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					Mathematics Curriculum Guide
             Peter Hansen
          SED 720 Dr. J. Cooks
           December 3, 2003

                        “Writing about the Problem-Solving Process to
                              Improve Problem-Solving Performance”
                                                   By Kenneth M. Williams
                             The Mathematics Teacher, Vol. 96, 3 March 2003


     This article describes and summarizes a problem solving study carried out
     in a Community College math classroom in Michigan. The study divided
     the class into two groups, the Treatment group and the Control group.
     Throughout the course of a semester, both groups were asked to work on
     the same, non-routine math problems. The Treatment group was asked to
     write about their problem solving strategies and difficulties with the
     problem in addition to solving it. The Control group was asked only to
     solve the problems and did not have to provide any additional information.
     The study found that the Treatment group out performed the Control
     group on tests that were administered throughout the semester. The
     conclusion is that writing about mathematical problem-solving will help
     students improve their problem-solving performance.
Literacy Connection

       This article connects mathematical problem-solving to literacy writing
       skills. Traditionally math classes do not require much writing at all.
       However, the research in this article suggests that incorporating more
       writing into a moth class will help to develop students’ mathematical
       performance. Writing about math can help students to organize their
       thought processes in order to more clearly understand a concept.


       The result of this study was that students who wrote about their problem
       solving processes became better math problem solvers. This is significant
       because it shows that, “When students explain mathematical concepts in
       writing, they learn to clearly and accurately communicate; and by…
       writing about these concepts in their own words, they may better
       understand and better remember them” (187). One can then conclude that
       writing about math should have more presence in today’s classrooms.

                      “Knowing and Teaching Elementary Mathematics”
                                                              By Richard Askey
                                                    American Educator, Fall 1999


       This article by Richard Askey is a commentary about a book with the
       same title by Liping Ma. The book and this article argue that in order to
       teach math more effectively, elementary school teachers should study
       math more in depth. The article advocated developing elementary school
       teachers’ numeracy so that they will understand math in a more
       comprehensive way. The article suggests that elementary school math is
       not basic math as many believe, but that in fact it is much more profound.

Literacy Connection

       One of the strongest ideas that this article promotes is using word
       problems to understand math. The article suggests that having students not
       only work through word problems, but to create their own word problems
       as well. Creating a word problem is challenging because the students are
       actually working backwards, from answer to question, rather than the
       traditional route of start to finish. Creating a word problem has many
       connections to literacy. Students will have to rely on their reading and
       writing skills, in addition to math skills, in order to complete the problem
Significance of Research

       One major issue facing math educators today is that many students enter
       High School with weak math skills. Part of this problem is that many
       elementary school teachers also have a difficult time with math. This
       article seeks to change this reality. It is very possible to teach pupils math
       at a young age, however, the teachers too must possess this knowledge
       before any significant progress is made.

                                     “Messy Monk Mathematics: An NCTM
                                                Standards Inspired Class”
                                                                By Larry Copes
                                  The Mathematics Teacher, Vol. 93, 4 April 2000


       This article is based on a transcribed tape from a High School mathematics
       class. Larry Copes is a guest speaker for a day in this particular classroom.
       Copes uses this class period to introduce students to a constructivist math
       teaching approach. He enters the class and presents the students with a
       problem-based situation. He poses one open-ended question (which
       covertly involves mathematical reasoning) to the class and then facilitates
       a discussion about the problem. By the end of the class period, the
       students together have constructed the necessary knowledge to solve the

Literacy Connection

       The connections to literacy in this article are the conversations and
       discussions that evolved as a result of the open-ended question. The
       students were engaging in discussions that required a considerable amount
       of logical expectations. The students also formed foes and allies
       throughout this exercise. This whole class exercise helped students to
       build on their strengths through group work and to build their
       interpersonal skills through thoughtful discussion. Both social interaction
       and the ability to work with others are powerful literacy skills to possess.


       The main significance of this article is that mathematical skills can be
       developed through constructivist teaching methods. On paper, this was a
       dream-like, perfect day in math class. In practice, this exercise was not
       only fun for the students, it also promoted some very high-level thinking
       and reasoning skills. This article sends a very positive message that says
       math classes conducted in this manner are just as credible, if not more, as
       classes taught in the more traditional manner.

                  "The Mathematical Miseducation of America's Youth"
                                                          By Michael T. Batista
                                      Phi Delta Kappan, Vol. 80, 6 February 1999


       This article provides an in-depth view into why and how American
       students are failing to adequately learn mathematics. Batista outlines the
       current debate about mathematical instruction, gives reasoning behind
       each side of the debate and then expounds on how and why math should
       be taught using reform approaches. Batista lambastes opponents of reform
       as supporting their arguments with "hearsay, misinformation and
       sensationalism". The author concludes that constructivist teaching
       strategies are the most beneficial methods of instruction for teaching math.

       Batista believes that in order to develop powerful mathematical thinking
       skills instruction should support a student's "personal construction of
       ideas". This type of instruction Batista states, "Will encourage students to
       invent, test and refine their own ideas, rather than to blindly follow
       procedures given to them by others".

Literacy Connection

       Mathematics taught using traditional methods of 'drill and kill' is the
       antithesis of literacy across content. Teaching math via reform approaches,
       most notably constructivist methods, allows students to use many of the
       literacy skills that they are learning in other classes. Reform approaches
       ask students to critically think about what a mathematical problem is
       asking, and to use their existing knowledge to logically solve the problem.
       These approaches often ask students to write about the strategies used to
       solve the problem and to support their answers with additional


       Undoubtedly a mathematical crisis exists today in American schools.
       American students consistently score lower on math tests than most other
       developed countries. This article points out that this is because math is
       traditionally taught in the U.S. in a manner that focuses on computational
       skills and imitating demonstration. Math is much more abstract than this.
       In order for American youth to adequately understand mathematics, the
       entire American public must be reeducated to understand the validity of
       reform approaches to teaching math. These approaches will provide the
       understanding needed to think of math in a logical manner, rather than just
       manipulating symbols.

                “The Myth of Objectivity in Mathematical Assessment”
                                                           By Lew Ramagnano
                               The Mathematics Teacher, Vol. 94, 1 January 2001


       One argument put forth against alternative assessment in mathematics is
       that it lacks objectivity. This article addresses that argument by pointing
       out that objective assessment in mathematics is just a myth. The author
       gives examples of graded, traditional assessments where score givers
       cannot agree on which marks to give a student. Ramagnano suggests that
       through alternative assessment teachers can consistently assess students’
       conceptual understanding and reasoning ability.

Literacy Connection

       This article advocates assessing students’ mathematical understanding via
       open-ended problems, performance tasks, writing assignments and
       portfolios. All of the above alternative assessment strategies are well
       connected to literacy. Each strategy asks the students to use a variety of
       skills (writing, reading, creativity, discussion etc.) to convey what they
       know and are able to do. These strategies ask students to explain, graph
       and give supporting examples, all of which require many literacy skills.


       Often mathematical assessment asks students for single numerical answers
       or simply to solve a problem. This type of familiar assessment fails to
       recognize the meaning of methods and often does not yield an accurate
       picture of what a student knows. Alternative assessment strategies can be
       aligned with in-class instruction to give the teacher evidence of the
       students’ level of understanding. These strategies by-pass the myth of
       objectivity by providing consistent assessment that directly addresses the


       This is a hands-on lesson that asks the students to come up with the
       number Pi by measuring different circular objects in the classroom. Pi can
       be found by measuring the circumference of an object and then dividing it
       by the measurement of the diameter. Students in this lesson will discover
       why Pi is in the formula C=(pi)d.

Positive Aspects

       This lesson allows the students to make some discoveries for themselves.
       Mathematical discovery is very empowering and very effective way to get
       students to understand mathematical ideas. This lesson also stresses group
       work and hands-on activities. Both of the aforementioned help students to
       relationally understand mathematics.

Where Improvement Is Needed

       The lesson plan leaves a lot of open spaces and is very short on details.
       For example, the plan states, "Then the teacher can lead the students to see
       how we get the formula". This seems incomplete. The lesson jumps right
       from measurements to how to get the formula. More exploratory questions
       before the formula may result in the students actually getting the formula
       rather than the teacher leading them to it.


       I advocate having the students answer some questions before leading them
       to the formula. The questions can ask about student observations involving
       their measurements. Do the students see any patterns? If you organize
       your measurements what do you notice about their ratios? Questions of
       this nature, allow the students to discover for themselves that pi is equal to
       the ratio C:d.
Slope Relationships in Parallel and Perpendicular Lines


       This lesson plan seeks to help students to discover that the slopes of
       parallel lines are equal and the slopes of perpendicular lines are negative
       reciprocals. The students are partly doing a hands-on lesson where they
       are asked to make many observations about lines that they have drawn
       using index cards. The students use the two parallel sides of the index card
       to draw parallel lines, and the perpendicular sides to draw perpendicular
       lines. The students are asked to make inferences about the measure of the
       slope in these particular cases.

Positive Aspects

       The most positive part about this lesson is that students are repeatedly
       asked to write about the observations that they are making. Through this
       writing the designer of the lesson hopes that the students will discover the
       measurement characteristics of the respective slopes. This is a very open-
       ended approach to lead the students to discovery. This method will have
       much more meaning than just telling the students what the slopes have in
       common. Another positive aspect is that the students are working in
       groups throughout this activity. The students will be able to collaborate
       about their observations.

Where Improvement is Needed

       One of the main ideas of this lesson is for students to identify that parallel
       lines have equal slope. It will be difficult for the students to discover this
       when the parallel lines that they draw, using the two sides of the index
       card, are so far apart. The students may have difficulty making the
       connection if they cannot immediately see that the lines are parallel. The
       lesson seems a bit zealous with its time estimates as well. Anytime
       students are graphing something, there is the potential that the lesson may
       run long.


       The major adaptation that I would make is that I would have the students
       cut the index cards in half. This way the students will more readily be able
       to identify that the lines are parallel and subsequently have the same slope.
       Perhaps another adaptation that may solve the time factor is to have half
         the groups work on parallel lines, and the other half work on perpendicular
         lines. This way the time of the lesson will be shortened and also the
         students will get a chance to present their findings to the other half of the

                             Leap Year Day


         This lesson has to do with the idea behind leap year. In the design of this
         lesson plan it says that it is an internet, interactive lesson because it uses
         information off a internet web-site. The lesson asks the students to answer
         12 questions based upon a historical account written about leap year day.
         The questions are mostly historical, but some questions do ask for
         mathematical reasoning.

  Positive Aspects

         This lesson plan is dealing with a relatively easy (not to mention
         interesting) mathematical situation. The whole idea of Leap Year and
         Leap Year Day is baffling to many people. In fact, I would guess that less
         than have of the American population has any idea why we have a Leap
         Year Day. In this respect I think the lesson plan idea is original and an
         interesting topic.

  Where Improvement Is Needed

         The lesson plan poorly addresses the mathematical idea behind Leap Year
         Day. The way that this lesson plan is presented seems to be more like a
         social studies lesson. There are many questions asked about who
         calculated something and what sort of instrument did he/she use. I enjoy
         history immensely and I feel that in order to fully understand something
         you must gain as much background information as possible. However, I
         feel that this lesson over emphasizes historical background which causes a
         sacrifice in learning the math behind Leap Year Day.


         First I would start by making hand-outs of the Leap Year information. The
         current lesson uses this information from the internet and asks that the
         students have internet ready computers on hand. This is often unrealistic,
         but no-matter because a hand-out of the information solves this potential
       problem. Additionally, I advocate asking more open-ended mathematical
       questions about Leap Year and Leap Year Day. For example, 'What do
       you think would happen if Leap Year Day was changed to the end of
       April?'. 'What would be the outcome if it was decided to have Leap Year
       every two years rather than every four?'. These types of questions allow
       the students to showcase what they know, while at the same time the
       teacher will be clued in to whether or not the students understand the
       mathematics behind Leap Year.

                            The Million $ Mission


       This is a lesson plan designed to get students to identify and visualize
       exponential growth. The lesson plan launch is a question that asks the
       students which monthly allowance plan they would choose, a million
       dollars at the end of the month or the combined sum of one cent on the
       first day and double your salary everyday after that for one month.

Positive Aspects

       This lesson plan successfully asks students an open-ended question. The
       students are given no formulas to plug and chug, conversely they are
       asked to come up with a formula themselves. I really like the idea of
       students thinking for themselves. Additionally, I like that this lesson plan
       is designed in a context that the students are familiar with, money. Pretty
       much everyone wants to figure-out how they can get the most money in a
       given situation, this problem address that quest. The lesson plan also
       includes some exploratory questions at the end. These questions ask the
       students to consider different rates of growth when answering the original
       question. These last few exploratory questions will allow the students who
       are performing well a chance to dig even deeper.

Where Improvement Is Needed

       The best way to improve this lesson is to change the wording of the initial
       question. The way that the initial question is asked, either answer will give
       the students a considerable allowance for one month. The students do not
       really care then that the exponential answer is so much more. Also, the
       lesson plan is lacking in detail.

       First, as was suggested above, I would change the initial question. I would
       present the question so that the students would receive the allowance only
       if they chose the plan that would allow them more money. Additionally,
       this lesson seems like a really good lesson to do in groups. The lesson plan
       does not suggest group work. Working in groups would allow the students
       to converse a lot about which plan is better and why. The students would
       also be able to create strategies together to figure out which plan is more

                          Pythagorean Theorem


       I chose this lesson because the Pythagorean theorem is one of the
       cornerstones of mathematics and there are smorgasbords of differing ways
       to teach this topic. This particular lesson plan first introduces the theorem
       by giving some examples (it notes “Like a ladder against a wall”). Then
       the class breaks into groups with each group creating examples of their
       own that depict the Pythagorean theorem. Next the class comes back
       together and together they solve an example from each group. (The above
       is pretty much the entire lesson plan)

Positive Aspects

       The most positive aspects to this lesson are the group work creating
       problems, and the whole class then solving these problems. It is extremely
       challenging to create math problems, however, I feel that creating the
       problems gives a person a deeper knowledge of the mathematical concept
       being studied. The whole class working together to solve the student
       created problems will be an excellent exercise. The students will have a
       deeper connection to each of the problems, and therefore will gain more
       from solving them.

Where Improvement Is Needed

       The introduction is what needs the most improvement in this lesson. The
       introduction is lacking in specific detail. I feel that the more preparation a
       teacher has, the less likely the teacher is to flounder. Just writing in your
       lesson plan ‘do some examples’, may lead to trouble. It would be better to
      carefully choose some examples, and even write in your lesson plans how
      you will introduce the problem and how you will go about solving it. Will
      you call on students? Will you rely on some prior student knowledge?
      What numbers will you use? These are all questions that should be
      answered before the lesson is started.


      The main thing that I would adapt in this lesson is filling in the detail for
      the introduction. For example, there are certain numbers that can be used
      in the Pythagorean theorem that are easier to compute than others. I would
      write this numbers into my lesson plan. Additionally, I would choose a
      problem that was somehow related to the students. For example, if the
      author really wants to use a ladder in the problem, well, have the ladder be
      leaning against the side of the school, or against a hallway wall so the
      custodian can fix a light bulb that has been broken for two weeks. I would
      try to make this problem fit into the students world in order that they make
      some attempt to do the same when they create their own problems.
                                    One and One

      Using tree-diagrams, students will be able to accurately predict outcomes of
      situations that involve probability.

      Overhead Projector
      Nerf Hoop and Nerf Basketball
      12 Calculators
      Transparency Explaining One and One
      2 Blank Transparencies

        Do not let students get carried away with the nerf basketball. Use these tools only
        for the first few minutes of class, then put the ball away. Make sure everyone
        understands what a one and one situation entails. (Have the nerf hoop in place
        before the students enter the room)

      NCTM: Applying mathematical concepts to a real-world situation to draw
      NCTM: Communicating mathematical ideas in written and oral formats.
      NCTM: Ability to reflect on the process of mathematical problem solving
      NCTM: Understand and apply basic concepts of probability
      NCTM: Select and use appropriate methods to analyze data

        Anticipatory (5min)
        Ask for five volunteers. Have the volunteers come to the front of the class.
        Explain that each of these students now has a one and one situation and that each
        gets their chance using the nerf basketball (put up overhead that explains one and
        one if necessary). Write each of their names on an overhead and keep track of the
        amount of points each volunteer makes. Have each volunteer take their shot(s).

   1. (2min) Present the students with this situation and question: Say that I am a 60%
      free-throw shooter. In a one and one situation how many points am I most likely
      to score? Each student write down what they think and why.
2. (2min) Ask a person from each group how many points they think I am most
   likely to score. Mark on the overhead what each groups thinks (most likely
   students will predict one point).
3. (4min) Ask the entire class if they have any ideas how we can figure this out
   mathematically or otherwise? (Let the students try to come up with methods, if
   none suggest anything like a tree diagram, introduce it one step at a time).
4. (3min) Start by asking the students what can happen with my first shot? (I can
   make it, or I can miss it). Next, start the tree diagram on an overhead.
Sample Start
         Shot 1


5. (5min) Now ask the students what will happen next? The students should be able
   to put in words that, if the first shot is missed- there are no more attempts. If the
   first shot is made- then I get one more shot. In this step the students should be
   able to put into words the entire tree diagram. If not I will give them assistance so
   that we create a diagram like the one below on the overhead.

            make        shot 2      make
shot 1

 miss                 miss

6. (2min) Let the students know that what we have created is called a tree diagram
   and that each segment of the diagram is called a branch.

7. (5min) Now ask the students to write this tree diagram in their notes. Include in
   the diagram the number of points that will be scored for the appropriate branch on
   the diagram. (Walk around making sure all diagrams are correct).

8. (12min) Ask the students in their groups to use the diagrams that they have
   created to find out how many points I am most likely to make (0, 1 or 2) when I
   take 100 shots (use calculators). (Walk around the room again helping) The
   students should create a diagram that looks something like the one below (next
   page). The students should summarize that zero points is the most likely (40
   cases) and that one point is the least likely (24 cases).
   1st 100 shots                2nd 60 cases                36 cases

   40 cases                 24 cases

          0 pts                    1 pt

   9. (2min) Ask the students to look back to what they predicted at the beginning of
       class to be the most likely. Where they correct?
   10. (3min) Refresh students knowledge of representing a percentage as a decimal.
       Ask the students how to represent 60% as a decimal? Help the students to recall
       that to represent a percentage as a decimal, just divide the percentage by 100. Ask
       them what 40% is as a decimal?
   11. (5min) Now ask the students to go back to their tree diagrams and attempt to fill
       in the percentages as decimals in the correct branches. Help the students by
       walking around to each group.
   12. (5min) Ask the students what they notice about the total cases for 0, 1 and 2
       points and the decimal points that they have filled in on their diagrams. Point
       students in the right direction so that they can see that another way to figure out
       the total amount of cases for 0,1 and 2 points is to multiply the total # of cases by
       each of the branches along the path to the desired amount of points.
   13. (5min) Ultimately help the students to see that if they multiply the decimal points
       (%'s) along the path to the desired outcome (0,1 &2 pts) they will find the
       percentage that each different case will occur. 0=.4=40%, 1=.6 x .4=24%, 2=.6 x

Follow-up Activity
      The follow-up activity is a challenging HW problem for the students to work on.

                  HW: LeBron plays for a professional basketball team. When LeBron
                  shoots free-throws in a one and one, he shoots 70% on the first shot, and
                  80% on the second shot.
                     1. Write down what you think is the most likely amount of points that
                          LeBron will score in a one and one situation. Explain your answer.
                     2. Calculate the percentages for how often LeBron will score 0, 1 and
                          2 points. Show your diagram if you make one.
                     3. Write a paragraph comparing and contrasting your answer for #1
                          and #2.

Literacy Aspect
       The two most evident literacy aspects in this lesson come from the group work
       and the HW. The group work has the students all working together to articulate
       what is happening mathematically in this one and one exercise. I think that this
        aspect of the lesson (group work) really helps the students to work on their
        interpersonal skills. Strong interpersonal skills are invaluable for an individual to
        possess both in and out of class. Two of the questions from the HW ask the
        students to write about the problem and compare and contrast their answer and the
        mathematical answer.

       There is built in assessment during this lesson and an assessment exercise for
       HW. Throughout this lesson the students are asked to give their opinion on which
       is the most likely case. As the lesson progresses the students opinions should get
       closer and closer to the correct mathematical answer (If not, then they are not
       understanding or I am doing a poor job). The HW assesses the students
       knowledge of the tree diagram in that there are two different percentages in the
       problem so the students must understand how to put the percentages in the correct
       place on the diagram to answer the second question successfully.

                            Understanding Graphing
      Students will become familiar with different types of graphs. Students will
      demonstrate this familiarity by examining different graphs and constructing their
      own graphs.

      12 Calculators
      48 Sheets Graphing Paper
      24 Rulers
      Overhead Projector
      3 Prepared Transparencies

        Make sure that the students do not take to much time perfecting their graphs.
        Understanding the graphs is more important in this lesson than pretty graphs.

      NCTM: Construct and draw inferences from charts, tables and graphs that
      summarize data from real-world situations.
      NCTM: Use representations to model and interpret physical, social, and
      mathematical phenomena
      NCTM: Create and use representations to organize, record, and communicate
      mathematical ideas
   Anticipatory Set: (8 min)
     Show transparency 1 on overhead projector (grades and studying)

       Transparency 1          How to get Better Grades

          Grade Point Avg.

                                  # of Hours Spent Studying Each Week

          1. Ask a student to read the title of the graph and the labels on each axis.
          2. Ask the students to explain to me what this graph is illustrating.
          3. Explain to the students that in many of life's situations one thing is
             dependent on another. The variable that is dependent on something is
             called the dependent variable. The non-dependent variable is called the
             independent variable.
          4. Ask the students to explain to me out loud, which variable is dependent?
             Which is independent? Why?
          1. (1min) Show the students another graph example (transparency 2) death
             and smoking.
          Transparency 2
                                       Death and Smoking

                 Avg. age of death

                                     # of Cigarettes Smoked per Day

          2. (2min) Ask a student to read the title and the labels.
          3. (3min) Ask the entire class, 'What information do we learn from this
             graph'. Let several students explain what they think.
           4. (3min) Ask the class which is the dependent variable? Which is the
              independent variable?
           5. (1min) Show Students Transparency 3.

       Transparency 3

                              Gas                                 Distance

  Amount of gas      A    B                     Distance
     In tank                          C        From Home

                    Distance From Home                       Time Elapsed Since Car
                                                                 Left Home

           6. (12min) Ask the students in their groups to analyze each of the graphs.
               Write answers to each of the following questions. Gas graph: Explain what
               this graph is portraying ? And explain what they think is happening at A,
               B and C. Distance graph: Explain what this graph is portraying. And at
               what part of this graph was the car moving fastest? Why?
           7. (3min) Call on one group to explain and answer the questions to the Gas
           8. (3min) Call on another group to explain and answer the questions to the
               Distance graph.
           9. (15min) In-class exercise. Ask each of the six groups to create two graphs
               on their own using the graph paper and rulers. The graphs do not have to
               have numbers. The graphs should have a title and have their axes labeled.
               There should also be a written description of what the group is trying to
               portray with their graphs and which variable are dependent and
           10. (8min) Ask for two groups to volunteer to share with the class the graphs
               that they have constructed. Explain what their graphs are portraying and
               why they chose to graph that particular information.

Extension Activity
      The extension activity is HW for each student. Each student should create their
      own graph that represents some social situation that concerns them (Ex. The
       smoking vs. death graph). The students must create a title for their graph and label
       the axes. Additionally they are required to write a paragraph that explains their
       graph and why they chose to graph that information.

Literacy Aspect
       In this lesson several students are asked to read out loud. This lesson also asks the
       students to use their communication skills in their groups. Additionally, the
       students are asked to write about the graphs in this lesson and also for the HW.
       The writing that the students will do for this lesson will help them greatly.
       Oftentimes students have difficulties writing about mathematics. Fortunately, this
       is a great tool to really get the students thinking, and also to get them to practice
       their writing skills. Some students are also asked to do mini-presentations in this
       lesson. I like this aspect of the lesson because it helps students work on their oral
       presentation skills and it also helps us build classroom community.

       The students will be assessed during this lesson based on the types of responses
       that they give with regard to the questions about the transparencies. The most
       crucial assessment tool in this lesson is the graphs that the students must create in
       their groups. These graphs will provide the teacher with information about the
       students understanding of dependent and independent variables and how to
       represent them graphically. This is a good group activity because it challenges
       them to come up with examples/information on their own.

                                  The Game Show
(This is an altered and adapted Curriculum and Instruction lesson plan from early-on in
this semester.)
        Students will demonstrate the ability to use algebra tiles to represent the
        distributive property. Additionally, building from the game show situation, the
        students will begin to identify an algebraic equation that shows the distributive

      Overhead and Transparencies (at least 4)
      Prepared Game Show Transparency
      Algebra Tiles (Instructor and all 6 Student Groups)
      12 Calculators
      The anticipatory set provides a great start to this lesson, however, make sure that
      the students do not forget about the original game show problem throughout the
       lesson. It is important to keep reminding the students of how they figured out that
       Latisha was slighted; this is directly applicable to achieving the lesson objective.

      NCTM: Generating expressions
      NCTM: Working toward solving equations
      NCTM: Ability to reflect on the process of mathematical problem solving
      NCTM: Represent and analyze mathematical situations using algebraic symbols
      NCTM: Understand the meaning of equivalent forms of expression
      NCTM: Use symbolic algebra to represent and explain mathematical relationships

 Anticipatory Set (5 min)

       Latisha was on a game show where before her last turn she had won; $15,000, a
       new motorcycle, 2 tickets to see “Fifty-Cent” in concert and 5 new portable CD
       players. For Latisha’s last turn she landed on ‘double your winnings’ and she
       answered the question correctly. Latisha was the big winner and she even doubled
       her original winnings. A week later she received her winnings; $30,000, a new
       motorcycle, 2 tickets to see “Fifty-Cent” and 10 new portable CD players.
            Pose the above scenario to the students on an overhead. Ask the students if
               Latisha should be upset. Encourage much dialogue.
            Explain to the students that the lesson today will give them some algebraic
               techniques to figure out how Latisha was slighted.

       As we move through this lesson every group of students has their own algebra
       tiles as well as the instructor. The groups are expected to follow along with their

   1. (4 min) Pass out algebra tiles to each group of students. On the overhead show the
      students the example below, multiplying with algebra tiles (Two large squares
      placed next to each other to create a rectangle). This is the preliminary step to
      learning grouping with algebra tiles. Explain to the students that:

              1)The dimensions of this rectangle are—X by 2X.
              2)Because two large squares comprise the area, the area is—2X2
              3)The area can also be written as a multiplication problem using the row
              and columns. This is equivalent to the dimensions of the rectangle—
                                X       X

                                  X2       X2
2. (5 min) With algebra tiles on the overhead place three large squares next to each
   other (Each group is doing the same with their tiles at their tables). Ask the
   students to work on the following questions in their groups.
                                                                            X X X
               1. What are the dimensions of the rectangle?
               2. What is the area of the rectangle?                   X
               3. Write the area as a multiplication problem.

3. As the students work on these questions I will walk around the room providing
   assistance. I will then call on volunteers to explain the answer to each question.

4. (8 min) Grouping with tiles. Using two rectangles and six small squares, show the
   students three different ways to group the shapes (shown below). Explain that we
   read and write the number of rows first, and the content of the rows second (‘2X’
   means two rows of X). (The students have been introduced in a previous lesson to
   the dimensions of algebra tile rectangles, small squares and big squares). Example
   at top of next page.

    2X+6 or X+X+6                 2(X)+2(3)                    2(X+3)
   or X+X+1+1+1+1+1+1
     Total Area (no groups)    Two rows of X and             Two rows of X+3
                                Two rows of 3

     I will be at the overhead with tiles as the groups work together with their own

5. (4 min) In order to reinforce grouping with tiles, next we do another example as in
   Part 4. This time use four rectangles and eight small squares. Again show the
   students the different ways that the tiles can be grouped together (just as in Part
   4). Together with help from the students, I will write on the overhead 1) how each
   grouping is represented algebraically, and 2) how to write each group in written
   words. Again, the students are working with their own tiles as well.

6. (6 min) On an overhead write the two expressions; a) 3(X+4), and b) 3(X)+3(4).
   Ask each group to make two figures with their algebra tiles, one that represents
   the expression in (a) and another for (b) (the groups are doing this together, I am
   not leading them from the overhead now). Each student should sketch the figures
   in their notes with the corresponding algebraic expression.
   7. (3 min)Next ask each group to compare the areas of the two sketches that they
      drew for part 6. Ask each group to write an algebraic equation that states the area
      relationship between the two sketches. Ask one student volunteer to write and
      explain the equation on the board that represents this relationship.

   8. (3 min) Tell each student to write this equation in their notes and explain the
      equation using their own words. Inform the students, and have them write in their
      notes, that this relationship is known as the “distributive property”.

   9. Now ask the students to do more problems themselves in pairs (exact problems
      listed below). First two examples with algebra tiles. In each example the student
      pairs should, a) show two ways to group the tiles together, b) write the algebraic
      expression that corresponds to each group, and c) write the expressions in the
      form of an equation that represents the distributive property.
              First problem: 3 rectangles, 6 squares
              Second problem: 4 rectangles, 12 squares

       Now two more problems (given below, next page) this time with algebraic
       symbols, students still in pairs. Pairs should rewrite each expression as an
       equation that represents the distributive property.
              First Problem: 5(X+3)
              Second Problem: 8X +24

   10. (8 min) Return to Latisha’s problem. Ask the students how what we have been
       doing relates to Latisha’s game show experience? Have the students represent
       Latisha’s situation in two ways (like (a) and (b) in section 6) in their groups and in
       their notes.

   11. (3 min)Write on the board the expression: a(b+c). Ask the class for another way
       to write this expression based on what we have been doing in today’s lesson.
       After some discourse rewrite the example as an algebraic equation (distribute) and
       label the equation “The Distributive Property”. Ask all students to write this
       example in their notes and to explain it using their own words.

Follow-up Activity
      (5 min remaining) Ask one volunteer to explain in their own words the
      distributive property. Keep calling on students until there is a satisfactory
      description. For HW each student should answer the following:

       HW Assignment:

           1. What exactly was missing when Latisha received her prizes from the game

           2. Using what we learned about the distributive property make an equation
              that represents what Latisha should have won on the game show. The
               equation should show how to double all of Latisha’s initial winnings to
               equal the total winnings she should have received in the mail. (You may
               create variables for each item that she won, for example- A=$15,000, B=1
               Fifty Cent Ticket, etc.).

           3. Explain in your own words how Latisha’s game show experience relates
              to the distributive property.

Literacy Aspect
       There are three major literacy aspects to this lesson. The first one is that the
       students are asked to do some mini-presentations in this lesson. These
       presentations involve the students either coming to the overhead to say and write
       what they did in their group, or to explain aloud their thinking to the class.
       Another literacy aspect comes from the HW assignment. Question 3 asks the
       students to explain in their own words basically the entire point to the whole
       lesson. Lastly, another literacy aspect of this lesson is that students are asked to
       draw deeper mathematical connections to Latisha’s game show experience.

       My intention is for the students to ultimately see the connection between the
       distributive property we defined in class and the game show situation. This
       connection is desirable for two reasons: 1) It may help the students to relationally
       understand the distributive property, and 2) The game show puts algebra into a
       ‘real-life’ setting.

       I ask for many volunteers to come up to the board during this lesson. This will
       help me assess how the students are doing throughout the lesson. Additionally, I
       think that the students will learn well from their peers in their groups and at the

       The HW will be collected the next class period to help me immediately assess the
       lesson. I will use the HW to ascertain how well the students understand the
       distributive property. The HW responses should help me decide how much extra
       attention to give to the game show in the next lesson.

                                  Talking Algebra
      Students will apply the general strategy for solving linear equations in one
      variable. The students will also demonstrate an understanding of transforming
      algebraic equations into written word phrases.

      Overhead projector
         Prepared Transparencies

         Make sure to inform the students that this lesson will involve some calculations
         and a lot of class participation. Express to them that they have already done this
         type of activity in their heads. Today they will reproduce their thought process
         onto paper.

      NCTM: Working toward solving equations
      NCTM: Ability to reflect on the process of mathematical problem solving
      NCTM: Communicating mathematical ideas in written and oral formats.

         Anticipatory: (5 min) Ask the students for an example of an equation, any
         equation. Write this equation on an overhead and ask the students, ‘what exactly
         does this mean’? Have the students discuss this in their groups. Ask them to
         prepare a response from each group.
   1.    (2 min) Ask the students what types of answers they wrote. (Anything is
         desirable, but what I ultimately want is for the students to understand in their own
         words what each symbol means).
   2.    (2 min) Ask the students what is clear/unclear about this task. (Answers will vary,
         ultimately I want the students to translate algebraic notation into written word
   3.    (1 min) Inform the students that by the end of today's lesson we will all feel more
         comfortable translating algebra into written word form.
   4.    (10 min) Using the overhead, introduce the students to the steps involved to solve
         a linear equation, and do a sample problem together as a class. This first example
         will involve a zero y-intercept for the sake of ease. Use y=2x or something
         similar. The students have not seen equations yet with two variables. Explain to
         them that they must still solve using the same rules they used for one variable.
   5.    (7 min) Students get into their groups and each group works on an example
         problem. The students are free to make up the equation they want to solve,
         however, I should monitor to make sure that the students do not make their
         equation to difficult.
   6.    (7 min) Two representatives from each group will show the class how to work
         through their example problem. The students will do this at the overhead.
         Encourage questions from the class.
   7.    (7 min) Have each group write a summary of their logic (an outline of the steps
         they used) in solving the linear equations. The students should use as much detail
         as possible in their descriptions.
   8.    (1 min) Show students original equation form the beginning of class and ask the
         same question from the start of the lesson, ‘In your own words, what does this
         equation actually mean’?
   9. (7 min) Write the equation large on the board and have student volunteers come to
      the board to write the literal meaning of each term and operation. Meanwhile,
      assist any students having difficulty.

Follow-Up Activity:
      (Remaining 5 min and HW) Students in groups will create two linear equations.
      Students must write the equations in algebraic form and written word form. The
      students must also solve the equations for the variable they created. Although this
      assignment can be worked on in your group, each student must turn in work.

Literacy Aspect:
       The idea behind translating algebraic equations into written word phrases, is to
       help the students better understand the algebraic representation. Mathematical
       symbols and variables often seem so foreign to students. The exercise of actually
       writing out what an equation is saying helps the students to think about the
       notation and symbols in a different manner. The exercise will also give them
       some writing practice. Additionally, the translation helps them express what they
       know in a more common language.

      It should be kept in mind that throughout this lesson it is important to spend equal
      time on the written phrase section and the steps-to-solving section. Students need
      to understand what all the algebraic notation means literally, in order to be
      successful in algebra. I will assess the students through this lesson by
      participating in the group conversations. The HW will also be a good indicator as
      to how much each individual student understands about today’s lesson and its

                                How Far is That?
      Using what they already know about the Pythagorean theorem, students will
      develop the distance formula for finding distance between two points.

      Graphing Paper (48 sheets)
      Rulers (12)
      Calculators (12)
      Blank Transparencies (2)
      Prepared Transparency (School)
      Overhead Projector
      NCTM: Applying mathematical concepts to a real-world situation to draw
      NCTM: Communicating mathematical ideas in written and oral formats.
      NCTM: Ability to reflect on the process of mathematical problem solving

        The first part of this lesson will be review for the students on the Pythagorean
        theorem. The second part may be difficult for the students. They are asked to
        make a rule (formula) for distance based on the Pythagorean theorem. However,
        many may have difficulty making the connection on their own. Encourage the
        students to work together and only give extra information when it is absolutely

  Anticipatory Step (5min)
       Start out by asking the students for an example situation where it may be
       necessary for someone to find out how far it is from one point to another.
       Encourage answers of any sort (Answers may vary, but many will resemble
       distances with respect to traveling or a map). Next, ask the students how they may
       figure that distance if something lies between them and the other point so that
       they cannot measure it directly. Again encourage all answers. Ask if anyone has
       any mathematical intuition that may solve this problem? (Do not give any
       information about the distance formula) Explain to the students that the goal of
       this lesson is to find a mathematical explanation to solving for distance between
       two points.

   1. (8min) First the students will review the Pythagorean theorem in their groups.
      Give the students the following situation along with transparency one below. If I
      am standing on top of the school, and Javier is looking at me from the street, what
      is the distance between Javier and I. Explain how you found your answer. What
      method did you use?

   Transparency 1


   2. (5min) Ask the class for their answers to the school problem. Try to call on
       enough students so that at least one person from each group offers information.
   3. (1min) Continue talking about the Pythagorean theorem until the following
       information surfaces: 1) The Pythagorean theorem necessitates a right triangle, 2)
       The sum of each leg squared will equal the square of the hypotenuse.
   4. (2min) Explain to the students that the distance formula we seek involves using
       the Pythagorean theorem and two points from the coordinate plane defined by
       (X1, Y1), (X2, Y2).
   5. (8min) In groups, ask the students to find the distance between the points (4, 3)
       and (7, 7). Each group should describe in writing what they did to find the answer.
   6. Walk around encouraging students. Look for a group that understands the
       problem well enough to present it to the class.
   7. (3min) Ask one group to show the rest of the class how they found the answer.
       Let the rest of the class ask the group questions.
   8. (6min) Do one more example, this time on the overhead. Use the points (1,2) and
       (6, 14). Call on students to lead the class through the example. Have the students
       clearly articulate why they are doing each step.
   9. (12min) Another problem to do in groups. Suppose (X1, Y1) & (X2, Y2) are the
       coordinates of two points. Ask the students to generalize what they did in their
       groups before and what we did on the overhead, to create a set of instructions or a
       formula that gives the distance between the two points.
   10. Walk around the class helping and encouraging the groups. Give hints if
   11. (8min) Check back in with the groups. Call on different groups to articulate their
       formula or set of instructions. Continue until we have written the distance

Follow-up Activity
      The follow-up activity builds on today’s lesson. Below is the HW question that
      will be passed out to each student.

              Will your generalizations (the distance formula) that we discovered in
              class still work if any of the points used involve negative numbers? Use
              examples in your answer and explain your examples/answer in detail.

       This HW exercise will motivate the students to look deeper into the mathematical
       concepts behind the distance formula.
Literacy Aspect
       Throughout this lesson the students are asked to work together in their groups.
       This is a tough lesson. Besides finding the distance formula, the students should
       also realize that working together through these examples they can count on each
       other to add their individual knowledge to the learning process. Working in
       groups models life outside of the classroom where people work together in their
       jobs and to work for social change. This lesson also stresses the importance of
       written explanations in the problem solving process. The students will be using
       their writing skills to describe their logic throughout this lesson.

      The mathematical ideas in this lesson will be used throughout the students’ high
      school math experience. It is important while walking around checking the group
      productivity to make sure that the students are working in the right direction
      toward creating the distance formula.

       The assessment aspect of this lesson will be revealed as I walk around the room
       checking the students’ progress. There is a natural progression that this lesson will
       take, from the Pythagorean theorem, to broad distance generalizations to the
       distance formula. The students’ understanding will be assessed in each of these
       steps while they are working in their groups. The telltale sign of understanding
       will come when the students are presenting their generalizations. The HW will
       also reveal the students’ deeper understanding of the mathematics behind today’s
                     Interactive Math Program (IMP)

     IMP is a mathematical reform curriculum program for high school. IMP textbooks
     are centered around larger unit problems with ensuing class work that is designed
     to assist the students in solving the unit problem. IMP is different from tradition
     textbooks in that it is basically doing math backwards. IMP starts out with
     problem situations and then moves the students toward making their own
     generalizations about the problem. In this way students are creating formulas and
     theorems on their own, rather than just memorizing what a book or teacher says.
     In fact, IMP books are basically void of any theorems or formulas.

Positive Aspects
       IMP challenges students to think. Through the unit problems and the POW's
       (problem of the week) the students discover math rather than mimic and
       memorize teachers and formulas. This is important because it allows students to
       better understand mathematical concepts and these concepts will stay with them
       longer. IMP also involves a lot of reading and writing which gives students any
       needed practice in these other literacy skills. IMP also allows for a multicultural
       perspective within a textbook. Many problems and units are designed to include
       events and/or ideas from many different cultures. Additionally, even if the
       curriculum that your district uses in different from IMP, IMP can be a source for
       many different and new problems.

Negative Aspects
       The one major critique of the IMP series is that it has a lot of emphasis on
       reading. Often students do not possess the reading skills necessary to get them
       through IMP books. Teachers often must modify the lessons so that reading does
       not have such a strong emphasis.

How I would use this in my classroom
      If a school does not have the IMP series for curriculum textbooks, there are still
      many ways that one can use them in the classroom. IMP is full of great POW's
      and interesting problems related to every high school math subject (and more). I
      would supplement the curriculum that my school uses with examples from the
      IMP series, and also adopt the strategies in IMP of working math backwards.
                                   Math Forum

     The Math Forum web page is an excellent resource for students, teachers and
     mathematicians. Math Forum, by Drexel University, has many different aspects.
     A math educator may visit the web site to read the current newsletter, to view the
     problem of the week, to participate in the discussion section or even to post a
     question about a math problem. Math Forum has excellent lesson plans posted on
     the site and many different projects to sift through listed by mathematical subject.
     There is even a link to career and job information. Basically, Math Forum is the
     first place a math teacher would turn if he or she needs any sort of mathematical

Positive Aspects
       The site is easy to navigate and the mathematical coverage it provides is immense.
       Some of the real positives are the lesson plans and the project ideas, however, my
       favorite part of math forum is the discussion section. The discussion section
       provides ongoing discussions between whomever feels like joining (usually it is
       math teachers and mathematicians). There are multiple discussions taking place
       about different topics facing math in general today. For instance, there is one
       discussion that has been taking place for months about the importance of an
       increased emphasis on the history of math in current math curriculum.

Negative Aspects
       One aspect of the Math Forum is known as Ask Dr. Math. Here visitors can pose
       questions about math problems, and the folks at Math Forum will answer them for
       you. However, the turn around time for answering the math questions is often
       several days, and from what I have seen is not all that comprehensive.

How I may use it in my classroom
      The strongest way that I feel I can implement the Math Forum into my classroom
      is by using the lesson plan examples and the project examples. There are many,
      many lesson plan examples for every mathematical subject, and every
      mathematical teaching style.
                             Graphing Calculators

     Graphing calculators are calculators with programs built in that allow users to
     graph functions. Most of these calculators allow graphing multiple functions at
     the same time on the same set of axis's. These calculators have built in functions
     that allow students to enter trigonometric functions, exponents, multiple variables
     and many other mathematical programs. These calculators are compatible with
     many overhead projectors allowing easy presentation in front of an entire class.

Positive Aspects
       These calculators allow students to graph many equations and functions that
       would otherwise be tedious done by hand. Students can quickly graph functions
       which allows them to change the functions and compare and contrast the different
       graphs all on the same set of axis's. The many functions that graphing calculators
       have built in (sine, cosine etc.) allow students to do accurate computations
       without having to reference many different charts.

Negative Aspects
       There are two main setbacks to graphing calculators; cost and user friendliness.
       Graphing calculator costs can easily exceed $80 making them out of reach for
       many students. Graphing calculators have so many buttons that it makes it
       difficult to learn how to use. Teachers may have to spend a day or so just to
       familiarize the students with the calculators different functions, even then, the
       teacher must frequently introduce new calculator capabilities to the students.

How would I use this in my classroom
     It is important for students to learn how to graph for themselves, however, it is
     also important not to ask the students to graph by long-hand too often. I will ask
     the students to use graphing calculators as much as possible. Very seldom will the
     students have to make a graph of their own (unless they are being tested on
     graphing) and therefore the use of graphing calculators will allow students to do
     their work more accurately and more timely. I write graphing calculators into
     many lessons and ask the students to use them to compare and contrast many
     different functions.

     This is the website for the National Council for Teachers of Mathematics. The
     website provides comprehensive information about NCTM standards. All the
     standards are listed on this site, and explained in thorough detail. There is also
     much information on current events in mathematics, such as exhibits,
     conferences, and current research findings. A membership to NCTM also
     provides a subscription to the Mathematics Teacher, a scholarly journal about
     current topics and events in mathematics. The most useful information on this
     website is the standards listed from K-12.

Positive Aspects
       The comprehensive information about the standards in K-12 mathematics can
       assist teachers greatly. The standards are not only listed, but explained well too.
       There are also many problem examples that show how to model problems after
       the standards.

Negative Aspects
       The most difficult part about this entire site is actually finding the standards. A
       visitor must navigate at least two pages before finding the actual standards.
       Sometimes the standards are listed in the middle of a page, or at the bottom of a
       page, in this way a visitor must scroll down in order to find the standards. This
       makes finding the standards difficult.

How I may use it in my classroom
      As everyone knows, standard are deeply ingrained in teaching. The NCTM
      standards for mathematics are quite different than the California standards in that
      NCTM standards are much more broad and loosely outlined. In this way it is
      easier to create lesson plans based on NCTM standards. I plan to use the website
      extensively when modeling my lesson plans to make sure that standards are

                          The Geometer's Sketchpad

     The Geometer's Sketchpad is a computer software program that allows students to
     draw and create geometric figures. GS is an exploration tool that assists students
       in exploring math in a hands-on manner. Students are able to compare, contrast,
       measure and manipulate different shapes in order to make general conjectures
       before attempting a formal proof. Geometry is not the only subject that GS is
       capable exploring, students can also use GS across the math curriculum to study
       and better understand algebra, trigonometry and calculus.

Positive Aspects
       One positive aspect of GS is that it allows students to understand and explore
       math in ways not possible with traditional tools such as protractors and rulers.
       Students gain a better understanding of mathematical properties by analyzing
       shapes and problems. GS makes this easy and fast. Students need little training on
       how to use the software, all that is needed is an instructor with some experience.

Negative Aspects
       Although GS can be hooked up to an overhead projector for all to see, it has its
       most influence when it is used directly by the students. This means that in order to
       use GS a classroom must have many available computers or must make frequent
       (if not daily) trips to the computer lab. The fact that often times computers are
       scarce in today's classroom makes GS only available to certain schools.

How I would use this in my classroom
      I would use GS before students are working on proofs. GS is a great tool for
      exploring shapes and then making generalizations about them. In this way,
      students can actually make their own geometric proofs by examining shapes
      closely. The idea is to show the students actual geometric proofs after they have
      created their own proofs. This way students will actually be 'doing' mathematics,
      and they can compare and contrast their own proofs to the proofs of

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