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Dr. David Ashley, [email protected] math.smsu.edu/faculty/ashley.html
Dr. Lynda Plymate, [email protected]
Department of Mathematics
Southwest Missouri State University
Springfield, MO 65804
NCTM’s Assessment Principle Internet Environments “Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.”
Assessment should enhance students' learning.
(e.g. CAS in Algebra and Calculus)
Martha made the pattern shown below on her TI-89 calculator. One line in this pattern has equation y = x - 1. Determine the equations of the other 7 lines in this pattern. Use the numerical limits on the x-axis and y-axis as references for your lines.
The following three graphs describe two cars, A and B. One line in this pattern has equation y = x - 1. Determine the equations of the other 7 lines in this pattern. Use the numerical limits on the x-axis and y-axis as references for your lines.
Complete each of the following tasks.
You and your partner are to use SketchPad and construct a Rhombus using three different techniques. Using a text box next to your rhombus describe how it was constructed. Make measurements to prove it is a Rhombus.
Using your sheet of definitions, investigate whether the rhombus is a square, parallelogram, rectangle, trapezoid, trapezium, or a kite include measurements that support your statements.
Construct the diagonals of the rhombus. What you can say about the diagonals that you can back up with measurements (Are the diagonals congruent? Do the diagonals bisect each other? Are the diagonals perpendicular? Does each diagonal bisect two angles of the quadrilateral?). What is true concerning the four triangles formed from the construction of the diagonals (are the triangles congruent, similar, equal in area)?
Construct the midpoints of all four sides. Construct a quadrilateral formed by the mid points. What type is this quadrilateral? Find the midpoints of each side of this quadrilateral and connect them to form a quadrilateral. Continue this process until you discover a pattern, or the sides of the quadrilaterals become to small to measure. Back up your conjectures with measurements. How does the area of each quadrilateral relate to the original figure you started with?
What can you determine concerning adjacent angles and corresponding angles for the original Rhombus.
How may lines of reflective symmetry exists. How many degrees of rotational symmetry?
What is the sum of the measures of the interior angles?
Explore: Tell me all you can about Rhombuses!!!!
The box plots shown below represent the ratings given to the 257 episodes, in the seven seasons, of Star Trek: The Next Generation (top plot is season 1 and bottom plot is season 7). These ratings, with 1 as the best and 257 as the worst, were determined by Entertainment Weekly magazine personnel. Use the center and spread in these plots to defend your choice of the “best season” for this program.
WORKING WITH EXPONENTSDr. Lynda Plymate
The men from the White House
Grade level: 4th grade
Subjects: Social Studies/ Language Arts/ Math
Materials needed: Internet articles from www.mtrushmore.nethttp://library.thinkquest.org/T0211461/history
The students will understand the importance of Mt. Rushmore.
The students will understand why each of the four Presidents was chosen to be on Mt. Rushmore.
The students will practice using ratios.
The students will practice and improve their writing styles.
Ask the students if anyone knows what Mt. Rushmore is. Wait for their response.
Ask the students if anyone knows who is on Mt. Rushmore. Wait for their response.
Have students speculate why each of those Presidents was chosen. Write their answers on the board.
Explain that we are going to learn about the sculptor, how big they are, how long it took, and why those Presidents were chosen.
Pass out copies of the article from www.mtrushmore.net
Read the article Mt. Rushmore Trivia to the class.
Have discussion on what we just read.
Pass out copies of the article from http://library.thinkquest.org/T0211461/history
Have students who are comfortable and want to read aloud, read the article Mt. Rushmore: Presidents on the Rocks.
Stop at the end of each section and have discussion about what was just read.
to toe, they would be 465 feet.
I found 'Geometry with Quilting' to be a fun project that effectively utilizes geometry
skills in making something practical. This project allows a lot of creativity and is a great way
to learn and apply geometric knowledge for visual/spatial learners such as myself. Before
working on this project, I had no idea how much geometry was used in quilting. The various
quilt patterns out there are amazing and really show ingenious uses of various shapes. I feel
that this project would be a great way to show children in the classroom a practical way to use
geometry outside of the school setting. Not only is the project fun, but it also can be used as a
creative lesson for children who don't necessarily master objectives easily from a textbook
format. It can be used to teach objectives about reflective symmetry, measuring and finding
midpoints, the different types of triangles, angles, and other important geometry skills.
Overall, this project is a great way to enhance the teaching of geometry.
While this project definitely has many advantages in the classroom, there are also
some drawbacks to teaching it. Some children, most likely boys, may not appreciate the
project or lessons because quilting seems to be such a feminine hobby. Second, this project
and a thematic unit may be too time consuming with all of the other objectives required in
today's classroom. To understand the unit and adequately cover the geometry in this project,
more time is required than would be if simply teaching terms and information out of a
textbook. When state standards require many objectives to be mastered in a short school year,
projects like this can put teachers behind in their teaching schedule. This seems to be a
problem in every subject, though, and I feel this lesson is worth the time and should be
incorporated when teaching geometry at the elementary level.
I have thoroughly enjoyed our Math 360 class this semester. The quilt project is a
useful endeavor because it is a "real world" teaching project. It demonstrates how
difficult that teaching effectively can be. With the tremendous pressure placed on
teachers today to fit more and more material in less time, we need more practice doing
interdisciplinary projects such as this one. I have enjoyed the teamwork aspects of this
class, although, frankly, I did not carry my own weight in this project. Sally deserves
most of my points for this project as she did most of the work. I offered help in the
design and some basic suggestions, but she was the star of our show.
The cons to the project were that quilts and learning about them would turn some
children off. I envision very few ten year old boys waking early with Christmas-like
excitement to hurry to get to school to learn about quilts. However, it is a perfect
example of how we must teach everything required, whether we like the material or not.
So, if we have to do it, why not make it fun?! Not all material will appeal to all children
or teachers. Another con with a positive spin is the potential of one student of a group
doing more work than the other. While it is "not fair ...boohoo " ...it is a fact of life that
we must step up to the plate sometimes and carry someone else. In our professional lives
(and our personal lives as well), there are times we will get the short end of a teamwork
arrangement. We have two choices. ..pout or try harder. In our group, Sally tried
harder. It was not that I didn't want to work hard or was bored with the class. I have
three kids (8, 2, 3 months) and a wife to support, and I work fun time.
Although this project took more work than I was anticipating, I think we came up
with a final product that can be useful in a future classroom. I thought the amount of
work was reasonable and it was easy to split between two teammates. I thought the
lesson plans were more effective as far as a future teaching tool, because we came up
with lessons that I would actually use in my classroom. That was the major pro of this
project for me: that I was putting the work into a product that will be useful later on.
That also leads me to the major con of the project, which is that I felt the quilt
making itself was very time consuming and I don't think that quilt will be something we
will use again. Also, I think for this level of learning, we as college students did not
necessarily need to be convinced of the geometric properties of quilts, which seemed to
be the point of making the quilt ourselves. I think that simply designing a quilt block by
using Sketchpad would have been sufficient to prove that we understood how a quilt fits
in with math. However, we as teammates managed the workload so that the quilt making
itself was not overwhelming, and we ended up with a neat design that has a lot of
geometric properties as well as arithmetic properties, so it could be used to demonstrate
how the Fibonacci sequence works at a higher level of education, perhaps.
I thought that this was a unique and interesting project that made me think about
quilts in a new way. I think that it could be modified to be more useful by requiring more
lesson plans, to create a more comprehensive unit, and a less emphasis on the
construction of the quilt itself.
Year anticipating, I think we came up
$0.00THE STORY OF TIM and TOM AT 9%
Tim and Tom are interesting characters. Some people think there is an important moral to their story. Tim and Tom were twins. They both went to work at age 20 with identical jobs, identical salaries, and at the end of each year, they received identical bonuses of $2000. However, they were not identical in all respects.
Early in life, Tim was conservative and was concerned about his future. Each year he invested his $2000 bonus in a savings program earning 9% interest compounded annually. Tim decided at age 30 to have some fun in life and he began spending his $2000 bonuses on vacations in the Bahamas. This continued until he retired at 65 years old.
Tom, on the other hand, believed in his youth that life was too short to be concerned about saving for the future. For ten years, he spent his $2000 bonuses on vacations in the Bahamas. At age 30, he began to realize that some day he might not be able to work and then would need funds to provide for his support. So he began investing his $2000 bonuses in a savings program earning 9% compounded annually. This continued until he was 65 years old.
Although separated for a few years, they were joyfully reunited at age 65 at a family reunion and exchanged many stories of the events in their lives. Eventually the conversation got around to retirement plans and savings programs. Each brother was proud of his savings and showed the other a spreadsheet describing his savings activities and accumulations. But they were amazed! Tom had made many more $2000 deposits than Tim. Yet,Tim had accumulated almost $200,000 more than Tom. How could that be? Using a spreadsheet fill in the chart and answer the questions.
Requirements anticipating, I think we came up: You and your partner are required to design and build a birdhouse from a eight foot by 10 inch by 1 inch piece of exterior grade lumber. You are to supply a set of design prints showing a top view, front view, back view, and side view of your design and how the cuts can be made from the piece of wood. You are to compute the surface area and volume for your birdhouse. You are to construct a scale model (out of a material of your choosing and the scale cannot be 1:1) of your group’s design. Your group will make a presentation (no longer than 3 minutes in length) to the class showing/describing your design, the amount of surface area and the structure’s volume, how you would mount the bird house, how you protect the box from predators and any other neat features of your project.
Your project is to visit three sites on the list and evaluate how well they measure up to the NCTM standards. A grading rubric is provide to assist you that was developed in MTH 479 by Emily McDaris. Some of these sites require you to explore further for a good evaluation. You should check out the site and visit as many links listed as you deem appropriate to get a good feel for what the site has to offer. You are to type a one page doubled spaced 12 pitch font with one inch margins summary for each of the three sites that you choose to visit. Your evaluation should state a score based on the grading rubric along with an explanation for the grade. Each site should be classified as to who you believe could use it (teachers, students or both) and what grade level it is appropriate for. Also state what you believe the general purpose is of the web site (lesson plan research, entertainment, student explorations, demonstrates the usefulness of mathematics), and what discipline does it emphasize. i.e. This site shows a high school student how you can use mathematics in art and architecture.
Once you have evaluated three of the sites listed you are to search the web and find an additional two web sites that you believe would be great sites under the NCTM standards and you would be willing to recommend to your peers. I want you to submit sites you believe are great sites to send a student to when they ask, “When am I ever going to use this stuff?” You are to write a one page summary for each site which addresses why you think this is a worthwhile web site and what grade level and discipline is united with mathematics.
Development: You are develop a 20 to 25 minute lesson involving using the internet and technology to reinforce concepts that have already been learned by Shae Johnson’s 4th grade class. The class will be divided in to two groups of teams, A and B. If you are identified as an A team you will choose a one of the concepts from the A list and you and your partner will develop your lesson from that concept. If you are identified as a B team you will choose one of the concepts from the B list and you and your partner will develop your lesson from that concept. Your lesson should be developed around the technology and should be student centered not teacher centered. It should last about 20 minutes and be detailed enough such that I could follow it and teach your lesson for you.
Actual Teaching: You will have the opportunity to present your lesson to two 4th graders during the 20 – 25 minutes. You are to actively involve them and have your lesson be student centered not teacher centered. Keep you students involved and on task. I know that you will not have much time but try and find out what they understand and do not understand about your concept. At the end of 20 to 25 minutes you will get a new group of two students and you will repeat your lesson.
Interactive JAVA explorations and lesson plans
Power Versus Exponential Functions involving using the internet and technology to reinforce concepts that have already been learned by Shae Johnson’s 4Lynda Plymate
f(x) := x a g(x) := a x
f(x) = x 2g(x) = 2 x
Requires time to reflect and reason
Non-routine problem solving
(time to build a working strategy)
Requires work with technology
6. The length of time it takes to paint the gymnasium changed as the number of people painting increased.
The Boston Marathon is one of the world's best-known foot races. Winning time in the Boston Marathon has decreased as runners get faster. The table shows time of the winning man, in minutes, for years 1959-1980. (Unlike women's winning time, men's winning time has not improved much since 1980, so that data is not given.)
i) Create a scatterplot to demonstrate winning time for the 1959 143
given years 1960 141
ii) Find r and r2. Then discuss what each number tells you about 1962 144
the relationship between men's winning time and year. 1963 139
iii) Find the least-squares regression line; and then carefully plot 1966 137
it on your scatterplot. Be sure to include and name at least 2 1967 136
specific points on your line. 1968 142
iv) By how much on the average did the winning time improve 1970 131
per year during this period? 1971 139
v) Use your regression line to predict the winning time in 1990, 1973 136
a decade later. Is this prediction trustworthy? Explain. 1974 134
vi) Complete a residual plot for your data. 1976 140
A house has been invaded by 100 termites. The population of termites triples every two days. If the population reaches 800,000, the building will be in danger of severe structural damage. The house has also become infested with 2,000 cockroaches. This cockroach population doubles every five days. If the population of cockroaches exceeds 32,000, the house will be condemned. If the exterminator can only address one problem at a time, which is more pressing, the termites or the cockroaches? In your process of answering this question, you must also demonstrate (using pencil and paper manipulations rather than the calculator "solve" feature) the following 4 items:
i) the specific function which models the number of termites present
after t days, call it f(t) ;
ii) the specific function which models the number of cockroaches
present after t days, call it g(t);
iii) the number of days it will take for the termite population to reach 800,000;
iv) the number of days it will take for the cockroach pop. to reach 32,000.
A common test extra-sensory perception (ESP) asks subjects to identify which of four shapes (star, circle, wave, or square) appears on a card unseen by the subject. Consider a test of n=10 cards. If a person does not have ESP and is just guessing, he/she should therefore get 25% right in the long run. So, the proportion of correct responses that the guessing subject would make in the long run would be p = 0.25.
Complete 100 simulated tests in which a subject guesses randomly on each of the 10 cards. In your simulation be sure to decide whether each card was guesses correctly; find the percent of cards guesses correctly by each subject; sort the 100 percents in ascending order; find the mean and standard deviation for your distribution of 100 tests; prepare a table showing the number of tests resulting in sample proportions falling in increments of 10% relevant to your data (like 0-10%, 10-20%, 20-30%, 30-40%, … ); and finally prepare a column graph to display those counts.
i. Turn in enough printed pages to convince me you have successfully generated random numbers, identified whether the subject's responses are correct, sorted sample proportions, found the requested statistics, table and graph.
ii. Referring to your simulated results, what percent of your subjects guessed correctly on 3 or more of their questions (which is better than random guessing)?
iii. How many of the 10 cards would a subject have to identify correctly before you would say that less than 5% of all guessing subjects would do that well or better?
This is a problem for you and your lab partner to work on together. You are to use Sketchpad to solve this problem. You are given a line segment AB that is one of the diagonals of a square. Your job is to construct the square from the line segment you have been given and explain your reasoning. What you must do is open sketchpad and construct a line segments AB and then copy it to use as your basis for the construction. Then with that information construct the square, provide an explanation of what you did, and print your results.
You and your partner given a line segment AB that is the diagonal of a rectangle. You are also given line segment DE that is one of the sides of your rectangle. Your job is to construct the rectangle from the two line segments you have been given and explain your reasoning. What you must do is open Sketchpad and construct a line segments AB and then DE. Copy these segments and use your copies as the working segments. Then with that information construct the rectangle and provide an explanation of what you did. Print your results. must complete 2 problems from 1 through 3 (15 points each) using Sketchpad.
I feel that taking a quiz with a partner together on the computer is a good way to test for acquired skills while promoting teamwork at the same time. It allows the two people that have been working together and discovering new things to put their minds together and be able to solve a problem. However, looking at this issue as a whole, there are some pros and cons for it. Some of the good things that come out of team quizzing are that it relieves some of the quiz anxiety , it allows for a pooling of resources and it promotes teamwork and social skills. By doing this type of quizzing students could possibly be challenged with harder problems than they could be otherwise. Some of the drawbacks are the dependency of one partner on the other and not being able to work together well. It would also be bad if the two people partnered up were both lacking in mathematical skills, because then neither one of those people would have sufficient help from a knowledgeable partner. In general, though, I do feel that it enhances mathematical learning through its teamwork approach in problem solving skills. It lets the two people learn strategies and ideas from each other, instead of just the teacher.
To me, this approach does seem to be fair if it is used on a limited basis. I do not feel that this would be the way to go all the time because there is still a great need to measure individual achievement. A good balance between individual and team quizzes would probably work best because it promotes two different styles which everyone can experience. To sum everything up, this quiz approach is very connected to what we are doing in class. It follows right along with the concepts of discovery and teamwork/self-guided learning.
I really don't care for the quizzes that involve two people on the computer using Sketchpad. I feel it is not a reliable source to base a student's ability .Time is limited and it's a mad rush just to get the problem done. There is really no time for learning by doing. One individual is working the computer while the other is watching, lost or trying to help but unable to get their message across to their partner. The solution is found but does not always have an explanation that works. Learning is occurring but it tends to beconfusing. I feel it could enhance the mathematical learning of an individual taking the quiz but not the learning of a pair taking the quiz together.
I consider the system of allowing two people to take a quiz together on a computer to be acceptable. In life outside of the math classroom, most students will have access to other people and sources of information for concepts they do not understand, so why make the classroom any different? I think working in pairs on quizzes is a good way to enhance mathematical learning, as students can gain a lot of knowledge from their peers. I can honestly say that I've learned more about sketchpad functions from my partner than I would have learned reading through a book on my own. I think every person has their own unique skills to offer, and we learn well when we share our expertise with others. I also believe it is fair for two people to take a quiz together because they both get the same grade. This, in my opinion, is enough to keep both partners motivated to contribute to the quiz. If one person doesn't help, they at least see the other person performing the functions and hopefully learn something from them. I feel that my partner and I have a lot to teach each other. I find it very helpful to have another person contributing ideas on how to solve problems on a quiz, even when we don't have the right answers. The practice of using two people to take a quiz together on a computer is not terrible, and I will most likely use it in my own classroom in the future.