Category Archives: ESEC508M

Activity File

This assignment was to create a set of classroom activities, including any worksheets required for these activities. As I look back at these activities three years later, I’m actually pleased. I really like the Remote Algebra activity. Can’t wait to use it!

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Journal entry: “Importance of notes and beginning of each class session”

March 6

I’m going to focus my notes more next year. Students slavishly copy down whatever is on the board, after they confirm that you want them to, to the detriment of learning, in many cases. They don’t hear the message, and they get frustrated when the text or image they’re copying down disappears a couple seconds too early. One girl told me today during open house that “the slides sometimes go by too fast,” and her mother pointed out that the girl is a perfectionist, to the point of being unwilling to abbreviate. I always tell the kids to abbreviate, and I even point out (using proofreader’s marks) what to leave out. I’ve been giving them too much credit. They need to be told exactly what to write, and it needs to be a reasonable amount. (I learned to be explicit earlier this year, after the 901st time I heard “do we write this down?” I thought my answer — “if it appears on the board, write it down” — was good enough, but there are things I want to post on the board because I want them to hear and see them but not write them. So, it’s clear we need to help 8th graders take notes. We’ve been encouraging Cornell Notes in our school, and we’ve considered requiring them. I think this would be a good thing. In terms of my Powerpoint slides, it might be a good idea to have way fewer slides, with each slide containing three sets of items: the message the teacher wants you to hear, the Cornell Notes question pertaining to this message, and the abbreviated way to write down the answer to the question. I’ve listed these items in chronological order, so an example would be a slide about perpendicular lines and their slopes:

  1. [I say, and they read,…] When two lines are parallel, their slopes have a product of -1. In other words, take the slope of line a, multiply it by the slope of line b, and if you get -1, then the lines are perpendicular.
  2. [In left margin…] How can you tell if two lines are perpendicular, if you know their equations?
  3. [Main body of slide…] slope of line a times slope of line b = -1
  4. [left margin…] What’s a shorter way to write this?
  5. [Main body of slide…] ma * mb = -1

March 7

I finally figured out what’s wrong with the beginning of my classes: wasted time. Pencil requests, paper requests, other missing items, seating-chart changes, casual chit-chat, insults across the room, and now that my seating charts are a little bit out of date it takes me five minutes to dodge the requests while finding each student so I can take roll. OK. So, I just need to keep the seating chart current and enforce radio silence for the first two minutes. I’ve attacked the problem this week with a clear goal, posted each day, of readiness. Most days this is 90% readiness: 90% of the students should be seated, silent, and facing forward, with pencils and paper ready. If they don’t make this goal, they lose our “Festive Friday” reward period. They’ve been very enthusiastic about making this goal, encouraging each other to SIT DOWN AND GET READY! I’ve been a bit lenient this week, so next week I’ll crack down, and one or more of the classes will probably miss their mark. I drop a pink card on the desk of anyone who isn’t ready; it’s amazing how they can protest – while walking to their desks immediately after the bell rings – that they “were ready” while it says on the overhead that they must be seated. I think I’ll take away their privilege in five-minute increments so they don’t lose hope. (The reward time has been used for origami projects and videos. I’m showing music videos by Michel Gondry – director of “Eternal Sunshine of the Spotless Mind” and “The Science of Sleep” – and I explain that they are “math-appropriate” because they deal with musical patterns.)

March 8

What can I do better? I spent some time prepping for the next lesson by studying the “note-taking guide” produced by the publisher for this section. I see this: vocabulary, a couple examples, some formal rules and two checkpoints. This won’t work for my kids at all. The formal rule — “for all real numbers a, b and c, if a > b, then a + c > b + c” — is totally true and indisputable. The kids will copy it down, but they won’t comprehend a bit of it. I’m going to present a number of examples in a table; the table will be oriented around the solution of each inequality – column heads “a,” “b,” “inequality,” “inequality (verbal),” “operation,” “solved inequality,” “solved inequality (verbal),” “sample solutions” — with a tiny introduction. The introduction will be meaningless, but the kids will quickly make sense of the examples and will volunteer to complete the table. I can then transition to numerous examples of homework problems.

What I’ve under-emphasized, as I did last year, is the importance of repetition. However, I think what caused me not to give the students enough opportunities to repeat homework exercises was a good motivation: students need to understand the concepts with greater depth. They tend to look at an exercise and launch into it based on its appearance, in spite of the instructions. (I’ve played with this by giving them quizzes that have instructions unrelated to the problem: “Which number is larger? 7 + 5”)

Perhaps a better way to teach algebra properties to the students who tend not to immediately grasp the abstract idea is, rather than to present a rule, to teach it backwards; that is, if they are given that “x > 7” then they should be claim without doing any arithmetic that “x + 2 > 7 + 2.” From there, they should be able to write “x + 2 > 9.” Would it be helpful or just confusing to then use the subtraction property to undo this work?

March 9

Practiced for the chapter test today using XMG Football. Students play football – boys against girls – on overhead, moving ball based on their ability to solve problems determined by roll of dice. Nice game, with lots of opportunities for volunteers: dice roller, marker mover, referee. Some students complained about unfairness when the referee didn’t recognize their hand as first up, but the biggest problem was that one group of kids lost interest and just started chatting. I’ll anticipate this next time and find a way to keep the whole room focused. (I’ve seen this whole class be focused on a competitive game, so I know it can be done.)

Another approach to using this game is to get a set of six or eight placemat-like football fields and have groups play. I would then have to try to prevent “shrinkage” (theft), especially with this crew, which has already stolen from me two calculators, a laser pointer, and the rubber feet from the bottom of my stapler.

Speaking of that, one thing I will do differently next year is to start out secretly not trusting the kids with anything other than paper and to slowly give them more freedom to work with my possessions. Apparently, by the end of the year there is more of a relationship, and you should be able to trust them with more things.

March 12

Next year I plan to poll some teachers at the site I’m at (wherever it turns out to be) and implement a strong, helpful, consistent homework policy. My students still think it’s okay to turn in homework that looks like this:

  1. A
  2. C
  3. B

No kidding. That’s what I got today.

Have I ever told them not to do that? A bit. Not enough. Today, I said, “Well, I really want you to be sure to go ahead and write the whole question and answer down.” I think I’ve been letting their whining influence my willingness to stand firm.

Which reminds me of another thing: Whining should automatically cause the work to be harder, not easier. I’m not being consistent when I tell the students to let one person at a time have the floor, to treat each other with respect, to try their best, etc., and then to not only allow whining, but to respond to it as if it’s part of a civilized classroom discussion.

So, homework needs to include the following characteristics:

  • Not too much.
  • Enough to get good practice.
  • Published assignments, on the web and in a predictable location in the classroom.
  • Either directly tied to the day’s lesson, or clearly review of an earlier concept.
  • Well-known formatting rules.
  • Regular communication with parents about homework, including good and bad news.
  • Accountability.

We’ve made some progress on the formatting and the predictable location. (I still have one guy – crack baby? — who hardly seems with it most days and who never remembers to write his assignment down. I remind him, but it often turns into a discussion about which required materials he’s missing that day.) My students are currently getting away with all kinds of stuff on the homework. Part of the cause of this is it’s impossible to collect, grade, record and return nightly. I collect packets every two weeks. This is only marginally acceptable as a way to keep the students accountable for their homework.

March 13

Not to get too sentimental or idealistic, but I am totally stoked about August. Got an offer from [other district] at Saturday’s job fair and am hoping for one to match from [other other district]. Wherever I end up, it will mean that I will have ample time to prepare for the first day of school – for the first time. (This year I came in three weeks late; last year I came in as a long-term sub with one weekend of warning.)

Surviving at the front of a classroom is all about setting up systems that work: Where does the paper go? How does the discipline work? What tasks are shared out to the students, and how is that managed? Etc., etc., etc. Just to start a short list, here is what will happen in August:

  • Acquire an LCD projector, from the district, from a friend, from eBay, whatever.
  • Purchase and label trays and other storage for the movement and temporary storage of paper: students’, mine, parents’, others. This will include scratch paper, graph paper, writing-assignment blanks, standard outbound memos, etc.
  • Establish long-term storage for student-discipline folders, student papers, assignments (including blanks and keys), memos, reference material, etc.
  • Establish digital-storage for student info (custom database or open-source database or something built in to district office-automation system).
  • Establish, approve (with school administration), and document classroom rules and procedures.
  • Establish and document classroom policies and consequence systems. Include “daily data display” (lesson name, number, goals, standards, homework, etc.) on small whiteboard or portion of class whiteboard.
  • Design rewards cards, lunchtime-homework-opportunity cards, and other reusable paper.
  • Write, approve, copy and store syllabi.
  • Acquire forms from administration for discipline.
  • Set up the classroom: desks, projector, trays, boards, teacher desk, posters, tables, forms, signs, supplies, etc. Wash windows, walls, boards, desks, tables. Vacuum floors. Clean ducts.
  • Confirm all necessary network access: Internet (district computer as well as laptop). Design seating-chart blanks for substitute folder. Write welcome letter for substitute folder.
  • Check wardrobe for ample summer/winter wear.

I am hoping to have a classroom where the kids are busy the entire time – and don’t notice it – and where early on the temptation for idle moments will be discouraged through non-punitive backup activities. I want to take the first two weeks of class and establish a healthy classroom community and procedures, including formatting of work. I realize this is optimistic, given the realities of pacing guides, but I’ll see how much I can eke out of the system.

Journal entry: “A more positive environment?”

 

February 15

I am trying to assign regular writing assignments to all classes. One paragraph, consisting of five sentences, including a topic sentence, three supporting sentences, and a conclusion. Complete sentences only. “Are fractions easy or difficult?” “Was yesterday’s lesson successful?”

I’m happy with the results. Most students required very little explanation to understand the assignment. Even more noteworthy, students had opinions about the assign itself; one girl insisted that these were not “essays” since they were only a paragraph, so I now refer to them as “mini-essays.” A few students cut corners, but most students really warmed to the opportunity to share their thoughts.

Ultimately, I hope to have the kids tell me what they get and don’t get about a lesson’s concepts, giving them access to higher-order learning modes.

February 16

A more positive environment? I work on this all week long. I have rewards cards (“operate the projector,” “one homework freebie,” “one minute tardy pass”) for good deeds or especially insightful questions. I give out gold stars and “Wow!” stickers. I give an EC point to anyone who points out (after silently raising his or her hand and waiting to be recognized) a typo or math error I’ve made on the whiteboard. I have a 15-minute period of music, videos, and origami at the end of each week – “Festive Friday” – set up as a reward for my classes which maintain a consistently high level of readiness and homework completion during the week.

So how can I create a more positive environment? I am working on “compassion-ifying” all my student interactions: corrections, suggestions, consequences and other behavior discussions need to be always motivated by concern for the student, even if I secretly would rather not deal with a given student ever.

February 19

For my 8th-grade Algebra classes, I’m developing a post-testing (probably May) project that will give the “non-math” kids an opportunity to succeed in math. I’m going to describe the optimal angle for solar cells at our latitude and then have the students survey the roofs in the school. (There are at least five different roof angles on our campus.) They will be measuring and computing slope. They will be required to present their results on a science-fair board, giving them all kinds of graphic-design opportunities. They will learn a bit of science and a bit of math. The rubric will include language-arts requirements.

Last year for the year-end project in my Geometry class, we made “creatures” out of rectangular prisms, cones, spheres and cylinders. They had to build the creatures, make up stories about them, measure all of their dimensions, and compute their surface areas and volumes. It was a huge success. They kids had fun, cooperated with each other, said they learned a lot and wished they could have done it earlier. It would be great to create something similar with this project.

February 20

This week I’m trying to come up with corrective actions for a student who makes inappropriate comments in the classroom. Danny has been doing this since we met in September. He has improved; he no longer chooses words such as “Jap” and “faggot,” words which got him a referral and us a meeting with his mother and sister (who had been told I was picking on him). However, I still hear “shut up” several times a week. Danny and his friends apparently say this to each other a lot. Danny apologizes to me now, sincerely I think, but continues to say “shut up.” Sometimes he says it to someone on the opposite side of the classroom.

I don’t really want to keep writing Danny up for these things. I’m trying to minimize the amount of paper I submit to administration. I’m looking for a small reminder, some kind of levy which Danny will have to fork over but which is outside of the usual escalating intervention matrix. We were collecting money for a cancer-charity campaign, and I was thinking of asking Danny for a quarter or two for each “shut up.” (This would not be “required” but would instead be a voluntary choice that Danny could choose in lieu of the regular escalating consequences.) The campaign is over, but I could still collect money toward a pizza party or something – after checking with the administration.

February 26

Today’s assignment was right out of the book, a review of the basics of the coordinate plane. I’m trying to help the kids get more involved with the material, especially in a verbal way. I required them to copy all instructions from the book and to underline several key terms, including ordered pair and coordinate. I good-naturedly chastised the room for verbal imprecision. I’m trying to deal with behaviors such as citing a single number or pointing at the board (when I tell someone to identify a point) instead of just naming an ordered pair, hedging verbal bets (“the y-intercept, or whatever…” and saying “and then” instead of “equals” or “over”), and hazy justifications for steps in solving equations.

February 27

I have a couple students with some issues. One boisterous girl who was belligerent and confrontational in September lately seems to be working hard to get my approval each day, at least when she’s with us. Her flakey attendance got worse lately – she recently disappeared for a week – and I found out today that she’s started having auditory hallucinations, voices that tell her to kill stray animals and classmates in ways that are detailed and bloody. The counselor is working hard to get her some mental health help. I’ve decided to give her a break on my usual lunch-”invitation” (detention) for a while: I’ll invite her in if she doesn’t bring in homework, but with no pressure, just as an opportunity to help her stay caught up with the rest of the class.

Another girl has some anger things going on, mixed in with complicated family dynamics. She also disappeared for a while. I found out yesterday that she was in the hospital for a week after slashing her forearm with a big piece of glass. She has been trying to strike up conversations with me, and I decided to let her visit once to talk during lunch. (I checked with the administrators this afternoon, who convinced me it’s probably not in my interest to do this again, even though I kept my classroom door open.)

I will find ways to make this girl more comfortable in the classroom, through errands and other responsibilities, regular greeting, and queries during the lesson. I’ll also be sure not to try to “rescue” her, as we’ve been trained to do in our CSUSB classes.

Why do group activities falter?

The assignment was, “Make a short list of what’s wrong with group activities.” My short list:

Reasons Why Group Cooperative Activities Falter

  1. A group’s threshold of distraction is at least as low as its most “distractable” member.
  2. Success is absolutely dependent on practicing the specific activity. If you haven’t practiced enough, groups and individuals will be confused about the process.
  3. Not all students are as motivated as all other students. This can create frustration and resentment between them in both directions.
  4. If roles are not clearly defined, students will be confused about their roles.
  5. Students must know how to turn off discussions instantly, or else time gets wasted during transitions.

 

Lesson: Whole-Group Interactive Lecture Mathematics Lesson

Objectives

At the conclusion of this lesson, the student will be able to:

  • Choose between point-slope and slope-intercept forms of linear equations as a starting point, given resources such as slope, one or more points, and intercepts.
  • Substitute and simplify chosen form and then convert to standard form.

Subject and ELL Standards

  • CA State Algebra Standard 7.0: Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.
  • Strategy 5b: Use multiple media to support concepts.

Materials

  • LCD projector, with remote
  • OpenOffice presentation software
  • OpenOffice presentation files
  • Overhead projector
  • Transparencies
  • Whiteboard
  • Pull-down eraseable coordinate plane

Introduction (Anticipatory Set)

  • Post three homework problems on the board (from PDFs supplied by textbook publisher) and ask students to write down, in complete sentences, what the given info is for each problem and what information the problem is asking for. Provide examples of “given info,” recalling earlier lessons’ examples such as y-intercept, slope, a point (or two points).
  • Randomly call students (using deck of cards or dice) to read answers. List answers on board, without commentary, under headings of “given info” and “type of answer.”

Teacher Activities (Instruction)

  • Quickly review prior lesson’s activity: building a table used to choose between two linear-equation forms and converting to slope-intercept (if in point-slope) and then graphing the result.
  • Announce that today’s activity will be very similar, but instead of trying to get all information into slope-intercept, we’ll be converting to standard form. Announce that there will be a short quiz at the end of the activity, and students will be turning in their work.
  • Display sample homemade student worksheet on LCD projector. (See illustration.)
  • Demonstrate populating first row, using first set of given information: m=-2, b=3. Complete row will look like this:


 

given info. m point b S-I/P-S? sub/simp. –> standard
m=-2, b=3 -2 —- 3 S-I y = mx + b
y = -2x +3
y = -2x + 3
+2x +2x
2x + y = 3

 

  • Demonstrate second row, using given info: m = -2, (1,3). This time, point-slope will be the initial form of linear equation, but the process of conversion to standard form (after the distribution step) will be reminiscent of row 1.
  • Give students two more rows to work on, supplying a slope and intercept for one and a slope and non-intercept point for the other. Guide them through this practice.
  • Survey students to recall ideas from previous lessons: What if there is no explicit m (students are given two points)? What if there is no explicit b, but the point contains an x-coordinate of zero? Conversely, what if there is an x-intercept given? How do they choose between initial forms? (Students should say that S-I is preferred, for its ease of graphing.) During these questions, call on students randomly (using dice or deck of cards), giving the class time to consider and discuss answers before rolling, and allowing no “I don’t know” answers. (Students who say “I don’t know” should be given time to discuss the question with neighbors.)
  • Give students two more rows to work on independently, supplying a slope and the point (0,1) for the first (students should recognize the number 1 as the y-intercept) and two points for the second (students should realize that they will need to compute m using the slope formula, which they have previously studied).
  • Administer quiz: three more questions, on new paper. Students should work alone.

Accommodations for ELL/SDAIE/Exceptional Learners

  • Use of LCD projector.
  • Use of white board.
  • Use of multiple colors of text on LCD presentation; use of multiple colors of marker on white board.
  • Speak slowly, clearly.
  • Use of gestures/expressions.

Student Performance

  • Students build semantic concepts table which leads them through the steps of choosing an initial form of linear equation, substituting the given info, simplifying, and converting to standard form.
  • Students use new paper and build table which they use for quiz. After the quiz, students hand in work for grading by teacher.

Assessment

  • Students hand in quizzes for grading by teacher.

Teacher Reflection

The work itself represents a lot of steps for some students, manageable by few. (This is a “foundations” class; I believe many of these students do poorly in algebra because they cannot focus on a problem longer than two steps. This was demonstrated in this exercise, and it’s demonstrated dramatically when solving linear equations: they can do a two-step equation, but when it comes to equations requiring distribution or collecting terms on one side, they seem to run out of steam almost before they even begin.)

Students occasionally needed reminders of tracking negatives and distributing. I also ran into the classic questions about how to handle more than two factors at once. (When point-slope equation has a fractional m, students have been taught to “clear the fraction” by multiplying both sides of the equation by the denominator. This confuses students who are drawn to next-door factors in a construction like this: ½ (x-3) (2). The students don’t see the cancellation of the 2 and the ½ but instead want to distribute one or the other – or both! – of these. Teacher should take care to write the denominator-canceling integer next to the fraction.)

A student who is conscientious failed to plan and ended up squeezing a lot of work into a single line on her lined paper. The teacher should anticipate this and tell students to allow five rows for each problem.

I have trained the class to recognize b (given either as “b=2” or as the point “(0,2)”) and to use this as their criterion for deciding whether to begin with point-slope or slope-intercept. However, some of the students in this exercise failed to recognize the “hidden” b and chose point-slope when they could have worked with slope-intercept instead.

The formatting of these papers represents the typical poor quality I see each day. I’ve spent a lot of time explaining how the papers should look. I may devote a whole day to this in the future. It’s hard to resist telling the students that this work is insulting to me. Since the students can’t all be counted on to actually draw the table, next time I would stop at this point and have students hold up working showing table before proceeding.

Illustration: Projector Slide demonstrating table students are expected to draw and populate.




Journal entry: “Hijacking the class”

February 5

The students reminded me today of the math classroom’s absolute total dependence on verbal skills.  We are studying linear equations, and we just moved from slope-intercept form to point-slope form.  The students are beginning to be comfortable with the instruction, “Write down point-slope form.”

Some of them can use this on their own in response to my prompt, “What do you do next?”

But the book is inconsistent in the meaning of “form,” and the students sense my discomfort.  The template or model equation itself — y – y1 = m(x – x1) — is described as “point-slope form”; at the same time we say that a specific equation — such as y – 5 = 3(x – 4) —  is written in in “point-slope-form.”  I don’t know how I would teach this differently next year, and we’re too far into the process for me to make a big deal out of it right now.  I need to keep looking for these things (and find solutions for them) that make the students whine out loud, “Why does math have to be so complicated?”

February 6

Period 3 had 100% homework today.  All students brought in homework.  I was astonished and praised them way too long.  Some apparently went home with tales of their weird math teacher.  (I even heard of parents who responded to these tales with their own praise of their children.)  It all felt really good.

Students had been showing me a reasonable rate of return, maybe 60-70%, and a few had excuses occasionally.  A few however, were just never bringing in homework and merely shrugged when I asked them why.  A couple weeks ago, I cracked down: no homework → lunch detention; skip lunch detention → official after-school detention.  I found that one total homework abstainer (after he got done complaining about my calls home and his detentions) is now bringing in his homework every day and even seems to be understanding more of what’s going on.

So, I learned from the students that even though they complain about having to work, they also are happy to be learning.  This is what we want, since when they’re learning, they’re engaged and not screwing around.  Plus, they’re learning, since they’re not screwing around.

February 7

The principal visited today on a pre-announced observation.  I was surprised to learn from defiant Frank that not everyone fears the principal.  After a couple reminders to turn and face forward, and then my next classroom-intervention-matrix step of a writing assignment, Frank began complaining about being singled out.  He reached the point of demanding to see the assistant principal and even the principal, whom he knew was right there in the room.  I spoke to him calmly but firmly at each stage, and I think it may have felt to him that he was being singled out.  I believe, however, that this is someone who is usually not denied anything.  Or maybe it’s the opposite, and he feels like the classroom is one place where he can make his own decisions.

Nevertheless, I was forced to deal with his hijacking directly, and eventually I had to ask him to leave.  The other students were complaining about his behavior, and after the class was over, the principal told me I had done the right thing. Regardless, it would have been nice to have an errand to send Frank on that would have allowed him to calm down without having to get in trouble.

February 8

Sean wrote in today’s mini-essay that the lesson hadn’t gone well.  He had trouble focusing (which was reflected in his disruptive behavior) and wrote that this was because he hadn’t gotten enough sleep the night before.  I now casually ask him, nearly every day, if he got plenty of rest last night, as a reminder to him to do so and as a way for both of us to recognize the cause if he becomes belligerent.

I learned to remember that behavior is not spontaneous and usually not premeditated.  I have tried since this discussion to be ready with questions about the students’ preceding 24 hours – enough food?  enough sleep?  enough peaceful opportunities to do homework? – whenever issues begin to arise.  This feels like a great use of my time and worthy investment toward classroom sanity.

February 12

I began using my 30-sided die today for random selection during lessons.  I learned from students today the following: this is a great technique for equity.  The first couple (or ten) times, they think you’re kidding.  Then they begin to figure out that the teacher is serious when he says, “Answer this question.  [Pose question.]  Turn to your neighbor, and discuss this for 30 seconds, and be ready to answer when I roll the die.”  Be sure they learn that you won’t let them off the hook with the response, “I don’t know.”

I’m happy with using this technique in just about any lesson now.  I knew, of course, that many students were hiding behind the hand-raisers and troublemakers and consequently not learning a thing. But I was letting the momentum of a marginally functional classroom keep me from fixing this problem.  So, the random-selection technique is working great.  And students are even learning about probability first-hand: “Yes, ladies and gentlemen, the purple die will give each of you an equal opportunity to show us how much you’ve learned!  And the purple die doesn’t even remember that it chose Maria a minute ago: it might just choose Maria again.  She still has the same chance as everyone else!”

February 13

Terrible day.  My students taught me today, among other things: don’t let them use the stapler (someone stole the rubber feet off the bottom), don’t let them have paper clips (they folded them into dangerous macelike balls), don’t let them out of their seats (they chased each other around), don’t let them have rubber bands or paper (somebody got hit in the eye with a folded missile from across the room).  Probably better not to let them have any freedom at all.

More realistically, I need to do a better job of limiting the choices the classes make (especially my difficult afternoon periods), and I need to simultaneously tie the students’ choices to their freedom.  I have an official “Festive Friday” each week: 20 minutes of origami, music, reading.  I can (and have) taken this privilege away from individual classes for misbehavior during the week; however, it’s been way too subjective.  I need to make yet another system – a numeric scale to tie Festive Friday to various behaviors in the class, including punctuality (see February 12), attentiveness, respect, and homework completion.

I told the class at the end of the period that I didn’t think it had gone well at all and asked them for their opinions.  They agreed by consensus that they had had too much freedom.

Lesson: Cooperative-Group Mathematics Lesson

 

Objectives

At the conclusion of this lesson, the student will be able to:

●      Write a linear equation, given various configurations of points, slopes and y-intercepts

Subject and ELL Standards

●      CA State Algebra Standard 7.0: Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

●      Strategy 5b: Use multiple media to support concepts.

Materials

●      LCD projector, with remote

●      OpenOffice presentation software

●      OpenOffice presentation files

●      Overhead projector

●      Transparencies

●      Whiteboard

●      Pull-down eraseable coordinate plane

Introduction (Anticipatory Set)

●      Post slope-intercept form on the board.  Query students in detail on what it means to answer a homework questions which prompt: “Write an equation in slope-intercept form.”  Students should recognize that they are filling in the missing numbers (m and b) and are retaining the variables x and y along with the organization of the equation.

Teacher Activities (Instruction)

●      Introduce “Making the Equation” game.

  1. Four “games”comprise complete set of activities. Fourth game is a culmination of the skills practiced in the three preceding games.
  2. There are three “hands” in each game.
  3. Each team gets a plastic bag containing 24 “resource cards”: colored pieces of paper containing the data needed to create a linear equation.  In the Yellow Game (yellow paper pieces), a resource card contains a slope and an ordered pair; in the Green Game, each resource card contains a slope and a y-intercept; in the Blue Game, each card contains two ordered pairs; in the Pink Game, the cards are a mixture of contents from the three preceding games.  (In this game, students must decide what to do with the resources and what form of a linear equation to start with.)
  4. In each hand, teams pick a resource card, write down the resources in the indicated area, write the template equation – either y = mx + b or y – y1 = m(x – x1) – and then substitute the resources and simplify if appropriate.
  5. At end of each hand, teams trade and grade.

●      Display instructions on LCD projector.  (See attached.)

●      Display sample homemade student worksheet on LCD projector.  (See attached.)

●      Demonstrate sample exercise on LCD projector.Guide students through folding two papers each into four sections: “taco” fold and then “taquito” fold.  Use this folded paper for the worksheets.

●      Guide students through each game.

Accommodations for ELL/SDAIE/Exceptional Learners

●      Use of LCD projector.

●      Use of white board.

●      Use of multiple colors of text on LCD presentation; use of multiple colors of marker on white board.

●      Speak slowly, clearly.

●      Use of gestures/expressions.

Student Performance

●      Students fold homemade worksheets.  Write heading info in top section of first page.

●      Students play one game at a time, selecting three resource cards and writing one equation each. After completing each game (three equations), students stop and await direction.

●      Each hand involves 1) copying resources from resource card to appropriate field on worksheet, 2) writing down template equation (either point-slope or slope-intercept), 3) substituting, and 4) simplifying.  Students box final answer.

●      Students trade papers with neighbor teams and grade each other’s work, using answers on overhead projector.

Assessment

●      “Trade & Grade.”  (Teams trade with next-door teams and grade work.  Answers are displayed on overhead projector.)

Teacher Reflection

An exercise in frustration for all concerned.  Students used no initiative in deciphering instructions displayed on overhead and generally refused to hear instructions.  Teacher did not allow enough time for instruction, and did not manage time well enough to get through all four games. Teacher did not explain clearly enough that each student should maintain his/her own work, to be turned in and graded.

Many students believed they were done when they found a slope, failing to understand that finding the slope was the first step to substituting and simplifying the equation itself.  I have been working faithfully on the concepts of an equation (as a statement and as grammatical form which requires three parts), but the students are attracted less to abstract concepts than to procedural recipes.  I will continue to discuss this, probably through short-duration, high-frequency “equation? Or not?” drills.

Repeat the game later for better results.  (Students figured out the instructions through trial-and-error and dozens of redundant questions.)

On the plus side, most students were generally on-task, especially considering the distractability of the group in question.

 

Journal entry: “When didn’t you learn about input/output tables?”

 

January 24

Some lessons seem to spontaneously skitter off-track into a mud puddle of confusion.  It feels like the students are distracted, annoyed, and politely bored all at the same time.  “Just be quiet and let us get our backpacks and start cleaning up.”  Today is one of those.  It’s hard to tell whether they’re mad at me because I’m frustrated with them or the other way around.  Sometimes I try stopping instantly and asking them what they have planned for the weekend.  (It takes a minute for them to warm to the topic and then they won’t shut up.)

I’ve decided that a couple of the many causes of this kind of confusion include (1) they don’t see where we’re going or the context of where we are, and (2) they hate algebra and don’t know “when they’re ever going to use this stuff.”  Today, I try to head off the second problem by acknowledging that most people truly won’t do algebra every day for the rest of their lives, but we hope they get other things out of this class, such as the ability to think critically —  “how do you know that?” for example.  I even indulge in a little harmless indoctrination: “For example, how you know there are weapons of mass destruction?  How do you know sending 21,000 more troops will fix everything and everybody will get to come home?”

My solution to the second problem is to post a small whiteboard next to my main board with Today’s Questions on it.  These questions are today’s objectives, turned around: How can I write an equation in slope-intercept form, given a slope and the y-intercept?  What is slope-intercept form?  I tell the students that each day they should keep an eye on these questions, and by the end of the lesson, they should know how to answer them.

January 25

Today’s lesson was humbling.  The lesson itself was about the slope-intercept form of a linear equation, and part of the small-group activity was to build a table of values.  I demonstrated the process, gave the students specific instructions on what values to choose for x, displayed complete instructions on the overhead, and by third period figured out that nobody had a clue what I was talking about.  They were confused about the example on the board,  which I had clearly labeled “Example,” and half the class copied the x and the y values directly from it.  We studied input-output tables a month ago, so I’m a little surprised that the students acted like they had never seen these in their lives.

I spent a lot of my childhood in many different books, including my father’s old college textbooks.  One analytic geometry book in particular provided all kinds of activities; I made a lot of input-output tables in order to graph curves such as parabolas.  This concept was already familiar when I had to do it in various math and science classes.  It’s a real eye-opener to find dozens of kids who need to have this explained to them over and over.

As a result of this lesson, I added a mini-lesson on tables of values to my next lesson, and I’ll probably continue to do it.  Our district tests have given low priority to functions and relations, so this has indirectly contributed to our skimming over tables.  I’ll add some table building to my sub plans for my absence day (when I’ll be in San Diego for the math teacher conference).

January 26

I’ve stretched the slope-intercept lesson another day because of the class’s unfamiliarity with tables of values.  (They were meant to build a five-row table of values and then plot the points in order to draw a line.)

In addition to the problems understanding the use and purpose of a table of values, the students are still having troubles with slope.  A few of the more attentive students have seen the pattern where slope emerges from five points in the table: I’ve required the students to work with the x values of -2, -1, 0, 1, and 2, and this means the differences between subsequent vales in the y side of the table are each the slope of the equation.  It makes me wonder if (when?) I should do a mini-lesson on these things.  Clearly, I need to incorporate these tables in lessons at every opportunity.

Another teacher recommends adding a third column to input/output tables, between the x and the y column, which would be the expression equivalent to y.  So in the equation y = 3x + 4 there would be an x column, a y column, and in the middle a 3x + 4 column, where the substitution and simplifying happen.  It bothers me a bit because it seems to duplicate effort, screw up the peer/peer relationship between x and y, and feels like coddling.  I’ve resisted doing this for a couple lessons now.  The trouble with not doing this is I can’t really tell the students where they should be doing the evaluation for each row; it’s kind of just hanging out there with no home.  Happily, I think all my repetition of the steps of evaluating an expression is paying off: I say, “Next step?  Next, after we write down the expression?  I’m thinking of a word that starts with S?”  And the majority of the responses are “substitute,” outnumbering “simplify” and almost completely wiping out “solve.”

January 29

I’m really trying to anticipate and forestall this “I don’t really get this stuff” lazy-ass blanket response.  Half the time I hear this, it’s because someone hasn’t been paying attention.  Still my problem.  The other half the time, does that mean I haven’t been doing my job right?  I’m trying to give them context, motivation, a sense of esprit de corps.  I present our daily destination three different ways these days: questions on the Today’s Questions board, as goals (described on the main board in the housekeeping section and in the slide presentation), and as graphical shorthand (a graph symbol, an arrow, an equation – representing that we can derive an equation from a graph).  Still they say, “I don’t really get any of this stuff.”  I need to encourage the outgoing 10% to paraphrase our destination for their peers.

I gave a few token writing assignments and an easy quiz for the first week of the semester, so that a maximum number of students start off on the right foot.  Half of the students now have honest-to-god A’s; for the rest, I’m offering makeup assignments.  One of my troublemakers actually saw “A” next to his student ID and had to ask me to confirm what he was seeing, apparently a rare event in his life.  (Imagine a cartoon character rubbing his eyes and blinking in disbelief.)

January 30

Today’s lesson in conversion of point-slope form to slope-intercept went pretty well, other than the predictable roadblocks: fractions and integer arithmetic.  They brought a fear of fractions with them from elementary school.  I’ve tried mocking it, lightheartedly: “I know you all got up this morning, and the first thing you thought was, ‘Dude.  I hate fractions.’”  And I’ve tried “counseling”: “It’s going to be okay.  We’re not even doing anything with this fraction.  We’re just writing it down.  It’s not going to hurt. … There.  Now, did that hurt?” This is an ancient dilemma, and I know a lot of smart-ass comments, but I don’t know a solution.  I will leave them with a fraction-simplification worksheet while I’m away Friday, and I will prep them for this Thursday.  Maybe that will help.

And then there’s the lingering inability to work with negative signs during basic operations.  I’ve tried reasoning with them – “if two times two is four, then would negative two be the same thing?” – but they don’t have enough number sense to be persuaded by such logic. I probably need to add more justification to (what seems to me to be the “paperwork”) part of the exercises: simplification.  The student should be able to explain why -4 + -2 = 6, rather than just hearing me tell them it’s wrong.

January 31

Continuing discussion of point-slope form and conversion.  Apparently, some students think that when you subtract the same amount from each side of an equation, that’s just for the assignments where the book says “solve” or “isolate”; we’re now seeing students who can’t make the next step when all it takes to get to y = mx+b is to subtract 2 from each side.  On the other hand, I seem to be seeing fewer messed up distributions than a couple months ago.

Students continue to have trouble reading a graph.  The book presents a line with a clearly marked point (labeled) and arrows representing rise and run.  About half the class can identify the point, fewer can identify the rise and run, and fewer still get the slope from this.  I need to develop a mini-lesson on identifying these elements of the line of a graph.

Word problems

Are word problems a problem for your students? Mine too! I don’t know if I’ve solved the problem, but maybe identifying what the problem is is the first step to solving it.

Some examples:

  1. You ate 3 of the 8 slices of a pizza. You paid $3.30 as your share of the total cost of the pizza. How much did the whole pizza cost?

    Problem: “I hate fractions!”

  2. You are loading large pile of newspapers onto a truck. You divide the pile into four equal-zie bundles. One bundle weights 37 pounds. You want to know the weight x of the original pile. Write an equation which represents this situation. Solve this correct equation.

    Problem: Students in algebra are unaccustomed to (and uncomfortable with) thinking of equation themselves as answers. (To most 8th graders, the “answer” is the part on the right side of the equal sign.)

  3. You have one hour to make cookies for your school bake sale. You spend 24 minutes mixing the dough. It then takes 12 minutes to bake each tray of cookies. If you bake one tray at a time, write a model you can use to find how many trays you can bake during the hour.

    Problem: Students expect numbers within one problem to be presented in compatible units, rather than having to convert between minutes and hours.

  4. You live near a mountain bike trail. You can rent a mountain bike and a helmet for $10 an hour. If you have your own helmet, the bike rental is $7 an hour. You can buy a helmet for $28. How many hours do you need to use th trail to justify buying your own helmet?

    Problem: This problem is verbally beyond many 8th graders.

  5. The cross-country track team ran 8.7 kilometers in 42.5 minutes during their workout. Which equation could you use to find r, the team’s average running speed (in kilometers per minute)?

    Problem: Students are afraid of decimal numbers. Students are unfamiliar with kilometers.

  6. A video store charges $8 to rent a video game for five days. Membership to the video store is free. A video game club charges only $3 to rent a game for five days, but membership in the club is $50 per year. Compare the costs of the two rental plans.

    Problem: Students will have trouble with the apples-and-oranges comparison of the annual fee. Students will have to decode the meaning of “compare the costs.”