Category Archives: Geometry

Common Core Geometry Beta Version Finished – by Kirk

The first version of Common Core Geometry is now done and completely posted. It consists of 10 total units:

geometry-table-of-contents-screen-shot

The total text is now at 95 lessons with homework sets. I’m always happy if a curriculum is somewhere between 100 and 110 lessons. I’ve found over the span of my career that in a 185 day school year, the 100 lesson count is a good benchmark given all sorts of extra days needed for reinforcement, review, assessment, assemblies, fire drills, snow days/delays, Regents exams, and so many other things that stop us from teaching content.

I would assume as I now head into the Answer Key part of the process that I will likely add some additional lessons. There is no mention of surface area in the Common Core Standards or on the New York State Formula sheet, but I still think it is a topic worthy of inclusion in that last unit on Measurement. I’ve only included one lesson on radian measurement, but I believe that could probably use another day. I’m sure extra days on practice of proofs would also be helpful as well as reinforcement days here and there.

Of course, all of that may also come in our second version of the text. After the answer key is done, or partly done, we will begin work on the videos. We are planning on making these much better quality than those for Common Core Algebra I and Algebra II, so they may take some additional time to create. We anticipate beginning the videos in early February, but don’t have a completion date yet for those. We will have all materials ready for the 2017-2018 school year.

We invite teachers to use any of the Geometry lessons and homework sets we’ve posted. Because we have not begun the Answer Key, they are in their Beta version and teachers should work through them before giving them to students. We appreciate feedback both in terms of typos and in terms of mathematical content. I can promise we will correct the typos and will consider suggestions on mathematical content as long as they don’t radically alter the order or structure of the course.

Although Phase 1 of Common Core Geometry is now done, we have two phases yet to go (Answer Key and Videos). It’s about time I got started on Phase 2. I hope everyone had a great Thanksgiving and are ready to get the next month over with before the big break. As always, you can email me with feedback at: Kirk@emathinstruction.com.

 

 

eMath October Newsletter

Here in the great Northeast, the leaves have changed and the days are getting colder. Since I last wrote, I’ve been hard at work on Common Core Geometry and the latest round of add-ons. Speaking of which, we just put them up on the site. Remember, links to them are at the bottom of each course page. You can also click on any of the blue links below to be taken directly to the add-on pages.  As always, we base what we create on feedback we get, especially on Facebook and other social media. So, let us know where you want the emphasis to be placed.

For Common Core Algebra I Add-Ons this month, we have a Form B for the Unit #3 Assessment, a Unit #4 Progress Quiz, and a worksheet on turning visual patterns into arithmetic sequences. We’ve heard from a lot of you that you want more assessment, especially make-ups/Form B’s. With all of our Form B assessments, we attempt to make them mirror the original so that you have as much equity and cross comparison as possible. Unit #4 on Linear Functions and Arithmetic Sequences is a long unit. So, we created a mid-unit quiz that assesses through Lesson #7. We even included a Form A and Form B of the quiz. Finally, we’ve all seen kids struggle on standardized exams turning visual patterns into arithmetic sequence rules. So, I created a short worksheet with a bunch of these patterns for you to use for practice with your kids. This is especially good for a sub day or other time you need a quick resource.

For Common Core Algebra II Add-Ons this month, it is much the same as with CC Alg I. We created a Form B make-up assessment for both Unit #4 and Unit# 5. Unit #4, of course, is that beastly long Exponential and Log unit, so it may help a lot to have a make-up for that one. Of course it also doesn’t hurt to have a make-up for Unit #5 (Sequences and Series) either. We also added a new lesson! No video yet, but we now introduce Unit #6.Lesson #5.5.Using Structure to Factor. This was a lesson we felt we had to create based on some of the very complex factoring we’ve seen on the first two Common Core Algebra II Regents exams in New York State. I must say, I love this lesson and this factoring. It’s all mixed up and forces kids to think about larger patters with gcf’s, difference of perfect squares, and trinomials. Check it out if you have this subscription.

Finally, for Algebra 2 with Trigonometry Add-Ons, we offer two new Formative Assessments for Units 3 and 4. We never did write unit assessments for Algebra 2 and Trigonometry (our first course). So, that’s going to be a focus of the Algebra 2 with Trigonometry Add-Ons this year. We want to make sure that teachers who are using that course have access to quality assessments.

So, besides the add-ons, I’ve obviously been busy, busy, busy with writing Common Core Geometry and working on technical issues with our website. Since the last eMath Newsletter, we’ve put up three more units. Check out all of the materials we have up now under the courses tab:

 

table-of-contents-through-7

Only three more units to go!!! We are trying to get the rough draft of the entire curriculum done by the beginning of winter break. That’s when we will start production on the answer key and on the videos. They will go hand in hand and I will likely post videos to YouTube by the unit.  I suspect we will have Unit 8, on Right Triangle Trigonometry, posted some time next week. The last two units, on Circle Geometry and Three Dimensional Geometry, will take a bit longer to get done and up because it’s…

Conference Season!!! I will be in Rye on November 10th through the 12th for the AMTNYS Fall conference. I’ll be showing teachers how to create interactive lessons on Desmos to address Common Core Algebra standards. I’ll also be showing teachers how to use our new Efofex software line to create graphics for all sorts of fields (geometry, algebra, statistics, etcetera). Don’t miss out. On November 19th, I’ll be at the ATMNYC conference at Hunter College. I’m going to be talking about the thinking that goes into the new emphasis on transformations in Common Core Geometry. I’m really excited about the talk as I’ve never had a chance to speak to teachers about H. Wu’s work on rigid motions and how it leads to congruence in geometry.

O.k. Enough for now. As always, email me with questions, suggestions, or any issues you are having, Kirk@emathinstruction.com.  Have a great rest of your October and a safe and happy Halloween.

eMath September Newsletter

As Labor Day Weekend starts to fade in our rear view mirror and mid-September approaches, it’s time for the monthly eMath Newsletter. I’ve been really busy since mid-August, mostly filling orders, answering lots of questions, and helping folks troubleshoot our new online Answer Key Subscription Service. My apologies for those who had trouble setting up their accounts. As always, email me if you have any issues.

I’ve also been working hard in the last few weeks on the latest round of add-ons that come with our Teacher Plus Subscriptions. I just posted them on the site, so those that have that access (and those that don’t) should go over and check them out. As usual, I wanted to tell you a little about them in the newsletter.

For Common Core Algebra I (our most popular course), we have three/four new selections. We came out with a mid-unit progress quiz for Unit #2. We put up a Form A and Form B for your convenience. We created a worksheet that gives students more practice with linear word problems. This one should definitely be done after you have done Lesson #7 in Unit #2. It’s a really good sheet that can be used as extra credit or just more practice for struggling learners. Finally, we created a small worksheet to prepare kids for inequality work later in Unit #2. This sheet is great for a two year CC Algebra I course where kids have a really hard time comparing two numbers using the greater than and less than operators.

For Common Core Algebra II, I started off with one of my favorites, a Desmos Classroom Activity on Forms of a Line. Students are supposed to come out of Common Core Geometry with some exposure to both the slope-intercept and point-slope form of a line. This Desmos Activity allows students to practice in an interactive way with equations of lines in both forms. It could be used as extra credit, extra practice, or even to replace Unit #3.Lesson #3. Don’t worry if you’ve never done a Classroom Activity on Desmos. I’ve also created a detailed Teacher Direction sheet. Email me if you still have questions.

I’ve also created a mid-unit quiz for Unit #4 of Common Core Algebra II. This is a mammoth unit, so I thought having a quiz that covered the topics from the first seven lessons would be helpful. It’s not a long quiz, but it assesses all of the fundamentals of exponential functions. Finally, I added a brand new lesson to Common Core Algebra II on the asymptotes of exponential and logarithmic functions. The term asymptote does not arise in the Common Core Algebra II PARCC standards, but New York State put it on their June Regents examination in CC Alg II, so I thought it might make sense to have a lesson on these important graphical features. No video, yet!

Finally, Algebra 2 with Trigonometry. I feel like this is sometimes the forgotten child of the three courses. It has been all but phased out in New York State, and, yet, plenty of schools still use our text. We love the course and recognize the important differences and similarities between it and Common Core Algebra II. For this month, I’ve added three new resources for the course. First, I have a full (but somewhat short) Unit #2 Quiz (on linear functions). I also included a Unit #3 Mid-Unit progress quiz. Unit #3 is a long unit on quadratic functions and their algebra, so I thought it appropriate to see what kids know after the first eight lessons. Finally, I included a brand-new lesson in Algebra 2 with Trig on Factoring by Grouping. This lesson was long overdue.

Now that I’ve gotten the add-ons out of the way, let’s talk about Common Core Geometry. I’ve really taken a pause on that since late August. I’ve simply been too busy with running the business and writing the add-ons, but now I should be able to get back to writing it. I’m in the middle of Unit 5 right now (the first four units have been posted) and hope to have it finished by the end of next week. I’ll post all of the first draft pdf files at that point. I’m going to continue to write units and maybe record some videos (just to try out some new tech that I have). My goal is to be done with all lessons and homework sets in first draft form by December winter break. The answer key and videos will take some time as well, but, with some hard work, I will hopefully be done with it all by late winter/early spring. Only then will we have Common Core Geometry subscriptions and workbooks to sell at eMathInstruction.

For those of you already working with the curriculum, I image some will be almost done with Unit #1 soon. Unit #2 in CC Geo is on transformations and many of the lessons involve the use of tracing paper. We create our own because we couldn’t find anything on the market we liked. I’m hoping that by Monday of next week (9/19/16) we will have it on our site for sale. It will come in 50 sheet spiral packs that students can rip sheets from (for $5 each) or in stacks of 500 (for $20 each). Here are a few pictures of the tracing paper:

img_3283

img_3284

Finally, a note on  our new software products from Efofex. There’s been a lot of interest in the programs given their ease of use. I was talking to a teacher on the phone about Geometry just a couple of days ago and he was bemoaning how difficult making diagrams for geo can be. Efofex MathPack, one of the packages we sell, makes creating these things so easy that I’m chomping at the bit to get back to writing it all. I just wish the software would have been around back in 2005 when I first started work with the Arlington Algebra Project. Imagine how long this would take using the standards graphics on MS Word?

Right Triangle

And my own least favorite graphs to draw, exponentials:

Shifted Exponential

If you are interested in seeing what the Efofex software can do, try downloading a free 30 day trial of it. No muss, no fuss, and no spam if you decide you don’t like it. It just stops working. If you have any questions about it, don’t hesitate to contact me.

O.k. So, that’s about it for September. I hope that everyone’s school year is starting off well and the temperatures are cooling everywhere (I’m sure Phoenix is still quite hot my Arizona friends). I’ll be working hard on Common Core Geometry and more add-ons in the next month. Tune in for all the updates in the October newsletter. As always, contact me via email if you have questions or suggestions: Kirk@emathinstruction.com.

 

Common Core Geometry – by Kirk

So, the summer has been a great mix of working on Add-Ons for the three courses we currently offer and thinking a lot about Common Core Geometry. I’ve now completed the first four units and a rough course outline that’s been indexed to the CCSS Standards for CC Geometry (as defined by PARCC and refined by NYSED).

I’ve got all of the lessons for those four units posted at this point under our Courses section:

CC Geo Homescreen

You can find the course outline if you click on the Table of Contents and Standards Documents link. All of the lessons and homework sets posted for the first four units are in rough draft form only. All diagrams that occur in those files were created on our new Efofex software.

The units of the course (at least at this point) are:

Unit 1 – Essential Geometric Tools and Concepts

Unit 2 – Transformations, Rigid Motions, and Congruence

Unit 3 – Euclidean Triangle Proofu

Unit 4 – Constructions

Unit 5 – The Tools of Coordinate Geometry

Unit 6 – Quadrilaterals

Unit 7 – Dilations and Similarity

Unit 8 – Right Triangle Trigonometry

Unit 9 – Circles with and without Coordinates

Unit 10 – The Geometry of Three Dimensions

I don’t think the order of the units is anything too radical, but I wanted to discuss why I have them in this order and my overall thinking at this point. I do want to be clear up front that I believe geometry is the most tactile of all mathematical fields. It is about space and that deserves not just proof and algebraic problems, but also the tools of geometry, which include compass, ruler, straightedge, protractor, and, yes, tracing paper. We currently developing a really good tracing paper that we will begin to sell on our site right after school begins.

eMath_TracingPaper_02

Geometry is really all about exploring physical space using tools we develop in mathematics, whether that be transformations, Euclidean logic, or Cartesian coordinates. I began the course in Unit 1 by making sure that students have some of the very basics down, including a really good sense for circles and arcs of circles.

In Unit 2, I wanted to establish certain properties about space by using H. Wu’s rigid motion work with congruence. This is an awesome unit, but probably one of the newer ones in the Common Core. Within this unit, though, we use rigid motions to establish a number of important facts, such as SAS, ASA, and SSS criteria for triangle congruence, facts about the perpendicular bisector of a segment, and parallel line properties.

Unit 3 puts us back on more solid ground with Euclidean triangle geometry proof. I did bring parallel lines back into the mix so that students could also be exposed to AAS and HL methods of proving triangles congruent. We look at many of the classic problems of geometry in this unit, including why the angle bisector represents all points equidistant from the sides of an angle.

In Unit 4 we explore the beautiful world of constructions. I know constructions can be a mystery to students, a set of directions that they need to memorize to carry out the construction. I make sure within every lesson that the constructions have both purpose and are proven by using Euclidean triangle proof.

I haven’t created the remaining units, but want to explain how they will unfold (hopefully). In Unit 5, I am going to introduce coordinate geometry. I wanted to do this relatively early in the course for two primary reasons: (1) so students could have a break from the world of Euclidean reasoning and (2) so that we could use coordinate geometry as well as Euclidean geometry and Wu’s transformation work to explore quadrilaterals (Unit 6) and similarity (Unit 7).

Of course, once we’ve established the concept of similarity in Unit 7, we will naturally move to right triangle trigonometry in Unit 8. Right triangle trig is one of the most applied fields within geometry and we will make sure students understand its basis in similarity and then its application in the real world.

Unit 9, which will come towards the end of the third marking period, will challenge students will the geometry of the circle. But, at this point, we will have a good background in congruence, similarity, the distance formula, and other tools that will allows students to explore the geometry of circles in the Euclidean and Cartesian planes.

Finally, Unit 10 will concentrate on three dimensional geometry as well as measurement and geometric modeling.

I will continue to work on lessons (Unit 5 here I come) for the remainder of the summer and will then begin video work for the first few units once Labor Day is behind us and my own kids have gone back to school (Max and Evie). Hopefully by late September all videos will be done for Units 1 through 4.

Work will continue on the units with pauses in October and November for all sorts of conferences (AMTNYS, AMTNYC, etc). I think, given my current pace, that the first draft of all units should be done by Winter break. I would then expect videos and answer keys to be done with books ready to order by late winter or early spring.

Feel free to give me feedback if you try the materials. If you have questions about the order or any given lesson, just shoot me an email: Kirk@emathinstruction.com

Cover 4

 

 

 

 

 

Common Core Geometry – by Kirk

The summer has begun and so I have begun my work in earnest on eMathInstruction’s Common Core Geometry. I have wanted to write a text on Geometry for about as long as I could remember, but things just kept getting in my way (kids, teaching, algebra… you know, life).

Geometry is an interesting subject that has been studied for almost 3,000 years. Let that one sink in. Three thousand. Euclid’s Elements is regarded as one of the most important books ever written, and he composed that little tome about 2,400 years ago. And, yet, why do we study geometry? We can cite art and architecture, trigonometry, and measurement as needing geometry. But, still, why proof? Why axioms? Why the endless emphasis on terminology, theorems, and corollaries? Dr. Piers Bursill-Hall of the Cambridge Faulkes Institute for Geometry (yes that exists) said it eloquently:

“Since the ancient Greeks, geometry has been the paradigm of truth and ordered knowledge, of clear thinking and the rigor of absolute precision of thought. But, it’s real power is that it is also about the world around us. It seems that one of the most fundamental of human scientific intuitions is that the physical world is ultimately geometric and that to study geometry is in some sense to uncover some kind of ultimate essence of the physical world around us.”

Wow! Go Piers!!! I know even asking the question might seem a bit odd, given that we all studied geometry on our way to becoming math teachers. But, unlike algebra, much of geometry faded for me as I pursued my studies in higher-level math.

Add to that question, the confusion that comes from changes to the way we think about a Geometry course given the Common Core Standards, and I wanted to think hard about this course before I started writing. I love, love, love geometry and want to write its story in a way that engages kids and teachers. A way that is hands-on and accessible to all students, including those with disabilities. But, whatever this course will become must also meet those Common Core Standards as outlined by the PARCC organization (and “clarified” by various state agencies like NYSED).

Now, much of geometry in the Common Core hasn’t changed, but a very new piece of it is the emphasis placed on using transformations to establish congruence and similarity. Not just establish, but define congruence and similarity. This is all based on the wonderful work of Hung-Hsi Wu, of U.C. Berkley. If you want to really understand geometric thinking based on transformations, you need to read his works, especially Teaching Geometry According to the Common Core Standards. I’ve digested two of them (around 150 pages each) in the last few weeks.

I’ll be incorporating his ideas as well as classic Euclidean and Cartesian geometry into one large picture. I’ve already made progress and am hoping to release the first three units of the course by mid-August. Then, I will continue to post new material in rough draft form as the year progresses. There will be no products (answer key subscriptions and workbooks) for sale until the Spring of 2017, but there will be plenty to look at and use.

A warning to all geometry teachers, though: this course will require extensive use of rulers, protractors, compasses, and tracing paper. That’s just the deal with geometry. It’s hands-on; it’s interactive; it’s tactile. On each lesson and homework sheet, I’ll put icons that quickly show which geometer’s tools are needed. Don’t worry, we will be offering easy to use tracing paper in our eMath Shop eventually.

And it should be the funnest math class a kid takes. In my humble opinion, at least.

Here are some pictures:

Cover 4

 

IMG_3095

 

IMG_3093

Area of a Circle – by Kevin Hoang

So, a friend of ours from the great state of California, Kevin Hoang, sent this really neat Geogebra demonstration that a student made. It shows the classic slicing up of a circle into sectors and then rearranging them into a parallelogram like figure.

This is an amazing demo if you are exploring informal limit ideas in Common Core Geometry (yes I went there) to derive classic measurement formulas.

Thanks Kevin!

Area of a Circle in Geogebra

Geogebra Area of Circle