Unit 8: Data Sets and Distributions
Lesson 2: Statistical Questions

In this lesson, students continue to analyze questions and the kinds of responses they can expect from those questions. They begin to recognize variability in data and learn about statistical questions and how they differ from non-statistical questions. In order to define variability, students categorize data sets and name the categories to make use of structure (MP7) because they are seeking mathematically important similarities between the objects.

This short video and interactive assessment activity is designed to teach fifth graders about subtracting capacities in compound units (english units).

This short video and interactive assessment activity is designed to teach fifth graders about subtraction in grams.

This short video and interactive assessment activity is designed to teach fifth graders about subtraction in kilograms.

Unit: Area and Surface Area
Lesson 18: Surface Area of a Cube

In this lesson, students practice using exponents of 2 and 3 to express products and to write square and cubic units. Along the way, they look for and make use of structure in numerical expressions (MP7). They also look for and express regularity in repeated reasoning (MP8) to write the formula for the surface area of a cube. Students will continue this work later in the course, in the unit on expressions and equations.

Note: Students will need to bring in a personal collection of 10–50 small objects ahead of time for the first lesson of the next unit. Examples include rocks, seashells, trading cards, or coins.

This learning video presents an introduction to graph theory through two fun, puzzle-like problems: ''The Seven Bridges of Konigsberg'' and ''The Chinese Postman Problem''. Any high school student in a college-preparatory math class should be able to participate in this lesson. Materials needed include: pen and paper for the students; if possible, printed-out copies of the graphs and image that are used in the module; and a blackboard or equivalent. During this video lesson, students will learn graph theory by finding a route through a city/town/village without crossing the same path twice. They will also learn to determine the length of the shortest route that covers all the roads in a city/town/village. To achieve these two learning objectives, they will use nodes and arcs to create a graph and represent a real problem.

This short video and interactive assessment activity is designed to teach third graders about given the perimeter, find the side length and area - squares.

This short video and interactive assessment activity is designed to teach fifth graders about using the four operations with money - word problems.

This short video and interactive assessment activity is designed to teach fifth graders about multiplication and division with money.

Unit 8: Data Sets and Distributions
Lesson 17: Using Box Plots

In the previous lesson, students analyzed a dot plot and a box plot in order to study the distribution of a data set. They saw that, while the box plot summarizes the distribution of the data and highlights some key measures, it was not possible to know all the data values of the distribution from the box plot alone. In this lesson, students use box plots to make sense of the data in context (MP2), compare distributions, and answer statistical questions about them.

Students compare box plots for distributions that have the same median but different IQRs, as well as box plots with the same IQRs but different medians. They recognize and articulate that the centers are the same but the spreads are different in the first case, and the centers are different but the spreads are the same in the second case. They use this understanding to compare typical members of different groups in terms of the context of the problem (MP2).

Unit 8: Data Sets and Distributions
Lesson 18: Using Data to Solve Problems

This lesson is a good opportunity for students to use the information they have learned in the unit and apply it to different situations, but may be shortened to fit time constraints.

In this lesson, students compare the center and spread of different distributions. They determine what these different measures (mean and MAD or median and IQR) represent in context. They select an appropriate representation for the distribution based on the structure of the data, an appropriate set of measures of center and spread, and interpret their meaning in the context (MP4).

For students who are curious why we are asking them to compute measures of center and variation by hand when computers would be more efficient and accurate, tell them that understanding the meaning of the values and knowing what questions to ask are skills computers have not yet mastered. By practicing with calculations on small data sets, students are becoming familiar with these measures as well as questioning skills so they can correctly interpret results from computers in the future. If students do not raise the question themselves, this point may be left until the topics are revisited in later grades.

Unit 8: Data Sets and Distributions
Lesson 5: Using Dot Plots to Answer Statistical Questions

In this lesson, students continue to use dot plots to develop their understanding of center and spread—by identifying values of center, describing spread, comparing centers and spreads of different distributions, and making use of the structure of the distributions (MP7) to understand them in the context of situations (MP2). In future lessons, they will make their descriptions and analyses more precise, as they learn about measures of center and spread.

Unit 8: Data Sets and Distributions
Lesson 7: Using Histograms to Answer Statistical Questions

In this lesson, students create, read, and interpret histograms (MP2). They characterize the distribution displayed in a histogram in terms of its shape and spread, and identify a measurement that is typical for the data set by looking for the center in a histogram (MP7). Students also use histograms to make comparisons and to better understand what different spreads and values of center mean in a given context.

Unit 8: Data Sets and Distributions
Lesson 12: Using Mean and MAD to Make Comparisons

In this lesson, students continue to develop their understanding of the mean and MAD as measures of center and spread as well as interpret these values in context. They practice computing the mean and the MAD for distributions; compare distributions with the same MAD but different means; and interpret the mean and MAD in the context of the data (MP2).

This short video and interactive assessment activity is designed to teach second graders about using symbol maps with compound units.

Students develop a physical understanding for the meaning of equality by trying to find equal lengths using rods.

This short video and interactive assessment activity is designed to teach second graders about visual differences in length (english units).

This short video and interactive assessment activity is designed to teach fifth graders about visual objects: comparing length.

This short video and interactive assessment activity is designed to teach second graders an overview of capacity (metric units).

Challenged with a hypothetical engineering work situation in which they need to figure out the volume and surface area of a nuclear power plant’s cooling tower (a hyperbolic shape), students learn to calculate the volume of complex solids that can be classified as solids of revolution or solids with known cross sections. These objects of complex shape defy standard procedures to compute volumes. Even calculus techniques depend on the ability to perform multiple measurements of the objects or find functional descriptions of their edges. During both guided and independent practice, students use (free GeoGebra) geometry software, a photograph of the object, a known dimension of it, a spreadsheet application and integral calculus techniques to calculate the volume of complex shape solids within a margin of error of less than 5%—an approach that can be used to compute the volumes of big or small objects. This activity is suitable for the end of the second semester of AP Calculus classes, serving as a major grade for the last six-week period, with students’ project results presentation grades used as the second semester final test.