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Mathematics Curriculum Guide By 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 Summary 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. Significance 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 Summary 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 successfully. 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 Summary 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 problem. 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. Significance 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 Summary 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 information. Significance 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 Summary 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. Significance 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 topic. http://www.highland.madison.k12.il.us/jbasden/lessons/pi_3_14159265358.html Summary 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. Adaptations 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 http://www.taskstream.com/Main/main_frame.asp Summary 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. Adaptations 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 class. Leap Year Day http://www.lessonplanspage.com/MathScienceCIOLeapYearDayWebsite48.htm Summary 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. Adaptations 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 http://math.rice.edu/~lanius/pro/rich.html Summary 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. Adaptations 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 beneficial. Pythagorean Theorem http://lesson.taskstream.com/lessonbuilder/lesson_builder/published_lesson.asp?q yz=xvFFYBMgEKhekXDWoDE&cybrary=1&viewing=aqznzfzwhvcyca Summary 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. Adaptations 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 Objective Using tree-diagrams, students will be able to accurately predict outcomes of situations that involve probability. Materials Overhead Projector Nerf Hoop and Nerf Basketball 12 Calculators Transparency Explaining One and One 2 Blank Transparencies Notes 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) Standards NCTM: Applying mathematical concepts to a real-world situation to draw inferences. 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 Steps 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 Make Shot 1 Miss 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 2pts 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 .6=36%. 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. Reflection/Assessment 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 Objective Students will become familiar with different types of graphs. Students will demonstrate this familiarity by examining different graphs and constructing their own graphs. Materials 12 Calculators 48 Sheets Graphing Paper 24 Rulers Overhead Projector 3 Prepared Transparencies Notes 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. Standards 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 Steps 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 graph. 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 independent. 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. Reflection/Assessment 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.) Objective 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 property. Materials Overhead and Transparencies (at least 4) Prepared Game Show Transparency Algebra Tiles (Instructor and all 6 Student Groups) 12 Calculators Notes 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. Standards 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 Steps 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 tiles. 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(2X)=2X2 X 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 tiles. 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 show? 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. Reflection/Assessment 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 board. 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 Objective: 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. Materials: Overhead projector Prepared Transparencies Calculators Notes: 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. Standards 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. Steps: 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 phrases) 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. Response/Assessment 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 concepts. How Far is That? Objective Using what they already know about the Pythagorean theorem, students will develop the distance formula for finding distance between two points. Materials Graphing Paper (48 sheets) Rulers (12) Calculators (12) Blank Transparencies (2) Prepared Transparency (School) Overhead Projector Standards NCTM: Applying mathematical concepts to a real-world situation to draw inferences. NCTM: Communicating mathematical ideas in written and oral formats. NCTM: Ability to reflect on the process of mathematical problem solving Notes 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 necessary. Steps 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 100ft 70ft 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 necessary. 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 formula. 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. HW 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. Response/Assessment 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 lesson. Interactive Math Program (IMP) http://www.mathimp.org/ Summary 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 http://www.mathforum.org/ Summary 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 information. 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 http://www.ti.com/ Summary 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. NCTM http://www.nctm.org/ Summary 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 included. The Geometer's Sketchpad http://www.keypress.com/sketchpad/ Summary 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 mathematicians.

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