, but I want more class discussion.  To this end I will eventually restrict how much speaking I will do in a class.  Students should at times struggle as well as be led.

       The skills of mathematics, especially Algebra are important. List them:  (I'm thinking properties of equality, distribution and composition)  Anyone, play soccer or baseball or football?  Make me a list of the rules of each of these games - who knows I might need one in time. Do you like the game? --if you're going to do this, don't read ahead yet --  (************************************************************************************************************************************************************************************************************************************************** Are the rules the game?  What purpose do the rules have?  What SHOULD happen when someone breaks the rules?

      A mistake that many smart people make is not looking for the creativity in math.  It is undefinable that there is math in art Donald in Math Magic Land .  Soon enough I will show you that there is wonderful art in math.  (I've got 2-3 art projects in functions)

      Lastly, I have a challenge to you.  (Commence with Bionic Bee.)

      ((I have a great problem which I know 5th graders can solve in about 5-10 minutes, high school students, alg2 and above, usually take at least the class period and benefit from completing it as homework, and the one time I gave it to college calculus students I had phone calls begging for help....it might make a good task here))

The Bionic Bee  (I play a clip from Wild,Wild West with Will Smith)
(my apologies, I can't really draw it out here and honestly drawing it out is important)

Two trains have managed to find themselves, though unaware, on the same track-heading towards each other.  You're a spy, and while you cannot stop the trains, you do have one trick up your sleeve.  The Bionic Bee!  You launch it down the track, speeding toward the other train.

It is capable of travelling 560 miles per hour, in a straight line, and also capable of instantaneously changing its direction 180 degrees (I do give the high school-ers and college students the angle as pi if I'm feeling mischievous).  It has a full array of sensors, but it also has a programming flaw - once it does a 180 it will continue making that same degree change.

The two trains are 980 miles apart (I've thrown Km in from time to time, but only on the 5th graders and guess what?  It doesn't hurt their time....)  One train is a new, green-energy steam train.  It cruises along at 170 mph.  The older diesel train barely manages 110 mph.  Neither train knows the destiny that fate has in store for them.

It is now 2:15 pm.  What questions do you have at this point?  (I normally just ask at what time do the trains collide and how far will each have traveled to get there?).


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Sadly, though this isn't a lesson which I don't already know.  I'm thinking for the 1st week of precalc it might make a good challenge.  I just need to be on the lookout for content and level (to borrow from mostly every video game) appropriate, and definitely nontrivial.

Standards of Mathematical Practice

I was one of the many people unable to attend the NCTM conference in NOLA.  Instead I followed along with many of my favorite people on twitter as they shared their impressions and opinions of the various presenters and programs.  It was interesting hearing from in many cases both presenters as well as people who attended the presentation.

@woutgeo (Avery Pickford) commented that there are 385 content standards occupying 60 pages and 8 practice standards in 3 pages.  If the practices standards are so important, why are they given so little focus.

I said that it is much easier to assess, check off the boxes, the content standards than it is to evaluate (with fidelity) the usage and performance with regard to the practice standards.

I have started to work on coming up with a couple of ways to address this problem.   The first one is that I made up a sheet on which I start a problem and ask the students to identify what practice(s) were used to get to that point in the problem.  The students then need to complete the problem and reflect on what practice they used to finish it.

The problems do need to be carefully considered, as I don't want a problem which only uses a single practice and has a good place to split it so that the work still assesses whether the student understands what they are doing.

Here is the first one I used with my students this spring.  I have a couple ideas for other problems to use, but I haven't done them yet.

Practice Standards Assignment

I am happy with the results I got from this first assignment.  I am required to have students complete a math-based essay as part of the final exam given each trimester.  I hope to use one of these at the end of this school year as the required essay.

Practices student work

Made For Math Monday *Rational Expressions*

Been a while since I've written one of these.  I am finding that my students are slowly, but surely getting the concept of identifying discontinuities in graphs of rational expressions.  We've been looking at them pretty piecemeal up until now, though.

I wanted to get them practicing and its always more fun when there is folding and coloring involved, so we made cootie catchers!  I used a template from Teach Inspire Prepare Students (TIPS) which was a lifesaver as I am really not good at getting word to do what I want with regard to writing sideways in text boxes.

When completed the cootie catcher is assessed by simply looking at the colors each of the innermost pieces are colored.  I am including the file I made (from the earlier mentioned template) as well as the instructions and a scanned completed page.

Here is the pre-solved cootie catcher Cootie Catcher Rational Discontinuities.  Below you can see a completed version.  I liked having the students do different colors for the various discontinuities because high school students always love an excuse to color something in (and it makes it sooo much easier to see how the students have done).

As an aside, primarily because I started this posting 3 weeks ago, I can say that the group of students with whom I used this activity did particularly well on the portion of the rational expressions assessment which dealt with discontinuities.

Thoughts on a precalc unit

Being my first time teaching this, I am still trying to conceive how I want to teach this class.  I was given a "map" on how to cover the standards (all the plus standards in CCSS for secondary math).  I am moving things around, but its still a work in progress.  The unit I will be describing is definitely not the first, nor should it be the last one in the class.   This is out of what I'm envisioning as 6 units.

CPVM

 Complex-Polar-Vector-Matrix

Complex Numbers and Graphing

          All basic operations, conjugates and graphs.  Review distance formula in relation to Pythagorean Theorem.  (Which is reviewed in a previous unit).  Lots of standards to be covered.

Polar graphing
          Not many standards here, but wonderfully fertile ground for what is to come.   Keeps students using and thinking about trig.   Pythagorean Theorem makes another appearance, keeps the students grounded with a comfortable concept.  

**shameless plug here:  Global Math Department did a week on the Pythagorean Theorem a few weeks ago and I left with at least 5 good ideas for using it to help my students think more mathematically.  

Vectors:  good number of standards here too.  More Pythagoreas here too.   Many of my students see vectors first in physics, but not all.  Students helping others at its greatest.   I'm tempted to challenge 3 students (in a class of 27-30) to be the helpers and offer them each class average on the quizzes plus 20 points, for their grade on those quizzes.

Matrices:  our students see matrices in Alg1, Alg2 as well.  Basic operations and solving matrix equations without inverses is well covered.  So is systems.  At this level we are doing row operations (gaussian/Jordan elimination) and understanding these ideas.  Also a really silly standard about writing a vector as a matrix is required.

Currently, this unit contains the most standards of any of my units, by a high margin.  It also was one of my students better units.   Looking back I did parametric equations very poorly.    I know I need to do more then just mention them, but I just haven't found the hook or way of engaging them.  

My thought is that this unit should take 4-5 weeks.  Each topic should include a quiz and 1-2 days per topic where formative assessment should drive the discussion.   (I teach in a school where this class is done over 2 trimesters and the class meets 5 days a week with 72 minute classes).

One last thing...  I wonder if the administration would accept that being able to most effectively teach vectors would require purchasing for me, and my classroom;a new pool table, balls and cue sticks.   It can't hurt...