Pictures from Proportions

I wrote the blog on Proportions at home, and had my notes at school, so I thought I’d add a few pictures.

This is what I wrote for the kids to compare the Original Proportion, the Proportion with common denominators or common numerators, the Problem we had to solve to make equivalent ratios:

Here is the same sheet, now the kids told me you could cross multiply the original proportion (or simplified proportion) and get the same problem we solved from the common num/denom problem:

This is from the next day. The numbers 1, 2, and 3 are the order I introduced the problem. First we reviewed how to solve proportions (first look for a relationship vertically or horizontally, then if there isn’t a convenient one they can cross multiply), then looked at proportions with more than one term in a position, and finally written problems. For the written problems they needed to 1) write a word ratio to organize their thinking, 2) write their proportion, and 3) solve (any way they chose) and write a word answer.

Finally, here is some of their work (they told me what to write) solving the problems from section 2 in the picture above. Some kids simplified vertically, others simplified horizontally, but they still got the same answer!

There you go!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Proportions

I’m doing a long term sub job at my old school in 6th grade, math & science. Next week is the last week of school – I’m happy and sad about it. I will miss talking to my friends there, but I’m glad I’m not going to be teaching full time anymore. But this post is about proportions!

Chapter 14 is Proportions. I went through the chapter and decided how I wanted to teach it, starting with defining a proportion as 2 equivalent ratios, with an = between them. Why do some kids want to write the ratios without the = ? Anyhow, next we worked on solving the proportions by finding relationships, vertically or horizontally. (I was so surprised when an algebra student told me he was simplifying his proportion horizontally. This was probably 7 – 9 years ago, and I ALWAYS simplified vertically, like simplifying a fraction. When he told me he was doing it horizontally I’m sure I thought, “You aren’t allowed to do that!” His dad had told him. I do remember I said OK, then wondered if it always worked. I was so good at following and teaching the procedures then.) Anyhow, now I teach the kids to look vertically or horizontally for a relationship. (But not diagonally! I tell them when you’re crossing out the numbers to rewrite the simplified numbers, you’ll also put a line through the = if you’re doing it diagonally – which now makes it “not equal”. I hope that makes sense to them.)

The next day we were going to solve proportions that didn’t have a convenient relationship. The book has the Cross Product Property with the example using a, b, c, & d, showing how the property works and that’s supposed to explain everything. In reality I think most teachers show the kids how to cross multiply and there isn’t a lot of discussion of WHY we can use the Cross Product Property – and the example with only variables is NOT helpful. So I decided we were going to solve the proportions by making common numerators or common denominators. We’d already discussed how proportions are equivalent ratios, so now we would just need to make them equivalent.

For example: 2/3=x/4 can be rewritten with a common denominator of 12. (I can’t figure out how to copy in a fraction from Word, so I’m just being lazy and using the /.) So now the proportion is 8/12 = 3x/12. Our denominators are the same. What do we need to do to make our numerators the same? We need to solve the equation 8 = 3x!

Here’s another example: 4/5 = 9/x. This was probably the 3rd example we did in class. Can we make common denominators? NO? What can we do? A student said we should make common numerators! (What? That’s not how we solve fractions….) So we did. We said the new proportion was 36/45 = 36/4x. So now the numerators are the same, how do we make the denominators the same? Solve the equation 45 = 4x!

We did 4 problems together then I rewrote what we’d done as a summary page, just showing the original proportion, the proportion with equal numerators or denominators, and the problem we had to solve at the end. I asked the kids to look at the original proportion, and the problem we had to solve at the end – thinking quietly for one minute, then talking to their groups. It was so awesome, it was totally quiet, then when I said “Okay, you can talk to your groups” it just exploded. It seemed like they couldn’t wait to share what they saw. What they saw was they could get the “problem” by “cross multiplying”. What?? I had them explain each one to me, while I used different colored highlighters, highlighting the diagonal number in the proportion, then drawing an arrow to that term in the problem. It was a very colorful page!

Fast forward to the review day and we talked about WHY they could cross multiply to solve a proportion. I wanted them to be able to tell me that it was because if we rewrote the proportion with equal numerators or denominators the piece we needed to solve in that proportion would be the problem we’d solve by cross multiplying. We could do it because cross multiplying was a short-cut to getting common numerators or denominators. I also asked this question on their test. I think around 75% get it. Now can they keep it?

I was really happy that I didn’t just teach them HOW to do it, they figured out that part on their own. Mostly!

We also solved problems from situations (word problems). I want them to write a “word ratio”, then the proportion, then solve it and give me a word in the answer. So not just x = 8, but x = 8 penguins. I did one problem with them, talking about each part, defining the Word Ratio as “the way to organize your proportion”, then gave them another problem to do and walked around checking that they had the word ratio and the proportion. How they solve it is their choice. I’m walking around and see one girl making Ratio Tables (which the class is AMAZING at by the way, great job Mrs. Marien!) instead of writing a proportion. You can definitely solve the problem this way, but I wanted to see that she understood how to write a proportion! I walk around, check each child’s work, then go back and do 2 more problems with the class (calling on students for each step). We’re done, here’s 7 more for you to do…check with your tables as you go.

It is so interesting to me that I can do 3 problems with the class, then give them 7 more to do, and as I walk around I’ll see kids doing whatever they want to get the answer. It’s not all about the ANSWER!! I’m walking around checking, helping, making sure they’re writing an = between the ratios and writing their word ratios. One boy was on his last problem when I got to his table….yes, he was this far along because he hadn’t written a single word ratio. Not for my examples or for the other 7. He didn’t feel like it, he could do it without them. I told him on the test I’m going to ask him to write word ratios, and they help him organize his numbers in the proportion. (It’s actually helpful! Don’t just do it because I’m going to test you on it, do it so you get the problem right!) Another boy – next table – isn’t writing any proportion, he’s just solving the problems using division – they only thing on his paper is his arithmetic. He’s my boy who thinks the goal is to get the answers ASAP, then be finished and bug other kids. And the answers don’t have to make sense. But I digress…

Anyhow, I think I did a good job teaching proportions without telling them HOW to do it. They can look at the proportion and see their own relationship, write a new one with common denominators, cross multiply to solve everything…it’s their choice.

They didn’t ALL do well on the test; my boy without any word ratios got either a 64 or 61%. But, one class had 5-100%’s and 5-99%’s (they got 1/2 off something), with 66% getting 90% or better. The other class only had 1-100%, but they had 56% with 90% or higher (yes, my no word ratio boy is in that class). One thing I need to do is connect how they can write a proportion from the Ratio Table. Many probably have already connected it, but we did a problem the other day with a Ratio Table and I thought they’d see they could go out of the table and write the problem as a proportion – but “that’s not how you do a Ratio Table”! Hopefully I can connect this before school is out (on Friday!!!) – actually I connect this, hopefully the kids can connect this if I make a point to show them! As the goal is for the kids to see how all the math flows together – what they do isn’t a bunch of unconnected concepts!!!

The Generic Rectangle

It’s been a long time since I posted a blog – I retired from teaching Middle School in June, 2017, but have been subbing for one of the teachers at my old school for a few days. I decided I wanted to write about the progression in her algebra class using the Generic Rectangle because I think it is so amazing, and the topics just logically flow together so intuitively.  Then I read my last post – and my algebra class had been using the Generic Rectangle! So some of this is also in my last post, but… it’s been a while, and I want to write about the whole thing. Maybe you will want to do it in your algebra classes!

When my friend began teaching her kids to multiply binomials, she used the Generic Rectangle. It’s called generic because the sizes of the boxes in the rectangle aren’t scaled to size, you just draw a rectangle and split it into 4 boxes. Before using the Generic Rectangle I would tell the kids to do the Double Distributive Property, but both of us didn’t say FOIL to our classes. This year her classes had a hashtag “#NOFOIL”. The book says FOIL.

Anyhow, she taught the kids to multiply binomials (or a monomial times a binomial, or binomial times trinomial) using the Generic Rectangle:

So now I’m subbing and it’s time to factor trinomials. The book has separated factoring into 2 sections: trinomials with a leading coefficient of 1, and trinomials with leading coefficients not 1. I decided to teach them together, since we’d be using the same procedure we used in multiplying – the Generic Rectangle.

To start the day we did a warm up/review of “Diamond Problems” – since most of the kids had used the CPM books last year and had done lots of these problems. I usually call it the “X” (because I’m too lazy to make a diamond around the X). They were quick to remember they’re looking for 2 numbers that will multiply to the top number, and add to the bottom number:

We also did a quick review of multiplying to remember where we put the terms on the outside of the rectangle, and how we get the terms in the trinomial.

Then I had them look at the rectangles again, and notice something about the products of the diagonals:

(Is this a coincidence? Nope. If you have a b c d outside the rectangle and multiply to get the inside boxes, then multiply the diagonals, you end up with abcd as both products.)

Now we could start factoring. I wrote down the problems, and had the kids tell me where to put the first and last terms in the Generic Rectangle. Then I drew an X and asked them what would go in the top… and they knew they could multiply the diagonal terms they had since it would be the same product as the diagonal terms they were looking for. Then I asked what term should be in the bottom of the X and they told me the middle term of my trinomial, since this is the sum of the diagonal terms they’re missing:

They then figured out that little puzzle, what multiplied to the top and added to the bottom terms, and filled in their Generic Rectangle. Now it was time to “undo” the rectangle! This is the most fun because it’s a little logic problem.

We begin by pulling out the GCF of 2 of the terms, either horizontally or vertically, their choice. This example shows me doing it vertically:

Here’s my thinking I did aloud with the class’s input (I didn’t draw different boxes):

Box on Upper Left: What is the largest term we can pull out of 2x^2 and 1x? x. Since we are doing the vertical terms, put the x on top of this column.

Box on Upper Right: What would I have multiplied x by to get 2x^2? 2x. So I write  it outside, to the left.

Box on Lower Left: Now what would I multiply 2x by to get -10x? -5. Put this in the upper right space.

Box on Lower Right: Ok, what do I multiply -5 by to get -5? 1. This goes under the 2x, on the left.

So what are your factors? (2x+1)(x-5)! It’s so intuitive, the kids are good at thinking about these little puzzles, and they’re just undoing what the had been doing to multiply. We did about 8 problems in class, some with leading coefficients not 1, some with leading coefficients of 1, and the factoring was NO BIG DEAL. They didn’t have to do something different, or another step when the leading coefficient was not 1.

NOTE: We did factor out any GCF of the trinomial before doing the generic rectangle, and factor out a -1 if the leading coefficient was negative, and this resulted in a little confusion because some kids factored out the GCF then used the original trinomial terms in the Generic Rectangle. But that made for a good discussion, and helped with clarification. I think this is normal, some kids are just working and not really paying attention, so their “errors that happened because they really weren’t paying attention” help the class have a better understanding. Another result that helped with understanding was when a student wrote a fractional term outside of the Generic Rectangle – we multiplied out their factors and the integer factors and got the same trinomial. I asked them to just use integers for their factors. A couple of students had trouble factoring because they were taking the GCF of the “column” and writing it next to the “row” rather than on top of the “column”. So we did an example of that which was helpful. We also had a couple of students not copy the problem correctly from their trinomial into their rectangle…so it wouldn’t factor…but once they saw that they’d copied it wrong it wasn’t the Generic Rectangle’s fault anymore.

The following week I was back and it was time for Factoring By Grouping. They had a Trimester Test the first day I was there, and I was thinking about the lesson. It hit me that we should be able to use the Generic Rectangle to factor 4 terms (I don’t remembered if I taught this last year using the CPM Algebra book, I think we just had 4 terms and didn’t even worry about Factoring By Grouping) so decided to try it using the book’s examples. I drew the rectangle and filled in the terms, top row then bottom row, the same order as listed in the example. This worked great if there was only one variable, but the order of the problem in the book didn’t work when there were 2 variables. So I thought about it and rearranged the terms – which I did sometimes when Factoring By Grouping – and the Generic Rectangle worked great!

Class started and this is what we did:

Boom. They’re done. “Why was it even an issue? It’s the same thing!”, said the kids. “Yeah, I know, but the book has it as a separate section, because it’s not using the Generic Rectangle”, I said. “So… let’s do Factoring By Grouping, because some of you might like to do it instead, and I want you to know what it looks like.”

In my mind, the problem with Factoring By Grouping is that it is not intuitive. It doesn’t flow for the kids, because of FOIL. We are having the kids multiply in their heads, so they don’t see the second step. These are typical questions I would get when I’d taught it as another procedure to do, in the past:

When I learned how to factor we did the double distributive property to start, and wrote out the numbers we’d be multiplying. Actually that’s complete BS, I have no idea how I originally learned it. I do know we used Guess and Check to factor, and some problem were so annoying because the A and C terms had too many factor choices. But if we HAD written out the second step instead of doing it in our heads, then Factoring By Grouping would make sense. Here’s how I showed it to the kids – by doing the Double Distributive Property to multiply the factors, then showing Factoring By Grouping and highlighting the lines that are the same in both. The first example without the highlighting is just showing Factoring By Grouping and my thoughts on why it’s confusing:

We did some problems with 2 variables so they would see that they needed to think about the placement in the rectangle. It was great, in both classes the kids put the terms into the rectangle the same way I had done when I was looking at it. And we couldn’t factor it. So I asked another way we could arrange the terms and then we got it. I told them sometimes they’d have to think about where to put the terms. They would put in one term, factor it, and then figure out what other term would have that factor. They’d write that down and go from there. It wasn’t a big deal, I think they liked “solving the puzzle” of which term to put where! (This is the second half of the page – the “multiplying using FOIL doesn’t” is from the last example.)

Some kids liked the Factoring By Grouping because it’s less to write down (that’s what they said), but when I walked around the class most of the kids were using the Generic Rectangle to factor the 4-term expressions.

That’s it. I wish I had known about the Generic Rectangle when I learned how to factor and multiply as a child. I wish I had known about it when I taught algebra for all those years, instead of teaching a different method for each type of expression. Seriously, for the basic ones like  x^2 + 8x + 15  I’d tell the kids to set up their parentheses (x +    )(x +   ), leaving a blank space for the numbers and to “think about what multiplies to 15 and add to 8. If they needed help I’d set up the X (I had been shown how to do this, I didn’t connect it to anything, so didn’t connect it for the kids either, it was just a trick they could do), and put the 15 in the top and the 8 in the bottom – no 15x^2 or 8x, just the numbers. Then the kids would take the numbers on the side and fill them into their parentheses.

But if they had a problem with a leading coefficient that wasn’t 1, then I’d show them the “X-Box” method. But this time they’d have to remember that the top number in the X was really “a times c” (I did tell them this for the other type of problems, but since “a” was always 1, it didn’t wasn’t significant to them.) Then they’d have to make a box and put the “a” and “c” terms in diagonally, and the other boxes got the numbers from the sides of the X. Then they’d pull out the common terms and use them to fill in the parentheses of the factors, but sometimes make mistakes with the negatives and what went where. But since I just knew it as a trick, I didn’t tie it to multiplying the factors at all.

If they messed this up, then we would show them other ways to do it, like by making it 4 terms, using the sides of the X as the middle terms, and factoring out a GCF, or maybe just having them guess and check, because “the more they did, the better they’d get”. That’s how I learned it, right? And then I’d teach them another procedure to use when the problem had 4 terms – Factoring By Grouping.

But then I saw the Generic Rectangle method of factoring at a session at CMC, and it blew my mind. I remember coming back to my school and saying, “If we taught the kids to multiply the binomials this way, they could factor it with the Box and it would make sense! It wouldn’t be a trick!”  My team didn’t want to change, so for a few years I didn’t either. But I eventually did!

And if this makes sense to you (hopefully it does) you can change how you teach it too.

But you need to buy in 100%, you can’t teach multiplying using FOIL, and factoring the simple problems with “think of what multiples to this and adds to that” or the X, and then pull out the Generic Rectangle for the “hard ones”. If you do this the Generic Rectangle will just be another procedure. It won’t flow. It won’t connect the multiplying with the dividing (AKA factoring). And the kids won’t see the beauty and connectivity of the algebra, they’ll just get frustrated trying to remember which procedure to use with each type of problem.

As Jo Boaler says, “Viva the Revolution!”

Days 119-127, March 7-17

Algebra CP – We finished our chapter on exponential functions and had our test. Most of the kids are doing really well, some are better at finding the constant multiplier (b) when given a table than if they are reading it in a situation. They just want to say, “Oh, 6.25% increase? That means b must be 6.25.” What? No! Add that to your 100%, then change it to your decimal. We’ll still work on it, and the homework still has them finding the exponential functions so I think it will be okay.

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!

 

Days 113-118, Feb 27 – Mar 6

I know I’m totally messing up the weeks, but I wanted to write about what we did today!

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!

Days 105 – 112, February 13-24

Algebra 1 – We are still working on exponential functions and the kids are getting really good at writing the equations for the graphs, and using their calculators (or phones) to find values. Last week we worked on a lot of little problems, then made a web seeing that we could make tables, write equations, make up a situation and draw a graph from a problem. The only thing we didn’t think we were that good at yet was writing the equation from the graph, but I think if we had actual points that it wouldn’t be a problem. This week we’ve worked with compound interest vs. simple interest and made step graphs comparing the amounts, and compared interest compounded annually to quarterly and monthly. In one class we even looked at daily compounding. Then for decay we played a theoretical penny game (I forgot to count out groups of 100 pennies for the teams, although I did have pennies at school!) where we tossed 100 pennies on a table and took out all of the tails each time. The class calculated how many times they could toss the pennies until there was only 1 left. We also looked at the equation and talked about how many there would be at time 0, which reinforced that something (other than 0) to the 0 power is 1. We figured out how many pennies there would be at time -1 or time -2 too. Another problem dealt with Carbon 14 dating, and there we looked at time -1 and decided that even though we could find the “number” it wasn’t really relevant since a person only has 100 grams of carbon when they’re alive. We did a little more exponent practice, changing negative exponents to positive in various situations, then, since we had about 15 minutes and I thought they were pretty confident in the equations….played Grudge to work on homework problems! In my second class we only had 8 minutes to play Grudge, but again, it had everyone involved and they used the time well. Fun times!

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.

Last Week’s Alg Pictures!

I forgot to insert the pictures of our first data gathering. Here’s the wall with the 5′ – 12′ distances measured out. We had 4 stations. That’s a yardstick on the wall!

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!

Days 65 – 104, Jan. 30 – Feb. 10

I know, it’s another 2 weeks. My goal is to be able to maintain this for the rest of the year. I definitely have a new appreciation for people that write 180 blogs!

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!!

Days 87 – 94 Jan 17 – 27

I forgot to write a post last week, so here’s 2 weeks worth of learning fun!

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!

 

 

 

Adding my Twitter Account info!

Follow @debboden