Posted on

Reordering N-Gen Math 8 and Middle School Videos

We’ve been getting great input this year on our new middle school courses that are aligned to the new Next Generation Math Standards (coming next year to New York K-8). I’ve had some great conversations with teachers who are using the courses. One of the things I heard from multiple teachers was that they would like to see algebra moved to the beginning of N-Gen Math 8 so that equation solving could be a greater part of the geometry units.

One of the new skills that students learn in 8th grade algebra is how to solve equations with variables on both sides. Since our original ordering had this skill taught in Unit 4, we didn’t incorporate it into our Units 1 through 3 (on geometry). We decided to remedy that and move our algebra unit to Unit 1 and then incorporate those algebraic skills into our geometry units (which became Units 2 through 4). Because of this reordering, we had to remove some problems that had been in the algebra unit, but we were able to beef up the algebra in many of the geometry problems.

We also created some new lessons in geometry based on feedback. We created an additional lesson in the Tools of Geometry on angle terminology associated with parallel lines (alternate interior, exterior, same side, corresponding, etcetera). We added a few lessons in our unit on Transformations. One of these looks at congruent angles formed by parallel lines through the lens of rigid motions. The other lesson comes at the end of the unit and gives students lots of practice using algebra to model geometry problems.

One of the very nice consequences of the reordering is that our unit on similarity now comes immediately before the unit on linear functions and graphs of lines. Given that in 8th grade we justify the slope of a line based on similarity, this flow of topics is much more natural. I think the changes to N-Gen Math 8 will make it more complete and will strengthen both the algebra and geometry skills of the students. I’d like to thank Elizabeth Agen of Golding Middle School and Beth Goldberg of Linden Avenue Middle School for the feedback on the order of the topics. I had some very good back and forth discussions with both teachers that really got me to think harder about the best sequence of topics in the course.

We’ve also begun to record and post the videos for the middle school courses. It’s going to take us quite some time to finish them as we can only do so many in a day and there’s other work to be done. My hope is to have them all recorded and posted by next spring. We’ll be recording more videos next week and then will need to take some time off for the fall AMTNYS conference up in Rochester. For now, keep watching as we continue to post the first few units of all three courses.

Posted on

TI-Nspire Directions for Algebra I and Geometry by Jeanne Oliveira

Our good friend and longtime contributor, Jeanne Oliveira, from Germantown CSD has created an amazing collection of TI-Nsprire directions that go with the eMath Curriculum for both Algebra I and Geometry. Some of these we’ve posted in the past. But now we have the full collection. These include both PDF versions of the instructions and TNSP versions. I’m going to post them by unit in zip folders for easier download.

Algebra I:

eMathInstruction Alg I Unit 01 Nspire Instructions

eMathInstruction Alg I Unit 02 Nspire Instructions

eMathInstruction Alg I Unit 03 Nspire Instructions

eMathInstruction Alg I Unit 05 Nspire Instructions

eMathInstruction Alg I Unit 06 Nspire Instructions

eMathInstruction Alg I Unit 07 Nspire Instructions

eMathInstruction Alg I Unit 08 Nspire Instructions

eMathInstruction Alg I Unit 09 Nspire Instructions

eMathInstruction Alg I Unit 10 Nspire Instructions

eMathInstruction Alg I Unit 11 Nspire Instructions

 

Geometry:

eMathInstruction Geometry Unit 05

eMathInstruction Geometry Unit 06

eMathInstruction Geometry Unit 07

eMathInstruction Geometry Unit 08

eMathInstruction Geometry Unit 09

eMathInstruction Geometry Unit 10

 

Jeanne spent a lot of time on these and plans to do a set for Algebra II as well. We’d like to thank her for sharing these materials with the wider math education community. Thanks Jeanne!

Posted on

Regents Live Review 2019

In just two weeks students all around New York State will sit down to take their (Common Core) Algebra I Regents exam. As we’ve done in the past two years, I will be doing a Live Review session for each of the three major high school math Regents exams (Algebra I, Geometry, and Algebra II).

I will be going over questions from past Regents exams. The question packets for each Regents are given below. Fair warning: they are extremely long (around 20 pages each)!!!

Common Core Algebra I Live Review Problems 2019 (pdf)

Common Core Algebra I Live Review Problems 2019 (MS Word)

Common Core Geometry Live Review Problems 2019 (pdf)

Common Core Geometry Live Review Problems 2019 (MS Word)

Common Core Algebra II Live Review Problems 2019 (pdf)

Common Core Algebra II Live Review Problems 2019 (MS Word)

We likely will not make it through the entire packet on any one of the nights. I think it really would take about 5 hours to do a complete review.

The reviews will be shown on YouTube Live (Kirk Weiler Channel) and Instagram Live (follow @emathinstruction). The video will be much better on the YouTube Live platform. We mainly will have Instagram Live going for students to comment (whatever it takes to get them to the review ;-).

Can’t wait to “see” you all there.

Posted on

eMath May 2019 Newsletter

Wow! Is it May already? Right about now teachers in New York are finishing curriculum and heading for the big push to the New York State Regents exams in mid June. We are here to help with more add-ons and Live reviews in June. Before we get to those, let’s discuss the latest and last round of add-ons for this academic year.

In Common Core Algebra I Add-Ons we bring you the Form C Assessment for Unit 10 (Statistics) and Unit 11 (A Final Look at Functions and Modeling). These two assessments mean that we now have three unit tests per unit in Algebra I. We’ll try to add a fourth next year.

For Common Core Geometry Add-Ons we have an introductory lesson for Unit 10 (Measurement and Modeling) along with Unit 10’s Form B Assessment. To begin, we have created a Lesson 0 for the unit on Rates, Densities, and Conversion Factors. We felt that there was quite a bit of work in this unit on applying concepts like density. This lesson reviews these various ideas early in the unit. Use this for kids who struggle with ratio and rate concepts. Our Form B for Unit 10 now means we have two unit tests per unit in Geometry. More to come.

Common Core Algebra II Add-Ons this month also bring two Form C Assessments. We now have a Unit 12 (Probability) and Unit 13 (Statistics) Form C. It’s especially nice to have more statistics questions for this very challenging unit.

Finally our Algebra 2 with Trigonometry Add-Ons also feature two additional assessments, both Form B. First we have the Unit 12 (Statistics) Form B and then we have our Unit 13 (Sequences and Series) Form B.

Lots of assessments this month!

Now for the fun part. We are yet again holding our YouTube Live and Instagram Live Review sessions this year for the New York State Regents Exams. For each exam we will be going over old Regents test questions for 3 hours. Here’s our schedule for this year:

Algebra I (Algebra 1):  Tuesday, June 18, from 6:00 to 9:00 p.m.

Geometry: Wednesday, June 19, from 6:00 to 9:00 p.m.

Algebra II (or Algebra 2): Thursday, June 20, from 6:00 to 9:00 p.m.

We will be broadcasting on our YouTube Channel (click link) and on our eMathInstruction Instagram Live account. YouTube will be much better quality, although commenting will be allowed on Instagram (only go there if you can stand constant comments from teenagers).

We’ll put out more information in the coming two weeks, including the problem sets we will work through at each of the sessions. For now, have a great Memorial Day weekend!

Posted on

eMath December 2018 Newsletter

Ah, its the last week of classes here in New York before the big break. Kids and teachers here are restive, but happy that one of the longer break is on the horizon. Just last minute tests, office parties, and school concerts to get through. Maybe our hypercube logo in a Santa hat is just what you need:

Or maybe you also just need that vacation. I know I do.

We’ve got the typical add-ons to discuss but first I wanted to update all of you on the latest timeline for New York State switching to the Next Generation Math Standards. If you are not teaching in New York, you may want to skip to the add-ons. However, if you have been confused about when the Next Generation standards are going to be implemented, read on.

As most teachers in New York know, these standards are very close to the Common Core Standards. But, like many states, the political backlash against those standards, earned or not, has made them untenable, at least in name. The timeline for adoption of these standards, of course, is dictated on when they will be assessed. That timeline has been fairly transparent for quite some time at the K-8 level. Here’s the graphic and link from the NYSED.gov website:

http://www.nysed.gov/curriculum-instruction/next-generation-learning-standards-and-assessment-implementation-timeline

The most important part of this graphic is the final piece (with my horrible highlighting):

At eMath we have been working under the assumption that the changes due to the Next Generation Standards in Algebra I, Geometry, and Algebra II would all go into effect in the Fall of 2020 and then assessed for the first time in June 2021. NYSED hasn’t updated that graphic on their website for quite some time (at least 6 months) and to my knowledge has not made any solid decision. But, NYSED is speaking to math teachers, especially in the form of responding to question from the Association of Math Teachers of New York State (AMTNYS). Here is the relevant question (from AMTNYS) and the reply (from NYSED):

I love the fact that “new” is in quotes. Here is the full document:

2018 AMTNYS Math Questions

So, it looks like at the very earliest the changes at the high school level will be assessed in June of 2022 (which seems like a century away to me). And it seems like that will only be for Algebra I. So, those of you holding your breath for 3 by 3 systems to be taken off the New York State Algebra II Regents exam could be waiting until the 2023-2024 school year. Let that sink in for a minute (or more).That’s five years and countless numbers of Regents exams between now and then.

The upshot of all of this for users of eMathInstruction courses is that we will not introduce these changes to our texts until at least the Fall of 2021. This, of course, is subject to change should NYSED decide to implement changes to the assessments earlier, but that would be disastrous for them.

O.k. Onto the add-ons for the month.

In Common Core Algebra I Add-Ons, we bring you two Form C assessments, one for Unit 5 (Systems of Linear Equations and Inequalities) and one for Unit 6 (Exponents, Exponents, Exponents). We will continue to put our these additional assessments year in and year out since we get a lot of positive feedback on them. Of course, it never hurts to have makeups and alternate exams.

In Common Core Geometry Add-Ons this month we have two sets of practice problems. First, we have a Unit 5 Coordinate Geometry Formula practice sheet. This is a short set of problems that give students a chance to practice what they’ve learned about the three fundamental coordinate geometry formulas (slope, midpoint, and distance). This is a good extra assignment for the average student. For Unit 6 we bring you a nice set of quadrilateral practice problems. These aren’t proofs, just algebraic and numerical problems that ask kids to use facts about the various quadrilaterals they’ve been studying.

For Common Core Algebra II Add-Ons we again bring you two Form C Assessments. We bring you the third version of the Unit Assessment for Unit 5 (Sequences and Series) and the third for Unit 6 (Quadratic Functions and Their Algebra).

Finally, for Algebra 2 with Trigonometry Add-Ons we have an assessment and a practice set of problems. We bring you the Form B Assessment for Unit 6 (Polynomials and Rational Functions). We also have another resource for Unit 7 (The Circular Functions). We created a set of fairly basic trigonometric graphs (no horizontal shifts) and ask the students to create their equations based on either the sine or cosine functions. This is a nice sheet to give kids to practice immediately after they have learned all of the basic sinusoidal graph parameters.

So, in other eMathInstruction news, we will be paying a visit to Edward R. Murrow high school in Brooklyn, New York this spring for our annual school visit. I’m excited to meet the students and teachers at Edward R. Murrow and amaze and awe them with some math that’s more fun than serious. Next month we will have some exciting news regarding new courses coming to eMath just in time for those Next Gen Standards I was talking about before. So, I’ll leave you with this teaser of an image:

Have a great holiday season! Enjoy the break, the rest, your family, and your friends. 2019 here we come!

 

Posted on

eMath August 2018 Newsletter

Wow! Is it already August 15th? For those of us living in New York, that generally means we have just over two weeks of summer break left before school begins. How time flies.

Speaking of the waning summer, if your school is still looking to purchase workbooks for this coming school year, make sure to do so as soon as possible. We’ve still got quite a bit of them in stock now, but by next week, the delivery times will start to push past Labor Day weekend. For New York City teachers, we now have all of our workbooks, including Common Core Algebra I in Spanish, on ShopDOE and the FAMIS ordering systems. Find all of the important information using the document below:

eMath Instruction ShopDOE and FAMIS List 2018

Now that the school year is almost upon us, we have new additions to each course. As always, please feel free to send us suggestions about what you would like to see as monthly add-ons to various courses. Always best to catch us with your suggestions at least a month ahead of time. Assessments are always in demand and we have a bunch for you this month.

In our Common Core Algebra I Add-Ons, we bring you a Unit 3 (Functions) Quiz. This quiz goes through Lesson 3 of the unit and we provide you with both a Form A and a Form B version of the quiz. This is the first year of our Common Core Geometry Add-Ons. For the Geometry Add-Ons this year we want to make sure you all have a Form B test for each unit. So, for this month we give you a Unit 1 and Unit 2 Form B assessment. These mirror assessments can be used as makeup for the typical exams or just as replacements.

For our Common Core Algebra II Add-Ons, this month we bring you a Unit 2 Quiz (Also Functions). This quiz should be given after Lesson 3. We provide both a Form A and Form B version of this quiz. In Algebra 2 and Trigonometry Add-Ons, we provide a Unit 1 and Unit 2 Form B quiz. In Alg2/Trig, the first two units are very short, so we did not write full unit assessments for them. Here we provide a Form B for the “quizzes” for each of these units.

This year we are going to have our eye on the changes coming due to the Next Generation Standards in New York State. For those outside of the state (or those out of the loop), New York decided, like many states, to massage the Common Core Standards and given them a less politically charged name (they’ve even taken it off of the NY standardized tests). These standards do not go into effect at the K-8 level until the 2020-2021 school year. They will first be tested at the 3-8 levels in the spring of 2021. Here is a graphic of the timeline:

Here’s a link to that page:

Next Generation Math Standards Implementation Timeline

Now, I get a lot of questions about when we will be incorporating the Next Gen changes into our Alg I/Geo/Alg II series of curricula/books. Well, New York State has been vague, at best, about when the changes will be implemented at the high school level. BUT, if you look at the fine print of the Timeline graphic above, you see:

Two things to note here: (1) NYSED has not yet decided on when these standards will be implemented, so we must assume the status quo in terms of the assessment content and (2) the implementation of the Next Gen standards on tests will not happen before June of 2021.

We plan to really start modifying our high school texts later this academic year and will continue to create lessons that fit these standards as add-ons. We do not plan on actually changing the workbooks until the 2020-2021 academic year. We also hope to release something for kids younger than 9th graders for the Next Gen standards as well. More on that later this year.

That’s it for now. Continue to enjoy the rest of summer. Get out to your local fairs, waterparks, and festivals. As for me, I’m looking forward to taking a long weekend to camp with the kids.

Posted on 2 Comments

eMath May 2018 Newsletter

Well, Spring has definitely sprung here in the great Northeast. As I sit here massive thundershowers are hitting us. We even have a tornado warning! I grew up in Illinois, where the flat terrain essentially guaranteed that tornadoes were commonplace. But, surrounded by mountains (or large hills as some would insist) here in Red Hook, we don’t tend to see twisters here.

But, enough about the weather (actually I’m going to come back to it eventually). We are heading into the homestretch of the school year. With roughly a month to go before standardized testing sets in, teachers should be heading into full on review mode. To that end, let’s discuss the add-ons for Common Core Algebra I, Common Core Algebra II, and Algebra 2 with Trigonometry. Last year for each of these courses we put out short (10 points each) Unit Review quizzes for each unit. For both CC Algebra II and Alg 2 with Trig we put multiple quizzes out for the longer units as well. We released these quizzes between the April and May add-on rounds from last year.

This year we are releasing the Form B (or makeup) quizzes for all three courses. We organized the quizzes from last year and rereleased them as Form A. We then also released Form B as new content. For each course, both documents, Form A and Form B, contain quizzes for all units. These quizzes will help spot check the students as you conduct unit review. Since Teachers have access to the Word document as well, you have the flexibility to combine quizzes as you like and need to.

We will start up with more add-ons in August! Including add-ons for Geometry.

In other exciting news, eMathInstruction will host Live reviews for the New York State High School Regents exams!!!

Last year we live streamed reviews for CC Algebra I, CC Algebra II, and CC Geometry (in that order). We did this using only Instagram Live, since that was where we thought we could reach the most students. By the time we did our last review, CC Geo, we had over 20k viewers. Granted, some of them probably weren’t students at all and left as soon as they realized it was a big, old geek (me) talking about math, but still! Over 20 thousand unique viewers. Even if 5,000 were students, that would be great.

This year we hope to live stream it on Instagram Live, Facebook Live, and YouTube Live. We still have to work out the details on that. But, the dates and times have been set:

Algebra I Review: Monday, June 11th, 6 until 9 p.m.

Algebra II Review: Wednesday, June 13th, 6 until 9 p.m.

Geometry Review: Monday, June 18th, 6 until 9 p.m.

Kids will be able to tune in via Instagram Live (at least) and have a chance to comment (briefly) at certain times during the video. They will need to follow:

@emathinstruction

 

Last year we streamed to my own personal account (@kirkweiler if you want to see me post pictures of my garden), but this year we are only Instagram Live Feeding to @emathinstruction. Have your students follow that to watch. Once we know how to watch on YouTube Live and Facebook Live we will let people know via Facebook.

Finally, back to the weather! In my experience, many math teachers are weather junkies. We like both the predictable, science side of it as well as its probabilistic nature. A neat website to help you visualize the weather patterns, especially air velocity, is a site called Ventusky.com.

It’s a super cool site where those little streaks you see in the image actually move on the screen relative to the velocity of the wind currents. It is especially impressive when you are looking at a hurricane. Here’s a hurricane like pattern off the coast of Japan (thankfully not an actual hurricane/typhoon).

So, check it out if you have the time. It is a CPU drain, so just keep that in mind and close whatever browser window you use to look at it, either on your computer or phone, once you’re done.

That’s it for me for now. We’ll get back to you all with way more news about the Live events as we get closer. Make sure to announce them to your students so they can have a night before (literally) review. You never know who it might push past the passing line.

 

 

 

Posted on

eMath April 2018 Newsletter

Well, it’s April 16th, so the weather in New York is now turning sunny and spring is in full tilt with the flowers blooming, the birds singing and …

Actually, I’m just kidding. I woke up to snow this morning. Still, not all is dreary. We are quickly approaching the final stretch of the academic year (you are likely just starting the fourth quarter). We have been busy at eMathInstruction working on a variety of different resources, updating our website, and creating add-ons for the courses. Let’s get right into those.

For Common Core Algebra I Add-Ons this month we have added a quiz and a new lesson. For Unit 10 (Statistics) we have added a Form B mid-unit quiz. Last year we added the original quiz and this year we have a makeup or Form B of the quiz. These are always handy when it comes to students who are absent or even if you want to give Period 1 the Form A and Period 2 the Form B. We always try to make them as similar as possible, while still making the questions unique. For Unit 11 (the final unit) we added Lesson 6.5 on Additional Piecewise Functions work. In this lesson we have piecewise functions that include linear, quadratic, and root functions. I still believe that the idea of piecewise functions is challenging enough so that it should be as basic as possible, i.e. just piecewise linear, but clearly the makers of standardized tests disagree with me as they often will include pieces that are non-linear.

For Common Core Algebra II Add-Ons we bring you two activities centered on the probability and statistics units (Units 12 and 13). For Unit 12 (Probability) we have created a Die Rolling Probability activity. We have the students use the random number simulator on the calculators to simulate rolling two die and summing their rolls. We then have them investigate the empirical versus theoretical probabilities associated with this experiment as well as answer a variety of conditional probability questions. There is no associated homework with this activity. In Unit 13 (Statistics) we created Lesson 4.5 on Sampling a Population. We created a data set of 200 values that students then randomly sample from. Students calculate sample means and standard deviations from their samples and then compare these to the population statistics. Like the Die Rolling Activity, this lesson has no homework associated with it.

In our Algebra 2 with Trigonometry Add-Ons this month we have a quiz and another lesson. For Unit 10 (Exponential and Logarithmic Functions) we offer up a mid-unit quiz that covers all of the content through Lesson #6 (Graphing Basic Logarithms). In Unit 11 (Probability) we created Lesson 7.5 on More Binomial Probability Practice. In this lesson we use the calculator to find binomial probabilities and cumulative binomial probabilities for a variety of applied problems.

In other exciting news, eMathInstruction will be at the national NCTM conference next week!!!

We will have a booth set up where you can come and chat and we will also be presenting on Friday afternoon. Drop by and say hi if you are at the conference.

In Geometry news, Version 2.0 of our workbook is now for sale. This new book has the Unit Reviews now included at the end of each unit, so its quite a bit longer than the original book. We’ll begin to add-on to this course starting in August. As always, if you have suggestions in terms of what you’d like us to add, please reach out (make up tests, mid-unit quizzes, extra problem sets, extra lessons).

We’ve been doing some major behind the scenes work on our website recently. We added a second server which does two things for us. First, if the primary server were ever to “crash” then the second server would keep the site up and running. But, even better, the second server allows us to “load balance” the traffic to the site. Hopefully this means shorter page and download times. Now, with all major website work, there are always hiccups in the system. Most of you probably have not had any trouble, but some of you, when trying to go to our main site, have experienced errors such as a “failure to redirect” or “emathinstruction.com” sent an invalid response or even scary messages like “corrupted content error.” I’ll be honest, I don’t really understand what’s happening. But, apparently a one-time fix of clearing the computer’s cache gets the job done. How to clear cache is very browser dependent (Chrome, Firefox, Edge, Internet Explorer, Safari …). The University of Wisconsin (go Badgers) has an awesome site with links to every major browser and operating system:

How to Clear Web Browser Cache (University of Wisconsin)

You know cache is the issue if you can get to our site through one browser but not through another one.

Have a great rest of your April. I hope that spring has arrive for you soon, if it hasn’t already shown up (no sign yet here).

Kirk

 

Posted on

eMath March 2018 Newsletter

Good morning everyone. I’m writing today from my house in Red Hook, New York, just a hundred miles or so north of New York City. We’ve had three nor’easters in the last week with another possible one next week. You might call this serious winter. We just call it March. And since it is mid-March, it’s time to put out the eMath March Newsletter. We’ve got plenty to report and add-ons to discuss, so let’s get to it.

This month for our Algebra I Add-Ons we bring you a new lesson in statistics. Unit 10.Lesson 4.5 is on the subject of outliers, how to identify them and what the effect of removing them is on the statistics for a data set. We stick to the standard definition of an outlier being any data value  ± 1.5IQR below or above the first or third quartiles, respectively. On the New York State Regents Exam in Algebra I, we’ve repeatedly seen multiple choice exam questions that require knowing this definition, for example #16 on the January 2018 exam:

To rule out (2) as the correct choice, think about how much work must be done. They must calculate Q1 and Q3, 41 and 68, then the IQR (27), and then 1.5*IQR (40.5). Subtracted and added to Q1 and Q3 gives you a “non-outlier” range of 0.5 to 108.5. Notice how this makes the 120 clearly an outlier but the data point of 0 is barely an outlier. So, it’s pretty important to know this technical definition of an outlier (which is not universally accepted) in order to not choose (2). Given the rather subjective nature of the correct answer, i.e. (1) that the mode is the “best” measure of central tendency, it is important to know why each of the other statements is true. Don’t get me wrong, I don’t think it is unreasonable for kids to know this, but I think that this is the type of curricular detail that needs to be explicitly spelled out as a piece of knowledge kids must learn.

We created two new videos for Common Core Algebra I. One of them goes with our add-on lesson Unit 8.Lesson 7.5.Linear-Quadratic systems. We also did a short video on a graphical reason for why the method of completing the square works. Both of these videos will be posted soon with a QR code added to the lesson when finished (likely today or Monday).

For Common Core Geometry, we have finally finished our Unit Reviews and Unit Assessments with Unit 10 – Measurement and Modeling. We have a fantastic Review with lots and lots of modeling problems, including additional density problems. These Reviews and Assessments finish Version 1 of our Common Core Geometry curriculum. We will begin releasing add-ons (additional lessons, activities, and assessments) beginning in August of 2018.

For our Common Core Algebra II Add-Ons, we’ve created a lesson, a mid-unit quiz (form B), and a video. Let’s start off with Unit 11.Lesson 8.5.More Work Graphing Sine and Cosine. This is a huge set of problems that give kids basic work graphing the sine and cosine function including amplitude, midline, and frequency (no horizontal shifting). We also give them a variety of curves and have them come up with the equations. This is an excellent problem set to give to kids over spring break (coming soon hopefully to a school near you). We also created a Form B of our Unit 11 (Trig) mid-unit quiz. We reposted the Form A quiz and you now have two similar quizzes to test their knowledge half-way through the Trigonometry (Circular functions) unit. Finally, we created a video to go with our Unit 10 lesson on the Sum and Difference of Perfect Cubes. I must say I’m disappointed that this topic got approved for the Next Generation Learning Standards at the Algebra II level:

I find almost no utility in having kids memorize:

x^3+y^3=(x+y)(x^2-xy+y^2) and x^3-y^3=(x-y)(x^2+xy+y^2)

I do think they are interesting patterns to study. I think the connections that can be made between these problems and the imaginary roots of quadratics are quite interesting. But, I see no utility in any type of realistic math problems where memorization of these patterns is helpful. I think it just takes up time and mental space that could be spent better elsewhere. What are your thoughts?

Finally, we have our Algebra 2 and Trigonometry add-on of the month. This month we’ve added an additional lesson in Unit 10 (Exponential and Logarithmic Functions). We’ve put in Lesson 4.5 on Additional Exponential Modeling. In this lesson we look at how to transition between different time units when modeling something using exponential growth and decay. This is a particularly nice lesson from an applied perspective.

That’s it for add-ons, but I’d like to keep discussing a bit more. Let’s begin with a cool 3D visualizing program for Geometry teachers that many of you likely have if you bought a PC within the last 3 years. All Windows 10 computers (and I believe some Windows 8) come with a program called 3D Builder:

Just type its name into your Search bar on a PC to see if you have it. Now, this program wasn’t installed because Microsoft thinks we are all Geometry teachers. Nope! This is a program designed to allow you to create 3D printer files. But, that doesn’t mean it isn’t great for 3D modeling. For example, remember the water-tower problem on the first CC Geometry Regents exam:

Now, imagine making the problem come alive by showing the students not just a model of it on 3D Builder but also that model taken apart into the three component pieces:

One of my favorite parts of this program is its Split command under the Edit Menu:

This function’s purpose is to slice a 3D object using a plane so that you only keep part of it to print. BUT, Geometry teachers can use it to show kids cross-sections of any orientation. With our cone example, here is an example of it showing a horizontal cross-section:

Notice those rotation arrows? They allow you to rotate the slicing plane into any orientation. So, if you wanted to see a classic vertical cross-section you could just rotate the plane 90 degrees to get:

Or, my favorite, the elliptical cross-section formed when slicing with an inclined plane:

Once you’ve rotated the plane, you can then move it left, right, up and down, allowing you to show kids how cross sections can change as you move the slicing plane in various directions. The Next Gen standards for Geometry state that kids will need to be able to visualize cross-sections of common solids, even with planes that are not horizontal or vertical:

Will they have to recognize that certain cross-sections of cubes are hexagons? Inquiring minds want to know.

Well, I think that’s about it for 3D Builder. I’ll be discussing this program and Tinkercad, one of my favorite online 3D modeling programs in additional posts. Only a few years ago, visualizing 3D geometric concepts would have been very difficult. Now, programs to do so are not just easy to find, but completely free.

We have some exciting changes coming to the website in the coming months, including a new way of organizing the Add-Ons and the Assessment Items. But, more on that in the April newsletter. For now, happy Pi-Day, happy March, and enjoy your spring break if you still haven’t been on it yet.

Posted on 3 Comments

eMath January 2018 Newsletter

Hello all! We are just about at the mid-point of the academic year, which means it is very, very cold in upstate New York. We are working hard at eMath this January to bring you new materials and to continue to update older ones. Let’s get right into the add-ons for this month.

For Common Core Algebra I Add-Ons this month we bring you a new lesson and an additional problem set, both from Unit #8 on Quadratic Functions and Their Algebra. First, we have added a lesson on solving Linear-Quadratic Systems Algebraically and Graphically. This is a topic that some already do in Algebra I and others will do once the New York State Next Generation Standards kick in a few years from now. We thought we’d get a jump on those and add this lesson now. We’ve also added a problem sheet with extra Quadratic Word problems (technically Lesson 9 in this unit). I think you can never go wrong with more quadratic word problems as kids need as much practice with these as possible.

In Common Core Geometry, we continue with the Unit Reviews and Unit Assessments. This month it is Unit #8 on Right Triangle Trigonometry. We’ve put together a great packet of Review problems and a good assessment that should help you assess your students’ knowledge of this important subject. It looks like at this pace we will be publishing the last Unit Review and Assessment in March (Unit #10 on Measurement and Modeling). That will round out Common Core Geometry for this academic year.

For Common Core Algebra II Add-Ons we have two new lessons in Unit #9 on Complex Numbers. We’ve always felt that this unit was a bit brief so we added lessons that we think would be great for enriching the unit. The first lesson (#2.5) is on the Division of Complex numbers. This is a challenging topic both conceptually and mechanically. The second lesson (#5) is on the Complex Plane. Graphing complex numbers and finding their modulus (absolute value) are emphasized in this lesson.

Finally, for Algebra 2 with Trigonometry we bring you a nice add-on for Unit #8 (Trigonometric Algebra). We created a nice review set of Trig Equations that emphasize all of the equation solving techniques in this unit, including basic equations, quadratic trig equations, equations involving trig identities, and equations that need to be solved graphically. This is a great problem set to use as additional review at the end of this unit.

That’s it for now. Here at eMathInstruction we’d like to wish everyone a happy and thoughtful Dr. Martin Luther King Jr. Day. As we go through some rough times in our country and we question our basic identity on a daily basis, we hope that this day above all others makes us reflect on the commonalities that bring us together rather than the differences that seem to be tearing us apart.