Algebra Outline

1st Quarter

Day 1 W (25 minutes class): Students fill in opening day first_day.pdf Day 2 H After seeing how much trouble the students had with coming up with a number for annual salary and how easily thechose numbers that were in millions we decided that before doing the Wealth Distribution activity which asks them to estimate population and wealth we would try to put some of the large number in perspective. The students worked on:
  1. How many days are there in one million seconds?
  2. How many days are there in one billion seconds?
  3. If we lined up all the people in the world into a line, how long would it be?
  4. If a plane flies 500 miles per hour, how long would it take to fly over this line from first to last person?

In question 2. students who finished early were asked how many days, or years that was.

In question 3. in one class students were already given that there are 6,828 million people in the world. In other they were not. In each class the students needed to estimate how much space each person would take up in the line.

In the conversations several things came up that yielded the following homework assignment:
  1. How many centuries are there in one trillion seconds?
  2. How many times will our line of people go around the Earth?
  3. Bill Gates makes $9 per minute (this tid bit was supplied by one of the students). What is Bill's annual salary? Before and after taxes (30%).

Day 3 M The students were given a pretest (we wanted to see if anyone is placed incorrectly, as well as to have a basis for measuring students' progress throughout the year). The test is here pretest.pdfAs the students finished they were given a worksheet with addition review -- Alg_AdditionReview1.docx.We had only few minutes in one class and none in the other to talk about their work on the review sheet. The reason is that Emina forgot to discuss the previous day's homework, while Brian spent some time going through it and hence had less time at the end of the class. Both classes were asked to finish the sheet at home and we intend to start the class discussing it, then moving onto Wealth Distribution Activity. Day 4 W Wealth Distribution Activity

The class began by discussing integer addition. After we finished the review, we started the activity. Both classes received only first page. The first class took too long to come up with names of continents, and had a huge trouble coming up with estimates of population and wealth. The students were hesitant to estimate as well as to work together (they were put into groups of 4 which may have been a mistake). We had a short discussion of why the task seemed complicated, as well as what knowledge we have could have helped us with our estimates. The homework assigned was to finish filling in the table, either by estimating or actually looking up this information. The second class filled in continents in group discussion, and they remained working in pairs on estimates. More students were filling in their estimates, and several pairs have completed questions 1 and 2 (without doing the chip part). Homework assigned was to answer questions at the bottom of the page.

Day 5 F Wealth distribution continued: Both classes continued working on the activity. We finally started working with chips. Students are very reluctant to talk with each other, very slow with getting the work done. Several groups were on task and discussing the problems they had while others claimed they did not know what to do. Once we talked about it, my asking questions and them answering, I would move to the next group, but after returning to them several minutes later, they would still be at the same spot claiming not to know what to do. Generally, holding hand and prodding would produce results, but independent work was not overwhelmingly accepted and practiced. Both classes were pulled together and one of the groups presented their findings. E's class went on to receive a second page and has started to work on it. Pretty quickly most people started asking how to find a percentage. My understanding is that they were supposed to have that as their background knowledge. B's class did not have quite as much time at the end, and has not started the second page. i also forget to mention that all my beautifully prepared materials did not get to be distributed properly and appropriately since I forgot to mind the clock (well, I didn't but I overestimated amount of speech I can squeeze into three minutes). Since the students claimed they couldn't remember how to find percentages, Brian thought that it would be a good idea to have a review of ratios and percentages at the beginning of next class.

Day 6 W Review Alg_%andProp_Review.pdf
In E's class this took entirely too long. The warm up lead nicely to the review however, the interest level waned as the class kept going on. Initially, I thought some of the students were done and this was the reason they did not seem engaged. However, it appears that several did not bother to write down work that they were not able to do on their own. I am not a fan of this system, and even though I do not think I talked too much at a time, I felt like I was dominating discussion and students were not participating enough. About 20 minutes before the end or 25, it was clear that there needed to be change of pace. B suggested that I give them a three minute break and then finish off, but I decided that I'd rather bring them back into activity by doing 10 chairs of inequality: USWealth.pdf I decided to have the first row and two volunteers turn toward the rest of the class and we went through questions similar to the ones in the file. The engagement of students was increased considerably and they were answering questions. However, when asked to estimate amount of money each family would receive if the wealth of the US were distributed fairly, couple of kids took a long time to say anything. We waited and we got them all to speak. B chose to take a different path and do the review with a short break in the middle.

Day 7 F The plan is to finish the activity and administer a short quiz. I am about to write it and will post it momentarily. -- Well, the quiz did not happen. But we did finish the activity. We had a discussion about why North and South America were replaced by US&Canada and Latin America. After the warm up that served as a review of percents and few remarks, which in my opinion were not well done on my part, students filled in their tables, each row was assigned one continent and they consequently shared their results for others to use. They pulled their continent from a hat and in their seat were required to individually figure out how many classmates are on that continent and how many cookies they will receive (these were previously done with colored chips). Once the students solved the problem and shared (it took a lot longer than I may lead you to believe here), the students formed groups according to their continent and each group was given a bag with appropriate number of cookies and were told to split it amongst themselves. We had 25 students in Asia with 10 cookies, 6 students in Africa with 1 cookie and 1 student in US&Canada with 12 cookies. After short discussion students were assigned homework (last page in the worksheet).

Day 8 T (9/14) I am behind on recording the events, as I am on many other things (it is 9/24 as I type this). I will try to do this by memory, but it will be hard.

After a warm up, we wanted to start discussing quantities. E started off by giving a definition and asking the students to read it and discuss with their neighbor about what it meant and how they understood it (you can see the definition in the linked page here:

Quantities and relationships

In hindsight it appears that the fact that words "quality" and "quantity" appear so many times and in context slightly different from what students use it in their every day world, there was general confusion about what it is that we were talking about. While our ultimate goal was for them to realize that some quantities can be measured while others have to be computed and therefore depend on others, what seems to have been students conclusion is that there are some qualities of the objects that can be measured or counted while others can not. We worked on Qualities_quantities.pdf in pairs then shared some of the qualities they have come up with. A lucky break was had when one of the students listed area as a measurable quantity, but we were at the end of our time and had not finished the discussion. Their homework was to come up with three more in each category *measurable/countable or not. Students had an opportunity to show what they learned quiz1.pdf

Day 9 H (9/16) Warm up consisted of percentage problems which majority of classes do not understand. I think we have tried all possible interpretations and representations of the percentages -- the problem seem to be: a) they are skill problems, therefor uninteresting hence students do not engage when we talk about them, b) general not understanding of proportions, c) general not understanding of multiplication of fractions (by this I mean of fractions and integers, although I suspect the more generally this statement is true). We continued discussion of measurable versus computable quantities. We collected some more of the homework qualities and discussed them. We had an engaged group while we were talking about measuring speed of wind, which was generally thought to be impossible to measure, and we tried to get to the fact that it could be computed. We moved onto discussing area and how the way the students thought it could be "measured" was really using a calculation to find area after we measured the sides of a rectangle. I do not think we made it clear that there is a "relationship" to be described - definitely not the dynamic view of the relationships. But we managed to get to the fact that it could actually truly be measured with the help of the square foot tiles on the floor of the classroom. However, all that discussion proved not to have had the impact I was hoping for after I had read all their homeworks for that day Qualities_quantities_homework.pdf
As the final task for the day the students were asked to graphically represent their distance from home in a given day -- p.s. "in a given day" is a phrase that is not understood by them, as it changes throughout the day. I told them about my day and how my distance changed and said that my picture should reflect these changes. Almost exclusively we got maps as a product of this assignment. Two students showed their examples and in both we commented how we could not distinguish their distance from home because they gave us their location as opposed to distance. In fact that was generally true -- although I do not believe we made this point well. I drew my graph and asked students to take a minute to see what that picture tells them. One of the students gave the explanation that is recorded in the following image:
Students received the following homework: relating_graphs_events_hwk1.pdf relating_graphs_events_hwk1.docx

Day 10 M (9/20) So we finally started talking about sketching relationships:

Sketches of relationships -- interpret graphs, graph relationships given verbally

In class we worked on relating_graphs_events.pdf -- students seemed to be little more engaged, although their interest wanes when discussion starts. They seem to be more comfortable with graphing then with arithmetic although the idea that the graphs represent a relationship between specific quantities that have to be listed on the graph is somewhat illusive for some. For instance, time-speed graph is often interpreted as time-elevation graph (a decreasing graph means that a person is going downhill). I noted this on their homework and we discussed it in some detail next period. We also discussed the following problem: pools_problem.pdf - the first two questions presented no problem, but students did not understand what the third question was asking them. We discussed one what the question meant, and how we might figure out the answer. Students noted that it would be best if we knew some numbers. I wanted them to think about how they could do it without numbers, and we talked about making the level of water in each pool the same and seeing which pump would finish first. I was hoping that we could also use the same time period and see which one would have had to pump the most water, but I wanted to see if they could do that themselves, so I asked them to think about it at home. We also had a period of working on the review sheet *although sometimes they aren't as quick to start, they seem to be very comfortable with this format and prefer it to class discussion* Alg_Review1.pdf which they self graded at the end of the period and submitted as part of homework.

Day 10 W (9/22) Today the plan was to go over some of the homework problems, finish the discussion of the pump problem, work on 01-InterpretingGraphs.pdf and have an opportunity to show what we've learned quiz2.pdf. As many times things did not work out as planned. Homework problems took too long, I forgot to talk about the pumps/pools problem. As we were finishing the graphs the interest seemed to dwindle, so I decided to switch gears and talk a little about input/output tables, which was something we planned to assign for homework but wavered since it seemed we had too much to cover. We did two examples: I'll take a number, multiply it by 3, then add 2. Students were asked for inputs and we generated outputs. First 10 inputs were all whole numbers. Once we asked for more exotic ones we got a negative number. Brian had to ask for a fraction. A student was asked to generate a recipe and we got a very similar recipe: take a number, multiply it by 4, add 5. We generated couple of outputs, then I asked if there was an input that would give me 0 as an output. We had a nice little estimate/compare/correct/evaluate/compare/correct process going on. Brian thought it was important to have the students write down what this process entailed, and in the process we were left with only 15 minutes to do the Opportunity which I thought was insufficient. We postponed it for Monday, and worked on Interpreting graphs. Students homework was to finish that as well as Input_Output_HW.pdf

Personal note: I, sadly, do not feel like our grand plan is being adequately executed. It seems to me that, although somewhat planned, things are happening pretty haphazardly. They take too long, and I often find I do not do what I intend to. Another thing that is fairly upsetting is the sheer volume of papers that we hand out to students. I lose track of it, and am surprised when they do, too. I get that they only get one piece and I'm taking care of 34 and who knows, but still. I wish there were a better system for doing this. If I had them copy all the graphs and questions, things would take infinitely longer. I am going to use -- well I was supposed to use today, and it already slipped -- to plan several days so that it is clear already how we are moving. Another note, and this is of course expected and clear, that looking through all the homeworks and making notes takes a lot of time. I am also slightly disheartened by the fact that I do not have enough time in class to talk to get to know kids very well.

Personal note 2: I decided that I would plan out the whole week. And I did. And I failed to implement my plan on Monday. Then I modified it so that Wednesday and Friday would follow what happened on Monday. On Wednesday that plan failed again. I will now modify the plan so that it can fail again on Friday.

There are several things that seem to me to contribute to the general feeling that things aren't going well:
  • students do not engage -- problems are too boring? too hard? too uninteresting?
  • students are not focused -- some do not take notes even after they were told to take notes, things take too long and they lose focus, material is boring?
  • students claim not to know what to do -- there is a fair amount
  • students are done quickly and bored while waiting for others to finish -- these ones usually aren't the problem, but i hate them being bored. this is easily fixable, i intended before to have more problems for them. the problem was that we get preoccupied by students who "need" help and don't get to them to give them extra problems.
  • students do not volunteer answers -- this is getting better.

I am not, as I hope one can see, try to blame it on students. I am willing to take responsibility. On occasion it seems though that we have a vicious cycle and it's hard to break out.

I fell behind in writing. I am attaching the documents we used for now, and will try to keep up with writing.













Describing relationships with...

  • words -- have been
  • numbers -- have been
  • expressions for simple ones -- recipes
  • graphs
    • categorize, notice different families
    • concepts: rates of change, intercepts (within graph and table)
    • skills: graphing, constructing tables

  • possible patterns:
    • linear lots!
    • quadratic: area of squares, acceleration due to g, fencing, triangular numbers, square numbers,
    • cubic: volume
    • exponential: growth models, interest

BIG GOAL: recognize constant rates of change!

arithmetic (reviewed throughout quarter)