A blog by a veteran math teacher who tries to learn new things. Most posts are about teaching math and problem-solving, but sometimes, it's just about life.
It was Meg's puppy/kitten/unicorn post and Sarah's kitten blog that got me thinking about creating a list of common student mistakes, and when my Algebra 2 Honors students came in at the beginning of the year making the same ones over and over, I thought it was time to take the "unicorn" by the horn.
Here is my Don't Kill a Kitten handout for Chapter 1 in Algebra 2 Honors. Whenever a student begins to solve a quadratic equation by setting it to the constant and factoring out an x (the horror!), I have them go to their handout and star the mistake they are making. The hope is that it will really stick with them. So far it's working. I may make one for every chapter.
I totally stole this Headbanz idea from Mary Bourassa, and I took many of the equations from Meg Craig's Speed Dating handout. I had leftover stock card and ribbon from old projects and I used a box cutter to make the slices on either side of the stock card to put in the ribbon. (Note that we do not teach students transformations with both a horizontal stretch/shrink and slide until we get to trig functions.)
My rules: (I have never really played Headbanz, but here's what I did)
1. Once you have the headbanz on (ask someone if it's right side up), you may ask one person a question.
2. The person who answers CANNOT speak! (They can move their arms in the shape of parent functions, for example.)
3. That student now asks you a question in the same way.
4. Each of you moves on to a new person and continues in this fashion until you know the equation.
5. When you think you know the answer, come to me and I will tell you if you are right.
This was really fun to watch. They could ask, do I have a vertical stretch? Then next student, what is the vertical stretch, and that student could show with their fingers. They could not, for example, say, what is the number inside of the function? So they had to use math language like: reflection, horizontal stretch, shrink, translation left, and right.
It was very fun, and the students recommended that I keep the rules the way we did them. However, I definitely would do this DAY 2 instead of day 1 when I taught the lesson because they did not all have time to process the information enough on day 1.
Space Race from Quizlet
The day before I taught transformations, I taught parent functions. For homework, I assigned this Space Race from Quizlet. I told them to work on it for at least 10 minutes and email me a snapshot of their highest score. Kids love it, and they really learn the parent functions better than any other way I have used in the past. I tell them to study the phrases and equations as they are on the page you see when you open the link, because when they click on space race in the upper right hand corner, the graphs or names fly across the screen and they have to type it exactly as they see it...try it! There are high scores, and I give a prize for the student with the highest score in each class. Desmos needs Space Race, am I right??
Notes:
Here are my notes for parent functions (this includes piecewise functions) and then transformations of functions.
And my notes from Transformation of functions...I actually use the Chalkboard font, but here it looks much more playful...not sure why it changed!
Speed dating:
I am going to use this on Monday after we go over homework. (we have a 30 minute period) by lining up desks and having students sit across from each other. I will give them a few minutes a problem and then have one side shift to the left (with the last person coming around to the first desk.) They will first have to check their answer from the last "date" with the new "date," and then they can move on. I will try out the datexx timer I just bought. Did I see that from Julie? or Meg? I love twitter.
Homework and answers:
Who did I get this worksheet from??
Explain Everything Video:
As I said, I didn't think my students had enough time to process, so I made this video on Explain Everything, which I blogged about once here (my first blog ever.) I uploaded it to Google Classroom so they could watch it this weekend if they needed any help. It is NOT perfect...and I don't necessarily recommend that you use it...but I am showing it to show how easy it is to do--you can make a quick video to recap what you did and send it out to students.
My rule for videos (as you can see)--make it once. Don't keep trying to fix what you did or what you said or it will take forever. Just make the correction and move on...unless REALLY wrong...I hope I did not make any mistakes that I didn't fix!
And that's it for transformations 2015. Any other transformation ideas?? Please send them along!
Our math teacher created a new course elective at Saint Andrew’s School called Honors Problem Solving Seminar. The class is interesting and has taught us everything from grit to fun little math tricks. So far we have discussed visual and figurative numbers, which are numbers that can be represented by a regular geometrical arrangement or sequence of evenly spaced points. They are most commonly expressed as regular polygons, for example, triangles, squares, pentagons, hexagons, etc. and also known as polygonal numbers. The class was a nice addition to the schedule providing us with a very relaxed way to enjoy math. This is what we have learned in our first big unit about figurative numbers.
(Note: this blog was collaboratively written by the entire class, originally in a Google Doc. All handouts are posted in my blog here.)
We first started with oblong numbers. Oblong numbers are the number of dots that can make up a rectangle with its length one more than its width.
To get to an explicit formula that can represent the development of the patterns of the number of dots on each rectangle, we first started by observing the rectangles. The first one consists of two dots; two for its length and one for width. The second has three for length and two for width. The third has four for length and three for width. We found out that as the pattern keeps going on, the new rectangle has one more dot on both its length and width than its previous term. So first, we constructed a recursive formula, using the number of dots on the previous term to define the number of dots in the current term. Since the length and width are both added by one each time, we thought that the current term is the resulting number of rows would be n and the resulting number of columns would be n+1.
Thus, the explicit formula for oblong number is n(n+1)
Then we were given triangular numbers. Triangular numbers are the dots grouped together that make up an equilateral triangle. Initially, we were trying to come up a formula based on the pattern and number of dots in each term: 1, 3, 6, 10... Obviously, the pattern does not appear directly based on the number of dots in each consecutive term. However, with Mrs. Winer’s small hint of drawing a diagonal, we found out that each term of the triangular number is exactly half of the oblong number. So the formula is basically the formula for oblong number divided by 2.
The explicit formula for triangular number is n(n+1)/2
Now we can use what we have learnt from the oblong and triangular numbers to apply it to finding the sum of n counting numbers. If we pay attention to the triangular number, we will notice that the formation for triangular number is:
So we can find out nth triangular numbers means the sum of the nth counting numbers.
Now, for the sum of nth even numbers, (not counting zero), the first even number, 2 is two times bigger than one, the second even number, 4 is two times bigger than two, and the third 6 is two times bigger than 3.The pattern is the same for their sums. While the sum for all the counting numbers are triangular numbers, by studying the even numbers, we learned, that oblong, the combination of two triangular numbers, is the sum for the even numbers. We also can find it out by investigating the oblong numbers diagram.
After learning about the sum of n even counting numbers, we asked what the sum was for n counting odd numbers. We found out that you could find the sum of the odds just as fast as the sum of the evens. By looking at the diagram below (square numbers), we can conclude that the explicit equation for the sum of odd numbers is n2because in the diagram the number of dots in each picture is a perfect square of the nth term.
Moving right along, we now faced the challenge of pentagonal numbers, which served to extend and solidify the concept of triangular numbers.
In a pentagon diagram, we can divide it into three parts (shown in the picture): red, blue and green. Count the dots in each part, and we can see the red part and the green part are the same; they are both the nth triangular number. In the middle, the number of dots in the blue section is also triangular number, but instead of nth term, it is the (n-2nd) term. Then we add all three parts together, but we count the first dot at the bottom twice, so we subtract 1. Simplify the equation, and we get the pattern for the pentagonal numbers.
Seeing that we were able to complete each task given so far, Mrs. Winer then challenged the class with yet another problem that once again would require an understanding of both the oblong and triangular numbers. She asked us,
“ What’s the difference between the sum of the first 2015 even counting numbers and the sum of the first 2015 odd counting numbers.”
At first the class was very confused and did not know how to solve this problem, but after much hard work, our class proudly could say that we discovered not one but TWO! very different methods to solve the problem.
Solution 1
First line up the first 5 even and first 5 odd numbers
2 + 4 + 6 + 8 + 10
1 + 3 + 5 + 7 + 9
Now you can see that the even numbers are always one greater than the odds. Since there are 2015 numbers, and each time the evens are one number more than the odds, the total difference will be the 2015.
Solution 2
This method uses direct substitution to find the first n counting even and odd numbers.
“N^2+N,” is the equation for sum of the even numbers and “N^2” for the sum of the odd numbers. The difference between the first n even and n odd counting numbers is the difference between the nth oblong numberN^2+N and N^2:
N^2+N - N^2
The class then used direct substitution to solve the problem
(2015^2 + 2015) - (2015^2)= 2015
Although the two methods are very different, they both work in finding the answer. The class was split over which one they thought was easier.
According to Pythagoras,“everything is related to mathematics. Numbers are the ultimate reality, and through mathematics, everything can be predicted and measured in rhythmic patterns or cycles.” So far, Pythagoras has been right about everything we have learned in this class and it is cool to see how everything comes together. Although there may be different ways to solve a problem, when it comes to math there is no ambiguity and everything eventually will fit together to form one solution.
I use the screen shots all the time--especially for student notes or while blogging. Here is how you screen shot and save to the clip board. Press these buttons below and the cross hairs will appear. Just drag what you want saved and then paste it where ever you want it.
For blogging, I prefer Command + shift + 4 because the photo will be saved to your desktop for easy import.
You may want to have a folder on your desktop labeled “screen shots,” and drag them in there so your desktop is not cluttered. To do this, go to your desktop, two finger tap, new folder.
Did you ever think about what the first few letters in Mu Alpha Theta spells? I love when my kids discover this on their own. I also love these tattoos that I got from http://www.tattoofun.com/. I could see using them as prizes in math class, but I actually got them for my math club Mu Alpha Theta, which I wrote about here. This is my fifth year advising the club, and we are the biggest club in the school. We will start meeting once a week again this Friday, and we have a lot of new students who are interested...I am really looking forward to a great year. It is such a fun club, and it has definitely given me so much joy. I highly recommend that everyone start one at their school! Students elected a President, Vice-President, Treasurer, Secretary, Historian, and Social Media Specialist (aka PR). We gave the tattoos out at the Club Fair, and students will likely wear them at our competitions next semester. Go Mu and Go Scots!
Last year, when I first started toying around with technology in the classroom, I printed QR Codes on stickers for Back to School Night. The codes had all the information parents needed from me: email address, phone number, office hours, etc. I typed "Thanks for popping in!" next to the QR codes on the stickers, and placed them on uncooked popcorn bags. I blogged about that here.
Parents told me that they really liked it, which was great. However, I have really grown technology-wise in the last year, thanks to Philips Exeter, TMC, ISTE, and of course, MTBoS. So this year, I used a thinglink (if you don't know what this is, basically, you link sites or pictures or text to a spot on a picture that you download). I made two of them...one for all classes, introducing myself to parents, and one that had a picture of each particular class.
This year, when parents came in, I had this picture on the smart board:
NOTE: IT TAKES A SECOND TO LOAD...WAIT FOR THE LITTLE BUTTONS TO SHOW UP!
Slide your cursor over the picture, and it should show little icons that you can click on. Just in case, click here to see it live. It's very cool! First, I went to http://ipiccy.com/ and created a shadow picture of myself. Then I imported a beautiful picture of our campus behind me (see the rainbow?? Thanks to JSA for that pic). Then, through thinglink.com, I made questions about myself where when you touch most of them, things open up. (I got the shadow and puzzle idea from a blog years ago...I wish I knew whose it was!)
First, I did not realize how small the text was on this picture until right before the parents came in, so I made the parents come up to the smart board...and that was actually a highlight of the night. They were much more participatory. After we went through each thinglink and answered the riddles/math problems (the answer to the puzzle, btw is Back To School Night, just in case), I then put up a second one:
(Original link is here.) Keep in mind that the real picture had all the kids in it, but I took it out for blogging! Each thinglink that I touched had to do with my extra help hours, calcchat.com (which I love), the school website and google classroom, and then how I use visible random grouping, desmos.com, and goformative.com (all of which I blogged about here and here.) The parents truly loved seeing how things have changed in the classroom since they were students, and they were genuinely excited for their children. I love that you can link to text, a picture, or a website immediately. The 10 minutes were up before I knew it. Just keep in mind that you can get a free trial from Thinglink, but otherwise it costs $35 a year for educators.
I will definitely do this again next year, but by then, who knows what the latest technological sites will be? Or how much better will these sites? It's crazy to think how far I have gotten in just one year, but I promise you, it gets exponentially easier with time.
I just wanted to thank Heather at Global Math for finding my first Back to School Night blog and asking me to participate in a webinar next week. Unfortunately I can't, so I decided to write about it instead.
Now that I do Visible Random Grouping everyday, which I wrote about here, I had to come up with an efficient way to hand back homework and assessments (I should probably state here that I collect homework every day, but I may save that discussion for another blog). I used to try to place the papers on the assigned desk of each student before class. But now that the groups are random every day, it is not easy to return these graded assignments efficiently.
I found some pocket over-the-door shoe holders, and I attached index cards to them with students' names. After I grade any formative or summative assessments, I stick them in each cubby, and I can do this after school or whenever I have the time. Students are excited when they come in to see that they have "mail." They know this is the first thing they should go to before they find their seat.
A major rule is that students are not allowed to look in anyone else's cubby. I fold assessments and place them grade down so it would be hard to peek at someone else's grade.
I think next year, I will have them design their own index cards. I did that a number of years ago when I first used the shoe holders, and students would actually write notes to other students and put them in cubbies as well--even from other classes! I'm not sure why I gave them up for a few years and put them in storage; I think it was because it does take a bit longer than to put them out on the desks. One advantage though is that now you can do it at your leisure, and not just right before class.
Finally, I had another idea: if I catch a student looking at their phone in class, it goes in his/her cubby for the rest of the period. Oh--and I alphabetize the index cards...but I had to switch a few tall students with shorter ones, so keep that in mind if you do this!
It is also good for when students are absent and you have a handout for them--just put it in their "mailbox." And if I need to see a student after class, I can just leave them a note. How many times I have forgotten to tell a student I need to see them, only to remember right after they walked out the door?!
I really don't know many of them personally. I mean, I feel like I know them, but they don't really know me. I read their very relatable posts and sometimes even read through their twitter banter at night before going to bed. I may make a small comment here or there, but mostly I just favorite their tweets and save their blogs and post them on my High School Math Lesson Plans Pinterest board for later use or to share.
Who am I speaking of? Those in the MTBoS = Math Twitter Blogosphere. You can google anything you're looking for in the MTBoS here, or you can subscribe to Bloglovin.com and get them delivered right to your proverbial door, or in my case, iPhone (you can save them here in categories as well.)
You know that song, It's the Most Wonderful Time of the Year? I am such a dork. I actually hum that to myself in the morning, replacing the word year with "day," as I sip my coffee and read through blogs before work. And yesterday I threw my lesson out the window in my Honors Problem Solving Seminar when I read the blog posted on BetterQs, written by Dan Anderson. He wrote about this Ted Talk below. WATCH IT!! It's fantastic, and not only did my kids love it (it's counterintuitive--the best kind of learning occurs, in my opinion, when it is--be sure to stop the video when it says so you don't get the answer), but so did the students and colleagues that came in during a routine tour of the school. In fact, my colleagues were still working on it at lunch and did get it (with much joy) when I gave them Dan's hint: What have you tried that hasn't worked? Great question. I will be using that question and the bridge puzzle, I am sure, for years to come in my teaching career. I will give it to my other classes when they need a brain break (though truly, it is no break for the brain!!)
I took what I saw on the MTBoS and added some personal flair, I guess you could say, because for homework, I gave them the assignment below:
And then they have to fill out this form for their e-journal:
I am looking forward to their responses (and perhaps hearing from their parents or teachers.)
The week before, the same thing happened. I was looking through tweets this time. My former colleague and newly discovered twitter buddy and problem solver extraordinaire, mrdardy, suggested that perhaps I should tweet up with Wendy Menard, who also has a problem solving class. (Love that he felt like a matchmaker! It's the beauty of the MTBoS.) We shared ideas, and I saw a retweet:
And of course I had to watch this awesome video about a math major who explains why math is scary, but why you shouldn't be scared of math...it is fantastic and I showed it to one class so far, and my students agreed with all of it...especially the students who love math that get the crazy looks from people when they tell them math is their favorite subject! (PS: to Wendy...I think I suggested for you to watch this with your class, when now I realize I got this from you!)
And then there's Sarah at http://mathequalslove.blogspot.com/. You can get a new idea or 12 every time you read her blog. But I saw her tweet the other day, and I can't wait to do it in my class!
I could go on and on. That's why the article that Glenn Waddell entitled NYT, YOU BLEW IT really resonated with me. Why pay teachers for their ideas when you can give and get for free through the MTBoS? My teaching has completely changed in the last year that I have discovered the rich, limitless resources out there, and I know it's helped a few of my colleagues, too. I signed up for Teachers Paying Teachers initially a few years ago, and followed some blogs from there, too. But I felt like I was reading a sales pitch every time, and I quickly removed them from my list of blogs. It's not that I don't like what they are doing, it's just not for me. Good for them for making money. But I'm just fine sharing and gaining these incredible ideas for free.
Each and every day in the MTBoS, I get a new idea. Which means each and every day, my students get to try out something new that hooks them in that I never would have done to begin with. And no, I'm not losing class time. I'm gaining student interest in math. And that is something that can never be lost.
