250 words, no more than 500· Focus on what you learned that made.docx

250 words, no more than 500
· Focus on what you learned that made an impression, what may
have surprised you, and what you found particularly beneficial
and why. Specifically:
· What did you find that was really useful, or that challenged
your thinking?
· What are you still mulling over?
· Was there anything that you may take back to your classroom?
· Is there anything you would like to have clarified?
Your Weekly Reflection will be graded on the following criteria
for a total of 5 points:
· Reflection is written in a clear and concise manner, making
meaningful connections to the investigations & objectives of the
week.
· Reflection demonstrates the ability to push beyond the scope
of the course, connecting to prior learning or experiences,
questioning personal preconceptions or assumptions, and/or
defining new modes of thinking.
BELOW ARE LESSON COVERED
· This week's investigations introduce and explore one of the
most common distributions (one you may be familiar with): the
Normal Distribution. In our explorations of the distribution and
its associated curve, we will revisit the question of "What is
typical?" and look at the likelihood (probability) that certain
observations would occur in a given population with a variable
that is normally distributed. We will apply our work with
Normal Distributions to briefly explore some big concepts of
inferential statistics, including the Central Limit Theorem and
Hypothesis Testing. There are a lot of new ideas in this week’s
work. This week is more exploratory in nature.
Goals:
· Explore the Empirical Rule
· Become familiar with the normal curve as a mathematical
model, its applications and limitations

· Calculate z-scores & explain what they mean
· Use technology to calculate normal probabilities
· Determine the statistical significance of an observed
difference in two means
· Use technology to perform a hypothesis test comparing means
(z-test) and interpret its meaning
· Use technology to perform a hypothesis test comparing means
(t-test) (optional)
· Gather data for Comparative Study Final Project.
·
DoW #5: The SAT & The ACT
Two Common Tests for college admission are the SAT
(Scholastic Aptitude Test) and the ACT (American College
Test). The scores for these tests are scaled so that they follow a
normal distribution.
· The SAT reported that its scores were normally distributed
with a mean μ=896 and a standard deviation σ=174
· The ACT reported that its scores were normally distributed
with a mean μ=20.6 and a standard deviation σ=5.2.
We have two questions to consider for this week’s DoW:
2. A high school student Bobby takes both of these tests. On the
SAT, he achieves a score of 1080. On the ACT, he achieves a
score of 30. He cannot decide which score is the better one to
send with his college applications.
. Question: Which test score is the stronger score to send to his
colleges?
· A hypothetical group called SAT Prep claims that students
who take their SAT Preparatory course score higher on the SAT
than the general population. To support their claim, they site a
study in which a random sample of 50 SAT Prep students had a
mean SAT score of 1000. They claim that since this mean is
higher than the known mean of 896 for all SAT scores, their
program must improve SAT scores.
. Question: Is this difference in the mean scores statistically
significant? Does SAT Prep truly improve SAT Scores?
.

Investigation 1: What is Normal?
One reason for gathering data is to see which observations are
most likely. For instance, when we looked at the raisin data in
DoW #3, we were looking to see what the most likely number of
raisins was for each brand of raisins. We cannot ever be certain
of the exact number of raisins in a box (because it varies) no
matter how much data we gather. But, we can estimate a likely
value or range for the number of raisins and determine the
empirical probability that a box of raisins would be in this
range.
In Activities A & B, we explore the probabilities seen in a
particularly symmetric distribution.
.
Inv 1, Activity A: Probability in a Distribution
Excerise A1: Relative Frequency
Histograms
Relative frequency histograms provide information about
probabilities. Consider the following relative frequency
histogram for the number of raisins in a 1/2 ounce box of Brand
B raisins
(The probability for each interval in the histogram is displayed
at the top of the bar for that interval.)
(a) Describe the shape of the distribution.
(b) Suppose I pick a random box of Brand B raisins. Based on
this data, what is the probability that the number of raisins in
the box is:
· greater than 28?
· less than or equal to 28?
· exactly 28?
· greater than 32?
· between 26 and 30 (inclusive)?
Answers: i) 32%; ii) 69%; iii) 23%; iv) 0%; v) 71%

A common way to "divide up" the histogram is to use the
standard deviation as a ruler. We will consider the data in
groups that are one, two and three standard deviations from the
mean. Let's start by looking at the group of data that is within
one standard deviation of the mean:
One Standard Deviation from the Mean:
We can add these values to our graph to see the group within
one standard deviation of the mean:
Exercise A2:Use the above histogram to:
(a) determine the probability that a random box of brand B
raisins would fall between 25.8 and 29.4 (within one standard
deviation of the mean)
(b) calculate the values that are two standard deviations away
from the mean. Sketch them on the graph.
(c) determine the probability that a random raisin box would
fall within two standard deviations of the mean.
(d) Repeat steps b and c for raisin boxes that fall within three
standard deviations from the mean.
(e) Summarize your work from this exercise in a table like the
one shown below:
Percentage of Observations
Brand B Raisins
Within 1 standard deviation of the mean
Within 2 standard deviations of the mean
within 3 standard deviations of the mean

Exercise A3: The following distribution shows a relative
frequency histogram of students scores on a Math placement
exam. Like the Brand B Raisins distribution, it is a roughly
symmetric, mound-shaped distribution.
(a) Determine the probability that a random student's score
would fall within one standard deviation of the mean; within
two standard deviations of the mean; and within three standard
deviations of the mean.
(b) Add your findings to the table you started in Exercise A2,
by adding a column for Math Placement Exam Scores:
Percentage of Observations...
Brand B Raisins
Math Placement Exams Scores
within 1 standard deviation of the mean
Our findings illustrate a “big idea” called the Empirical Rule:
THE EMPIRICAL RULE
(68-95-99.7%)

In "special" symmetric, mound-shaped distributions, about 68%
of the observations fall within one standard deviation, about
95% fall within two standard deviations, and about 99.7%
(nearly all) all within three standard deviations.
Exercise A5: Practice with the Empirical Rule
Each of the distributions shown below is a symmetric, mound-
shaped distribution with a mean of 0 and a standard deviation of
1. Use the empirical rule to determine the percentage of
observations represented by the shaded area on each
distribution.
(answers follow)
1. Between -2 and 2
2. More Than 2 or Less Than -2
3. Greater Than -1
4. Greater than 1
Answers: (1) 95%, (2) 5%, (3) 84%, (4) 16%
·
Inv 1, Activity B: The Normal Distribution
The "special" symmetric, mound-shaped distributions that
follow the Empirical Rule (like the ones you looked at in
investigation 1 are called Normal Distributions. Normal
distributions are a family of distributions with very specific
properties, though the way most of us think of them is as a "bell
curve." The key properties we think of for a normal distribution
are:
· One peak in the middle (mean)
· Symmetric about the mean
· Follows the Empirical Rule: 68-95-99.7
These are not the only defining characteristics of a normal
distribution. Normal distributions are defined by equations,

which dictate specifics about the shape of the mound, how it
curves, and the areas beneath the different sections. However,
for most situations you will encounter, mound-shaped
symmetric distributions can be considered to be nearly normally
distributed.
Why are they called normal? For starters, many variables in
many different contexts follow the normal distribution, making
this distribution the typical, expected or "normal" pattern. When
we talk about skewed distributions, we usually mean "skewed
from the normal."
The samples you worked with in Investigation 1 (the raisin
boxes or the math placement scores) are displayed with
histograms. The histograms are roughly symmetrical and
mound-shaped. They approximate the theoretical normal curve
that represents the entire population. The graph below shows
the theoretical normal curve superimposed on the distribution of
the Brand B raisin data:
Notice that this sample of 22 raisin boxes nearly fills the area
beneath the theoretical curve. The normal curve models what we
think the true population distribution should be.
The histogram shows the distribution of the actual sample data.
As such, its mean and standard deviation are statistics,
represented by μ and s, respectively.
· The normal curve shows the distribution of the theoretical
population, so its mean and standard deviation are parameters,
represented by μ and σ.
Complete the following activities about Normal Distributions:
Exercise B1 Watch the video clip entitled, "4. Normal
Distributions" in the Annenberg Series Against All Odds.

As you watch, take notes on the following questions:
· What is a density curve? How is it related to a histogram or
other display of a distribution?
· What properties does a Normal Curve have?
· What are some variables that are normally distributed? What
is it about them that makes them normally distributed?
· Why are the Normal Curves considered to be a family?
· What is the formula for standardizing an observation? Why
would you want to standardize?
·
Inv 1, Activity C: Applying the Empirical Rule
Normal distributions are entirely defined by the mean and
standard deviation of the distribution. When we know these two
values, we have the full picture of the distribution of a variable
for the population. From this, we can sketch the distribution and
determine the likelihood of specific events occurring (using the
empirical rule and other methods.) This is the purpose of the
following example:
Examples:The Scenario
At Lesley Middle School, all students were timed when they ran
a 100m dash. It was found that the data was normally
distributed with a mean of 17.2 seconds and a standard
deviation of 2.1 seconds.
· What range of times would you expect to be typical for the
100m dash at this school? (let’s say that “typical” would mean
at least 95% of students would be in this range)
· A student ran the 100 m dash in 10.9 seconds. How likely is it
that a student could run the 100 m dash in 10.9 seconds or less?
First, Picture it
In this scenario, we know everything we need to know to get a
picture of this data. So, let's make the graph. We know it is
normally distributed - this gives us the general shape The graph
need not be perfectly to scale, but it should show the inflection
points – places where the curvature switches. Recall from the
Annenberg video that inflection points are found 1 standard
deviation from the mean.

Next, label the mean and the values of the points that are 1
standard deviation above and below the mean of the graph.
Lastly, label the points that are 2 and 3 standard deviations
from the mean.
Then, Apply the Empirical Rule to answer the questions:
1. The Empirical Rule states that 95% of the data will lie within
2 standard deviations of the mean. For this scenario, that would
be roughly between 13 second and 21.4 seconds.
So we can say that a typical range of times for middle schoolers
at Lesley Middle School is between 13 and 21.4 seconds,
because 95% of all students would have times in this range.
2. Times that are as fast or faster than 10.9 seconds would be
more than 3 standard deviations from the mean.
From the Empirical rule, we know that 99.7% of the data is
within 3 standard deviations of the mean. This means that 0.3%
is within the two tails. So the likelihood of having a time that is
10.9 second or less would be HALF of 0.3%, or 0.15%. This is
very unlikely – Perhaps this student is a truly exceptional
athlete; or perhaps there was an error in recording this time? A
probability this low warrants extra attention.
Exercise C1: SAT scores from DoW #5: In DoW #5, we are
given thatthe SAT (Scholastic Aptitude Test) scores are
normally distributed with μ=896 and σ=174.

(a) Sketch a normal curve for this distribution, with three
standard deviations from the mean marked out.
(b) What range of scores is "typical" for this test (with “typical”
meaning 95% students score in this range)?
(c) What is the probability that a student would score above
1418?
(d) What is the approximate probability that a student would
score between 1200 and 1400? Explain how you found this
answer.
Exercise C2: ACT scores from DoW #5: In DoW #5, we are
given that ACT (American College Test) scores are normally
distributed with μ=20.6 and σ=5.2.
(a) Sketch a normal curve for this distribution, with three
standard deviations from the mean marked out.
(b) What range of scores is "typical" for this test? (with
“typical” meaning 95% of students score in this range)
(c) What is the probability that a student would score below
15.4?
(d) What is the approximate probability that a student would
score between 15 and 30? Explain how you found this answer.
Exercise C3: Bobby’s Test Scores: In DoW #5, we learn that
Bobby scored 1080 on the SAT and 30 on the ACT. Consider
your work in Exercises C1 and C2. Which score do you think
Bobby should send to his colleges? Why?
Post your response to Exercise C3 to the Discussion Thread “
DoW#5: Bobby's Scores” by Tuesday, 10 PM EST. Review the
posts of others and make at least two follow-up posts by
Thursday, 10 PM ESTDo Not Read On until you have completed
these exercises: the answer to Exercise C3 is discussed in
Activity D.
·
Inv 1, Activity D: Beyond the Empirical Rule
In Exercise C3, Bobby’s scores were not an even number of
standard deviations away from the mean. This makes comparing

them (and determining the probabilities associated with those
scores) difficult (though not impossible). There are ways of
determining the probabilities when the desired values are not 1,
2, or 3 standard deviations from the mean. We will explore one
such approach in this investigation.
Recall that all normal curves are members of the same family.
By changing the mean, you can alter the location of the curve
(shift it left or right). By changing the standard deviation, you
can alter the height and width of the curve. These properties
allow us to compare values from different curves by
standardizing the data.
Consider the graphs below, showing the normal curves for the
SAT and ACT from the last investigation. Added beneath each
is a new scale, showing the number of standard deviations from
the mean.
This rescaling allows us to compare values for each of the
distributions by looking at how many standard deviations the
values are from the mean. Now, we need to calculate how many
standard deviations Kathy's score and Bobby's score are from
the mean of its distribution. We do this using the
standardization formula:
Where x is a specific observation, μ is the population mean and
σ is the population standard deviation.
This shows that Bobby’s ACT score is a greater number of
deviations from the mean. On a standardized normal curve
(where μ=0 and σ=1), we can compare the two scores:
Two ways you can think of this are:

· The percentage of students scoring Bobby’s score (or higher)
on the ACT is less than the percentage of students scoring
Bobby's score (or higher) on the SAT.
· Bobby’s ACT score is higher than Bobby's SAT score, relative
to their respective means.
So, the ACT score is the stronger college test score!
Z-scores are useful for comparing data; in addition, they are
useful for finding probabilities that we cannot find with the
Empirical Rule alone. Suppose we wanted to know the
probability that a student would score as high as Bobby, or
higher:
From the Empirical Rule, we know that 47.5% of students
would score between 20.6 and 31.0 (2 standard deviations). So,
we could estimate the percentage above Bobby’s ACT score of
30 to be a little more than 2.5% (50% - 47.5%).
We can determine a more exact percentage using the graphing
calculator. The graphing calculator can calculate the probability
that a z-score is between two values on a standard normal curve
using the function normalcdf:
Press [2nd][VARS] and select Normalcdf(
To determine the probability that a student scored higher than
Bobby on the ACT (above 1.81 standard deviaions) we are
looking at the part of the graph between 1.81 and positive
infinity. So, enter:
normalcdf(1.81, 99999) = 3.5%
Notice that the second value, 99999, is an arbitrary large
postive number. Since the area under the curve becomes so
small as the z-scores get larger and larger, this value will give
an accurate percentage. We have used the calculator to
determine that the probability that a student would score as high
as Bobby on the ACT is 3.5%. In other words, P(z>1.81)=3.5%
An alternate option to the graphing calculator is this online

convertor.
Since we are only interested in greater than choose one tailed.
Insert 1.81 for the z score. Click submit and you will get the
same answer.
Exercise D1. Use the graphing calculator to determine the
probability that a student would score as high as Bobby on the
SAT.
Exercise D2: Use the graphing calculator to determine the
following normal distribution probabilities:
(a) P(z<1.28)
(b) P(z>1.28)
(c) P(z< -2.25)
(d) P(-1.1<z<1.1)
Answers: D1) 14.5
Answers: D2. a) 90.0% ;b) 10.0%; c) 1.2%; d) 72.9%
The calculator function normalcdf( ) will also calculate
probabilities without the use of a z-score. Let's revisit Bobby’s
ACT score one more time. He scored a 30 on the ACT, which
has m=20.6 and s=5.2. We can use the graphing calculator to
find the probability that a student would score 30 or higher on
the ACT without first finding a z-score. To do this, enter:
normalcdf( minimum, maximum,m, s)
normalcdf( 30, 99999, 20.6, 5.2)
This gives the same percentage we found earlier using Bobby’s
ACT z-score (3.5%).
Another option for this is to use this link.
You will insert 20.6 for the mean, 5.2 for standard deviation,
and 30 for x. This will give you a cumulative probabilty of
.96467. The percent above this is nearly the same answer of
3.5%
Exercise D3: Use the graphing calculator to recalculate Bobby’s

SAT probability, as we just did for his ACT probability.
·
Investigation 2: Testing Hypotheses
The second part of DoW #5 asks us to evaluate a claim based on
sample data: is the sample of scores from SAT Prep students
different from other SAT scores, in a way that is more than just
random variation. This question is asking us to take what we
know about SAT scores – they follow a normal distribution,
with a mean of 896 and a standard deviation of 174 – and infer
whether or not a sample mean SAT score of 1000 is “not
normal”. Certainly, students score higher than 1000 all the
time…but the question is,
“How likely is it that a whole group of 50 students would have
an average score that is a full 100 points above the mean of
896? Couldn’t this just be due to normal variation?”
This investigation looks at how to address this question
statistically.
·
Inv 2, Activity E: Sample Means are Normal
Exercise E1: Consider the question posed above: “How likely is
it that a whole group of 50 students would have an average
score that is a full 100 points above the mean of 896? Couldn’t
this just be due to normal variation?”
What are your initial thoughts on this question? Do you think
SAT Prep’s results support their claim that they raise SAT
scores? Record your initial thoughts in your journal.
In the SAT Prep claim, they refer to the mean of a sample of 50
scores. It is hard to compare a mean of 50 scores to the mean
for the whole population. To better understand this situation, we
need to look at LOTS of samples of 50 scores from the whole
population, and get an idea of what their means look like.
Exercise E2: Complete the Central Limit Theorem document.

In this exercise, you will run a virtual experiment to collect
random samples of 50 SAT scores and calculate their means.
This will produce a distribution of sample means, that you can
compare to the sample mean for SAT Prep.
Exercise E3: In Exercise E2, you answered the question, “do
you think there sufficient evidence to conclude that SAT Prep
improves SAT scores, as they claim?”
Post your response to this question to your group’s DB for DoW
#5:SAT Prep by Friday, 11:59 PM EST.
Review the responses of your group. Consider your work in the
remainder of this investigation. Post at least three meaningful
responses to your group by Sunday, 11:59 PM EST.
In these exercises, you created a portion of a Sampling
Distribution for the Means: a distribution of the means from lots
of different samples. The true sampling distribution for the
means would have EVERY mean from EVERY possible random
sample of 50 test scores. The distribution in Part II, with 355
scores, is very close to the true sampling distribution for the
means:
You probably noticed that this distribution has nearly the same
mean as the SAT scores population mean score (896). However,
its spread is MUCH MUCH smaller. In fact, the standard
deviation of the sampling distribution is a fraction of the
original standard deviation (174). This should make sense – in
the population, individual students will have scores that are
very high and very low. But, within a sample of 50 students,
those highs and lows will “average out” bringing the overall
mean of the sample closer to the true population mean.
This observation reflects a major theorem in statistics – the
Central Limit Theorem.

The Central Limit Theorem states that if you have sample sizes
larger than 30, the sample means will be nearly normal
distributions.
· The mean of the sampling distribution will be the same as the
population mean.
· The standard deviation of the sampling distribution will be
population standard deviation divided by the square root of the
sample size.
Note: One powerful part of the Central Limit Theorem is that
the population does NOT have to be normally distributed itself
to have a sampling distribution that is normally distributed
For our example, the SAT Prep sample is larger than 30 (it is
50). By the Central Limit Theorem, all samples of size 50 will
have means that follow a normal distribution. This normal
distribution will have a mean of 896 (the same as the mean for
the population of SAT scores). BUT, the standard deviation will
be much much smaller: 174/sqrt(50) = 24.6. This tells us that
the mean scores for groups of 50 students will not vary much -
they will be very close to the center of the distribution.
Population of SAT Scores
m = 896 s = 174
Sampling Distribution of Means
from samples of
50 Scores
s = 24.6
With the SAT Prep sample, we can now calculate the
probability of a random sample of 50 SAT scores having a mean
of 1000:
Normalcdf( 1000, 99999999, 896, 24.6) = 0.00001
This is an incredibly low probability – it is highly unlikely that

the sample of SAT Prep students is “the same” as other samples
of 50 students.
·
Inv 2, Activity F: Hypothesis Testing
The process of using statistics to test the validity of claim is
part of a branch of statistics called Inferential Statistics. Up to
this point, our course has focused primarily on Descriptive
Statistics.
Descriptive Statistics are used to summarize and describe the
data. Tools like histograms, box plots, mean, median, and
standard deviation all describe and summarize. Descriptive
statistics allow us to describe the distribution and make
comparisons among distributions.
Inferential Statistics is a use of statistics to make an inference
beyond what is immediately known from the data. Sometimes
this involves estimating the true value of a population parameter
(like estimating the true mean number of raisins in a ½ oz box
of raisins). It also involves using statistics to support or
disprove a hypothesis, as we did with the claim by SAT Prep.
The process of evaluating a hypothesis is called Hypothesis
Testing. In Activity E, you looked at an overview of the
process: we started with a claim, we found a way to calculate
the likelihood of the claim, and we used this to determine
whether or not the claim was probable. The actual process of
Hypothesis Testing is more technical. We will touch the surface
of this concept in the next two activities.
Exercise F1: Watch the video 25.Tests of Significnace in the
Annenberg Series, Against All Odds.
As you watch, take notes on:
· The Steps of hypothesis Testing
· What is a p-value?
· What is meant by statistically significant?
· What is the difference between a one-tail test and a two-tail
test.

· Optional Informationon Tailed tests
The Steps of a Hypothesis Test:
· State a Null Hypothesis: H0
This is generally a statement that is assumed to be true or based
on a known truth.
· State the Alternative Hypothesis: HA
This is the statement you are trying to support
· Determine the desired level of significance a (alpha)
This is the probability level that would allow you to reject the
Null Hypothesis with Confidence.
· Calculate the p-value
This is the probability of your observed results
· Compare the p-value to the level of significance a
If the p-value is below the level of significance, then the H0 is
rejected in favor of the HA; if not, H0 remains the operating
hypothesis. It is important to note the H0 is never PROVEN. It
can be supported by the data or discredited by the data.
Likewise, HA is never PROVEN. It can become the new
hypothesis, because the old hypothesis is discredited. This is
not proof of certainty.
Example: Perform a hypothesis test for our work with SAT Prep
in DoW #5
· H0: The sample mean score of the SAT Prep students is the
same as the population mean score for all SAT students.
H0:
· HA: The sample mean score of the SAT Prep students is
greater than the population mean score for all SAT Students
HA: Note: We had three choices for the HA. The sample mean
() could be greater than the population mean, less than the
population mean, or just not equal to the population mean. We
chose greater than here, because the claim is that SAT Prep
improves the scores.
· The level of significance can vary from situation to situation.
A strong level of significance is ( 0.5%).

In this setting, we mean that IF the probability of the sample of
50 SAT Prep students having a mean score of 1000 is LESS
THAN 0.5% under H0, we would consider it to be statistically
significant and that H0 would be rejected.
· Calculate the p-value for the sample mean.
We did this when we talked about Central Limit Theorem
earlier. Recall the sampling distribution:
s = 24.6
Based on this, we calculate the probability of getting a sample
mean at or above 1000:
p =Normalcdf( 1000, 99999999, 896, 24.6) = 0.00001
This means that if the sample of 50 SAT Prep scores truly were
the same as every other sample of 50 scores, the likelihood of
getting a sample mean as high as 1000 would be 0.00001or
.01% – highly unlikely!
· Since the p-value (.0001) is well below the level of significant
(0.5%), we reject the null hypothesis in favor of the alternative
hypothesis. We now assume that the mean score of the SAT
Prep students is higher than the mean SAT score for the
population.
·
Inv 2, Activity H: Hypothesis Testing
Evaluating a hypothesis test can be done using the graphing
calculator or online tools. There are two hypothesis tests to be
familiar with for testing sample means. This Activity introduces
you to these two tests.
Exercise H1: Complete a hypothesis test comparing the Sample
Mean for SAT Prep to the known population mean for the SAT
using Hypothesis Test Mean. This video describes how to do the
T-test on a TI-84 or 83.

This link provides a calculator for the Hypothesis Test Mean.
http://easycalculation.com/statistics/hypothesis-test-population-
mean.php
Exercise H2: Complete a hypothesis test comparing two sample
mean SAT Scores using Hypothesis Test 2 Means. This video
describes how to do the T-test for two means on the TI-84 or
83.
Diversity in the
Key Assignment
Mathematics Classroom
EQUITY IN MATHEMATICS EDUCATION
Research Assignment
Select a subgroup from the NCTM’s recommended subgroups
list located near the bottom of this page. It is suggested that
you choose a group with which you have little or no
professional experience. Acquaint yourself with some of the
literature that supports work with learners who identify with
your chosen group. Supporting articles may be found at
http://www.nctm.org/search.aspx?c=all&q=Equity by entering
the name of your desired subgroup in the Search Box.
1. create a list of 10 (or more if you are so inclined) important

considerations that would aide in guiding teachers in their work
with diversified math learners within their classrooms
2. This list should be carefully crafted, articulate and
comprehensive
◦The list should reflect the overarching themes of the articles
(i.e., what did you highlight, what are the salient themes?)
3. Provide a short (paragraph) reflection on the importance of
such an activity and the relevance this has in your role as an
educator
•Your Key assignment should include:
◦A brief introduction about the topic you chose and why (just a
few sentences)
◦A bulleted list (of your 10 or so items) (This can also be in
table form, etc)
◦A closing paragraph that serves as a reflection (importance of
such an activity and the relevance this has in your role as an
educator)
◦List of references
The NCTM Subgroups are:
· AT-RISK LEARNERS
· ENGLISH LANGUAGE LEARNERS
· GENDER EQUITY
· GIFTED STUDENTS
· MINORITY STUDENTS
· SPECIAL NEEDS/DISABILITIES

Please refer to the Equity in Mathematics Education: Research
Assignment rubric for additional details about the scoring
criteria.
Rubric for Key Assessment Equity in Mathematics Education-
Research Assignment
Course:
(Diversity in the Mathematics Classroom)
Standard / Competency addressed
Criteria
4
Distinguished
3
Proficient
2
Developing Skills
1
Unsatisfactory
Common Core State Standards
Standards for Mathematical Practice #5: Use Appropriate Tools
Strategically.
A mathematically proficient student at various grade levels is
able to identify relevant external mathematical resources, such
as digital content located on a website, and use them to pose or
solve problems. The student is able to use technological tools to
explore and deepen his/her understanding of concepts.
Student supplies additional addresses of at least 4 websites
chosen to augment information gathered on the NCTM website.
A clear and concise explanation of why these websites were

chosen, is provided.
Student supplies additional addresses of 2 websites chosen to
augment information gathered on the NCTM website. A suitable
explanation of why these websites were chosen, is provided.
Student supplies additional address of 1 website chosen to
augment information gathered on the NCTM website. Explains
why this websites was chosen, but offers limited detail.
Student supplies no additional addresses of any website that
would augment information gathered on the NCTM website, or
offers no explanation as to why any website was selected.
ELA
College and Career Readiness Anchor Standards for Writing #7
Produces clear and coherent writing in which the student
extracts an overall thesis from the research.
Student clearly, with clear and coherent writing, extracts an
overall thesis from the literature.
Student adequately extracts an overall thesis from the literature,
and presents it in an organized fashion.
Student presents an underdeveloped review of the academic
literature while making an inadequate attempt to extract an
overall thesis.
Student presents a deficient review of academic literature, and
makes no attempt to extract an overall thesis.
ELA
Summarizes the research while producing clear and coherent
writing.
Student clearly, coherently and adeptly paraphrases or
summarizes the research, while developing major ideas in
his/her own scholarly voice.

Student summarizes the research, developing major ideas with
good clarity.
Student is sometimes unclear as s/he summarizes the research
ideas from the literature.
In a manner that is unclear and incoherent, the student
summarizes the research from the literature.
ELA
Produce clear and coherent writing in which the thematic
connections between sources are drawn.
Student shows that thematic connections between sources are
clearly drawn and coherently explained.
Student identifies some of the thematic connections between
sources, and provides a thoughtful explanation of these
connections.
Student identifies some of the thematic connections between
sources, but provides an explanation that demonstrates little
thought towards understanding of these connections.
Student presents a poor understanding of thematic connections
between sources.
ELA
Produce clear and coherent writing in which organization, is
appropriate to task, purpose, and audience, by providing a list
of 10 considerations from the Equity subgroup.
Student provides a list of 10 considerations from the Equity
subgroup and demonstrates that each of the 10 considerations is
clearly distinguishable from the others.
Student develops a list of 10 considerations from the Equity
subgroup that was selected, but some of these exhibit

unnecessarily overlapping ideas.
Student develops a list of fewer than, or more than, 10
considerations from the Equity subgroup that was selected, or
exhibits a problem in clarity of expression.
Student develops a list of fewer than, or more than, 10
considerations from the Equity subgroup that was selected, that
are irrelevant or unsatisfactory overall.
ELA
Produce clear and coherent writing in which the development,
organization, and style are appropriate to task, purpose, and
audience, reflecting high academic standards.
Student carefully balances his/her own ideas with those of
others, while respectfully challenging “group-mates” to
maintain high academic standards.
Student integrates his/her own ideas with those of others, while
encouraging maintenance of high academic standards.
Student offers few ideas in the process of group assignment.
Relies on “group-mates” to do much of the work. High
academic standards are being compromised.
Student participation in group work is limited or unsatisfactory.
There is no evidence of high academic standards.
ELA
The student produces thoughtful feedback to peers’ suggested
revisions.

Student shows that evaluation of peers’ suggested revisions is
apparent. Through thoughtful integration of feedback, shows
expanded ideas.
Student incorporates peers’ suggested revisions without
compromising the thread of the initial ideas.
Student takes peers’ suggested revisions lightly, with little
thoughtful reflection.
Student demonstrates no evidence of feedback or contributing
toward peers’ suggested revisions.
ELA
College and Career Readiness Anchor Standards for Writing
#7
Student promotes non-biased language in countering another
group’s perspective, based on his/her own personal experience.
Writing is clear and concise with the writer incorporating an
active voice.
Student writes in a crisp, clear, and succinct manner,
incorporating an active voice. Appropriate use of pronouns,
modifiers and non-biased language is evident.
Student writes in a generally clear, but unnecessary manner
where, non-biased language is occasionally used to produce
words that counter a group’s perspective. Sometimes the
meaning is obscured.
Student writes in an unclear manner, avoiding non-biased
language, and using inappropriate or inaccurate words that
often.
Student writes in a manner that does not adequately support the
assignment, making no effort to avoid non-biased language.
ELA

Produce clear and coherent writing, with strong evidence of
being proofread, in which the development, organization, and
style are appropriate to task, purpose, and audience.
Student submits writing that is thoroughly proofread, with no
apparent errors.
Student submits writing that has been proofread in a cursory
manner (1 - 5 errors), but overall is unimpaired.
Student submits writing that has been poorly proofread (more
than 5 errors) that impair overall content.
Student submits writing that clearly has not been proofread,
rendering it unacceptable for graduate work.
Revised July 15, 2012

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