Category Archives: Lessons

Cost of paint … priceless!

Today’s class assignment…

How much will it cost to paint this room?

Image

Each wall: 13’ x 13’
Right window: 5’ x 7’
Left window: 6’ x 9’
Door (do not paint): 7’ x 3’
Cost of paint: $71.20 / gallon
Spread rate: 400 sq ft / gallon

That’s it. Pretty easy, right? Figure out the area of the walls, minus door and windows. Decide how many gallons of paint you need. Figure out the price of that much paint. Done.

Except! I didn’t give them the givens — none of them. I made them work in groups to develop their own questions, write them down, and hand them to me for answering. They had to decide for themselves what information they needed, and since my time was a limited resource, they had to try to do the whole thing efficiently. They found that sometimes their questions were too vague or general and I wouldn’t answer them. Sometimes they asked questions whose answers didn’t help. Sometimes I gave them answers that they decided were incomplete, and then they had to rephrase their questions.

It was incredible. Students were engaged. They were arguing with each other. The groups were being quiet because they were competing against each other.

Actual questions:
  • How big is the room? (No answer.)
  • What color is the rug? (No answer.)
  • What city is the room in? (Really?)
  • Do we have to paint the ceiling? (No.)
  • How tall is the room? (13′) How wide is the room? (13′ — two questions where one would do)
  • What are the dimensions of the windows? (5’ x 7’ and 6’ x 9’)
  • What is the cost of the paint? ($71.20 / gallon)
  • What is the cost to paint one wall? (Pick another question you can use to answer that yourself.)
  • How much wall does one gallon of paint cover? (400 sqft)

The coverage question usually appeared during the second wave of questions, after the students began to get a feel for the exercise. Students started by asking one or two questions, and then picked up momentum. One group of football players (who challenge my biases every day) came up with the most succinct list of questions the quickest, a list which nearly matched my list above.

I did not use the word area myself, but some teams discovered it and used it. I believe there was some learning there as the stronger players in the team helped the weaker players discover and adopt the word. (It’s much more intimate to use the word in the small-group context than to read it in a question that asks, “What is the area of a floor that is 8′ x 12′?”)

I reminded my students that “be less helpful” is my motto this year. This exercise was an example of that. It was wildly popular, and of course time flew.

This lesson and the phrase “be less helpful” were massively inspired by the great Dan Meyer.

Optional additions next time:

  1. Students pay for questions. Give out ten tickets to each team. Each question costs one ticket. At the end of the exercise, redeem tickets for points. (Give points for accuracy and speed as well.)
  2. Be sure to let teams announce their own results.
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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.




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.