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Mathematics TEKS Refinement 2006 – 9-12 Tarleton State University

Handout 1 Spaghetti Regression 3-6

Spaghetti Regression

Overview

Participants will investigate the concept of the “goodness-of-fit” and its significance in determining the regression line or best-fit line for the data.

Learning Objectives This activity supports Teacher Content Knowledge needed for A2D: The student is expected to collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations.

Background

Fitting the graph of an equation to a data set is covered in all mathematics courses from Algebra I to Calculus and beyond. The objective of this module is to explore the concept in-depth to provide understanding beyond that in ordinary secondary texts.

To enrich the study of functions, the TEKS call for the inclusion of problem situations which illustrate how mathematics can model aspects of the world. In real life, functions arise from data gathered through observations or experiments. This data rarely falls neatly into a straight line or along a curve. There is variability in real data and it is up to the student to find the function that best 'fits' the data. Regression, in its many facets, is probably the most widely use statistical methodology in existence. It is the basis of almost all modeling.

This activity supports teacher knowledge underlying TEKS A.2.D, wherein students create scatterplots to develop an understanding of the relationships of bivariate data; this includes studying correlations and creating models from which they will predict and make critical judgments. As always, it is beneficial for students to generate their own data. This gives them ownership of the data and gives them insight into the process of collecting reliable data. Teachers should naturally encourage the students to discuss important concepts such as goodness-of fit. Using the graphing calculator facilitates this understanding. Students will be curious about how the linear functions are created, and teachers should help students develop this understanding.


Transparency 1/Handout 2 Spaghetti Regression 3-7

Scatterplot



Activity 1 Goodness-of-Fit Objective: To Investigate the concept of goodness of fit and develop an understanding of residuals in determining a line of best-fit. 1. Examine the plot provided and visually determine a line of best-fit (or trend line)

using a piece of spaghetti. Tape your spaghetti line onto your graph. 2. Now let us investigate the “goodness” of the fit. Use a second piece of spaghetti to

measure the distance from the first point to the line. Break off this piece to represent that distance. Each person at the table must measure in the same way, so discuss the method you will use before starting. Repeat this for each point.

3. Line up your spaghetti distances to determine who in your group has the closest fit.

Determine the total error; i.e., total distance from your line to the data. Then replace the segments and tape them to your scatterplot.

Total error = _______



Measuring There are at least three ways to measure the space between a point and the line: vertically in the y direction, horizontally in the x direction, and the shortest distance from a point to the line (on a perpendicular to the line.) In regression, we usually choose to measure the space vertically. These distances are known as residuals. • Why would you want to measure this way? What do the residuals represent in relation to our function? Consider the purpose of the line and the following diagram. The purpose of regression is to find a function that can model a data set. The function is then used to predict the y values (or outputs, f(x) ) for any given input x. So, the vertical distance represents how far off the prediction is from the actual data point (i.e., the “error” in each prediction.) Residuals are calculated by subtracting the model’s predicted values, f(xi), from the observed values, yi.

Residual = )( ii xfy −


Handout 5-1 Spaghetti Regression 3-17

Activity 2 Objective: Investigate various methods of regression. Whose model makes the best predictions? Let us compare everyone’s lines using the residuals. Before we begin, we need to know the equation for your spaghetti function, f(x) = mx + b. Assume the lower left corner of the graph is (0,0). f(x) = __________________ 1. Enter your function at Y1= in the calculator. 2. Enter the actual data into L1 and L2. Put the x-values in L1 and the y-values in L2.

Make certain that the x’s are typed in correspondence to the y’s.

x 2 5 6 10 12 15 16 20 20 y 14 19 9 21 7 21 18 10 22

3. Place the predicted values, f(xi), created by your function, in L3. To do this, place

your cursor on L3 and enter your function, using L1 as the inputs of the function. (See below.)

FYI: Y1 can be found under [vars] → [Y-vars] → [1:function] → 1:Y1



4. Compute the residuals (the distances between the predicted values, f(xi) , and actual y values) and place them in L4. This can be done by entering L4 = L2-L3.

5. On your home screen compute Sum(L4). Record your group’s functions and the

corresponding sums.

FYI: Sum can be found under [2nd][stat] → [math] → 5:sum

Function Sum of the residual errors

• Examine your values in L4. What is the meaning of a negative residual in terms of the graph and in terms of the function’s predictions? What is the meaning of a positive or negative total for the functions in #5?



Examine the following student’s work.

• In L4 what is the meaning of 39.23? What is the corresponding value in your table? Describe its meaning.

• What is the meaning of a low total residual error? Is it a good measure of fit? Why or why not?



There are two possible ways to fix the above problem. One way is to take the absolute value of the residual; the other is to square the residual. Taking the absolute value of the residuals is synonymous with using our spaghetti segments to measure the vertical error. 6. Find Sum(abs(L4)). Record your group’s functions and the corresponding sums.

FYI: abs can be found under [2nd][0]

Function Sum of the errorresidual

• Compare with those in the class to determine who now has the lowest total error.

Note: The calculator’s regression method uses the squared residuals when measuring the goodness-of-fit of a regression line. Let us compare our lines of best-fit, using the squared residuals. 7. Find the total of the squared residuals by Sum((L4)2) . This is often referred to as

the Sum of the Squared Errors, noted SSE.

Function SSE



• Compare with those in the class to determine who has the lowest sum of the squared errors. Did the best line in the group change? Why or why not?

Let us compare our lines against the calculator’s regression line. 8. Use your calculator to compute the linear regression function, f(x) = mx + b.

f(x) = ___________________

9. Enter the function into Y1 and place the function’s predicted values f(xi) in L3, i.e., L3 =

Y1(L1). 10. Quickly, compute the sum of squared errors by using SUM((L2- L3)2). SSE = ________

• How do the functions in the class compare to this one?

11. Create a scatterplot and graph your group’s functions and the calculator’s regression

function. Examine visually the goodness of fit of each in regard to their SSE. At least two methods exist for evaluating goodness of fit: taking the absolute value of the residuals and squaring the residuals. Although taking the absolute value seems most intuitive, relying on squaring does several things. The most desirable one is that it simplifies the mathematics needed to guarantee the “best” line. (See the appendix.) In Activity 3, you can investigate how squaring the residuals when measuring our goodness-of-fit affects the choice of the regression line. Understanding what you are looking for is always the toughest part of any problem, so the hard part is done. You now know how to measure “goodness” of fit. We can also say exactly what the calculator means by the line of best-fit. If we compute the residuals (i.e., the error in the y direction), square each one, and add up the squares, we say the line of best-fit is the line for which that sum is the least. Since it is a sum of squares, the method is called the Method of Least Squares! This is the most commonly used method but, as we have seen, it isn’t the only way!



Activity 3 Absolute Value vs. Squaring OBJECTIVE: It is important to understand the effect squaring has on the residuals and the placement of a regression line. In this activity, we will use an interactive java applet to investigate several data sets and contrast geometrically and numerically the effect of using the square of the residuals vs. the absolute value of the residuals. 1. Place three points forming a triangle on the graph. Select “plot line” and place a

trend line on the graph. 2. Select “Draw residuals.” Using the handle points, adjust your line to visually

minimize the length of the residuals.

Select “Show Trend Line Equation.” ____________________ 3. Select “Draw (residuals)2.” Using the handle points, adjust your line to visually

minimize the area of the squares.

Equation of the line: ____________________ 4. Now select “Sum of the residuals” and adjust your line to numerically minimize the

|residuals|. Record the equation and total: ___________________ 5. Now select “Sum of the (residuals)2” and adjust your line to numerically minimize the

(residuals)2. Record the equation and total:_________________ 6. Create a situation where the sum of the squares is less than the sum of the absolute

value. 7. Create a data set in which the least absolute value and least squares methods agree

on the line of best fit. 8. Place the following ordered pairs (4, 1), (4, 4), (-4, 0), and (-4, -3) in the table. Find

the line of best fit for each method. • Compare and contrast these two methods.



• How does squaring the residuals affect how individual data points contribute to the total error? Does squaring increase or decrease the effect of an individual residual on the total error? • What is the effect of an outlier point on each of the possible trend lines for each method? Further investigation Another method for finding regression lines is Chebyshev’s Best-Fit Line Method, also known as the MinMax Method, which finds the line with the minimum maximum residual. Chebyshev’s evaluates each line based on its largest residual and takes the

line with the smallest (largest residual ) as the regression line. • Use Chebyshev’s method in the previous graphs to determine a line of best fit. How does it compare to the least absolute value and least squares methods? How it is affected by outliers?



Supplemental Material

Two ways to minimize the sum of the squared residuals

The key to solving this or any problem is understanding exactly for what you are looking. Our model, or line of “best fit”, bmxxf +=)( , will be one that minimizes the sum of the squares of the vertical distances between the actual points and the predicted ones, i.e., the residuals = )( ii xfy − . It can be written

2))(( xfyL −= ∑ or 2))((∑ +−= bmxyL .

What kind of equation is 2))((∑ +−= bmxyL ? That’s right, quadratic. And we actually know enough about quadratics from Algebra II to solve this problem. But, one of the key words in the above paragraph is minimize, which should also make you think Calculus! This gives us an easy alternative approach.

Let us examine this quadratic more closely.

2))((∑ +−= bmxyL

)222( 2222 ybymyxbbmxxm +−−++=∑

It may look daunting, but remember, m and b are the only unknowns here. x and y are just numbers supplied by each of the actual points in our scatterplot.

Expanding L farther, ∑ ∑ ∑ ∑ ∑+−−++= 2222 222 yybxymnbxbmxmL

(You might want to double check all this! Why let someone else have all the fun?)

Remember that the summations are just constants! So now we have a choice to use calculus to find its minimum or use Algebra II to find its vertex.

Let’ try the Calculus!

In calculus, the minimum occurs here where the derivative is equal to zero. Since we have two variables, m and b, we will want to take the derivative of each variable separately. (These are called partial derivatives.)

∑ ∑ ∑ =−+=∂∂ 0222 2 xyxbxmmL



∑ ∑ =−+=∂∂ 0222 ynbxm

bL

All that’s left is to solve this system of equations by elimination or substitution. Take your pick. Using substitution, b in the second equation looks easiest to solve for. So, we get

nxmy

b ∑∑ −= . Substituting for b into the first equation and simplifying, we get

22 )(∑∑∑ ∑∑

−

+=

xxnyxxyn

m .

And that’s it. Your calculator or computer just sums the x’s, the y’s, the xy’s, etc. and out pops the slope and y-intercept of your regression equation. It is not hard, but certainly tedious when done by hand. (You may wonder how we know it is a minimum and not a maximum. The second derivative is 2; a positive second derivative means it must be a minimum.) Let us try it with Algebra!

Here we go. Remember that we want to find the minimum of

∑ ∑ ∑ ∑ ∑+−−++= 2222 222 yybxymnbxbmxmL

and that all of those summations are just constants. Thus, L is a quadratic with respect to m or b. This can be seen easily by rearranging.

=)(mL ∑ )( 2x m2 + )22( ∑∑ − xyxb m )2( 22 nbyyb ++− ∑ ∑

L(b)= n b2+ )22( ∑∑ − yxm b )2( 222 ∑∑∑ −−+ yxymxm



Do they open up or down? The leading coefficients, ∑ 2x and n, are both positive, so the answer is up.

From Algebra II, we know the vertex of Ax2 + Bx + C occurs at AB

2− .

So m = ∑

∑∑ −−22

)22(x

xyxb =

∑∑∑ −

2xxbxy

, and

b = n

yxm2

)22( ∑∑ −− = n

xmy ∑∑ − .

Substituting one into the other, we get 22 )(∑∑∑ ∑∑

−

+=

xxnyxxyn

m and

b = 222

)(∑∑∑∑∑∑

+

−

xxnxyxxy

. This is exactly the same result as before.

Some Historical Notes

Who invented the method of least squares? It is not clear. Often credit is given to Karl Friedrich Gauss (1777–1855), who was first published on this subject in 1809. But the Frenchman Adrien Marie Legendre (1752–1833) published a clear example of the method four years earlier. Legendre was in charge of setting up the new metric system of measurement, and the meter was to be one ten-millionth of the distance from the North Pole through Paris to the Equator. Surveyors had measured portions of the arc but to get the best measurement for the whole arc, Legendre developed the method of least squares. He would probably use GPS today, but he was still amazingly accurate.

Where does the term "regression” come from? The term was first used by Sir Francis Galton (1822-1911) in his hereditary studies. He wanted to predict the heights of sons from their father’s heights. He learned that a tall father tended to have sons shorter than himself, and a short father tended to have sons taller than himself. The heights of sons thus regressed towards the mean height of the population over several generations. The term "regression” is now used for many types of prediction problems, and does not merely apply to regression towards the mean.


Handout 1-1 Understanding Correlation Properties with a Visual Model 3-42

ACTIVITY 1 This module opens with an explanation of the way that paired measurements can be plotted in two-dimensional space. Next, positive and negative relationships are discussed and participants are asked to predict values using a regression equation. It concludes with a discussion of outliers. PART A Consider the following. At approximately 6:45 a.m., Tuesday morning, Principal Espinoza saw something strange as he opened the backdoor to B. Wyatt High School. As he entered the hallway, he immediately discovered the broken glass from the classroom door. It was a 9th grade Math classroom. The computers were missing, the desks were overturned, and the prized school banner was torn from the wall. The perpetrators were long gone, but they had left something behind. Next to the desk, where Mrs. Joe’s computer once sat, was the imprint of a forearm on the board. When the police arrived, they immediately began to gather forensic evidence. Mr. Espinosa, knowing your love of CSI and Numb3rs, asks you to help gather data to help identify the bandits.

Bones of the arm can reveal interesting facts about an individual. But can they reveal a person's height? Forensic anthropologists team up with law enforcers to help solve crimes. Let us combine math with forensics to see how.

Collect data for 8 people.

Person Forearm Length (inches)

Height (inches)



1. From the table, describe any relationships you see between the variables forearm length and height.

Making a scatter plot can provide a useful summary of a set of bivariate data (two variables). It gives a good visual picture of the relationship between the two variables and aids in the interpretation of the correlation coefficient and regression model. The scatterplot should always be drawn before working out a linear correlation coefficient or fitting a regression line. A positive association is indicated on a scatterplot by an upward trend (positive slope), where larger x-values correspond to larger y-values and smaller x-values correspond to smaller y-values. A negative association would be indicated by the opposite effect (negative slope) where the higher x-values would correspond to lower y-values. Or, there might not be any notable linear association. 2. We will use the web applet Correlation for further investigation in the following

exercises. Enter the forearm length and height data into the table and examine the scatterplot.

In 1896, Karl Pearson gave the formula for calculating the correlation coefficient known as r. (To see it, select show equation for r.) He argued that it was the best indicator of linear relationships. It derives its name from linear, meaning “straight line,” and co-relation meaning to "go together." The drudgery of computing the correlation coefficient by hand is quite ominous. However, today’s calculators can easily compute r. It is often referred to as the Pearson Product Moment Correlation Coefficient. We can generally categorize the strength of correlation as follows:

• Strong |r| > 0.8

• Moderate: 0.5< |r |



If variables are strongly correlated, we often use one to predict the other. A gross example from forensic science is using the size and larva stage of maggots to predict time of death. Linear regression is the method used to create these mathematical prediction models. Given X, we can predict Y. If the correlation is high enough, record the function for the regression line.

3. Using the information you collected, try predicting the height of our assailant for Mr. Espinosa. A copy of the police imprint from our assailant is attached.

● What would increase your confidence in this prediction?

In real life, mathematics always begins with a question. What do you want to know? This is followed by data collection. If it is bivariate data, scatterplots are drawn to give the “big picture.” If the relationship looks linear, the correlation coefficient is calculated to quantify the relationship. If the r value is reasonable, a linear function can be found that is used to predict what has not been observed; in our case, the height of the assailant.



PART B – A Closer Look

Now let us look more closely at how we measure the strength of associations between data sets. The correlation coefficient can range from -1 to 1. ( ± 1 being a perfect linear correlation between the two variables.) If the variables are completely independent, the correlation is 0. However, the converse is not true since the correlation coefficient detects only linear dependencies between two variables. Let us investigate changes in our data set.

1. Click and drag one point of your scatterplot until the correlation is 0.3. Record the coordinates.

● Is the placement of this point unique? ● What does the new point represent in terms of the context? An outlier is an observation that lies an abnormal distance from other values in a sample. In a sense, this definition leaves it up to you, the analyst, to decide what will be considered abnormal. Before abnormal observations can be singled out, it is necessary to characterize normal observations. If the data point is in error, it should be corrected if possible. If there is no reason to believe that the outlying point is in error, it should not be deleted without careful consideration. ● Would you consider your point an outlier? Why? 2. Suppose a “mistake” was made. All the forearm sizes were reported in centimeters

(1 in. = 2.54 cm.), and all the heights were recorded in inches. A student tells you that the correlation will be too low saying that increasing the forearm data by a factor greater than 1 will spread the points in a graph. Do you agree with the student? How would you explore this issue?



● What do you suppose would happen to our correlation value if we changed to different height scale?

3. Delete the outlying point from your table. Now, add two additional points to make a

correlation of 0.99. Discuss the placement of your points.

The geocenter, also called the center of mass or centroid is the “average” point of the data. If we have the points (x1,y1), (x2,y2) (x3,y3), and (x4,y4) then the coordinates of the geocenter would be ⎟

⎠⎞

⎜⎝⎛ ++++++

44321,

44321 yyyyxxxx . The further a point is from the

geocenter of the data the more “leverage” it has. (Note: The regression line always passes through this point.)

Students often have a naïve sense of correlation. We should look to extend their understandings. Dynamic applications such as Geometers Sketch Pad and web applets open up new avenues for exploration and deeper understandings. By allowing students to explore and test their own conjectures, they take ownership of their mathematical understandings.



ACTIVITY 2

H.G Wells once said,”Statistical thinking will one day be as necessary as the ability to read and write.”

The goal of this activity is to gain an intuitive understanding of r. Using the web applet Correlation, scatterplots can easily be constructed. The dynamic nature of the applet allows you to see how the correlation changes when a data point is added or moved. PART A 1. Clear your table and place two points on the graph. Note the correlation. ● Would any two points have the same value? Explain. A student remarks that “when r is undefined, it means there is no linear model for the data.” Do you agree? How would you explore/explain this? 2. Make a lower left to upper right pattern of 10 points with a correlation of 0.7. 3. Make a vertical stack of 9 data points on the left side of the window. Add a 10th point

somewhere to the right and drag it until the correlation again reaches 0.7. Is its placement unique?

4. Make another scatter plot with 10 data points in a curved pattern that starts at the

lower left, rises to the right, then falls again at the far right. Adjust the points until you have a smooth curve with a correlation close to 0.7.

● Does any other curved pattern have this same correlation? ● What can you conclude about the numerical value of a correlation?



Juan and Kaylee have collected data for a class experiment. They correctly found an r-value of .15. Juan claims that no function will model the data. But, Kaylee says that the r-value is wrong because she has found a good one. Is this possible? How would you help these students? 5. Make a data plot with a correlation of 0 by placing 8 to 10 points on the graph. 6. Enter 4 points in the table to make a perfect rectangle. Note the correlation value. 7. Create several other data sets with a horizontal or vertical line of symmetry and note

the correlation value.

Let us take a closer look at the numerical value of r by investigating the equation that produces this quantity. n is the number of points.

8. Select “Show equation for r” and examine the formula to determine why and when the correlation is undefined. (Hint: use two points)

So far, we have developed some intuitions about r. Its formal definition is quite complex. However, r2 is much simpler. So we mention it here. r2 is the fraction of total variation in the y variable that can be explained by the regression equation. The rest of the variation is due to randomness or some other factors. For example, if the correlation coefficient is 0.7 then r2= 0.49 meaning that 49% of the variation in the y-variable can be explained by the regression equation. The other 51% is due to some other factors. How does this affect your understanding of how the strength of correlations are categorized in part A of activity 1? Consider some other “strong,” “moderate,” and “weak” r-values.



PART B – The r Game

One basic rule when interpreting the correlation coefficient is to “First look at the scatterplot to see if the relationship between variables is linear.” If it is, you may calculate the correlation coefficient. Always remember that a visual analysis of data is quite valuable in addition to a numerical analysis.

To understand r, it is important to understand how individual points affect the value of correlations. The relationship of outliers, leverage points and non-leverage points to the geocenter of a set of data are explored in this simple exercise. Use the web applet Correlation to practice creating scatterplots with a specific correlation. 1. Challenge your classmates to place seven points on the graph that have a correlation

of 0.7. 2. You are not allowed to delete or drag points once they are placed on the graph. 3. Price is Right Rules – The closest r-value without going over wins! Play several times varying the number of points and r-value.



ACTIVITY 3 - Correlation vs. Causation Objective To explore the relationship between correlation and causation. PART A In a Gallup poll, surveyors asked, “Do you believe correlation implies causation?” 64% of American’s answered “Yes” while only 38% replied “No”. The other 8% were undecided. Consider the following:

Ice-cream sales are strongly correlated with crime rates. Therefore, ice-cream causes crime.

If correlation implies causation, this would be a fabulous finding! To reduce or eliminate crime, all we would have to do is stop selling ice cream. Even though the two variables are strongly correlated, assuming that one causes the other would be erroneous. What are some possible explanations for the strong correlation between the two? One possibility might be that high temperatures increase crime rates (presumably by making people irritable) as well as ice-cream sales.

An entertaining demonstration of this fallacy once appeared in an episode of The Simpsons (Season 7, "Much Apu About Nothing"). The city had just spent millions of dollars creating a highly sophisticated "Bear Patrol" in response to the sighting of a single bear the week before.



Homer: Not a bear in sight. The "Bear Patrol" is working like a charm!

Lisa: That's specious reasoning, Dad. Homer: [uncomprehendingly] Thanks, honey. Lisa: By your logic, I could claim that this rock keeps tigers away. Homer: Hmm. How does it work? Lisa: It doesn't work; it's just a stupid rock! Homer: Uh-huh. Lisa: But I don't see any tigers around, do you? Homer: (pause) Lisa, I want to buy your rock.

Correlations are often reported inferring causation in newspaper articles, magazines, and television news. But, without proper interpretation, causation should not be implied or assumed.

Consider the following research undertaken by the University of Texas Health Science Center at San Antonio, appearing to show a link between consumption of diet soda and weight gain.

The study of more than 600 normal-weight people found, eight years later, that they were 65 percent more likely to be overweight if they drank one diet soda a day than if they drank none. And if they drank two or more diet sodas a day, they were even more likely to become overweight or obese.

Our students and the general public often take such relationships as causal. By no means does this state that diet soda causes obesity - but there is a strange pattern at play here.

A relationship other than causal might exist between the two variables. It is possible that there is some other variable or factor that is causing the outcome. This is sometimes referred to as the "third variable" or "missing variable" problem.

• What are some other possible plausible alternative explanations to our diet soda/obesity research example?

We must be very careful in interpreting correlation coefficients. Just because two variables are highly correlated does not mean that one causes the other. In statistical terms, we simply say that correlation does not imply causation. There are many



good examples of correlation which are nonsensical when interpreted in terms of causation.

For example, each of the following are strongly correlated:

• Ice cream sales and the number of shark attacks on swimmers. • Skirt lengths and stock prices (as stock prices go up, skirt lengths get shorter). • The number of cavities in elementary school children and vocabulary size. • Peanut butter sales and the economy. Two relationships which can be mistaken for causation are: 1. Common response: Both X and Y respond to changes in some unobserved variable,

Z. All three of our examples above are examples of common response.

2. Confounding variables: The effect of X on Y is hopelessly mixed up with the effects

of other variables on Y. When studying medical treatments, the placebo effect is an example of confounding. The placebo effect is the phenomenon that a patient's symptoms can be alleviated by an otherwise ineffective treatment, since the individual expects or believes that it will work. For example, if we are studying the effects of Tylenol on reducing pain, and we give a group of pain-sufferers Tylenol and record how much their pain is reduced, the effect of Tylenol is confounded with giving them any pill. Many people will report a reduction in pain by simply being given a sugar pill with no medication. False causes can be ruled out using the scientific method. This is done through a designed experiment.

In practice, three conditions must be met in order to conclude that X causes Y, directly or indirectly:

1) X must precede Y 2) Y must not occur when X does not occur 3) Y must occur whenever X occurs



Experimental research attempts to understand and predict causal relationships (X→Y). Since correlations can be created by an antecedent, Z, which causes both X and Y (Z →X & Y), or by confounding variables, controlled experiments are performed to remove these possibilities. Unless data has been gathered by experimental means and confounding variables have been eliminated, one can not infer causation.

Still the great Scottish philosopher David Hume has argued that we can only perceive correlation, and causality can never truly be known or proven.


Handout 4 Understanding Correlation Properties with a Visual Model 3-65

PART B - Headlines Consider the following headlines and their matching correlations from various sources. Accepted uncritically, each might be used to "prove" one’s point of view in an article. Within your group, brainstorm common causes or confounding variables. Write your ideas below and be prepared to share. Correlated variables Causation factors 1. Kids’ TV Habits Tied to Lower IQ Scores IQ scores and hours of TV time (r = -0.54) 2. Eating Pizza ‘Cuts Cancer Risk’ Pizza consumption and cancer rate (r = -0.59) 3. Gun bill introduced to ward off crime Gun ownership and crime (r = 0.71) 4. Reading Fights Cavities Number of cavities in elementary school children and their vocabulary size (r = 0.67) 5. Graffiti Linked to Obesity in City Dwellers BMI and amount of graffiti and litter (r =0.45) 6. Stop Global Warming: Become a Pirate Average global temperature and number of pirates ( r = -0.93)



Supplemental Reading NEW POLL SHOWS CORRELATION IS CAUSATION

WASHINGTON (AP) The results of a new survey conducted by pollsters suggest that, contrary to common scientific wisdom, correlation does in fact imply causation. The highly reputable source, Gallup Polls, Inc., surveyed 1009 Americans during the month of October and asked them, "Do you believe correlation implies causation?" An overwhelming 64% of American's answered "YES", while only 38% replied "NO". Another 8% were undecided. This result threatens to shake the foundations of both the scientific and mainstream community. "It is really a mandate from the people." commented one pundit who wished to remain anonymous. "It says that The American People are sick and tired of the scientific mumbo-jumbo that they keep trying to shove down our throats, and want some clear rules about what to believe. Now that correlation implies causation, not only is everything easier to understand, it also shows that even Science must answer to the will of John and Jane Q. Public." Others are excited because this new, important result actually gives insight into why the result occurred in the first place. "If you look at the numbers over the past two decades, you can see that Americans have been placing less and less faith in the old maxim 'Correlation is not Causation' as time progresses." explained pollster and pop media icon Sarah Purcell. "Now, with the results of the latest poll, we are able to determine that people's lack of belief in correlation not being causal has caused correlation to now become causal. It is a real advance in the field of meta-epistemology." This major philosophical advance is, surprisingly, looked on with skepticism amongst the theological community. Rabbi Marvin Pachino feels that the new finding will not affect the plight of theists around the world. "You see, those who hold a deep religious belief have a thing called faith, and with faith all things are possible. We still fervently believe, albeit contrary to strong evidence, that correlation does not imply causation. Our steadfast and determined faith has guided us through thousands of years of trials and tribulations, and so we will weather this storm and survive, as we have survived before." Joining the theologists in their skepticism are the philosophers. "It's really the chicken and the egg problem. Back when we had to worry about causation, we could debate which came first. Now that correlation IS causation, I'm pretty much out of work." philosopher-king Jesse "The Mind" Ventura told reporters. "I've spent the last fifteen years in a heated philosophical debate about epistemics, and then all of the sudden Gallup comes along and says, "Average household consumption of peanut butter is up, people prefer red to blue, and...by the way, CORRELATION IS CAUSATION. Do you know what this means? This means that good looks actually make you smarter! This means that Katie Couric makes the sun come up in the morning! This means that Bill Gates was right and the Y2K bug is Gregory's fault." Ventura was referring to Pope Gregory XIII, the 16th century pontiff who introduced the "Gregorian Calendar" we use today, and who we now know is to blame for the year 2000. The scientific community is deeply divided on this matter. "It sure makes my job a lot easier." confided neuroscientist Thad Polk. "Those who criticize my work always point out that, although highly correlated, cerebral blood flow is not 'thought'. Now that we know correlation IS causal, I can solve that pesky mind-body problem and conclude that thinking is merely the dynamic movement of blood within cerebral tissue. This is going to make getting tenure a piece of cake!"



Anti-correlationist Travis Seymour is more cynical. "What about all the previous correlational results? Do they get grandfathered in? Like, the old stock market/hemline Pearson's rho is about 0.85. Does this mean dress lengths actually dictated the stock market, even though they did it at a time when correlation did not imply causation? And what about negative and marginally significant correlations? These questions must be answered before the scientific community will accept the results of the poll wholeheartedly. More research is definitely needed." Whether one welcomes the news or sheds a tear at the loss of the ages-old maxim that hoped to eternally separate the highly correlated from the causal, one must admit that the new logic is here and it's here to stay. Here to stay, of course, until next October, when Gallup, Inc. plans on administering the poll again. But chances are, once Americans begin seeing the entrepeneurial and market opportunities associated with this major philosophical advance, there will be no returning to the darker age when causal relationships were much more difficult to detect.

http://www.obereed.net/hh/correlation.html


Handout 1-1 Understanding Functions 4-13

Understanding Functions

Objective To gain a deeper understanding of equations and functions and how they are related. Materials Pencil, markers (red, green, and blue), and a calculator with graphic and table generating capabilities. Introduction The intent of this exercise is to increase your understanding of functions. The function is the single most important concept in all of mathematics. It enables us to mathematically model and describe the world around us. Hence, to understand and use mathematics correctly, it is paramount that you understand functions. In this exercise, we use the process definition of the function, which we believe is the best suited definition of function for teaching and understanding algebra. The process definition treats the function as an input/output machine. This machine accepts inputs and produces well defined, corresponding outputs.



Hence, when working with a function, always be asking yourself: what are the acceptable inputs (the domain) and what are the corresponding outputs (the range)? Throughout this exercise, you will be asked to study in detail four representations of the function: the tabular, the graphical, the algebraic, and the contextual.

It is important to realize that none of these is really the function. These representations are just the most popular ways that we use to convey the idea of how functions produce outputs from inputs.



Activity 1 Equation: a question about functions Consider the following situation: Roger throws a ball of string upward from a downward moving construction elevator. The ball is modeled by the function -16t2+52t+140. The elevator is modeled by the function -10t+132. The units for this problem are in feet and seconds. 1) Write an equation to answer the question: When will the ball be at the same height as the elevator? 2) Solve this equation algebraically.



Comparison of functions: f(x) = g(x) Let us unfold this problem by considering it from the point of view of a comparison of functions. First, let us go through the standard algebraic steps used in answering the question: When will the elevator and the ball be at the same height? To find the input time when the two functions output the same height, place the functions equal to each other. Stage 1: -16t2 + 52t + 140 = -10t + 132 f1(t) = g1(t) The next algebraic step is to combine any constants, so subtract 132 from both sides of the equation. Stage 2: -16t2 + 52t +8 = -10t f2(t) = g2(t) The next algebraic step is to combine the first order terms, so add 10t to both sides. Stage 3: -16t2 + 62t +8 = 0 f3(t) = g3(t) The next step would be to factor and use the property of zero to find that the answers to this question, or the roots to the equation, are -1/8 and 4 seconds after the ball was thrown. In this problem, we are assuming that time is positive, so -1/8 is out of the domain of this problem. Hence the only solution (root) to this problem is 4 seconds after the ball was thrown.


Handout 2 Understanding Functions 4-17

Recording Sheet A

Now let us study the functions that were produced at each stage of the problem and bring in the other representations of the functions to help us analyze what these functions are really telling us. 1) On recording sheet B, use the functions from each stage to complete the tables, and then use the tables to answer the questions. 2) On recording sheet C graph the functions. 3) Use a green marker to circle all the roots/solutions that you find on recording sheet B and recording sheet C. 4) Explain contextually what the root/solution is telling you at each stage. Stage 1: Stage 2: Stage 3: 5) Use a red marker to circle all the zeros of the functions on recording sheet B; also circle all the x-intercepts of the functions on recording sheet C. 6) How do roots and zeros/x-intercepts differ? Explain. 7) Take a blue marker and on recording sheet C, draw a line vertically through all the roots. What stays the same and why does it not change?



Recording Sheet B



Recording Sheet C Stage 1 Stage 2 Stage 3



TEKS/TAKS Recording Sheet Connecting the language of functions with the TEKS refinement. ● What is an equation? ● What is a root? ● What is a zero or x-intercept? With your understanding of the language of roots, zeros, and x-intercepts, let's analyze some TEKS and TAKS. 1) Revise this TEKS item so that the language is correct. The TEKS, Algebra I (b) Knowledge and Skills (A.10) B, was changed. Before refinement, the TEKS item reads: The student relates the solutions of quadratic equations to the roots of their functions.



The TEKS item after refinement reads: The student is expected to make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function. 2) How does this compare to the revision you wrote? 3) Revise this TEKS item so that the language is correct. The TEKS, Algebra II (b) Knowledge and Skills (2A.6) C, reads: The student is expected to determine a quadratic function from its roots or a graph. 4) Revise this TAKS question so that the language is correct.

42 Which ordered pair represents one of the roots of the function f(x) = 2x2 + 3x - 20? F (-5/2, 0) G (-4, 0) H (-5, 0) J (-20, 0)


Handout 1-1 Picture This! 5-4

PICTURE THIS! Part 1 Diagrams

Which of these two-dimensional pictures could be used to represent a cube?



PICTURE THIS! Part 1 Diagram Solutions

Which of these two-dimensional pictures could be used to represent a cube?

Orthographic

Orthographic

Dimetric Perspective

Dimetric

Isometric

156o



PICTURE THIS! Drawings in Two-Dimensions

Use the definitions below to determine which of these descriptions best fits each of the sketches given in Part I. Perspective View: The technique of portraying solid objects and spatial relationships on

a flat surface. There are numerous methods of depicting an object, depending upon the purpose of the drawing.

Vanishing Point: In an artistic perspective drawing, receding parallel lines (lines that run

away from the viewer) converge at a vanishing point on the horizon line. This maintains a realistic appearance of the object depicted, even as the vantage point changes.

Orthographic Drawing: Ortho means “straight” and the views in orthographic drawings show the faces of a solid as though you are looking at them “head-on” from the top, front or side. Orthographic drawings are used in engineering drawings to convey all the necessary information of how to make the part to the manufacturing department. The line of sight is perpendicular to the surfaces of the object. Two conventions are used in technical drawings. These are first angle and third angle, which differ only in position of the plan (top, front, and side views). They are derived from the method of projection used to transfer two-dimensional views onto an imaginary, transparent box surrounding the object being drawn. This is not a distinction the state of Texas has seen fit to recognize in the TEKS.

appears as

right sidefronttop



A common use of a “top” orthographic view is the floor plan of a house or other building.

Axonometric drawing: A two-dimensional drawing showing three dimensions of an

object. The vantage point is not perpendicular to a surface of the object being drawn. Axonometric means, “to measure along the axes.” There are three types: isometric, dimetric, and trimetric.

Isometric Drawing: A two-dimensional drawing that shows three sides of an object in one view. The vantage point is 45° to the side and above the object being viewed. The resulting angle between any two axes appears to be 120° . Isometric means, “one measure.” In this view, 90° angles appear to be 120° or 60° .

Courtesy Electronic Arts, Inc.

Video games, such as SimCity 2000, frequently use isometric drawings

120°

120° 120°



Dimetric drawing: A two-dimensional drawing similar to an isometric drawing, but with only two of the resulting angles between the axes having the same apparent measure. Dimetric means, “two measures.”

Trimetric drawing: Again, A two-dimensional drawing similar to an isometric drawing, but with none of the resulting angles between the axes having the same apparent measure. Trimetric means, “three measures.”

X Y

Z

100°

130° 130°



PICTURE THIS! Isometric Drawings

Make orthographic drawings of each diagram. You should include drawings from the top, front, and right side.

Diagram 2Diagram 1



PICTURE THIS! Isometric Drawings Solutions

top front right sideright sidefronttop

The orthogonal drawings are the same! There are easy solutions to this dilemma. Try adding segments in the drawing to outline the cube faces seen in each view. Indicate different distances from the viewer to different parts of the orthogonal drawings by using different line weights. Faces nearest the viewer can be outlined with dark segments. Faces one cube farther away can be a medium weight and faces two cubes farther away can be a light or dashed segment. Alternatively, add segments in the top view to outline the cube faces and number the squares to indicate the number of cubes in the “stack.”


Handout 1 Texas “T” Activity 5-22

TEXAS “T”: Part I Using snap cubes with the edge of one cube representing 1 unit of length, build the first “T” like the one shown in the illustration. The “T” is 4 units tall and 3 units wide. Use the “T” to complete the table below:

What do you think will happen to the height of the “T” when the dimensions of each part

are doubled? ___________________ tripled?________________________

What do you think will happen to the surface area of the “T” when the dimensions of

each part are doubled? _________________ tripled?________________________

What do you think will happen to the volume of the “T” when the dimensions of each

part are doubled? ___________________ tripled?________________________

N = “T” NUMBER HEIGHT OF “T”

TOTAL SURFACE AREA

VOLUME

“T” #1


Handout 2 Texas “T” Activity 5-23

TEXAS “T”: Part I In your group, build at least the next two “Ts”, where n represents the height of the “T.” To build each “T”, take the dimensions of “T” #1 and multiply by n (n=2, n=3, n=4).

N = “T” NUMBER HEIGHT OF “T”

TOTAL SURFACE AREA

VOLUME

“T” #2

“T” #3

“T” #4

“T” N


Handout 3-1 Texas “T” Activity 5-30

TEXAS “T”: Part III One and Two-Dimensional Change

Thus far we have examined the effect on surface area and volume of a figure if all dimensions are changed by the same factor. What do you suppose will happen to surface area and volume if only one dimension is changed? For this exercise, let’s use a simpler figure. l = length w = width h = height Volume of a rectangular prism: V = lwh Surface area of a rectangular prism: SA = 2(lw) + 2(wh) + 2(hl) Volume: A. Change in one dimension:

Given a rectangular prism with a length of 2, width of 2 and a height of 1;

1. Without changing the length and width, change the height by a factor of n and

complete the following table.

n Volume

1 4 2 3 4 5 6

n

2. Enter the data into lists in your calculator as L1 and L2 and plot the data.



3. Write in words the pattern you observe in the volumes.

4. Write an algebraic function for the pattern.

5. Without changing the length and height, change the width by a factor of n and complete the following table.

n Volume 1 4 2 3 4 5 6

n



8. Why is the pattern the same as #3?



B. Change in two dimensions Given a rectangular prism with a length of 2, width of 2 and a height of 1; 1. Without changing the length, change the height and width by a factor of n and


n Volume 1 4 2 3 4 5 6

n

2. Enter the data into lists in your calculator as L1 and L3 and plot the data. 3. Write in words the pattern you observe in the volumes.


5. Write a function to predict the volume of our figure if all three dimension change

by factor of n.



Surface Area: A. Change in one dimension:

Given a rectangular prism with a length of 2, width of 2 and a height of 1;

1. Without changing the length and width, change the height by a factor of n and


n Surface Area 1 2 3 4 5 6

n

2. Enter the data into lists in your calculator as L1 and L2 and plot the data. 3. Write in words the pattern you observe in the surface area.

___________________________________________________________________

___________________________________________________________________

4. Write an algebraic function for the pattern._______________________________



B. Change in two dimensions: Given a rectangular prism with a length of 2, width of 2 and a height of 1;

1. Without changing the length, change the height and width by a factor of n and


n Surface Area 1 2 3 4 5 6

n

2. Enter the data into lists in your calculator as L1 and L2 and plot the data.


___________________________________________________________________

___________________________________________________________________

4. Write an algebraic function for the pattern._______________________________

Think about it: Does a one dimensional change in any 3-dimensional figure create a linear change in volume? What if the figure were a can?


Handout 1-1 What’s Your Problem? 6-25

Sample Assessment Items – Roger Throwing a Ball Yesterday we looked at a series of graphs and tables that modeled Roger throwing a baseball upward from a downward moving elevator. Based on that work, answer the following. The graphs and tables below represent Roger starting from a different height, throwing at a different initial velocity.

1. Match the graph with the appropriate table. 2. What are the roots, solutions shown? ____________ 3. When will Roger and the baseball be at the same height?

_______ 4. Equations have _______ or ________, whereas functions

have _______ or _______. 5. What is Roger’s new starting height? ______

___a.

i.

___b.

ii.

___c.

iii



Yesterday we looked at a series of graphs and tables that modeled Roger throwing a baseball upward from a downward moving elevator. Based on that work, Raul and Marco answered the following question. Who is right and why? 1. Given the table and graph representing Roger throwing a

baseball upward from a downward moving elevator, what are the roots, solutions shown?

Raul

Roots and solutions are the same as zeros. Y1 is zero at x = 0 and in between x = 3 and x = 4, so the roots are x = 0 and x is about 3.5

Marco

A root is the solution to an equation so we are looking for where Y1 = Y2. In the table, both Y1 and Y2 have the same values at x = 0 and x = 4. On the graph, Y1 and Y2 intersect at these same x values x = 0 and x = 4.

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What are 2 possible sets of lists that would produce the following graph:

Fill in the lists and note your window. Possibility 1: Possibility 2:



Rectangle RECT has an area of 4 cm2. It is enlarged to make the rectangle BIGS. BIGS has an area of 16. What was the scale factor used to enlarge RECT to get BIGS?



Sketch the view of a cube below, but make it appear three-dimensional by making some of the lines dashed.



Square SQAR has side length s. Choose your favorite scale factor to dilate SQAR to a similar square and label new square HUGE. Label the side lengths of HUGE in terms of s and find the area of HUGE in terms of s.



Write two quadratic functions with zeros x = 2, x = -3



A cube, P, has volume 1000 cm3. It is dilated by a scale factor of 0.5 to form a similar cube, S. What is the side length of cube S?



The clear cube shown has the letters DOT printed on one face. When a light is shined on that face, the image of DOT appears on the opposite face. The image of DOT on the opposite face is then painted. Copy the net of the cube and sketch the painted image of the word, DOT, on the correct square and in the correct position.

Discovering Geometry



The cube has designs on three faces. When unfolded, which figure at right could it become?




Find the equations of two linear functions that intersect at the point (2, 3)



Find the equations of two linear functions that do not intersect.



Write your two favorite linear functions with a y-intercept of 4



Write your two favorite linear functions with a slope of 4.



The green prism below right was built from the two solids below left. Copy the figure on the right onto isometric dot paper and shade in one of the two pieces to show how the complete figure was created.




Sketch the solid shown, but with the two blue cubes removed and the red cube moved to cover the visible face of the green cube.




Given f x( )= x2 + 3, find f 2( ) Cameron got the following answer. What do you think he did? What should he have done?

f 2( )= 2x2 + 6



The width of rectangle DAWN was enlarged by a scale factor of 2:3 to form a new rectangle RISE. What is the perimeter of RISE?

Justin’s work is below. What do you say to Justin?

10 to 15, 20 to 30 so 2(15) + 2 (30) = 30 + 60 = 90

90 cm

10 cm

20 cm D A

W N



Sanna solved the equation below as follows. Is she correct? If not, explain to her why.

x2 + 9x − 36 = 0x2 = −9x + 36

x2 = −9x + 36x = −3, 6



José answered the following question below. How do you respond to his answer? Include the correct response.

Yesterday we looked at a series of graphs and tables that modeled Roger throwing a baseball upward from a downward moving elevator. Based on that work, answer the following. The graphs and tables below represent Roger starting from a different height, throwing at a different initial velocity.

What are the roots, solutions shown? _______

“The graph intersects the x-axis around 4.5, so the root (solution) is x = 4.5.”



For the figure shown, Kamisha drew the following orthographic drawing. How would you help her fix it?

Front Side Top



Lakrea created the net below. She is convinced that it will fold into a cube. Will it? If not, how would you convince her otherwise?



McKay says that the scatter plot below shows positive correlation because “the data goes up” as he pointed up from left to right. What do you say to McKay?



Bo was absent yesterday. When you started telling him about the Roger-throwing-the-ball-elevator activity, he said, “What’s the big deal? Roots, zeros, solutions, x-intercepts – they are all the same thing.” How do you respond to Bo? A complete answer includes graphs, tables, equations and discussion.



Yesterday we added marbles to cups hanging by rubber bands. We measured 3 distances, graphed the data, and found trend lines. Two groups results are below.

Group A

Group B

1. What can you conclude about Group 1’s marbles and rubber band compared to that of Group 2?

2. What can you conclude about Group 1’s table and cup compared to that of Group 2?



Yesterday we graphed different classes heights and arm spans. Two different classes are represented below.

Group A

Group B

1. Name one difference between the people measured by Group A and the people measured by Group B and explain how you know.

2. Name another difference between the people measured by Group A and the people measured by Group B and explain how you know.



Marisol spilled soda on her homework. Fill in the two missing steps covered by the stain.



For a week we have been bouncing balls under motion detectors and finding function rules to model one complete bounce. Here is some similar data. What function rule would you write to model the first complete bounce (the one the trace cursor is on)?



If the ratio of the heights in two similar figures is m/n, the ratio of their perimeters would be ______________, the ratio of their surface area would be _________________ and the ratio of their volume would be ____________________.



One of the following solids cannot be represented by the orthographic drawings. Change the orthographic drawings so that they represent the misfit and not the others.

top front side

a

b

c

d


Handout 4 What’s Your Problem? 6-58

Three Problem Types – How to Write

Snap Shot Problems: What are two ideas, processes, or representations that students mix up? Juxtapose them and ask which is which. What part of a large activity can you grab to assess if students got the gist of the large activity? Un-Doing Problems: Can you start with the answer? Can you start in the middle? Can you change one constraint? Can you start with a different representation? Ask students to create or invent the beginning of a problem. Error Analysis What are the typical errors that students make? Pose an incorrect solution. Ask students to explain what went wrong. Sometimes show the incorrect process; sometimes just show the incorrect answer.


Handout 1 The Power of Creating 6-63

Sample Class Responses

y-intercept of 4 Function Rule Graph

1. y = 4 2. y = 4 − x 3. y = 4 + x 4. y = 3x − 4 5. y = −4 + 3x 6. y = 4 − 9x 7. y = 4 + 50x 8. y = 4 − 0.2x 9. y = 4 + 0.4x 10. 11. 12.

Slope of 4

Function Rule Graph 1. y = 4x 2. y = 2 + 4x 3. y = −2 + 4x 4. y = −15 + 4x 5. y = 12 + 4x 6. y = −8 + 4x 7. y = 7 + 4x 8. y = 0.5 + 4x 9. y = 18 + 4x 10. 11. 12.

TAB 1 - Refinement in the Mathematics TEKSWhat Are the Changes 9-12?Handout 1

TAB 3 - Algebra ISpaghetti RegressionHandout 1: Spaghetti RegressionTransparency 1/Handout 2: ScatterplotHandout 3: Activity 1 Goodness-of-FitHandout 4: MeasuringHandout 5: Activity 2Handout 6: Activity 3 Absolute Value vs. SquaringHandout 7: Supplemental Material

Understanding Correlation Properties with a Visual ModelHandout 1: Activity 1Handout 2: Activity 2Handout 3: Activity 3 Correlation vs. CausationHandout 4: Activity 3, Part B - HeadlinesHandout 5: Supplemental Reading

TAB 4 - Algebra IIUnderstanding FunctionsHandout 1: Understanding FunctionsHandout 2: Recording Sheet AHandout 3: Recording Sheet BHandout 4: Recording Sheet CHandout 5: TEKS/TAKS Recording Sheet

TAB 5 - GeometryPicture This!Handout 1: Part I DiagramsHandout 2: Drawings in Two-DimensionsHandout 3: Isometric Drawings

Texas "T" ActivityHandout 1: Part IHandout 2: Part IHandout 3: Part III

TAB 6 - AssessmentWhat's Your ProblemHandout 1-Sample Assessment Items-Roger Throws a BallHandout 2-Three Problem Types LabelsHandout 3-What's Your Problem? ItemsHandout 4-Three Problem Types - How to Write

The Power of CreatingHandout 1-Sample Class Responses

teks refinement and implications for the classroom · mathematics teks refinement 2006 – 9-12...

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