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Standard 4.4 Data Analysis, Probability, and
Discrete Mathematics
All students will develop an understanding of the concepts and
techniques of data analysis, probability, and discrete mathematics,
and will use them to model situations, solve problems, and analyze
and draw appropriate inferences from data. |
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Big Idea Data Analysis: Reading,
understanding, interpreting, and communicating data are critical in
modeling a variety of real-world situations, drawing appropriate
inferences, making informed decisions, and justifying those
decisions.
Big Idea Probability: Probability quantifies the
likelihood that something will happen and enables us to make
predictions and informed decisions.
Big Idea Discrete Mathematics: Discrete mathematics
consists of tools and strategies for representing, organizing, and
interpreting non-continuous data. |
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4.4.5 A.
Data Analysis |
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Descriptive Statement: In today's information-based world,
students need to be able to read, understand, and interpret data in
order to make informed decisions. In the early grades, students
should be involved in collecting and organizing data, and in
presenting it using tables, charts, and graphs. As they progress,
they should gather data using sampling, and should increasingly be
expected to analyze and make inferences from data, as well as to
analyze data and inferences made by others. |
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Essential Questions |
Enduring Understandings |
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- How can the collection, organization,
interpretation, and display of data be used to answer questions?
(4.5A4; 4.5A6; 4.5E1; 4.5E2; 4.5F1; 4.5F6) |
- The message conveyed by the data depends on how
the data is collected, represented, and summarized. (4.5A6; 4.5D6;
4.5E1; 4.5E2; 4.5E3)
- The results of a statistical investigation can be used to support
or refute an argument. (4.5D1; 4.5D3; 4.5D5; 4.5E2; 4.5E3; 4.5F6) |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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1.
Collect, generate, organize, and display data
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Data generated
from surveys |
Assessment of this CPI is generally within the
context of CPI 4.4.5A2. |
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2. Read, interpret, select,
construct, analyze, generate questions about, and draw inferences from displays
of data.
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Bar graph,
line graph, circle graph, table
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Range, median, and
mean
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Sample SCR Item: On five tests of 100
points each, José has an average of exactly 90. What is the lowest
score he could have made on any of the five tests? |
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3.
Respond to questions about data
and generate their own questions and hypotheses. |
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4.4.5
B. Probability |
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Descriptive Statement: Students need to understand the
fundamental concepts of probability so that they can interpret
weather forecasts, avoid unfair games of chance, and make informed
decisions about medical treatments whose success rate is provided in
terms of percentages. They should regularly be engaged in predicting
and determining probabilities, often based on experiments (like
flipping a coin 100 times), but eventually based on theoretical
discussions of probability that make use of systematic counting
strategies. High school students should use probability models and
solve problems involving compound events and sampling. |
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Essential Questions |
Enduring Understandings |
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- How can experimental and theoretical
probabilities be used to make predictions or draw conclusions?
(4.5D5; 4.5D6) |
- Experimental results tend to approach
theoretical probabilities after a large number of trials. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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1.
Determine probabilities of events.
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Event, probability of an event
· Probability of certain event is 1 and of impossible event is 0
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Sample SCR Item: Mike has a number cube
with the letter "M" on all six faces. What is the probability of his
rolling an "M" on his next roll? |
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2. Determine
probability using intuitive, experimental, and theoretical methods (e.g., using
model of picking items of different colors from a bag).
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Given numbers of various types of items in a bag, what is the
probability that an item of one type will be picked
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Given data obtained experimentally, what is the likely
distribution of items in the bag
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Sample SCR Item: If there are seven
marbles in a bag, three red and four green, what is the probability
that a marble picked from the bag will be red? (Answer: 3/7 or 3 out
of 7)
Sample MC Item: Cynthia has a bag of 10 marbles that contains 4
red marbles and 6 blue marbles. If Cynthia reached into the bag
without looking and picked one marble, what is the probability that
she would pick a blue marble?
a. 1/10
b. 2/5
* c.
3/5
d. 1 |
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3.
Model situations involving probability using simulations (with
spinners, dice) and theoretical models. |
Instructional Focus:
•This CPI is largely an instructional CPI. Assessment of this CPI is
generally within the context of one or more of the other content
CPIs. |
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4.4.5
C. Discrete Mathematics - Systematic Listing And Counting |
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Descriptive Statement: Development of strategies for
listing and counting can progress through all grade levels, with
middle and high school students using the strategies to solve
problems in probability. Primary students, for example, might find
all outfits that can be worn using two coats and three hats; middle
school students might systematically list and count the number of
routes from one site on a map to another; and high school students
might determine the number of three-person delegations that can be
selected from their class to visit the mayor. |
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Essential Questions |
Enduring Understandings |
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- How can attributes be used to classify
data/objects?
- What is the best way to solve this? What counting strategy works
best here? |
- Grouping by attributes (classification) can be
used to answer mathematical questions. (4.5E1; 4.5E3)
- Algorithms can effectively and efficiently be used to quantify and
interpret discrete information. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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Organized lists, charts, tree diagrams, tables
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Sample ECR Item: Out of six students who
have expressed interest in student government, represent all
possibilities for a slate of three officers, using a list, a chart,
or a tree diagram.
Sample MC Item: Four fifth-graders are scheduled to have their
picture taken as a group. If they are going to stand side-by-side,
in how many ways can they be arranged?
a. 8
b. 12
c. 16
* d.
24 |
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2.
Explore the
multiplication principle of counting in simple situations by representing all
possibilities in an organized way (e.g., you can make 3
x 4 = 12 outfits using 3 shirts and 4
skirts). |
Instructional Focus:
• This Content should be introduced at this grade level, but mastery
of the content is not assessed in statewide assessment at this grade
level. |
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4.4.6 D. Discrete Mathematics - Vertex-Edge Graphs And Algorithms |
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Descriptive Statement: Vertex-edge graphs, consisting of
dots (vertices) and lines joining them (edges), can be used to
represent and solve problems based on real-world situations.
Students should learn to follow and devise lists of instructions,
called "algorithms," and use algorithmic thinking to find the best
solution to problems like those involving vertex-edge graphs, but
also to solve other problems. |
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Essential Questions |
Enduring Understandings |
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- How can visual tools such as networks
(vertex-edge graphs) be used to answer questions? (4.5E1; 4.5E3)
- How can algorithmic thinking be used to solve problems? |
- Optimization is finding the best solution
within given constraints.
- Algorithms can effectively and efficiently be used to quantify and
interpret discrete information. |
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Areas of Focus/Cumulative Progress Indicators |
Comments and Examples |
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1.
Devise strategies for winning
simple games (e.g., start with two piles of objects, each of two players in turn
removes any number of objects from a single pile, and the person to take the
last group of objects wins) and express those strategies as sets of directions. |
Sample MC Item: Joe and Janet are playing a game in which
they take turns removing one or two counters from a pile. Whoever
takes the last counter wins. There are 4 counters left, and it is
Janet's turn. How many counters should she take?
*a. 1
b. 2
c. It does not matter; she will win.
d. It does not matter; she will lose. |
