A Jewish-Italian Hannakristmas

I'm in Cleveland at my dad's house for the annual event... the homemade gnocchi and sauce were great, the kids got lots of noisy plastic crap to forget about by tomorrow, and the vegan chocolate-peanut butter cake from Mustard Seed cafe was tasty: I had two, ok three, slices.

After desert, board games, and being shot at with nerf guns (which have gotten scarily high-caliber), I was talking with my 10-year-old half-brother about his school. Specifically, I wanted to see how he is doing in math. He is (not surprisingly) unable to articulate exactly what he is doing in math, so I was asking him specific questions to see what his math is like. Last year, I was surprised to find out he knew square roots already - I explained about cube and higher roots, and he picked it up instantly.

I wanted to see what he knew about fractions as a 4th grader at a typical Cleveland-area public school. I asked him if 1/2 or 3/4 was bigger.. way too easy. I asked him if 2/3 or 3/4 was bigger. He got it right, and quickly, but couldn't really explain why. I then asked him if 3/7 or 3/8 was bigger, and he said 3/7 immediately. I asked him to explain how he knew, and he looked at me like I was stupid, saying, "a seventh is bigger than an eighth, so..". I asked if he had worked with mixed numbers, and he hadn't, so I asked him to figure out what 3 1/4 - 1 1/2 is. He couldn't do it in his head, so I told him to get paper and draw a picture. That's all I said - he drew fractions circles correctly, crossed off a whole, the fourth, and then another fourth from a whole, and came up with 1 3/4.

I've only ever taught at DCP, so I don't have much of a frame of reference for knowing if he is above average or not, but this is the kind of thinking that all students must have to be successful in high school math. This is the kind of numeracy ability I want my students to develop; this way, when they get to a new problem, instead of giving up, they can reason it through and at least make progress. I struggle daily to get them to pay attention, to care, to think, to not give up when a problem is hard, and their mathematical progress is painfully slow. In a couple weeks, when we start reviewing for finals, and half the kids don't even remember what an integer is, it will be painful. But I know that, by the end of the year, most of my students will have improved their math abilities in many ways. Never as much as I want, but it will have to do! I just gave our grade-level equivalency test before break, and the median score has improved by 1.1 grade levels (from 5.9 to 7.0) and the average by 1.65 grade levels (from 5.76 to 7.42) since the summer. If I can squeeze that kind of growth or better out of them during the second semester, most will be in pretty good shape for next year.

A case study: freshmen's ability to listen...

(Background: cell phones are not permitted. If seen/heard, they are confiscated until parents pick them up.)

{Phone rings; I answer}

Me: C, they need you at the front desk with your cell phone, because your sister got hurt and they don't have your mom's cell phone number.

C: (Looking angry) But I don't even have a cell phone!

Me: They don't want to take your phone, they just need your mom's number.

C: But I don't have a phone!

Me: Just go...

(5 minutes later)

C: Mr. Greene, do you know why they wanted me? My sister hurt her leg, and they needed my mom's number!

Me: That's exactly what I told you.

C: You did?

More decimal and percent work

For the next lesson in Numeracy, I wanted to keep building students' ideas about what percents and decimals are, and how they relate to fractions. I've mentioned before that I am teaching the students to use bar modeling to solve word problems, but I haven't been posting the problems, and some examples would probably be nice. This week, I've started incorporating percents into the problems, which is, of course, throwing the students for a loop. But, they will get it eventually (and some already have), and I think that continually reinforcing the visual connection between percents and fractions is important. So here are the three problems I'm using this week:

Lesson 1

Diego and Dora both took a test in Algebra 1. Diego got 70% of the questions correct, which was 42 points. Dora did very well, and even got the bonus problem right, so she got a 105% on the test. How many points did Dora score on the test?

Lesson 2

By the end of tutorial, Mariana completed 45% of her homework. She spent 54 minutes working (the rest of the time, she was giggling with Gricelda). If she works at the same speed at home, how much longer will it take her to finish all of her homework?

Lesson 3

At the school dance, 70% of the students were girls, and the rest were boys. Ms. Vasquez wondered why there were so few boys there – she counted only 36 boys. How many students were at the dance in total?

We only do one problem like this per lesson (3 lessons per week - block schedule), because it really takes 15 - 20 minutes for the whole process (more when the students are unfocused) to play out. What's nice about these problems and this method is that it naturally connects percents to the work students have been doing for months drawing whole number bars (first) and then fraction bars. Right now, lots of students are still struggling, but I think it's more due to the proximity of vacation affecting their ability to care about math than a conceptual problem. We'll pick up with this after break as we review for finals, and I think it'll go better.

For this lesson, after finishing the problem solving portion, we did another class activity. I gave each group a set of 10 post-it notes with various decimals and percents, all between a pair of consecutive whole numbers. They had to stick their post-it notes to the board (where I had blue-taped up a long number line) drawing arrows with marker to indicate more precisely where the number should go. This only took about 5 minutes or so, and then I had everyone sit back down so we could evaluate how we did. I told them that they would earn 2 team points (whoopie!) for each number in the right place, and 1 bonus point (what can I say? Freshmen love their points!) if they could find a mistake in another team's positioning. We went through team by team, and I asked the class to point out any mistakes. The mistakes that were pointed out lead to additional discussions and modeling, until students seemed satisfied that everything was in the right place.

After doing this with my first period, I wasn't sure if the activity had been all that useful. I asked students if they found it useful (many did) and to share something that they had learned. This lead to some good questions and observations - the key one being that 2.45 is less than 2.5 (I still had the papers on the board from the previous lesson to refer back to) because of the value of each number; similarly, several realized that 2.45 is not more than 2.5 even though it has more numbers in it. Other students said that it helped them see better how a percent and a decimal could both be plotted together on the same line (yes!). So this made me feel better about the whole thing, and I didn't modify the lesson for the other periods.

Now that we've used manipulatives and done a couple of class activities, in tomorrow's lesson, students will be doing a worksheet I put together to get some solid independent practice time. They will start by drawing paper pieces (as in the last lesson) to represent a variety of decimals, fractions, and percents. Then, given a number in one form, they will have to write it in the other two forms. Finally, they have a bunch of problems where they must compare a pair of numbers - written in any form - to see which is larger.

The decimal point's job... where's the one?

This has been a long and stressful week, but it ended well yesterday. The biology classes were on a field trip, so my Algebra 2 class only had about 5 kids in it. I didn't bother trying to teach anything new - I just helped them with the homework, and then we spent the rest of the time looking at different problems they needed help with, and talking about stuff that was on their minds... some "remember when", since I taught most of them as Freshmen - they remember amazingly well things that I said over two years ago, as long as they are not math related, apparently... we talked through some of their fears about going to college... we reviewed fractions... I wish there were more times available to just sit and chat with students.

In my Numeracy classes, the goal of the day's lesson was to learn how the base-10 positional system works, with a focus on the decimal side of things. Specifically, I wanted them to be able to represent fractions and percents as decimals, and to understand how each decimal place relates to a specific base-10 fraction. This lesson went very poorly on Thursday with my first two classes, due to management issues (i.e. they were behaving terribly). Is there ever a point in time when you can get through a week without feeling like a first-year teacher again? Anyway, I didn't rewrite the lesson for my Friday classes, because I was confident that it was actually a good lesson, and worthwhile.. I just tightened it up, and made sure to stay on top of things behaviorally, and it really paid off. The discussions in both periods (and these are my two weaker classes) were quite rich and productive. On my board, I had taped up papers as in the graphic above, and told students that each smaller piece had been made by cutting the previous one up into 10 pieces. I asked how big each was, and they easily saw that each paper was 1/10 of the one to the left. I then asked which paper represented one whole. Some students thought it was the 10 big pieces, some thought it was the single full-size piece, some thought it was one of the smaller ones. They discussed it, and as I kept asking more students what they thought, in each class, several students started saying that any one of the pieces could be one whole. They convinced each other without much of my prompting - it was like magic! I then pulled out my magic Decimal Point cutout, and told them that the decimal point has one and only one job - to determine which shape is worth one whole. I taped the point up on the board (making the full sheet of paper the one whole for this lesson), and we then proceeded to label all the place values relative to one whole. All of a sudden, why the tenths are called "tenths" started to make sense for some students.

I showed what the fraction 71/100 would look like by taping up 7 strips and 1 tiny square in the right columns, and then translated this to the decimal 0.71. We did a couple more examples (I called students up to try some), and then they began asking about adding zeros at the end. Some thought it was ok, and others thought it would change the value. So I had them debate for a while if 0.6 and 0.60 were the same or not. Eventually, they were convinced that they were the same. So, to push them, I asked why 6 and 60 are not the same, which allowed me to show them that the 0's only change the value when they push a digit into a different place value. For each example, I also went back to the definition of percent (a fraction out of 100) to show that the tiny squares must therefore be considered 1% each. This made it easy to see why 0.36 would be the same as 36%, as they could see the 36 little squares on the board (comprised of 3 strips and 6 squares). It also made it easy to explain why 0.237 is the same as 23.7% - you have 23 little squares, and 7/10 of a little square.

I saw quite a few light bulb moments over the course of this lesson. I need to find a way to get my first two periods back on board now... We'll review this on Monday; I hope that, by the end of next week, students will all be able to compare and order decimals, and easily move back and forth between decimal, fraction, and percent representations. That would be quite an accomplishment, and a good way to go into break.