Institute Project Requirements page1 of 20

Institute Project Requirements:

Participants will select their own project topic relevant to their teaching scenario focusing on the Framework's Learning Standards and curriculum goals. (The topic can be at “Chapter” level, at “Section” level, or at higher/lower levels. Participants will create three coherent and synergistic components: Knowledge Maps, Toolkits, and Applications (items 1-3) for the project. In detail, they will:

  1. independently construct Knowledge Maps focusing on the relative connections among both concepts and, if necessary,procedures
  2. independently create Toolkits documenting common stumbling blocks in student learning and their remedies
  3. independently create their own Application problems (and solutions) for the project topics

Deliverquality, not quantity. The goal is to impress math colleagues with a professional report in 3-5 pages. Please also prepare a 3-5 minutes presentation for October. It is usually necessary to spend at least half of your time on the map and use the remaining time to go over the highlights of your project.

Peer review on SaturdayMay 14, 2005

Final Presentation and report due in October, 2005 (Second Follow-up date, TBA)

Please hand in reports as Microsoft WORD files for PCs, either on a floppy disk or as email attachments. Filename: 2005 MMSP 2X FIRST LAST - TITLE.doc, where “FIRST” and “LAST” are your first and last names and TITLE is the name of your project.

Notes: (1) The individual boxes in the Knowledge Map should be numbered and structured so that fundamental ones (with lower designated numbers) are near the bottom and advanced ones are near the top. In this convention, the majority of the arrows should point up. (2) Type the corresponding Frameworks item(s) at the bottom of the page. (3) Number each box for peer review. (4) Use yellow sticky notes to do your first draft. (5) Do a concept map, not a procedure map.

[FIVE SAMPLE REPORTS ENCLOSED]
[SAMPLE REPORT 1]

“Euclid”

MemorialMiddle School

Hull, MA

Mass Frameworks 8G3

“Euclid”

MemorialMiddle School

Hull, MA

Toolkit:

Stumbling blocks:

  1. Understanding Pi
  2. Confusing radius squared with radius times 2

Remedies

  1. Using various size circles students determine the radius and make a radius square. They then cut out more squares of the same dimensions to see how many radius squares it takes to fit into the circle. Students will come to the conclusion that no matter what size the circle is it always takes a little more than three of the radius squares to fit inside the circle. At this time the term pi is introduced.
  2. Drawing squares on grid paper to see the product of the length x width and compare that to a picture of the length times two.

Five squared

5 x 5=25

5 x 2=10

Application Problems

1. At Sal’s Pizzeria a medium pizza (12 inch diameter) costs $6.00 and a large pizza (16 inch diameter) costs $9.00. Which is a better deal? Explain how you know.

2. In Sherwood Forest if you cut down or damage a tree you must replace it with a tree or trees that will make up for the same area as the tree that was lost. A tree with the diameter of 40ft. was destroyed. The only trees available to replace the plant have a diameter of 10ft. How many trees will need to be planted to replace the destroyed tree?

[SAMPLE REPORT 2]

“Pascal”

W.S.ParkerMiddle School

Reading, MA

Explanation of the Box-and-Whisker Plot

Before a student can create a box-and-whisker plot they need to have a good base knowledge of a few main concepts. Students will need to feel comfortable ordering and comparing real numbers. It doesn’t matter if data is sorted in ascending or descending order and many times the use of a graphing utility or software can be quite helpful. Students will also need to know some basic statistical vocabulary such as quartiles and medians. Since the box-and-whisker plot is created primarily using medians, students should know what to do when dealing with both an odd and even amount of data. The mean is a concept used but only when students have to deal with even amounts of data.

Some places where students get hung up besides the obvious carelessness is statistical vocabulary. Many times the 3 M’s – mean, median, and mode, get confusing for our young mathematicians. A good review and some clever memory devices can help them differentiate between these terms. Another area of difficulty is just some basic ordering skills. Deciding which real number is greater than another. If the data is made up of natural numbers, this isn’t a problem but it does present a problem when the data is made up entirely of integers or real numbers.

I find another place where students tend to struggle is with choosing the intervals for the number line placed under the box-and-whisker plot. Again, a graphing utility can help but students need to know how to work with intervals and scales without a calculator before they try the technology route.

The box-and-whisker plot is not a hard plot to use with students but its lack of exposure in real life data and its current exposure on the MCAS make it a topic worth touching upon.

“Pascal”

W.S.ParkerMiddle School

Reading, MA

Knowledge Map for Creating Box-and-Whisker Plot

“Pascal”

W.S.ParkerMiddle School

Reading, MA

Toolkit

Stumbling Blocks/Common Mistakes

1.Losing data when ordering

2.Confusion on Median with even data

3.Confusion on the difference between Quartile 1, Quartile 3, Mean, Median, Mode and Range

Remedies

1.Count data items after ordering or check off data

2.Practice finding the average of 2 items

3.Practice statistical vocabulary

“Pascal”

W.S.ParkerMiddle School

Reading, MA

Application Problems

  1. The following houses are being sold in Methuen on Isoldmyhouse.com. Using the 6 houses below, create a box-and-whisker plot for the data.

$459,900

$389,900

$249,900

$379,000

$303,000

$39,000

.264
.288
.326
.293
.333
.322
.297
.285
.303
.195
.246
.247
.295
.235
.333
  1. The data on the right represents the Red Sox batting averages. Take the averages and create a box-and-whisker plot

“Pascal”

W.S.ParkerMiddle School

Reading, MA

Answers to Applications

Problem #1

Minimum = $39,000

Quartile 1 = $249,900

Median = $341,000

Quartile 3 = $389,900

Maximum = $459,900

Problem #2

Minimum = .195

Quartile 1 = .247

Median = .293

Quartile 3 = .322

Maximum = .333

[SAMPLE REPORT 3]

“Gauss”

MohawkRegionalHigh School

Buckland, MA

Final Report: “Unlocking Linear Equations” (Nov. 2001)

The knowledge map for understanding systems of linear inequalities (sometimes called “linear programming”) needs to cover many basic skills and concepts in elementary algebra. As the map shows, just below actually solving systems of linear inequalities the student must know how to solve systems of linear equations as well as how to graph linear inequalities. The student must know how to change equations into various forms in order to facilitate lining up the x’s and y’s in a system or rearranging the x’s and y’s in order to use one of the various methods in solving systems. Further below on the map the student must know how to solve linear equations in one variable, which means using all the basic skills of first-year algebra.

Taking a look at the right side of the map, just below graphing linear inequalities, the student must know how to graph linear equations. In order to graph lines from equations, the student must know how to graph single points and also know the concept of slope.

In fact, the concept of slope is foundational for understanding the concept of graphing lines from equations and also for understanding the concept of the various forms of equations—standard, slope-intercept, and point-slope. We can therefore see why the concept of slope is in the center of the map, making an important step before proceeding on to further topics.

Below the concept of slope and graphing lines is understanding how to graph single points. This skill requires knowledge of the concept of the Cartesian coordinate plane, the number line, and the real number system. Of course we could go on and on down the line back to when we first start understanding the concept of number and counting, but due to space limitations our map will break at elementary algebra.

As I developed the knowledge map I took careful consideration of Liping Ma’s distinction between “procedural understanding” and “conceptual understanding.”(pp. 22-26) I tried to construct my map on a totally “procedural” basis—what skills are needed in order to do other skills. But as I went through this process, I noticed exactly what Liping Ma brought out in her book: that you cannot completely separate the “procedural” from the “conceptual”—each one supports and connects with the other. “This suggests that in order to have a pedagogically powerful representation for a topic, a teacher should first have a comprehensive understanding of it.” (p. 83) Knowing this can help us as teachers to present subject matter in a balanced way. Not only must we show HOW to do something (procedural) but we must also “know why it makes sense mathematically”(conceptual).(p. 108) In Ma’s book the Chinese teachers showed evidence of their dual involvement of the procedural and the conceptual, and this strengthened their teaching.(p. 109)

The knowledge map also shows connections among the various skills and concepts. This aspect of the interrelatedness of topics—even on an elementary level—is forcefully presented in Ma’s book, and definitely demonstrated by the majority of the Chinese teachers. (p.116)

Knowing how topics connect and having a good conceptual understanding also provide teachers with a strong foundation with which to develop a useful toolkit when it comes to helping students avoid or correct common mistakes and stumbling blocks. Of course, laying a solid foundation at the beginning is the most efficient way to prevent mistakes in the first place—this it seems is already known by many of the Chinese teachers. (p.115)

Although she makes a strong case, Liping Ma seems to over-emphasize some points by giving an over-abundance of examples repeating the same thing—almost as if she was beating a dead horse. The book probably could have been shortened in these areas.

Certainly her demonstration of the relative lack of math skills and understanding by American teachers was crystal clear, and the reasons she gives for this lack of basic understanding brings up some interesting conclusions. She explains there is a fundamental difference in the educational systems of China and the U.S. Because the teachers were never taught concepts properly in U.S. schools, they in turn pass down their watered down understanding to a new crop of students who in turn will pass down their lack of understanding ad infinitum. The lack of enough “preparation time” by U.S. teachers is a big hindrance. I could not believe that the Chinese teachers only actually teach 3 or 4 classes per day, and then have the rest of the time to correct, tutor, talk with colleagues, and read, study and prepare their lessons in their own offices!!! There is a definite cultural difference in the way the teaching profession seems to be respected in China. A strong family life is probably a big contributing factor to this culture of respect for teachers and for learning. While family life in the U.S. appears to be continuing its downward trend, the social problems brought to the schools as a result seem to be increasing. Not only do American teachers teach anymore, they must also be parents, counselors, emotional supporters, and after-school supervisors or baby-sitters to a growing number of “hurting” children who are not taught basic social skills or values at home. In many cases these kids come from broken homes or unsupervised homes where parents are either too busy or just don’t care. The cry is more and more: “Let the schools take care of my kids.” It’s easy to see that the burden for bringing up the children socially and morally has fallen on the schools. Before any useful changes can come about in American education, the society’s culture must change, and the disintegration of family must stop.

“Gauss”

Mohawk Trail Regional

Buckland, MA 01337

KNOWLEDGE MAP FOR SYSTEMS OF LINEAR INEQUALITIES

“Gauss”

Mohawk Trail Regional High

Buckland, MA 01337

Toolkit

Stumbling blocks/common mistakes:

  1. Dealing with negative numbers when solving equations and when making an x/y table of values for graphing.
  2. Confusing x and y in graphing points and lines.
  3. Confusing vertical and horizontal lines and axes.

Remedies:

  1. Practice basic skills to lessen mistakes with negatives.
  2. Emphasize the slope-intercept form (y = mx + b) to help with graphing.
  3. Encourage labeling the axes and writing x and y over the numerical coordinates to lessen confusion.

Application problems (attach separate solution sheets)

  1. A Piano factory produces two kinds of pianos—grands and spinets. Factory can make at most 200 grands and 450 spinets. Cost per unit: $900 for grand, $600 for spinet. Profit per unit: $200 for grand, $125 for spinet. During one month factory has $360,000 to spend. How many of each piano should be made to get maximum profit?

2. A parking lot has space for cars and busses. Total lot is 600 sq. meters. A car needs 6 sq. meters and a bus needs 30 sq. meters of space. The parking attendant can handle no more than 60 vehicles. Each car is charged $2.50, each bus $7.50. How many of each vehicle should be parked to maximize income?

[SAMPLE REPORT 4]

“Laplace”

Brookline High

July 24, 2002

The Meaning of Slope

Part 1: Concept Map

Slope
Parallel
Rate of Change
Steepness
Perpendicular
Lines
Vertical
Horizontal
Exponential
Quadratic
Logarithmic
Rise/Run
Pos/Neg
Other Relations


Part 2: Toolbox

There are a variety of issues that arise when introducing the concept of the slope of a line to the class. I have developed a strategy that I believe addresses these issues effectively. Specifically, I use the students’ understanding of patterns to connect to the concept of slope. I begin by giving students a variety of sequences of towers that contain a different number of blocks such that the relationship between the tower number and the number of blocks in that tower is linear. Then we explore these patterns.

Issue #1: Students have difficulties in comprehending slope as rate of change.

As we explore the aforementioned patterns of towers, we discover that the difference in height of consecutive towers is constant. We call that the rate of change of the heights of the towers. Later on, I will tell them that another word for this rate of change is slope. As we move from towers of blocks to points on a line in the coordinate plane and are asked to find the slope of that line, I always go back to the idea of the tower sequences and remind the students that we are asking for the rate of change between successive towers (points).

Issue #2: Students never really understand the formula to find the slope of a line and, therefore, do not use it correctly.

During the exploration of the tower sequences I ask my students to find the rate of change of a sequence of towers in which they are given only the height of two non-consecutive towers. They will find the change in height between the towers and then find out how many “jumps” there are between the two tower numbers and divide the two to find the rate of change. Then, I give the students the same problem but rather than show the towers, I represent them in a coordinate plane. That is, I show two points in a coordinate plane and tell the students that the first coordinate is the tower “number” and the second coordinate is the height of that tower. Then we find the rate of change. At that point we can connect the two points with a line and talk about the connection between the rate of change and the slope of that line. Finally, as we abstract the problem completely, I will tell them that another term for the change in height between any two towers is the “rise” and the number of jumps between tower numbers is the run and, therefore, the slope of the line (rate of change) is found by dividing the rise by the run.

Issue #3: Students have difficulty identifying the sign of the slope of a line.

This issue is addressed very well using the tower sequences. If the heights of the towers are increasing the rate of change, and therefore, the slope, is positive. If the heights of the towers are decreasing the rate of change is negative.

Issue #4: Students don’t remember the slopes of horizontal and vertical lines.

We look at a sequence of towers that all have the same height. Since there is no change in the heights of the towers the rate of change is zero. This exploration gives an image of a horizontal line connected to a slope of zero. The “other one”, that is, the vertical line, will therefore, have an undefined slope. Furthermore, we explore a tower sequence in which there is one tower with lots of different heights. Since this has no meaning we say that there is no rate of change.

Part 3: Problems and Solutions

  1. In a sequence of towers in which the rate of change of the height consecutive towers is constant, the 3rd tower has a height of 13 and the 8th tower has a height of -2. What is the rate of change of the height of the towers?
  1. One way to consider the slope of a curve at a point is to calculate the slope of the line between the given point and another point near it.
  1. Find the slope of the curve given by the function f(x) = x2 between (3, 9) and the point (x1 , y1 ) which lies on the curve if x1 = :
  2. 4
  3. 7/2
  4. 13/4
  1. Describe the pattern you see in part a.
  2. Find the slope of the curve given by the function f(x) = x2 between (3, 9) and the point (x1 , y1 ) which lies on the curve if x1 = 3 + k.
  3. What happens to the slope of the line found in part c. if k is really close to 0.
  4. Find the slope of the curve given by the function f(x) = x2 between (n, n2) and the point (x1 , y1 ) which lies on the curve if x1 = n + k.
  5. What happens to the slope of the line found in part e. if k is really close to 0?
  1. For each function below, find the slope of the line found between (n, f(n)) and (n+k, f(n+k)).

Then describe what would happen to the slope when k is really close to 0.