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Select the cluster for resources on this page:

**Standards**:

- IC-A.1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
- IC-A.2. Decide if a specified model is
consistent with results from a given data-generating process, e.g.,
using simulation.
*For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model*?

**Technology-enhanced investigations:**

Stattrek.com Statistics Dictionary contains the following key vocabulary for this domain. Select the term from the Stattrek pull-down menu. Use with IC-A, IC-B:

experiment | observational study | population | mean |

margin of error | parameter | random sample | statistics |

significance level | randomization | sample survey | treatment |

Calculator.net: Sample Size Calculator

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

- Lesson set: Understand statistics as a process for making inferences: Seven video lessons: Distinguish between population and sample, take a simple random sample; take a systematic, stratified, convenience, voluntary sample; choose a sampling method given a situation. Aligns with IC-A.1.
- Lesson set: Understand statistics as a process for making inferences about populations: Four video lessons: Determine the population and parameter from a statistical question, take a random sample, obtain a random sample from a simulation, reduce variation by increasing sample size. Aligns with IC-A.1.
- Lesson set: Decide if a model is consistent with results: Five video lessons: Test a model using samples, using a simulation with random numbers, using a simple simulation, against given results; test a hypothesis for a population parameter. Aligns with IC-A.2.
- Lesson set: Decide if a specified model is consistent with results: Four video lessons: Compare theoretical and empirical results, explain and use the Law of Large Numbers, evaluate the effectiveness of a treatment, design a simulation. Aligns with IC-A.2.

MIT BLOSSOMS: Video lessons with additional teacher and learner resources. Descriptions are from the video summaries.

- Flu Math Games: "This video lesson shows students that math can play a role in understanding how an infectious disease spreads and how it can be controlled." Additional simulations are included. Aligns with Algebra standards SSE-B.3.c and REI-A.1; Function standards IF-C.8.b, BF-B.4.a, and LE-A-1.(a, c); and Statistics and Probability standards ID-B.6.a, IC-A.1, IC-B.4, CP-A.2, and MD-A.1.
- Averages: Still Flawed: "This learning video continues the theme of an early BLOSSOMS lesson, Flaws of Averages, using new examples—including how all the children from Lake Wobegon can be above average, as well as the Friendship Paradox. As mentioned in the original module, averages are often worthwhile representations of a set of data by a single descriptive number. The objective of this module, once again, is to simply point out a few pitfalls that could arise if one is not attentive to details when calculating and interpreting averages." Aligns with IC-A.2.

PhET Interactive Simulations: Plinko Probability. Per its description: "Drop balls through a triangular grid of pegs and see them accumulate in containers. Switch to a histogram view and compare the distribution of balls to an ideal binomial distribution. Adjust the binary probability and develop your knowledge of statistics!" Aligns with ID-A.1, ID-A.2, and IC-A.2.

Statistics Canada: Statistics: Power from Data: Sampling Methods. Use with IC-A.1.

Stat Trek: Tutorial: Simulation of Random Events

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 7 manipulatives addressing IC-A.1 and 2 manipulatives addressing IC-A.2: Among those are:

- Random samples and random permutations: "This demonstration is intended for its usefulness to instructors and students in elementary statistics to replace the use of random number tables." Aligns with IC-A.1.
- Decisions based on p-values and significance levels: Aligns with IC-A.1, IC-A.2.
- Confidence Intervals: Confidence Level, Sample Size, and Margin of Error: Aligns with IC-A.1, IC-A.2.
- Confidence intervals for the mean: Aligns with IC-A.1, IC-B.4.
- Hypothesis tests about a population mean: Aligns with IC-A.1, IC-B.5.
- The Power of a Test Concerning the Mean of a Normal Population: Aligns with IC-A.1, IC-B.5.

**Multiple Choice:**

Khan Academy: HSS-S-IC includes HSS-IC.A.1 and HSS-IC.A.2 multiple choice and constructed-response Common Core aligned problems.

Also see the Significance Tests (Hypothesis testing) unit with practice questions with videos.

- Valid claims: Aligns with IC-A.1.
- Simple hypothesis testing: Aligns with IC-A.2.

**Constructed-response:**

**Performance tasks:**

Illustrative Mathematics: Statistics and Probability:

- Strict Parents: This task aligns with standards IC-A.1 and IC-B.3.
- Sarah, the Chimpanzee: This task aligns with standard IC-A.2.

Mathematics Vision Project, Secondary 3 Student Edition:

- Module 8: Statistics: This module contains 8 classroom tasks. Task 5 and task 7 align with standard IC-A.1. Tasks 6-8 address IC-A.2. Additionally task 8 also addresses IC-B.3.

NCTM's Reasoning and Sense Making Task Library: Eruptions: Old Faithful includes the task overview, teacher notes for its use, and student activity sheet. Aligns with IC-A.1, ID-A.1, and mathematical practice standards 1, 3, and 5.

Statistics in Schools from the U.S. Census Bureau: Activities: Math: 9-12:

- Educational Attainment and Marriage: Testing a Correlation Coefficient's Significance. Students "develop, justify, and evaluate conjectures about the relationship between two quantitative variables over time in the United States: the median age (in years) when women first marry and the percentage of women aged 25–34 with a bachelor’s degree or higher. Students will write a regression equation for the data, interpret in context the linear model’s slope and y-intercept, and find the correlation coefficient (r), assessing the strength of the linear relationship and whether a significant relationship exists between the variables. Students will then summarize their conclusions and consider whether correlation implies causation." Aligns with standards ID-C.8, ID-C.9, and IC-A.1.

Related to the above task is the article: What Is The Aim Of Finding Correlation? Why Is It Used If Correlation Doesn’t Imply Causation? posted at ScienceABC. Author Argha Sengupta (2022, July 8) delved into Finding Meaning From Random Data: Exploratory Analysis and Correlation May Not Imply Causation, the latter of which might be owing to an influence of a third variable.

**Standards**:

- IC-B.3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
- IC-B.4. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
- IC-B.5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
- IC-B.6. Evaluate reports based on data.

**Technology-enhanced investigations:**

Khan Academy: Videos: Bernoulli distributions and margins of error. Use with IC-B.4.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

- Lesson set: Distinguish between surveys, experiments, and observational studies; relate randomization to each: Four video lessons: Conduct a survey and select a sampling method; develop and conduct a statistical experiment, an observational study; select a type of study. Aligns with IC-B.3.
- Lesson set: Recognize the purposes of and differences among research methods, including how they relate to randomization: Five video lessons: Distinguish between observational studies, surveys, and experiments, determine whether an investigation uses a simple random sample or a systematic sample, determine the differences between a cluster sample and a stratified sample, understand sources and types of bias, avoid bias. Aligns with IC-B.3.
- Lesson set: Use survey data to estimate means and proportions, develop a margin of error through simulation models, and evaluate reports based on data: Three video lessons: Determine the likelihood that a hypothesis is reasonable, approximating a sampling distribution from a simulation, decrease the size of a confidence interval. Aligns with IC-B.4.
- Lesson set: Use data from a sample survey and evaluate reports based on data: Six video lessons: Distinguish between descriptive and inferential statistics, between different types of survey research; generate survey data through simulations, use comparative box plots to compare data sets, interpret margin of error, construct and interpret a confidence interval. Aligns with IC-B.4 and IC-B.6.
- Lesson set: Use data from a randomized experiment to compare treatments; evaluate reports based on data: Five video lessons: Design and select a design method for a statistical experiment; compare experimental treatments by comparing box plots, state null and alternative hypotheses, compare treatments using a resampling strategy. Aligns with IC-B.5.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources. Description is from the video summary. Flu Math Games: "This video lesson shows students that math can play a role in understanding how an infectious disease spreads and how it can be controlled." Additional simulations are included. Aligns with Algebra standards SSE-B.3.c and REI-A.1; Function standards IF-C.8.b, BF-B.4.a, and LE-A-1.(a, c); and Statistics nd Probability standards ID-B.6.a, IC-A.1, IC-B.4, CP-A.2, and MD-A.1.

Shodor Interactivate: Statistics and Probability Making Inferences and Justifying Conclusions: Understand and evaluate random processes underlying statistical experiments. This includes 10 lessons and 22 activities with virtual manipulatives involving simulations and experiments.

Statistics How To: Misleading Graphs: Real Life Examples

Stat Trek: Tutorials on Surveys and Experiments: Surveys: Data Collection Methods: This tutorial presents four main methods of data collection: census, sample surveys, experiments, and observational studies; What is an experiment?; and Experimental Design in Statistics

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 1 manipulative addressing IC-B.3, 3 manipulatives addressing IC-B.4, and 2 manipulatives addressing IC-B.5. Among those are:

- Randomly filling an array: Aligns with IC-B.3, IC-B.4.
- Confidence intervals for the mean: Aligns with IC-A.1, IC-B.4.
- The Central Limit Theorem: "The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the size of the sample grows." Aligns with IC-B.4.
- Hypothesis tests about a population mean: Aligns with IC-A.1, IC-B.5.
- The Power of a Test Concerning the Mean of a Normal Population: Aligns with IC-A.1, IC-B.5.

**MMultiple Choice:**

Khan Academy: HSS-S-IC includes HSS-IC.B.3 and HSS-IC.B.5 through HSS-IC.B.6 Common Core aligned problems.

Also see the following practice questions with videos.

- Types of statistical studies: Aligns with IC-B.3, IC-B.6.
- Hypothesis testing in experiments: Aligns with IC-B.5.

Interpret graphs from the Earth Observatory at NASA. Analyze some temperature and precipitation graphs from different cities and match them up to the correct biome.

**Constructed-response:**

**Performance tasks:**

Illustrative Mathematics: Statistics and Probability:

- Strict Parents: Aligns with IC-B.3 and IC-A.1.
- Words and Music II: Aligns with IC-B.3.

Mathematics Assessment Project: Standards: High School: Statistics & Probability: Task 217: Interpreting Statistics: A Case of Muddying the Waters

Mathematics Vision Project, Secondary 3 Student Edition:

- Module 8: Statistics: This module contains 8 classroom tasks. Task 5 and task 7 align with standard IC-A.1. Tasks 6-8 address IC-A.2. Additionally task 8 also addresses IC-B.3.

Common Core Math:
Intro | HS Statistics & Probability Domain:
ID | **IC** | CP | MD |