Our current chapter is on Quadratic Functions. We’ve started factoring, and everything is through the generic rectangle (we did use algebra tiles initially, then moved on to the rectangle. I’ve used the generic rectangle before and the X to help factor before, but 1) I had never thought about the diagonals in the rectangle (see picture below – Casey’s Pattern) and 2) had never included the x’s when thinking about “what multiplies to this and adds to that”. CPM really ties everything together so well, so the generic rectangle and the X make sense, and there are patterns in them… honestly when I used them 8-10 years ago they were just “a trick”, or way to help factor, we didn’t explain any of the Why, just the How!

Math 7CP – Last week we did a couple of days review then took the Trimester 2 Benchmark from our district. We aren’t going in the same order as the book, and haven’t taught inequalities yet, however there were 5 problems about inequalities on the Benchmark. So, we gave the kids those answers, then calculated their scores out of 20 questions instead of 25 questions. Overall our kids did really well. Unfortunately, most of the questions were DOK Level 1 on the test, and we’ve really been trying to do more DOK Level 2, and even some Level 3 questions with our kids. But that’s why they thought it was pretty easy and did so well. We also looked at the Grade 7 Math Performance Task on the CASSPP Released Questions so we could talk about everything that is looked for. The kids are getting better at reading the questions to make sure they’re answering everything, and they really need to do that on the Performance Task! Hopefully the one on the actual test is as interesting as this one was so the kids get into doing it.

This week we started Unit 7, Geometry. We began with circles, and on 3.14.17 did the measuring the circle to find the diameter and circumference (using string and rulers) to derive pi. One group was really close with their measurements, others not quite as accurate, but our class average was 3.25, which is pretty good! The next day we did the Fractions Circle activity by Illuminations (scroll down to see this activity, it’s the second one). We did this last year too and really like it – it’s better than cutting the paper plate apart! I didn’t have enough time to have them understand how the “base” of the cut-up-and-glued-on-as-a-rectangle-shape was 1/2 of the circumference, so 1/2 of pi(d), which would be pi(r), since 1/2 of the diameter is the radius. So the next day we continued and I think it was a little clearer. One of our mixed review problems for homework on Wednesday was: “4. A bicycle wheel has a diameter of 60 inches. In one turn, how far does the wheel travel in feet and inches?” (it’s a really big bike LOL). I had RTI that day and a lot of kids were asking questions on that problem, so they totally weren’t relating the circumference to the distance. Soooo, as I was taking a shower the next morning I was thinking about what I could do to help them see this relationship. I thought about bringing in an orange and using a slice of it, but decided a banana would be better. This is what happened the next day: I had kids come up and explain how they did problems 1-3, then I talked about problem 4. I brought up a small knife and my banana and cut a slice about the width of my little finger from the center of the banana (one student says “you’re wasting food”). I told the class, “This is my wheel.” Then I marked a green line on the peel, and told the class, “OK, now we need to see how far the wheel travels until this green line is on top again.” So I rolled the banana slice along my paper until the green line appeared again, and made a line for how long that was (I’d marked where we started). Then I cut along the line, ate the banana slice (“No, I’m not wasting food”) and opened up the peel to show that it was the same length as the line! Then I made it a circle again! I think this helped the kids see that the distance traveled in one turn was the circumference. I ate a little more of the banana and other kids wanted a bite so the rest of the banana was disposed of without any waste. (HA! I showed you, you 7th grade boy!) I liked this idea, and I’d definitely do it again. We worked on circumference and area problems for the rest of the week, and took a little group quiz today. When we were doing the problems I’d give them the diameter or radius and have them draw me that circle, with the correct line. Then they had to write down the other measurement (d or r) and the formula for the Circumference and fill in the numbers and find the answer. Same for Area. I told them each problem would be 2.5 points – 1 point for the formula, one for the correct answer, and 1/2 for the measurement. So if they just write the answer they only get 1 out of 2.5 points. Today when going over the HW neither kid had the equation or the measurement. I had them write them on their paper, and reminded them that I need to see the formulas! The CAASPP expects the students to be able to “know and use the formulas” and doesn’t put them on the test (PARC does). After that we did a few more review questions then did the Historic Bicycle Task from MARS as a group quiz. I helped them convert the measurements some because they’d have to recreate it again in the last problem….it looks like about 1/2 understand so we’ll keep working on it!

Math 7 Accelerated – We also started a new chapter! Last week we finished Ms. Pac-Man on Tuesday (I was out at the UCSB Math Leadership Cadre) and then used grade 7 TEAM UP! cards to do review for a few days. We’re learning the 8th grade standards so the kids can be accelerated to honors algebra 1 next year but they are still going to be tested on the 7th grade standards so we need to review/teach some more before the testing starts at the end of April.

This week we started on our Data Analysis and Slope Chapter. The first couple of days we made and talked about circle graphs, categorical data vs. numeric data, and reviewed Box Plots (AKA Box and Whisker Graphs). Yesterday and today we started slope, revisiting y = mx + b and comparing and finding slope on a graph today. The groups worked pretty independently today and I walked around and joined a few of the groups. I think it was a good day for them because they got to work at their pace in their groups. Happy St. Patrick’s Day!

Here’s a picture of what my backyard looked like this morning. We’ve had so much rain and now it’s warm so there are flowers galore!

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Algebra 1 – We’ve been working on Exponential Functions this chapter, and it is amazing how CPM has it laid out. I was talking with Karen, who teaches one of the Algebra Honors classes, about a problem we’d done in class and she said they didn’t do anything like that in Honors. She said, “This could be a Performance Task for Honors!” (she likes the CPM problems), and it totally could. A regular problem for our class is extraordinary for the Honors Alg kids. The last couple of days we were “curve fitting” given 2 points of the exponential curve….not right next to each other. Initially we did it using a table, then we did it by substituting in the values of x and y into 2 equations, solving the equations for “a”, then setting the equations equal to each other and solving for “b”. Then we substituted the “b” value back into one of the original equations, found the “a” value” and wrote the equation. Today we went over one of the homework problems where they had to do the above method, then also solved it using the table. We ended up having to find the 4th root of 0.0256, or we just used rational exponents and raised 0.0256 to the 1/4 power! It was so cool. Today the lesson was solving a system of equations of exponential functions by graphing. It was about 2 cars that were depreciating in value and one that was appreciating, and which should we buy. We’ll finish it tomorrow. It’s so different, and so cool!

Math 7CP – Last week we finished up our Percents Unit. I made a Math Notes page (to copy the phrase from CPM) to sum everything up, and a few problems for practice. I told the kids they could use the Math Notes page (it’s not called this on the paper) on their test. I liked it a lot, I’m not sure if it helped though…. they either know it or they don’t, and I saw errors that wouldn’t have occurred if they’d looked at it! Today we played Grudge to review for our Trimester 2 Benchmark. The Benchmark is so basic compared to what we’ve been having our kids do….so hopefully they do well! That will be on Wednesday, and then I think we may do a Performance Task on Thursday – or we’ll do puzzles! This is the Puzzle HW we are doing this week.

Math 7 Accelerated – Last week we reviewed and had our Chapter 6 Test. We had the kids do a Concept Map tying all of the vocabulary together for this chapter. What I love about this as a review activity is the kids don’t just link the vocabulary words together, they have to write WHY they are connected! Here are some pictures:

Above shows the kids working (I just noticed they’re actually right next to each other, notice the boy with his hands on his head? hahaha) and below are my favorites:

I haven’t finished grading the tests yet (lazy this weekend) but I’ll finish them tonight.

Today we did Robert Kaplinsky‘s Ms. Pac-Man! Ideally we wanted to do it as a review activity BEFORE the test, but we had an assembly and a Rally and class times got messed up so we would have had to test tomorrow, and Tuesdays are short days (39 min periods) so we just decided to do it after the test. Here is the link to where I wrote about it originally, and here is the link to the game board, which I apparently didn’t link to when I wrote about it. I also played Mario Music while the kids were working! We’ll finish it tomorrow, but here are a few pictures of today. Most kids figured out their route before writing out the moves. It definitely went better this year than the first time we did it last year!

]]>Math 7CP – We are continuing with our percent unit. The class had a quiz last week that started out as individual, but a lot of the kids didn’t finish it. So the next day I had them work together at their tables, to help each other figure out some of the problems, and to check their work. Then I had them trade quizzes and correct (I had put 2 tables on it and decided that it would be a bear to correct….so I wasn’t lazy, I was using my resources LOL) them by circling any problem numbers that had different answers. I went over all of the quizzes and put on the scores. I feel pretty confident that they didn’t just copy each other on the second day because I was walking around, and it definitely helped their scores! But that wasn’t the fun day. We had a 4 day weekend due to President’s day(s) in our district, so when we came back on Tuesday we worked on the large whiteboards on the wall and did percent increase and decrease. I had done an example with the class and then used my iPad and Doceri to write new problems as I walked around. They had to show their work and used calculators. The kids like working on the big whiteboards, it was a minimum day (all Tuesdays are) so I think it was a good intro to percent change. The next day, I had a worksheet for them. We did the first one together, and then played Grudge for the rest. It was a great day. The whole class was involved and checking each others work, making sure they wrote “increase” or “decrease” and working hard to get finished with the problem so they could go erase another team’s X from the board. I play it by giving 4 X’s to each table, everyone at the table has to show me their work and complete answer before they can have 1 person go erase an X, and if they call me over and everyone has the wrong answer they 1) can’t go up that time and 2)have to correct it before they can do the next one. It was a 47 minute period and we finished with 4 minutes to spare. So we played Simon Says for 3 minutes. I’m Simon. If I had given the class the worksheet and told them to “work in their groups” some kids would have done it, but a bunch would have fooled around and not finished. I don’t play Grudge that often, but when I do the kids love it.

Math 7 Accelerated – We’re still working on Transformations, and have now included dilations. Monday the kids did an activity from CPM where they have a few shapes and have to follow directions to translate the shapes to make a picture. It becomes a rocket with the moon in the sky. I like that activity. After that they had a quiz on rigid transformation, then we started dilations. I like how CPM teaches it, there are a lot of thinking problems, then an activity where they distort the image by multiplying x and y by different factors. From dilations we move logically to similar figures! Today we are supposed to work on corresponding sides, but I’m going on our snowboard/ski trip (it’s the last one of the year) so I’m having the class do a transformations review with the teacher that’s subbing for me (Thank you Gila!!). The week wasn’t as fun as Math 7’s (no Grudge!) but still a good week.

]]>Here are some pictures of the kids gathering their data. The farther you stand, the more you see! What would be the measure on the “y-intercept”???

(it would be the width of the tube, or 1.75 inches!)

Finally, here are a couple of pictures of the kids working together on their partner quiz. The conversation is what is so awesome, they’re justifying their answers, questioning their partners, helping each other understand. It’s awesome!

]]>Algebra CP – It’s been an interesting 2 weeks. We started on Monday, January 30 by returning the chapter 5 tests – most kids did well but enough didn’t that I said we could have a retake after corrections were turned in and we made sure there was understanding. The retake was last Friday, Feb 10 and 4 kids took it. All of them improved their grades. I had said I would have 87% as the max, and 3 kids got 86% so that was pretty awesome! But back to Jan. 30….

We started chapter 6, which was Modeling Two Variable Data, by figuring out if a student would be able to watch a sold-out football game by looking through a pipe located at the end of one end zone. The kids used the cardboard from a paper towel roll to model the “pipe” and stood 5,6,7,8,9,10,11 and 12 feet from a wall, upon which I’d taped a yard stick (feet and inches due to football field measurements). They took turns looking through the tube at the different distances, and said how far they could see on the yard stick. One person recorded the data (distance from wall, amount that could be seen) for the group, then they came inside and all copied the data and graphed it. Some groups graphed it backwards (there still isn’t a firm connection between Left Column = x values, Right Column = y values), so had to redo it. Then the groups drew a line of best fit and calculated the equation of the line. I have them find the slope by making a little T-table and finding delta y/delta x , then plugging in one point to find the y-intercept (not the slope equation or point-slope form, although I do think point-slope form is a good beginning for vertex form).

The next day we learned about Residuals (actual value – predicted value), and then Upper and Lower Bounds. These are some of the topics that are new to Algebra that had been in Algebra 2, and I’d never taught them before, so I learned a lot too. The next lessons were graphing and we used Desmos on the iPads. CPM had the problem as a link so we didn’t have to put in the data, and there was a line that could be moved around to find the line of fit. Initially we ignored the outliers and found our line of fit, then the kids typed in the equation: y1 ~mx1 + b. Desmos subscripted it for them, and gave them a beautiful regression line. THEN, they could click on the button that said PUSH under the heading Residuals, and all of the residuals were graphed above or below the x-axis. AWESOME!! However, the lesson also had them finding the R-squared values….. and we didn’t do that… this seemed like enough new stuff for the day.

The next day I gave them a w/s that had the values graphed on it (same BB data) and had them draw another line of fit that included the outliers and we compared it to the line from Desmos. They had problem to do for homework and I told them to download the Desmos app. Some did, some didn’t. So the next day we did it in class and it was AWESOME and SO EASY!! So much easier than doing it on a graphing calculator!! (I think after a lot of me bugging them that most have downloaded the app on their phones.) So the next day we had a partner quiz. I used the same w/s with the BB data plotted on it, and put the table on it too, which had the actual values. Then I circled 2 points and asked them to find the line of fit through these points. I had drawn squares around 3 points and asked them to find the residuals. Then asked a general question about the residuals. Mostly good grades….they were seeing how much Desmos could help them, after doing it by hand again. The next day I went over the homework in detail, and just decided that we were done with this chapter – I doubt the other algebra classes will be doing R-squared or correlation coefficient so…. left it for them in algebra 2 (I know, I copped out). We had another quiz that they could do with a partner or individually. It had one question with data to plot, line to fit, equation to write, and other questions on systems, isolating the variable, etc. from previous chapters. In general they did well. Tomorrow we start Exponential Functions and I promise we’ll do the whole chapter!

OH! But the best thing about the last quiz was what I’d put for the Name____ line. A while ago I had read a blog (by John Scammell, but can’t find exact link) and he’d had listed all of these fun things to write instead of Name ___. So I had written “Hi! I’m _____. I like to _____.” I had some great responses. My favorite is “not take math tests”, but I also got “sleep, eat, breathe, acquire currency, dance, read paint, play hockey, boogie, move it, not have hw, play softball, skate, run/hurdle” and more. I also had some “do math, be in Mrs. Boden’s class, get A’s” and one group left it blank – WHAT?? This is the funnest part of the quiz and you left it blank??? Totally great to read.

Math 7CP – We finished our probability unit doing compound events. I don’t think we did enough of these, we should have had more hands on and drawn more tables. The kids wanted to add the probability of each item, like they did with “pulling a green or red marble”. I think I had them go to multiplying the probabilities too quickly because some kids understood but more wanted to just list each probability. (which of course I saw on the test…). Anyhow Monday and Tuesday (Jan. 30-31) we worked on compound events, then reviewed and had a group quiz on Thursday. Friday we went over it and did more problems on the large white boards. Monday we had the test, and that’s when I really realized we hadn’t done enough, so I highlighted the problems that had something that should be corrected and gave them back the tests on Tuesday. They had part of the period to make corrections (on a separate paper, staple to quiz) and then take a pre-test on Percents. Maddy had to do it for BTSA so I decided to do it too. I think 3 kids can find the percent of a number from a problem like “28 is what percent of 40”. I had my TA “grade” them and then I looked at them. Yes, we need to teach this as well as percent increase & decrease, mark-up, discount, tax and tip. Anyhow, after the corrections many kids got 100% which was cool. Some kids didn’t turn it back in, so they got the original score (I had written it down on a separate page, they didn’t know their scores at the time of corrections.) I bugged those kids the next day for their test/corrections but a few didn’t turn them in….and there’s honestly only so long I’m going to do it.

We started our Percents Unit. We thought about doing it from the 7th grade CPM book (we had used most of CPM for our probability unit) but we need to do more than in the book so are using resources we’d made and found last year. This year I decided to just teach “direct translation” as I honestly think using the proportions are something for them to memorize unless they really understand it. I will show them the option of writing the percent as a decimal or a fraction, but I want them to be able to figure out what they’re being asked in a problem, write that out in English, then translate it into an equation. So if they have the problem “Last year we had 972 students at Los Cerritos and next year we are projected to have 12% fewer students. How many students are projected to attend Los Cerritos next year?” I want them to know they either need to find 12% and subtract it “What is 12% of 972?” or find 88% of the students from this year “What is 88% of 972?”. Then translate those sentences to “x=0.12(972)” or “x=0.88(972)” and figure out their answers. When I started teaching 6th grade in 2000 this is how I taught it. I had a student come ask me “Why don’t you use “is over of equals percent over 100”? I had no idea what she was talking about. Apparently her mom was a math tutor and used this to help the kids. My fellow math teachers used it so I did too for years, but not anymore (I totally endorse Nix The Tricks! by Tina Cardone). Last year I used “part/whole = %/100 and talked about the right side being the percent written as a fraction so the equation would be a proportion, but I still think the kids memorized it. Which is why this year I’m teaching “write down the problem you need to solve, then translate the English to Math” (**of** means multiply, as in 6 x 5 means 6 groups OF 5, **is** means equals, as in 4 + 4 is 8, and when we see something we don’t know, as in “what number” or “what percent” then we write something to stand for the unknown. Like a variable! Like x!) So far so good…

Math 7 Accelerated – What stands out for me the most is the “discussions” I got to have with this 6th period, end of the day, 36 kids in the class, class. We also had our chapter test on Monday this past week and started Rigid Transformation. The first lesson has this totally FUN task: http://technology.cpm.org/general/keylock/ and the kids had to type it into the iPads. BUT, I’d written it as …general/key/lock/ so was getting an error. I had to run across to ask Maddy and when I was out of the room I heard a loud yell, and I confirmed it with my TA that someone had totally yelled while I was out. So I asked who did it, no reply (big surprise), then did the (not advisable) threat of “I will give the entire class a detention in 5 seconds if someone doesn’t tell me who yelled.” Luckily Laine didn’t want a detention and raised his hand to tell me he thought Payton did it. So I got Payton to admit it and told him I’d talk with him later. Then we did the TOTALLY FUN ACTIVITY – go try it! (I told Payton that if he’s going to act like a 12 year old (he’s 12) and do things without thinking (because they honestly don’t think) then he also needs to man-up and accept the responsibility of it. I told him if he would have just said, “I did it, I don’t know why, I’m sorry.” then I wouldn’t be giving him a detention. The detention wasn’t for yelling, it was for not telling me. He understood, I hope! And when I gave him the detention he asked if he could do 2 lunch detentions instead, I said No, because then his mom wouldn’t know that he had to stay after school. She still might not know, but I think he’s learned from it.) So that was Tuesday, then Thursday they were working in their groups and I was putting in attendance on the other side of the room and I heard a girl making strange singing?? noises (from across the room). I decided to “talk” to the class about appropriate behavior in my room, that we’d just had Payton yell when I was out, now Gaby was “yodeling??” when I was on the other side of the room. Come on! Friday I decided to make seat changes of a couple of groups, and Gaby is one of those affected…I had let them pick their groups and some made bad choices….I’m just being the adult and fixing it.

But other fun things did happen in class. We did one large problem together drawing a triangle (calling it the pre-image), then finding the image, another image and another image, all labeled with A, A’, A”, A”’. We also used patty-paper to do 90 degree counter-clockwise rotations, once around one of the vertices (3, -2) then around the origin too see how the triangle ended up in totally different places based on what was the point of rotation. I think this will be a fun unit. We’re going to end it with Robert Kaplinsky’s Ms. Pac-Man, so stay tuned (if you do go to that link and scroll to the bottom you can see how I implemented it with Maddy last year – Robert included my post!).

BYE for now!!

]]>Algebra 1 – We finished our chapter on Explicit and Recursive equations. I had never taught Recursive before (I may have said this…). Anyhow it was fun. Here is a question from the group quiz: “Jackie and Emma were discussing two sequences. Jackie was writing out the first ten terms of the sequence given by and Emma was writing down the first ten terms of the sequence, when the teacher asked them whether their two sequences had any terms in common.

*“Well, none of the first ten terms are the same,”* Emma said.

*“But that doesn’t mean there isn’t a term in common later in the sequence,”* Jackie responded. *“How can we check?”*

*“I know,”* Emma said quickly. *“If the graphs of the sequences cross, then they must have a term in common. The graphs of these sequences are linear, and since they aren’t parallel, they must cross! We’re done!”*

*“Wait a minute!”* Jackie exclaimed. *“I’m not sure that just because the lines cross, the two sequences must share a term. We might be overlooking something.”*

Help Jackie and Emma by discussing Emma’s conclusions and either explaining any errors she made or convincing Jackie she is right. Then, find any terms the two sequences have in common.”

We had compared functions and sequences – domains and graphs specifically. One of my classes said the point they have in common is the -17th term…every single person. My other class had 12 correct answers “terms aren’t negative, they start at 1…” and 17 that again said the -17th term. The questions on the back were equally good (there were 2), and it ended up that some kids got them all right, some all wrong, and no one in the one class got them all correct. Sooooo I passed them back the next day and had the groups help each other (class discussion in the one class) and they made corrections. Then they attached their team quiz and corrections to the individual test and I’ll count the 3 points on it!

Math 7 CCSS – We are working on probability. We started out last week with the MARS task “Counting Trees” where they are sampling and estimating. We did the “pre-task” individually with study carrels up for 12 minutes, then they discussed their methods in groups and with the class. I think it went pretty well. I liked the reflection where they had to talk about what their assumptions were. The class knew what they were assuming! We played Grudge for review for a quiz we had last Friday. I modified the rules (actually Maddy, my coworker did, I just copy her!) to these: 1) students work independently for 1 min (or so, depends on problem) 2) students share answers at table and discuss so everyone understands answer. 3) They raise their hands when the table is ready and I check that everyone has the work and answer, then 1 person goes up and erases an X from the board (I started with 4 X’s per table). If they don’t all have the work when they call me over they can’t go that time. Sometimes I’ll add an X to every team. Then we go to the next question. I think they really liked it, and they were all engaged so that they could compete as a table. I think it was the first time this class had done it, and I’ll do it again with them

When correcting the quiz I just highlighted anything they needed to correct (in this case everything I highlighted would have lost points, sometimes it’s just little things that I’d like them to fix, but they don’t know which loses points). I wrote down their score and posted it in the gradebook, but it wasn’t on their paper. They had to do corrections and turn it in again. I think they like doing corrections better with highlights rather than -1 written on the quiz. (In my algebra classes I was checking everything really well and turning it back if everything wasn’t correct (I admit I don’t do this every time) and I have kids on their 3rd try of correcting! (which is why I don’t do it every time…I know… I should….)). We’re using the CPM CC2 book for the probability lessons (Ch 1 and 5). I think they’re enjoying and learning! yay!

Math 7 Accelerated – We just started chapter 5 in the CPM CC3 book, which is systems of equations. I LOVE how CPM introduces systems! The chapter starts by teaching how to eliminate decimals and fractions from equations by multiplying. For fractions they call it Fraction Busters. The kids are getting better at it! I had an extra review/practice sheet for them, and it helped them and me (I was at my UCSB Math Leadership Cadre meeting on Tuesday, so had good practice. For systems, CPM starts with the Iditarod race! One girl has a dog team and starts in Fairbanks, her friend is on a snowmobile and starts in Nome. They’re given a graph with the different check points the girls have passed, and need to extend the lines to find when they meet up, who finishes first, and who is going the fastest. We discussed it as a class and the kids had really good answers. Today we did my favorite problem “The Chubby Bunny”. Who could stress about that?! We were writing equations from problems, basically in slope intercept form, then setting them equal to each other and solving for them. I LOVE how the kids start right off by writing the equations, they aren’t just given 2 random equations to “solve”. Tonight’s homework has 2 lines that are parallel – they graph them then solve algebraically. Wow! You get “no solution” when the lines are parallel! Yep, since “solution” is where the 2 lines cross, it makes sense that parallel lines don’t have a “solution”!

I put up the poems from math 7 on my door, and my friend Helen cut out letters for me. Here you go:

I’ve tried to clean the spots off my door! They’re kind of rusty or something.

It’s been a good 2 weeks!

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Algebra 1 – We are continuing with sequences this week. I love this first problem the kids had to do (this was 2 day lesson):

“Samantha and her teacher have been busy creating new sequence generators and the sequences they produce. Below are the sequences Samantha and her teacher created.**Your Task:** Working together, organize the sequences into families of similar sequences. Your team will need to decide how many families to make, what common features make the sequences a family, and what characteristics make each family different from the others. Follow the directions below. As you work, use the following questions to help guide your team’s discussion.

a. −4, −1, 2, 5, … | b. 1.5, 3, 6, 12, … |

c. 0, 1, 4, 9, … | d. 2, 3.5, 5, 6.5, … |

e. 1, 1, 2, 3, 5, 8, … | f. 9, 7, 5, 3, … |

g. 48, 24, 12, … | h. 27, 9, 3, 1, … |

i. 8, 2, 0, 2, 8, 18, … | j. , , 5, 10, … |

(1) As a team, initially sort the sequence strips into groups based on your first glance at the sequences. Remember that you can sort the sequences into more than two families. You will have a chance to revise your groups throughout this activity, so just sort them in a way that makes sense to start out with. Which seem to behave similarly? Record your groupings and what they have in common before proceeding.

(2) If one exists, find a sequence generator (growth pattern) for each sequence and write it on the strip. You can express the sequence generator either in symbols or in words. Also record the next three terms in each sequence on the strips. Do your sequence families still make sense? If so, what new information do you have about your sequence families? If not, reorganize the strips and explain how you decided to group them.(3) Get a set of resource pages, saving a copy of each of the tables and graphs for your team. Then record each sequence in a table. Your table should compare the **term number**, *n*, to the value of each **term**, *t*(*n*). This means that your sequence itself is a list of *outputs* of the relationship and the *inputs* are a list of integers! The first term in a sequence is always *n* = 1. Attach the table to the sequence strip it represents. Do your sequence families still make sense? Record any new information or reorganize your sequence families if nece

ssary.

ssary.

(4) Now graph each sequence on a the graphs your teacher gave you. Include as many terms as will fit on the existing set of axes. Be sure to decide whether your graphs should be discrete or continuous. Use color to show the growth between the points on each graph. Attach the graph to the sequence strip it represents. Does your sequence families still make sense? Record any new information and reorganize your sequence families if necessary.”

There was a lot of discussion on grouping, and some kids persevered on the rule and others had some troubles. We discussed the rules and I had students share how they figured out the harder ones. The next day they “learned the language” of sequences – arithmetic, geometric, sequence generator, common difference and were reminded of discrete vs continuous and domain. It was also new to write the sequence as t(n) = 3n+4, and to understand that n is the term number, 1st, 2nd, 3rd. The next day was explicit equations vs. recursive equations. It’s the first time I’ve taught recursive equations as this didn’t used to be in algebra 1! The kids didn’t see the point of it at first, why should they write an equation depending on the term before the current term if they could just write the rule? But…. a few days before we’d had the Fibonacci numbers in one of the sequences, so they remembered they couldn’t write a rule for it. However, they could write a recursive rule for it! Another great CPM Algebra lesson. Today we had a quiz, that was all on review material (which is given daily in the homework). I told them if they don’t do well on the quiz it’s because they aren’t doing their homework very well (CPM has help for every problem, with hints, for free. Some check their work, some don’t…) I’ll grade the quizzes this weekend and see how they do!

Math 7 – We started our Probability Unit this week. We’re using the lessons from the 7th grade CPM book (CC2). If you buy a Black Line Master from CPM you can make as many copies of it as you like – so since I had one we could copy the lessons for the kids (it’s in ch 1 and ch 5). And, since the homework help is FREE, they can get help if we assign those problems…which we aren’t since we do the weekly hw page which is AWESOME.

We have worked on experimental vs. theoretical probability, we investigated probability by making a spinner and then having each student spin the spinner (bobby pin) 10 times. We totaled the spins by color and compared the class results to the theoretical results. Today we wrote about the difference between experimental vs. theoretical probability, and then calculated the theoretical probability for different things if it was possible.

“Look at the situations below and decide with your team if you can find a theoretical probability for each one. If you decide that you can find the theoretical probability, then do so.

- Picking an Ace from a standard 52-card deck.
*Not*rolling a 3 on a standard number cube.- The chances of a thumbtack landing with its point up or on its side.
- Getting the one red crayon from a set of eight different-color crayons.
- The likelihood that you will run out of gas on a long car trip.”

Initially the kids said #2 was 1/6, then some said NO, it’s Not rolling a 3! so 5/6.

After that we worked on what would happen if we “modified the sample space”. They had a bag with different colored blocks in it and calculated the theoretical probability for drawing the different colors (naturally 1 bag was missing a green, and that was the group I called on for their first probability…. so it was 4/11 rather than 4/12… so i had to run across to where I’d left the box ‘o blocks to get another green!). Then I gave them another (identical) bag and they calculated the probabilities again. They were the same!

The initial questions in the lesson were “If you want to have the best chances of getting a red gumball from a gumball machine, is it better if the machine is full of gumballs or half empty? How do the chances of getting an ace in a deck of playing cards change if you have three or four decks of cards to choose from instead of only one deck? In this lesson, you will think about the size of the **sample space** (the collection of all possible outcomes of an event).” We will talk about this again on Tuesday (Martin Luther King, Jr. day is Monday) because it’s natural to think that if there are more you have a better chance. It was a fun start to the unit.

Math 7 Accelerated – We are finishing up chapter 4 in the CPM CC3 book so we were putting together the pattern, table, rule and graph. They graphed a line without an xy table. There was a lot of discussion on the “growth” (we haven’t said slope yet), and why if the pattern increases by 4 each time you graph that as up 4 and over 1. I wrote “change in y / change in x” then introduced it as delta y/delta x and we talked about and drew the “growth triangles”. It’s hard for me to remember to do the actual drawing “up and over”, I’m so used to just “counting” up and over. But I think it pushed the kids and that was good. Yesterday we went to the computer lab and the kids did Desmos’ Marble Slides-Lines. Last year we had done this at the beginning of the unit and the kids figured out “how” to make the lines move, but didn’t know “what” they were doing. This year it was better, but some were still not sure how to move the lines up and down. We’ll be doing a lot more with the lines, and I think we’ll go back and do Marble Slides again because most of the kids didn’t get to the “hard” ones. It was a great way to review slope and y-intercepts though! Today they had their group quiz. One of the problems gave them an equation then a tile pattern from a different equation. They needed to write the rule, draw 3 or 4 patterns, make a table and graph the line for each of the patterns, then tell me if they thought the 2 patterns would ever have the same number of tiles. Yep – we looked at it and said, “these are great!” So last night I did the quiz – and looked at the answer key. The second pattern was triangular (okay) and decreased (okay), removing one layer, or column each time. So pattern 0 had 7 columns, with 7, then 6, then 5, then 4 etc. tiles per column. Pattern 1 had 6 columns. I told them they needed to draw the pattern first – at least 3 more – then do the other representations. If they didn’t know how to write the rule they could describe what was happening. This is because the rule is [(x-8)(x-7)]/2. Yep. I think I’m going to change the instructions to tell them to describe the rule because I like having them do this problem….but so many were starting in slope intercept form, with -28 as “b”, since pattern 0 had 28 tiles! I’ll grade those this weekend too.

That was the week!

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We started out by representing exponential growth by looking at bunnies reproducing. There were several different situations that we looked at, drew patterns for and graphed. Next came dropping a bouncy-ball from different heights and seeing what the rebound rate was (it was raining that day, so we had to go into the “Jungle Hallway” – see the pics). We calculated the “rebound ratio” and figured out that this was the slope of the line.

The next day we used the bouncy-balls again, except this time they caught the ball at the top of the rebound, recorded that height, then dropped it from there again. This graph was ball bounce to bounce height – and showed exponential decay! It was pretty cool.

Math 7 – We decided to finish up proportions and come back later to equations again. We (awesome co-teacher-Maddy and I) need to do more work with using inverse operations to solve equations that have fraction answers. Our kids are really good at “undoing” or “looking inside” the equations to find the answer when it’s an integer, but we want to work more on just “using inverse operations to solve the equation” so they have another resource when the equation isn’t as “nice”. So… we did big white board review of proportions – we gave the kids a situation and they had to write the proportion from it and solve it – and equations, and some area and perimeter. (We started reviewing area and perimeter in homework, so put a perimeter question on our test.) Wednesday we used Mr. Orr’s awesome post on a site he found “Internet in Real Time”. The link to the original site wasn’t working so I found another one that showed the numbers changing (go look at his post!), then froze it at one second. The kids made up problems on the white boards (if there are 214 tweets per second, how many tweets would there be in a day?) and then walked around the room looking at the different questions. Then I called on a few kids and they picked one of the problems and we all solved it – and explained how they solved it. I NEVER had to explain how to figure out how many seconds were in a day! They said 60 x 60 x 214 because there are 60 seconds in a minute and 60 minutes in an hour, then times that by 24 for the hours in a day. WooHoo!! On Thursday we did some more proportion problems and then on Friday they took a team test. We started out by having the kids work in pairs, then they could check their results with the pair across from them. And, a lot didn’t finish, so they’ll finish today.

I think the reason they didn’t finish is because we are still discussing the homework everyday, but that is so great! This week one problem was to write a situation for 157 – 162 and the student that was explaining it said he lent his mom $157 and then spent $162 ( so -157 – 162) and got -319. Hands went up. He called on a student who said it wasn’t -157, just 157. The boy explaining said, “But you can’t take away 162 from 157, it doesn’t make since. You can’t spend more than you have.” I said “What if you reversed the problem?” So then he wrote -162 + 157 and said, “Oh, it’s -5.” I loved that 1) kids questioned his answer, and 2) he thought about it making sense when writing his story and 3) realized he could have reversed it. This is the best homework, and use of homework I have ever done.

Math 7 Accelerated – We also started a new chapter, Chapter 4 in the CPM Core Connections 3 book. It’s on patterns and equations, tables and situations, and tying them all together. The groups all had patterns that were all quadratic, and we were working on “seeing the pattern number in squares and rectangles in the patterns”. It was pretty cool seeing the different ways the kids “saw” the patterns. We spend a few days doing it – and used Fawn’s site www.visualpatterns.org for some extra examples. On Friday we started talking about y=mx+b and where the m and b are shown in the patterns. It is so much fun watching the kids “see” the growth.

]]>Algebra CP – We finished chapter 4. The week started by choosing the best method to solve systems problems, and writing systems from word problems. We had the group quiz on Tuesday, then went over it and did some more review and had the test on Thursday. Friday the kids did some test corrections and then did the Stacking Cups Activity that Fawn wrote about. Some of the kids remembered it from last year – everyone got the equation really well, I think it was better here than in 7th Accelerated (note to self).

Math 7CP – I didn’t get to post some of the “If I Were One Inch Tall” poems my kids wrote, so I want to post them now…

This week we worked on writing proportions from word problems, writing a little “word ratio” so it would be easier to see which numbers were representing which items. I still need to work more with them on why cross multiplying works – we did examples like 1/2 = 2/4 so they could cross multiply and see the results were the same, but I think next week I want do more with it – I think if we reduce them to the lowest terms it’s easier to see that we are multiplying the same things, so that’s why we get the same answers. We had a quiz on proportions this week and then on

Friday I had 2 different activities – Ultimate TicTacToe and Area Puzzles from a book I had bought (after seeing it on Twitter)…

Math 7 Accelerated – This week we had the chapter 3 group quiz, chapter 3 closure and the chapter 3 test. What a fun week! It was on equations so was really a review of the work we’d done at the beginning of the year, but there are still too many kids not thinking about what they’re doing and not distributing correctly or other errors. We’ll keep solving equations! On Thursday we did Robert Kaplinsky’s In and Out Burger activity and I think they liked it. On Friday the kids worked together to do problems from the Area Puzzles book (see above) and these problems definitely get challenging!

This was a short post because 1) I’m writing it 2 weeks late so don’t remember some of the things we did, and 2) it was the week before the break so it was tests and quizzes before the break! I did like that I found puzzles and fun math activities for the last day, I didn’t want to “have a party and show a movie” this year. This is the end of 2016!

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