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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 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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By the end of Grade 12: |
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1.
Use surveys and
sampling techniques to generate data and draw conclusions about large groups.
·
Advantages/disadvantages of sample selection methods (e.g., convenience
sampling, responses to survey, random sampling)
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2.
Evaluate the use
of data in real-world contexts.
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Accuracy and
reasonableness of conclusions drawn
·
Bias in
conclusions drawn (e.g., influence of how data is displayed)
·
Statistical claims
based on sampling
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3.
Design a
statistical experiment, conduct the experiment, and interpret and communicate
the outcome |
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4.
Estimate or
determine lines of best fit (or curves of best fit if appropriate) with
technology, and use them to interpolate within the range of the data. |
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5.
Analyze data using
technology, and use statistical terminology to describe conclusions.
·
Measures of
dispersion: variance, standard deviation, outliers
·
Correlation
coefficient
·
Normal
distribution (e.g., approximately 95% of the sample lies between two standard
deviations on either side of the mean)
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4.4
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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By the end of Grade 12: |
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1.
Calculate the expected value of a probability-based game, given
the probabilities and payoffs of the various outcomes, and determine whether the
game is fair. |
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2.
Use concepts and
formulas of area to calculate geometric probabilities. |
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3.
Model situations
involving probability with simulations (using spinners, dice, calculators and
computers) and theoretical models, and solve problems using these models. |
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4.
Determine
probabilities in complex situations.
·
Conditional events
·
Complementary
events
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Dependent and
independent events |
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5.
Estimate
probabilities and make predictions based on experimental and theoretical
probabilities. |
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6.
Understand and use
the “law of large numbers” (that experimental results tend to approach
theoretical probabilities after a large number of trials). |
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4.4
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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By the end of Grade 12: |
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1.
Calculate
combinations with replacement (e.g., the number of possible ways of tossing a
coin 5 times and getting 3 heads) and without replacement (e.g., number of
possible delegations of 3 out of 23 students). |
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2.
Apply the multiplication rule of counting in complex situations,
recognize the difference between situations with replacement and without
replacement, and recognize the difference between ordered and unordered counting
situations. |
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3.
Justify solutions to counting problems. |
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4.
Recognize and explain relationships involving combinations and
Pascal’s Triangle, and apply those methods to situations involving probability |
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4.4 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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By the end of Grade 12: |
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1.
Use vertex-edge graphs and algorithmic thinking to represent and
solve practical problems.
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Circuits that include every edge in a graph
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Circuits that include every vertex in a graph
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Scheduling problems (e.g., when project meetings should be
scheduled to avoid conflicts) using graph coloring
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Applications to science (e.g., who-eats-whom graphs, genetic
trees, molecular structures)
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2.
Explore strategies for making fair decisions.
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Combining individual preferences into a group decision (e.g.,
determining winner of an election or selection process)
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Determining how many Student Council representatives each class (9th,
10th, 11th, and 12th grade) gets when the
classes have unequal sizes (apportionment) |
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