Stumbling blocks/common mistakes: Dealing with negative ...
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 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
Deliver quality, 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 Peer Review. 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 Tuesday February 10, 2005
Final Presentation and report due on the second Follow-up Date (TBA), 2005
Please hand in reports as ONE Microsoft WORD files for PCs, either on a floppy disk or as an email attachment. Filename: First_Last-Title.doc, where “First” and “Last” are your first and last names; “Title” is the project title.
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 (related to the top box) 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 concept map, not procedure map.
[FIVE SAMPLE REPORTS ENCLOSED]
[SAMPLE REPORT 1]
“Euclid”
Memorial Middle School
Hull, MA
“Euclid”
Memorial Middle 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. Parker Middle 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. Parker Middle School
Reading, MA
Knowledge Map for Creating Box-and-Whisker Plot
“Pascal”
W.S. Parker Middle 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. Parker Middle School
Reading, MA
Application Problems
1. The following houses are being sold in Methuen on . 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 |
2. The data on the right represents the Red Sox batting averages. Take the averages and create a box-and-whisker plot
“Pascal”
W.S. Parker Middle 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 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?
2. 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.
a. 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 = :
i. 4
ii. 7/2
iii. 13/4
b. Describe the pattern you see in part a.
c. 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.
d. What happens to the slope of the line found in part c. if k is really close to 0.
e. 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.
f. What happens to the slope of the line found in part e. if k is really close to 0?
3. 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.
a. f(x) = ax2
b. f(x) = ax2 + bx
c. f(x) = ax2 + bx + c
Answers:
1. –3
2. a. i. 7
ii. 6.5
iii. 6.25
b. The slopes are approaching 6. Other answers possible.
c. 6 + k
d. The slope is really close to 6.
e. 2n + k
f. The slope is really close to 2n.
3. a. a(2n + k). If k is really close to 0, the slope is close to 2an.
b. 2an + ak + b. If k is really close to 0, the slope is close to 2an + b.
c. Same as b.
[SAMPLE REPORT 5]
“Noether”
Sacred Heart Intermediate School
Kingston, MA
Solving Systems of Linear Inequalities
This topic is one of my favorites. For the past four years I have taught Algebra I Honors. This is a great course to teach at Sacred Heart Intermediate School. It not only has a great deal of subject matter, but the age of the students lends to making this a fast paced, mathematics challenging course with great advancements possible for the student. This topic combines so many of the previous topics that it is a great end of the year topic to show the students how things tie together. The downside of this is that at the end of the year the students can be tired, ready for a break and feeling that they have done enough. When I bring out these problems, I get a lot of “When will I ever use this..”, “ Who cares about…..”, or “What kind of a job would you ever do this in…”. So, my challenge to my self was to develop a series of problems that if not useful in their lives than would at least be relevant to them. I took this from the very simplistic to the somewhat more sophisticated, to try and rope them in gently at the end of the year.
PROBLEM #1:
Mike would listen to CDs or watch DVDs 24-7 if he were allowed. But, even if he could forget about school, eating, sleeping or parents and their annoying desires to interrupt his free time with demands, there are still only 24 hours in the day. (So even if you watch DVDs all day, how many can you watch at three hours per DVD??) Write two statements using the inequality relationship and the knowledge that the average CD runs 80 minutes and the average DVD three hours that express the limits of time on the number of CDs(x) and the number of DVDs(y) that Mike can listen/watch in one day.
(Reminder, change hours to minutes to be consistent)
Graph these two statements on a coordinate system. What shape do you get? Label the corner points. Discuss the meaning of each of the corner points.
Even Mike is in the real world sometimes. He realizes that there have to be some limits on Nirvana. Although he wishes he could live this life, he realizes that he will have to restrict CDs and DVDs to “free” time. He thinks that on a school day this has to be 9(!!!!) hours. Write the inequality that best describes this limitation.
“Noether”
Sacred Heart Intermediate School
Kingston, MA
Why?????
Now, take this new system, graph it on a coordinate plane, and tell what shape you now have, list the corner points and explain their meanings.
Mike’s parents enter the scene. Totally unreasonable, they see some different limits on Mike’s free time. On a school day they see the total time to be used for CDs and DVDs as three hours or less. On a weekend they would up this to no more than 6. Using one of their restrictions at a time (first, school day, then, weekend) write the inequalities that best describe each scenario.
Finally, in the “real” world money is an issue. Each CD costs $15. Each DVD costs $20., Go back and describe what the cost would be at each corner point.
This topic has just given me the chance to bring together the coordinate plane, meaning of ordered pairs, horizontal and vertical lines, writing inequality statements from a word problem, algebraic manipulations, slope-intercept form, graphing systems of inequalities to create a region, and finding corner points of that region. The last step, finding the cost at corner points, gives me a chance to talk a little about the future math that they will take and max and min. The downside to all of this is that in reality the toolkit is enormous, if you really write down each skill they need and the common errors are also numerous.
The problems that books give on this topic may sometimes add to the outcry that these problems are of no use in the real world. My final goal here would be to develop a couple more problems in which my 8th graders could find some reality and reasonableness.
“Noether”
Sacred Heart Intermediate School
Kingston, MA
Eighth Grade: Algebra I, Honors Level
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Solving Systems of Linear Inequalities
Common Errors
Algebraic Manipulations
Sign of slope, y-intercept
Shading in inequalities
Identifying unknowns
Meaning of x-,y- axis
Toolkit:
Graphing
Slope-Intercept form
Meaning of inequality Horiz. & Vert. Lines
Definition of slope, y-interecept
Coordinate plane
Writing an equation in 2 unknowns
Writing an ordered pair
Putting an
Equation in
y- form
Slope-Intercept
Form of an Equation
Definition of
Slope
Horizontal and
Vertical lines
Graphing Equalities
Y = mx + b
Graphing Inequalities
Y < mx +b
Finding region of solutions
Quadrant location of the ordered pair
Understanding the meaning in the problem of the ordered pair.
Identifying the meaning of the unknowns and the information given in the problems
Translating word problems into linear inequalities
Think! Positive or Negative
Divide
Graph Points
Number of “jumps” between points
Change in height between points
Calculation
Addition
Multiplication Facts
Whole number and decimal place value
Vocabulary: radius, pi, diameter, circle
Division
Multiplication
Understanding of
Area
Exponents
Area of a square
Understanding of the relationship between the areas in the radius square to the area of the circle
Relate understanding of pi and the relationship between the radius square and the area inside the circle to create formula a=πr squared
Finding the Area of a Circle
Quartiles
Reasonability
Intervals & Scales
Dividing
Real #s
Median
Mean
Adding
Real #s
Rounding
Ordering Ascending or Descending
& Comparing
Real #s ()
Odd data
Even data
Creating a
Box-and-Whisker Plot
[pic]
[pic]
$0 $100 $200 $300 $400
Methuen Housing Costs ($1000’s)
On
.100 .150 .200 .250 .300
Red Sox Batting Averages up to July 2003
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