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Connected Mathematics (2004)
1. Prime Time
1.1. The Factor Game
Finding Factors with Area Models
Find factors of a number using an area model. Reshape the area rectangle to see different factorizations of the number. Find the prime factorization using a factor tree. 5 Minute Preview
1.5. Factorizations
Finding Factors with Area Models
Find factors of a number using an area model. Reshape the area rectangle to see different factorizations of the number. Find the prime factorization using a factor tree. 5 Minute Preview
1.6. The Locker Problem
Finding Patterns
Build a pattern to complete a sequence of patterns. Study a sequence of three patterns of squares in a grid and build the fourth pattern of the sequence in a grid. 5 Minute Preview
2. Data About Us
2.1. Looking at Data
Mean, Median, and Mode
Build a data set and find the mean, median, and mode. Explore the mean, median, and mode illustrated as frogs on a seesaw, frogs on a scale, and as frogs stacked under a bar of variable height. 5 Minute Preview
Mean, Median, and Mode
Build a data set and find the mean, median, and mode. Explore the mean, median, and mode illustrated as frogs on a seesaw, frogs on a scale, and as frogs stacked under a bar of variable height. 5 Minute Preview
Describing Data Using Statistics
Investigate the mean, median, mode, and range of a data set through its graph. Manipulate the data and watch how the mean, median, mode, and range change (or, in some cases, how they don't change). 5 Minute Preview
2.3. Using Graphs to Group Data
Stem-and-Leaf Plots
Build a data set and compare the line plot of the data set to the stem-and-leaf plot. 5 Minute Preview
2.4. Coordinate Graphs
Trends in Scatter Plots
Examine the scatter plot for a random data set with negative or positive correlation. Vary the correlation and explore how correlation is reflected in the scatter plot and the trend line. 5 Minute Preview
Solving Using Trend Lines
Examine the scatter plots for data related to weather at different latitudes. The Gizmo includes three different data sets, one with negative correlation, one positive, and one with no correlation. Compare the least squares best-fit line. 5 Minute Preview
3. Shapes and Designs
3.2. Building Polygons
Triangle Inequalities
Discover the inequalities related to the side lengths and angle measures of a triangle. Reshape and resize the triangle to confirm that these properties are true for all triangles. 5 Minute Preview
3.3. Polygons and Angles
3.5. Side-Angle-Shape Connections
Rotations, Reflections, and Translations
Rotate, reflect, and translate a figure in the plane. Compare the translated figure to the original figure. 5 Minute Preview
Classifying Quadrilaterals
Apply constraints to a quadrilateral, and then reshape and resize it. Classify the figure by its constraints. Explore the differences between the different kinds of quadrilaterals. 5 Minute Preview
4. Bits and Pieces
4.2. Comparing Fractions
Modeling Fractions (Area Models)
Model and compare fractions using area models. Set the denominators with the arrow buttons, and then set the numerators with the arrow buttons or by clicking in the models. Compare fractions visually, on a number line, or numerically using the least common denominator. 5 Minute Preview
4.3. Cooking with Fractions
Improper Fractions and Mixed Numbers
Represent a quantity given by a shaded region as an improper fraction and as a mixed number. Experiment with different shaded regions sliced differently. 5 Minute Preview
4.4. From Fractions to Decimals
Fraction, Decimal, Percent (Area and Grid Models)
Model and compare fractions, decimals, and percents using area models. Each area model can have 10 or 100 sections and can be set to display a fraction, decimal, or percent. Click inside the area models to shade them. Compare the numbers visually or on a number line. 5 Minute Preview
4.6. Out of One Hundred
Percents, Fractions, and Decimals
Compare a quantity represented by an area with its percent, fraction, and decimal forms. 5 Minute Preview
5. Covering and Surrounding
5.5. Measuring Parallelograms
Area of Parallelograms
Examine and manipulate a parallelogram and find its area. Explore the relationship between the area of a parallelogram and the area of a rectangle using an animation. 5 Minute Preview
5.6. Measuring Triangles
Area of Parallelograms
Examine and manipulate a parallelogram and find its area. Explore the relationship between the area of a parallelogram and the area of a rectangle using an animation. 5 Minute Preview
5.7. Going Around in Circles
Circumference and Area of Circles
Resize a circle and compare its radius, circumference, and area. 5 Minute Preview
6. How Likely Is It?
6.1. A First Look at Chance
Probability Simulations
Experiment with spinners and compare the experimental probability of particular outcomes to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview
6.3. Using Spinners to Predict Chances
Probability Simulations
Experiment with spinners and compare the experimental probability of particular outcomes to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview
6.4. Theoretical probabilities
Theoretical and Experimental Probability
Experiment with spinners and compare the experimental probability of a particular outcome to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview
6.5. Analyzing Games of Chance
Probability Simulations
Experiment with spinners and compare the experimental probability of particular outcomes to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview
6.7. Probability and Genetics
Mouse Genetics (Two Traits)
Breed "pure" mice with known genotypes that exhibit specific fur and eye colors, and learn how traits are passed on via dominant and recessive genes. Mice can be stored in cages for future breeding, and the statistics of fur and eye color are reported every time a pair of mice breed. Punnett squares can be used to predict results. 5 Minute Preview
7. Bits and Pieces II
7.1. Using Percents
Percents and Proportions
Find a part from the percent and whole, a percent from the part and whole, or a whole from the part and percent using a graphic model. 5 Minute Preview
7.2. More About Percents
Percent of Change
Apply markups and discounts using interactive "percent rulers." Improve number sense for percents with this dynamic, visual tool. Reinforce the original cost (or original price) as the baseline for percent calculations. 5 Minute Preview
7.3. Estimating with Fractions and Decimals
Estimating Sums and Differences
Estimate the sum or difference of two fractions using area models. Compare estimates to exact sums and differences. 5 Minute Preview
7.4. Adding and Subtracting Fractions
Fractions with Unlike Denominators
Find the sum or difference of two fractions with unlike denominators using graphic models. Find the least common denominator graphically. 5 Minute Preview
7.5. Finding Areas and Other Products
Multiplying Fractions
Multiply two fractions using an area model. Vary the vertical area to change one fraction and vary the horizontal area to change the other. Then examine the intersection of the areas to find the product. 5 Minute Preview
7.6. Computing with Decimals
Multiplying with Decimals
Multiply two decimals using a dynamic area model. On a grid, shade the region with width equal to one of the decimals and height equal to the other decimal and find the area of the region. 5 Minute Preview
7.7. Dividing Fractions
Dividing Fractions
Divide fractions using area models. Adjust the numerators and denominators of the divisor and dividend and see how the area model and calculation change. 5 Minute Preview
8. Ruins of Montarek
8.1. Building Plans
3D and Orthographic Views
Arrange blocks in a three-dimensional space so that the top view, front view, and side view match the target top view, front view, and side view. 5 Minute Preview
8.2. Making Buildings
3D and Orthographic Views
Arrange blocks in a three-dimensional space so that the top view, front view, and side view match the target top view, front view, and side view. 5 Minute Preview
8.3. Describing Unique Buildings
3D and Orthographic Views
Arrange blocks in a three-dimensional space so that the top view, front view, and side view match the target top view, front view, and side view. 5 Minute Preview
9. Variables and Patterns
9.1. Variables and Coordinate Graphs
Points in the Coordinate Plane
Identify the coordinates of a point in the coordinate plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview
Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. 5 Minute Preview
9.2. Graphing Change
Distance-Time and Velocity-Time Graphs
Create a graph of a runner's position versus time and watch the runner run a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the velocity of the runner. Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. Also experiment with a graph of velocity versus time for the runners, and also distance traveled versus time. 5 Minute Preview
9.3. Analyzing Graphs and Tables
Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. 5 Minute Preview
Points, Lines, and Equations
Compare the graph of a linear function to its rule and to a table of its values. Change the function by dragging two points on the line. Examine how the rule and table change. 5 Minute Preview
9.4. Patterns and Rules
Introduction to Functions
Determine if a relation is a function using the mapping diagram, ordered pairs, or the graph of the relation. Drag arrows from the domain to the range, type in ordered pairs, or drag points to the graph to add inputs and outputs to the relation. 5 Minute Preview
10. Stretching and Shrinking
10.1. Enlarging Figures
Dilations
Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in
10.2. Similar Figures
Similar Figures
Vary the scale factor and rotation of an image and compare it to the preimage. Determine how the angle measures and side lengths of the two figures are related. 5 Minute Preview
10.4. Using Similarity
Perimeters and Areas of Similar Figures
Manipulate two similar figures and vary the scale factor to see what changes are possible under similarity. Explore how the perimeters and areas of two similar figures compare. 5 Minute Preview
11. Comparing and Scaling
11.1. Making Comparisons
Part-to-part and Part-to-whole Ratios
Compare a ratio represented by an area with its percent, fraction, and decimal forms. 5 Minute Preview
11.2. Comparing by Finding Percents
Percents and Proportions
Find a part from the percent and whole, a percent from the part and whole, or a whole from the part and percent using a graphic model. 5 Minute Preview
11.3. Comparing by Using Ratios
Part-to-part and Part-to-whole Ratios
Compare a ratio represented by an area with its percent, fraction, and decimal forms. 5 Minute Preview
11.5. Estimating Populations and Population Densities
Estimating Population Size
Adjust the number of fish in a lake to be tagged and the number of fish to be recaptured. Use the number of tagged fish in the catch to estimate the number of fish in the lake. 5 Minute Preview
12. Accentuate the Negative
12.1. Extending the Number Line
Integers, Opposites, and Absolute Values
Compare and order integers using draggable points on a number line. Also explore opposites and absolute values on the number line. 5 Minute Preview
Rational Numbers, Opposites, and Absolute Values
Use a number line to compare rational numbers. Change values by dragging points on the number line. Compare the opposites and absolute values of the numbers. 5 Minute Preview
12.2. Adding Integers
Adding on the Number Line
Add real numbers using dynamic arrows on a number line. Find the sum of the numbers at the end of the final arrow. Compare the numerical calculation. 5 Minute Preview
12.3. Subtracting Integers
Adding and Subtracting Integers
Add and subtract integers on a number line using dynamic arrows. 5 Minute Preview
12.5. Coordinate Grids
Points in the Coordinate Plane
Identify the coordinates of a point in the coordinate plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview
13. Moving Straight Ahead
13.1. Predicting from Patterns
Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. 5 Minute Preview
13.2. Walking Rates
Direct and Inverse Variation
Adjust the constant of variation and explore how the graph of the direct or inverse variation function changes in response. Compare direct variation functions to inverse variation functions. 5 Minute Preview
13.4. Solving Equations
Modeling and Solving Two-Step Equations
Solve a two-step equation using a cup-and-counter model. Use step-by-step feedback to diagnose and correct incorrect steps. 5 Minute Preview
13.5. Exploring Slope
Slope
Explore the slope of a line, and learn how to calculate slope. Adjust the line by moving points that are on the line, and see how its slope changes. 5 Minute Preview
13.6. Writing and Equation for a Line
Point-Slope Form of a Line
Compare the point-slope form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview
14. Filling and Wrapping
14.2. Designing Packages
Surface and Lateral Areas of Prisms and Cylinders
Vary the dimensions of a prism or cylinder and investigate how the surface area changes. Use the dynamic net of the solid to compute the lateral area and the surface area of the solid. 5 Minute Preview
14.3. Finding Volumes of Boxes
Prisms and Cylinders
Vary the height and base-edge or radius length of a prism or cylinder and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of an oblique prism or cylinder to the volume of a right prism or cylinder. 5 Minute Preview
14.4. Cylinders
Prisms and Cylinders
Vary the height and base-edge or radius length of a prism or cylinder and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of an oblique prism or cylinder to the volume of a right prism or cylinder. 5 Minute Preview
14.5. Cones and Spheres
Pyramids and Cones
Vary the height and base-edge or radius length of a pyramid or cone and examine how its three-dimensional representation changes. Determine the area of the base and the volume of the solid. Compare the volume of a skew pyramid or cone to the volume of a right pyramid or cone. 5 Minute Preview
15. What Do You Expect?
15.1. Evaluation Games of Chance
Probability Simulations
Experiment with spinners and compare the experimental probability of particular outcomes to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview
15.2. Analyzing Number-Cube Games
Theoretical and Experimental Probability
Experiment with spinners and compare the experimental probability of a particular outcome to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview
15.3. Probability and Area
Geometric Probability
Randomly throw darts at a target and see what percent are "hits." Vary the size of the target and repeat the experiment. Study the relationship between the area of the target and the percent of darts that strike it 5 Minute Preview
15.4. Analyzing Two-Stage Games
Independent and Dependent Events
Compare the theoretical and experimental probabilities of drawing colored marbles from a bag. Record results of successive draws to find the experimental probability. Perform the drawings with replacement of the marbles to study independent events, or without replacement to explore dependent events. 5 Minute Preview
15.5. Expected Value
Probability Simulations
Experiment with spinners and compare the experimental probability of particular outcomes to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes. 5 Minute Preview
15.6. Carnival Games
Independent and Dependent Events
Compare the theoretical and experimental probabilities of drawing colored marbles from a bag. Record results of successive draws to find the experimental probability. Perform the drawings with replacement of the marbles to study independent events, or without replacement to explore dependent events. 5 Minute Preview
15.7. Analyzing Sequences of Outcomes
Independent and Dependent Events
Compare the theoretical and experimental probabilities of drawing colored marbles from a bag. Record results of successive draws to find the experimental probability. Perform the drawings with replacement of the marbles to study independent events, or without replacement to explore dependent events. 5 Minute Preview
17. Thinking with Mathematical Models
17.1. Linear Models
Linear Functions
Determine if a relation is a function from the mapping diagram, ordered pairs, or graph. Use the graph to determine if it is linear. 5 Minute Preview
17.2. Nonlinear Models
Direct and Inverse Variation
Adjust the constant of variation and explore how the graph of the direct or inverse variation function changes in response. Compare direct variation functions to inverse variation functions. 5 Minute Preview
17.3. More Nonlinear Models
Arithmetic and Geometric Sequences
Find the value of individual terms in an arithmetic or geometric sequence using graphs of the sequence and direct computation. Vary the common difference and common ratio and examine how the sequence changes in response. 5 Minute Preview
17.4. A World of Patterns
Distance-Time and Velocity-Time Graphs
Create a graph of a runner's position versus time and watch the runner run a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the velocity of the runner. Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. Also experiment with a graph of velocity versus time for the runners, and also distance traveled versus time. 5 Minute Preview
18. Looking for Pythagoras
18.1. Locating Points
Points in the Coordinate Plane
Identify the coordinates of a point in the coordinate plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview
18.3. The Pythagorean Theorem
Pythagorean Theorem
Explore the Pythagorean Theorem using a dynamic right triangle. Examine a visual, geometric application of the Pythagorean Theorem, using the areas of squares on the sides of the triangle. 5 Minute Preview
19. Growing, Growing, Growing
19.1. Exponential Growth
Introduction to Exponential Functions
Explore the graph of the exponential function. Vary the initial amount and base of the function. Investigate the changes to the graph. 5 Minute Preview
19.2. Growth Patterns
Arithmetic and Geometric Sequences
Find the value of individual terms in an arithmetic or geometric sequence using graphs of the sequence and direct computation. Vary the common difference and common ratio and examine how the sequence changes in response. 5 Minute Preview
19.3. Growth Factors
Compound Interest
Explore compound interest in-depth, from compounded annually to compounded continuously. In addition, compare the END POINTS graph, with dots that fit an exponential curve, to the ALL TIME graph, which has a more step-like appearance. 5 Minute Preview
19.4. Exponential Decay
Introduction to Exponential Functions
Explore the graph of the exponential function. Vary the initial amount and base of the function. Investigate the changes to the graph. 5 Minute Preview
20. Frogs, Fleas, and Painted Cubes
20.1. Introduction to Quadratic Relationships
20.2. Quadratic Expressions
Parabolas
Explore parabolas in a conic section context. Find the relationship among the vertex, focus, and directrix of a parabola, and how that relates to its equation. 5 Minute Preview
Quadratics in Factored Form
Investigate the factors of a quadratic through its graph and through its equation. Vary the roots of the quadratic and examine how the graph and the equation change in response. 5 Minute Preview
20.4. What is a Quadratic Function?
Quadratics in Polynomial Form
Compare the graph of a quadratic to its equation in polynomial form. Vary the coefficients of the equation and explore how the graph changes in response. 5 Minute Preview
21. Say It with Symbols
21.1. Order of Operations
Order of Operations
Select and evaluate the operations in an expression following the correct order of operations. 5 Minute Preview
21.4. Solving Equations
Solving Two-Step Equations
Choose the correct steps to solve a two-step equation. Use the feedback to diagnose incorrect steps. 5 Minute Preview
Quadratics in Factored Form
Investigate the factors of a quadratic through its graph and through its equation. Vary the roots of the quadratic and examine how the graph and the equation change in response. 5 Minute Preview
22. Kaleidoscopes, Hubcaps, and Mirrors
22.1. Three Types of Symmetry
Rotations, Reflections, and Translations
Rotate, reflect, and translate a figure in the plane. Compare the translated figure to the original figure. 5 Minute Preview
Reflections
Reshape and resize a figure and examine how its reflection changes in response. Move the line of reflection and explore how the reflection is translated. 5 Minute Preview
23. Samples and Populations
23.1. Comparing Data Sets
Box-and-Whisker Plots
Construct a box-and-whisker plot to match a line plots, and construct a line plot to match a box-and-whisker plots. Manipulate the line plot and examine how the box-and-whisker plot changes. Then manipulate the box-and-whisker plot and examine how the line plot changes. 5 Minute Preview
Box-and-Whisker Plots
Construct a box-and-whisker plot to match a line plots, and construct a line plot to match a box-and-whisker plots. Manipulate the line plot and examine how the box-and-whisker plot changes. Then manipulate the box-and-whisker plot and examine how the line plot changes. 5 Minute Preview
Trends in Scatter Plots
Examine the scatter plot for a random data set with negative or positive correlation. Vary the correlation and explore how correlation is reflected in the scatter plot and the trend line. 5 Minute Preview
24. Clever Counting
24.2. Opening Locks
Permutations and Combinations
Experiment with permutations and combinations of a number of letters represented by letter tiles selected at random from a box. Count the permutations and combinations using a dynamic tree diagram, a dynamic list of permutations, and a dynamic computation by the counting principle. 5 Minute Preview
24.4. Deciding Whether Order is Important
Permutations and Combinations
Experiment with permutations and combinations of a number of letters represented by letter tiles selected at random from a box. Count the permutations and combinations using a dynamic tree diagram, a dynamic list of permutations, and a dynamic computation by the counting principle. 5 Minute Preview
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Start teaching with 20-40 Free Gizmos. See the full list.
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All other Gizmos are limited to a 5 Minute Preview and can only be used for 5 minutes a day.
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About STEM Cases
Students assume the role of a scientist trying to solve a real world problem. They use scientific practices to collect and analyze data, and form and test a hypothesis as they solve the problems.
Each STEM Case uses realtime reporting to show live student results.
Introduction to the Heatmap
STEM Cases take between 30-90 minutes for students to complete, depending on the case.
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Multiple grade-appropriate versions, or levels, exist for each STEM Case.
Each STEM Case level has an associated Handbook. These are interactive guides that focus on the science concepts underlying the case.
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