We just completed a project in Algebra 2. Each student designed a picture made up only of lines and shaded regions. They had to then determine the equations of the boundary lines and the systems of inequalities that described each region. The drawings and equations/inequalities were all displayed today; to finish the assignment, each student had to pick someone else's project, copy down their inequalities, graph it, and see if they got the same picture.
Graphing systems of inequalities is always a challenging topic for my students; this project underscored just how difficult the topic is, as most students needed repeated explanations and examples, just to figure out what they were supposed to do. Though it took longer than expected (and time budgeted), it seems to have been worth the time. Ideally, this should be followed up by a linear programming unit, but we don't have the time this year. Here are some pictures:
And, a bonus picture from a few weeks back... each year we have a spirit week in April, and part of it is class competitions. Here is our media center all decked out for the event.
My students always complain that we don't play enough games in class. I know they love games, but most of the games I've seen are quite ineffective. "Showdown" is one of my favorite review activities for my older (more mature, more motivated) students in Algebra 2, but it doesn't work so well for my freshmen in numeracy.
To many of my students, tic tac toe is a riveting activity to be played surreptitiously during a dull lesson, or after a test. I thought I would capitalize on that, and so I present to you:
The students are broken into pairs that collaborate against other pairs. Each group of 4 is given a game board with several tic tac toe grids on it. One pair picks X and the other gets O. Turns alternate with each problem. When the problem is shown, both teams should work on it. If it is X's turn, if they are right, they get their square. If they are wrong, and the O's are right, the O's get to steal a square. (Students took a while to get this - at first they all thought it was unfair). When a game is won, the winning team gets a point. At the end of the activity, whichever teams won more games get a prize.
The benefits: - All students are engaged on every problem. Even if it's not their turn, they can steal if the other team is wrong. - Students have a partner to collaborate with, so weaker students are not put on the spot and can learn during the activity. - Pairs monitor each other for cheating - they can only get the square if they've shown their work. - Tic tac toe is the funnest game on the planet. Apparently. - Generic mechanical pencils in fun colors come 30 to a pack for $5. Great prizes! Mini candy bars work too.
Enjoy! Let me know if you play it and it works (or doesn't work!) for you.
The 10% cut in school funding in CA has really thrown the school for a loop, and we've spent the last couple of months figuring out how to respond in a way that is positive for both families and staff. It's been very difficult, but it has also helped spur us on to re-look at some of our practices and hopefully make some positive changes. I'll write more about the actual changes once they get finalized - everything is still in the discussion phase, although we are beginning to make some real headway.
Note to bay area math teachers (or those wanting to relocate to the 408): we are hiring! Check out the website or email me if you want more info.
One of the problems with block scheduling is that, when you lose a day of school, it throws your whole system off. Due to community day on Thursday, and spring break starting on Friday, periods 1 - 4 met twice this week, but 5-6 only met once. So, it was time for a slush lesson. Sorry, I mean "enrichment". I find these hard to do well, because if it is something worthwhile - such that you can justify spending 80 minutes of time with periods 1 through 4 - then you want the other periods to see it to. And if it isn't worthwhile, then why not just have a pizza party or something? But you'll never catch me throwing away a lesson like that. There's just no time to waste.
So I decided to experiment with correcting a linguistic problem that bothers me, but is not necessarily mission critical. That is the reversal of terms when saying division and subtraction problems out loud, confusing divided by with divided into, and my personal favorite, "subtract 7 to both sides". I know that part of the problem here is the somewhat arbitrary nature of prepositions, and I've been told that fluency with prepositions is one of the last things to develop when a person is learning a new language, and can take many years of practice. When students make these mistakes in class, I tend to repeat their words back to them, using the correct language, but not making a big deal out of it. My thinking here was that I could do a lesson on it, and then, when they make those mistakes in the future, I can just say "remember the correct way to say that?" and jog their memory, instead of launching cold into an explanation again and again.
I did the lesson. Nothing fancy - just some explanation, some practice, a little board wars (which I typically shun, but it's a slush lesson, so what the hey) and some delectable Easter candy prizes. Yesterday, the students were pretty non-enthused about working on prepositions (shocking, I know), and board wars was so-so, although there were quite a few kids who were very motivated to win the giant bunny lollipops. Today, I had some pig- and ducky-shaped candies to give away, and I think I struck gold, because the minute I showed them to my class, they freaked out and got super-focused. I don't really like bribery, but I think it's probably ok to break form on the day before vacation.
In any case, we were well into the first round of board wars when the phone rang. When I picked up the receiver, I heard some students say "Mr. Greene, we're in Algebra class right now and we have a question." I was pretty confused, until their teacher came on the line. He had them on speaker phone, and said, "My students are telling me that I'm not speaking like a mathematician." (Speak Like a Mathematician is the phrase I use with them for all matters linguistic.) They were all laughing in the background. I finally got what was going on, and said, "Hold on, let me put you on speaker phone here." When I did that, his class erupted in a cheer, which my class could hear, and they were shouting hellos back and forth (although nobody knew who was in each class). They quieted down, and I had them ask me the question - it seems that their teacher said "subtract by 7", and not only did they notice the mistake, they had enough confidence in themselves to call him on it. So I settled it for them, all the kids shouted goodbye to each other, and we went on to an excellent board wars competition.
Later, when talking about it with the other teacher, he told me that he had actually read a problem that said "reduced by 7", but the students swore he said "subtracted by 7" and he decided to play it up for them and call me since he knew I'd been working on it with them. Moments like that are really cool (and potentially powerful), and they can't really be planned out. I love when the last class before a break is a really good one.
p.s. Does anyone else remember the phrase "anywhere a cat can go"? I still remember it from 7th grade French.
p.p.s Funny cat videos. My classes loved these for the physical humor. But if you've ever had a cat, you'll see that the cartoonist captures their behavior really well. Enjoy!
I've been using Keynote this semester as an experiment, to see how it could work in my Numeracy class. So far, it's gone pretty well - especially after I bought a remote mouse so I could control it from anywhere in the room. Combined with the mini-whiteboards, it's been a really efficient way of getting students to do work. After presenting a concept, I can have them practice a few problems right away by showing the next slide, and having them work on their boards. There is no time wasted passing out worksheets. Also, I can make sure all students are focusing on a specific set of problems (versus on a worksheet, where they tend to start jumping around right away, based on what seems easiest). Then, I can show work/answers on the slide without having to pull out a transparency.
Since I've got the projector reserved and set up now, I can easily insert fun and interesting images, sounds, and video clips. I've recorded myself and other teachers singing little ditties (like the infamous "Don't add across"). I've started scouring YouTube for interesting stuff... though the ratio of total crap to interesting stuff is quite high, I've found a couple of gems. I even unearthed my old calculus professor from college, who recorded a "top ten algebra mistakes hit parade" as well as "all of calculus in 20 minutes".
So I'm in my fraction adding unit now, and we've been working with fraction circles to understand adding. Now, we're taking a break from that to do some work on prime factorization, reducing fractions by canceling common prime factors, and finding LCM. Once they get all this mastered, we can go back to adding fractions using common denominators. I hope they don't forget it all over spring break... I've always found it difficult to teach factors and multiples, and GCF and LCM because students confuse these concepts very easily. Part of the problem is their difficulty with the language of division. Just about every student I have says "divide 6 by 40" when they mean 40÷6. If I ask "does 3 go into 12?", they'll say yes. But they'll also say yes if I ask "does 12 go into 3?". (Aside: I think I'm going to devote an entire lesson to this issue - along with the whole "subtracted from"/"subtracted to" issue.)
In any case, I YouTubed LCM and GCF to see if there was anything interesting out there. I was surprised to find a method for finding both LCM and GCF at the same time using Venn Diagrams that I'd never seen before. It's mathematically equivalent to looking at the prime factorizations and picking the right factors, but it provides a nice structure for students to remember which is which. So I designed a lesson to practice finding factors and multiples, and then using this model to find LCM and GCF. It went quite well. I don't know how much will be retained over the weekend, but we'll practice more on Monday/Tuesday because I want them to have LCM down solid. Here are two of my slides, and then the original video I got the idea from.
In the past, I've given diagnostics before a unit so as to be able to compare pre- and post-instruction scores. Now, in the spirit of differentiation, I'm going to go one step further.
The next unit is about adding and subtracting fractions and mixed numbers. On my diagnostic, I wanted to see what percent of the students are still "adders-across" (#25 down: snakes that are bad at math). That would be 68/80, or 85%. The remaining 12 students could all do the basic algorithms, but most stumbled on the more complicated mixed number subtraction problem.
So here's the plan. In each class, I will assign one of the non-adders-across (NAA) to an adder-across (AA), tasking the NAA to help the AA learn over the coming lessons. If I see that they remain on task during practice time, the NAA will not have to take the quizzes, earning an automatic 100% on them. This seems reasonable, since they have already shown me they know the skill. Additionally, if the AA passes the quizzes (i.e. becomes an NAA!) then the NAA helper will earn some oh-so-coveted extra credit points. This way, the NAA has strong incentive to help, but there is no penalty if the AA doesn't make enough improvement.
Since almost no students showed mastery of the mixed number subtraction problems, every one will need to take that quiz when we get to it.
Now, the only thing that remains is to pair up the NAAs with the AAs effectively. I need to factor in personality, motivation, and so forth. Also, this experiment really highlights the imbalance between classes, even though we try to avoid any tracking (a constant difficulty in a small school). Here are the numbers of NAAs by period... Period 1: 5, Period 2: 4, Period 4: 2, Period 6: 1.
I've been using mini-whiteboards daily in my numeracy classes all year. Students use them most of the time, except when I have a worksheet for them to do (and even then, they tend to use them for scratch work).
Positives:
I can see, from anywhere in the room, what students are doing, and if they are on task.
Students enjoy writing on their whiteboards more than on paper.
Students don't have to waste paper for scratch work (this is especially helpful for those students who have still not mastered the art of bringing school supplies to class).
And I don't have to make worksheets for every single task either.
It makes collaboration easier during pair/group work tasks.
It's great for quick checks of understanding - put a problem up, students do it on their boards, and then immediately lift them up for inspection.
Negatives:
We burn through markers like nobody's business, and the ones that are low-odor cost about a buck a piece. I've tried the cheaper ones, but they run out really fast, or have fumes that cause much complaining of headaches.
Tables and hands tend to get really messy (for some students more than others...) Our beautiful white laptops are getting covered in whiteboard marker smudges.
"Mr. Greene, can I please go wash my hands???"
Some students not able to respect materials, destroying markers by pounding in their tips, or writing with them on paper till they run out.
Some students unable to stop drawing beautiful works of art when I am presenting material. Or maybe this is a positive because I can see that they are off-task, whereas if they were doing plain old paper-and-pencil doodling, I might not notice?
I was wondering if anyone had any ideas to help with the logistical issues of mess and expense? Remember the Magna Doodle?
Most of my numeracy students remember that helpful rule from middle school: "Multiplying by 10 means adding a zero", and so we get results like the title of this post. This is one of those fundamental place-value problems, the type of thing that betrays just how little some students really get about the number system. It's taken about two weeks of practice to get them comfortable with the idea of shifting the decimal place left and right (and remembering which way to shift it, depending on the operation).
We are also currently struggling with the issue of the missing decimal point... when there is no point shown in a number, where is it really? Some of my students still think that you put the point at the front of the number. Why do they think this? I'm not sure. Before break, we spent a whole lesson on what the decimal point means, and it seemed to go well. Since we've been back in the second semester, the question of where the missing decimal point goes has been asked and answered many times each class period. They are getting better at comparisons: if I ask them to compare 473 and .473, or .4 and .39, or .4 and .04, they are usually getting it right. And yet, when faced with the problem 473 ÷ 100,000, some students seem to forget it all and start with the decimal at the front of the 473 (or sometimes between the 4 and the 7), forgetting that this changes the value of the number.
No wonder scientific notation is such a bear to teach in Algebra... To reinforce both concepts, I've been teaching scientific notation (with positive exponents only) in this unit, and it's finally starting to work. From the start, my students could tell me that 10^6 was the number 1 followed by 6 zeros, but they couldn't see the relationship between the problems 9.02 x 10^6 (which was totally confusing) and 9.02 x 1,000,000 (which is finally becoming easy). Converting a number into scientific notation is starting to make more sense to them now, since I've finally figured out another flaw in some of the students' understanding: they don't really get the significance of the equals sign. I would show over and over why 302,000,000 = 3.02 x 10^8, and some kids just weren't catching on. But then, when I asked them what they would get if they multiplied 3.02 x 10^8, they were surprised to see that it was 302,000,000. I would get lots of "ohhhs" as they realized that the two parts of the equation had to be the same, and that you could multiply to check your answer. The main problem I still have is getting them to remember that the first part must be between 1 and 10. But at least we're making progress! Though we have been learning dividing by powers of 10 at the same time, I don't want to introduce scientific notation with negative exponents now (since they have never seen negative exponents before). I want to give this time to sink in, and maybe come back to it later in the year.
We have the rest of this week off for winter break; when we start next week, I think it's time to move on from this percent and decimal concepts unit and start in on fraction operations. 1/2 + 2/3 = 3/5, here we come! (One of my favorite things to show numeracy students is why this equation doesn't make sense.)
I am a math teacher in the Bay Area, California. Previously, I taught for eleven years at Downtown College Prep charter high school in San Jose, where the target student is low-skilled and of the first generation to go to college. Currently, I am teaching at Capuchino high school in San Bruno.
On this blog, I will post strategies that I am trying, and I would love to see strategies that others are using to help bring students up to grade level and teach them real math skills.
If you want more information on any of my lessons, feel free to email me at yochanan AT gmail.com. Also, if you have ideas to share, please comment.
?
