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- Mathematics: Precalculus
Arizona - Mathematics: Precalculus
High Academic Standards for Students | Adopted: 2018
RFR: : Reasoning with Functions and Relations
RFR.AF: : Analyze functions
RFR.AF.1: : Interpret parameters of a function defined by an expression in the context of the situation.
Absolute Value Equations and Inequalities
Solve an inequality involving absolute values using a graph of the absolute-value function. Vary the terms of the absolute-value function and vary the value that you are comparing it to. Then explore how the graph and solution set change in response. 5 Minute Preview
Absolute Value with Linear Functions
Compare the graph of a linear function, the graph of an absolute-value function, and the graphs of their translations. Vary the coefficients and constants in the functions and investigate how the graphs change in response. 5 Minute Preview
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
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
Exponential Growth and Decay
Explore the graph of the exponential growth or decay function. Vary the initial amount and the rate of growth or decay and investigate the changes to the graph. 5 Minute Preview
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
RFR.AF.2: : Sketch the graph of a function that models a relationship between two quantities, identifying key features.
Absolute Value Equations and Inequalities
Solve an inequality involving absolute values using a graph of the absolute-value function. Vary the terms of the absolute-value function and vary the value that you are comparing it to. Then explore how the graph and solution set change in response. 5 Minute Preview
Absolute Value with Linear Functions
Compare the graph of a linear function, the graph of an absolute-value function, and the graphs of their translations. Vary the coefficients and constants in the functions and investigate how the graphs change in response. 5 Minute Preview
Cosine Function
Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Exponential Functions
Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview
General Form of a Rational Function
Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview
Graphs of Polynomial Functions
Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview
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
Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
Logarithmic Functions: Translating and Scaling
Vary the values in the equation of a logarithmic function and examine how the graph is translated or scaled. Connect these transformations with the domain of the function, and the asymptote in the graph. 5 Minute Preview
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
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
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
Quadratics in Vertex Form
Compare the graph of a quadratic to its equation in vertex form. Vary the terms of the equation and explore how the graph changes in response. 5 Minute Preview
Radical Functions
Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview
Rational Functions
Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview
Roots of a Quadratic
Find the root of a quadratic using its graph or the quadratic formula. Explore the graph of the roots and the point of symmetry in the complex plane. Compare the axis of symmetry and graph of the quadratic in the real plane. 5 Minute Preview
Sine Function
Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Slope-Intercept Form of a Line
Compare the slope-intercept form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview
Tangent Function
Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview
Translating and Scaling Functions
Vary the coefficients in the equation of a function and examine how the graph of the function is translated or scaled. Select different functions to translate and scale, and determine what they have in common. 5 Minute Preview
Translating and Scaling Sine and Cosine Functions
Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview
RFR.AF.3: : Interpret key features of graphs and tables for a function that models a relationship between two quantities in terms of the quantities.
Absolute Value with Linear Functions
Compare the graph of a linear function, the graph of an absolute-value function, and the graphs of their translations. Vary the coefficients and constants in the functions and investigate how the graphs change in response. 5 Minute Preview
Distance-Time Graphs
Create a graph of a runner's position versus time and watch the runner complete a 40-yard dash based on the graph you made. Notice the connection between the slope of the line and the speed of the runner. What will the runner do if the slope of the line is zero? What if the slope is negative? Add a second runner (a second graph) and connect real-world meaning to the intersection of two graphs. 5 Minute Preview
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
Exponential Functions
Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview
General Form of a Rational Function
Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview
Graphs of Polynomial Functions
Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview
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
Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
Logarithmic Functions: Translating and Scaling
Vary the values in the equation of a logarithmic function and examine how the graph is translated or scaled. Connect these transformations with the domain of the function, and the asymptote in the graph. 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
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
Quadratics in Vertex Form
Compare the graph of a quadratic to its equation in vertex form. Vary the terms of the equation and explore how the graph changes in response. 5 Minute Preview
Radical Functions
Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview
Rational Functions
Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview
Standard Form of a Line
Compare the standard form of a linear equation to its graph. Vary the coefficients and explore how the graph changes in response. 5 Minute Preview
RFR.AF.4: : Use limits to describe long-range behavior, asymptotic behavior, and points of discontinuity.
Exponential Functions
Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview
General Form of a Rational Function
Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview
Graphs of Polynomial Functions
Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview
Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
Logarithmic Functions: Translating and Scaling
Vary the values in the equation of a logarithmic function and examine how the graph is translated or scaled. Connect these transformations with the domain of the function, and the asymptote in the graph. 5 Minute Preview
Rational Functions
Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview
RFR.AF.5: : Sketch the graph of all six trigonometric functions, identifying key features.
Cosine Function
Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Sine Function
Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Tangent Function
Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview
Translating and Scaling Functions
Vary the coefficients in the equation of a function and examine how the graph of the function is translated or scaled. Select different functions to translate and scale, and determine what they have in common. 5 Minute Preview
Translating and Scaling Sine and Cosine Functions
Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview
RFR.AF.b: : Students should have opportunities to… Calculate, interpret, and use average rate of change.
Cat and Mouse (Modeling with Linear Systems)
Experiment with a system of two lines representing a cat-and-mouse chase. Adjust the speeds of the cat and mouse and the head start of the mouse, and immediately see the effects on the graph and on the chase. Connect real-world meaning to slope, y-intercept, and the intersection of lines. 5 Minute Preview
Graphs of Derivative Functions
What does the graph of a derivative function look like? What can a derivative function tell you about the original function? What can't it tell you? Explore these questions for five different types of functions: linear, quadratic, cubic, absolute value, and sine. 5 Minute Preview
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
RFR.AF.c: : Students should have opportunities to… Use multiple representations of function models appropriately.
Absolute Value Equations and Inequalities
Solve an inequality involving absolute values using a graph of the absolute-value function. Vary the terms of the absolute-value function and vary the value that you are comparing it to. Then explore how the graph and solution set change in response. 5 Minute Preview
Absolute Value with Linear Functions
Compare the graph of a linear function, the graph of an absolute-value function, and the graphs of their translations. Vary the coefficients and constants in the functions and investigate how the graphs change in response. 5 Minute Preview
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
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
Exponential Growth and Decay
Explore the graph of the exponential growth or decay function. Vary the initial amount and the rate of growth or decay and investigate the changes to the graph. 5 Minute Preview
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
RFR.AF.d: : Students should have opportunities to… Work with different families of functions beyond linear and quadratic functions including but not limited to exponential, logarithmic, rational, polynomial, logistic, radical, and piecewise-defined functions.
Absolute Value Equations and Inequalities
Solve an inequality involving absolute values using a graph of the absolute-value function. Vary the terms of the absolute-value function and vary the value that you are comparing it to. Then explore how the graph and solution set change in response. 5 Minute Preview
Absolute Value with Linear Functions
Compare the graph of a linear function, the graph of an absolute-value function, and the graphs of their translations. Vary the coefficients and constants in the functions and investigate how the graphs change in response. 5 Minute Preview
Cosine Function
Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Exponential Functions
Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview
Exponential Growth and Decay
Explore the graph of the exponential growth or decay function. Vary the initial amount and the rate of growth or decay and investigate the changes to the graph. 5 Minute Preview
General Form of a Rational Function
Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview
Graphs of Polynomial Functions
Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview
Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
Logarithmic Functions: Translating and Scaling
Vary the values in the equation of a logarithmic function and examine how the graph is translated or scaled. Connect these transformations with the domain of the function, and the asymptote in the graph. 5 Minute Preview
Polynomials and Linear Factors
Create a polynomial as a product of linear factors. Vary the values in the linear factors to see how their connection to the roots of the function. 5 Minute Preview
Radical Functions
Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview
Rational Functions
Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview
Sine Function
Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Tangent Function
Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview
Translating and Scaling Functions
Vary the coefficients in the equation of a function and examine how the graph of the function is translated or scaled. Select different functions to translate and scale, and determine what they have in common. 5 Minute Preview
Translating and Scaling Sine and Cosine Functions
Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview
RFR.AF.e: : Students should have opportunities to… Graph rational functions including those whose graphs contain horizontal asymptotes, vertical asymptotes, oblique asymptotes, and/or holes.
General Form of a Rational Function
Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview
Rational Functions
Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview
RFR.AF.f: : Students should have opportunities to… Make sense of radian measure.
Radians
As factory belt operator, your job is to move boxes just the right distance on the belt, so they can be stamped for delivery. Your only controls are the radius and rotation of the belt’s wheel. How do you set these to get the distance right? See how this relates to arc length, and discover how radians help make this task easier. 5 Minute Preview
RFR.AF.g: : Students should have opportunities to… Use the unit circle as a tool to graph the trig functions in radians and degrees.
Cosine Function
Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Sine Function
Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Tangent Function
Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview
RFR.AF.h: : Students should have opportunities to… Develop fluency with the unit circle. Include opportunities beyond the special angles, for example, explain why sin(1.1) > sin(0.3) in radians.
Cosine Function
Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Sine Function
Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Tangent Function
Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview
RFR.AF.i: : Students should have opportunities to… Graph sine, cosine, tangent functions in radians and degrees and analyze/explain the characteristics of each.
Cosine Function
Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Sine Function
Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Tangent Function
Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview
Translating and Scaling Sine and Cosine Functions
Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview
RFR.BF: : Building Functions
RFR.BF.1: : Model relationships between quantities that require adding, subtracting, multiplying, and/or dividing functions.
Addition and Subtraction of Functions
Explore the graphs of two polynomials and the graph of their sum or difference. Vary the coefficients in the polynomials and investigate how the graphs change in response. 5 Minute Preview
RFR.BF.4: : Determine if a function has an inverse. If so, find the inverse. If not, define a restriction on the domain that meets the requirement for invertibility and find the inverse on the restricted domain.
Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
Radical Functions
Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview
RFR.BF.5: : Interpret the meanings of quantities involving functions and their inverses.
Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
RFR.BF.6: : Verify by analytical methods that one function is the inverse of another.
Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
RFR.BF.a: : Students should have opportunities to… Model real-world situations with the sum, difference, product, or quotient of other function models.
Addition and Subtraction of Functions
Explore the graphs of two polynomials and the graph of their sum or difference. Vary the coefficients in the polynomials and investigate how the graphs change in response. 5 Minute Preview
RFR.BF.b: : Students should have opportunities to… Describe relationships of quantities in functions and within a composition of those functions.
Absolute Value with Linear Functions
Compare the graph of a linear function, the graph of an absolute-value function, and the graphs of their translations. Vary the coefficients and constants in the functions and investigate how the graphs change in response. 5 Minute Preview
Exponential Functions
Explore the graph of an exponential function. Vary the coefficient and base of the function and investigate the changes to the graph of the function. 5 Minute Preview
General Form of a Rational Function
Compare the equation of a rational function to its graph. Multiply or divide the numerator and denominator by linear factors and explore how the graph changes in response. 5 Minute Preview
Graphs of Polynomial Functions
Study the graphs of polynomials up to the fourth degree. Vary the coefficients of the equation and investigate how the graph changes in response. Explore things like intercepts, end behavior, and even near-zero behavior. 5 Minute Preview
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
Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
Logarithmic Functions: Translating and Scaling
Vary the values in the equation of a logarithmic function and examine how the graph is translated or scaled. Connect these transformations with the domain of the function, and the asymptote in the graph. 5 Minute Preview
Radical Functions
Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview
Rational Functions
Compare the graph of a rational function to its equation. Vary the terms of the equation and explore how the graph is translated and stretched as a result. Examine the domain on a number line and compare it to the graph of the equation. 5 Minute Preview
RFR.BF.c: : Students should have opportunities to… Find an inverse algebraically, for example given y = f(x) algebraically find x = f^-1(y). Work with exponential and logarithmic functions, and quadratic and square root functions at a minimum.
Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
Radical Functions
Compare the graph of a radical function to its equation. Vary the terms of the equation. Explore how the graph is translated and stretched by the changes to the equation. 5 Minute Preview
RFR.BF.e: : Students should have opportunities to… Describe the meaning of f^-1(20) given a function f that takes hours as an input and gives miles as an output.
Logarithmic Functions
Compare the equation of a logarithmic function to its graph. Change the base of the logarithmic function and examine how the graph changes in response. Use the line y = x to compare the associated exponential function. 5 Minute Preview
RFR.IC: : Interpreting Conics
RFR.IC.1: : Model real-world situations which involve conic sections.
Circles
Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview
Hyperbolas
Compare the equation of a hyperbola to its graph. Vary the terms of the equation of the hyperbola. Examine how the graph of the hyperbola and its asymptotes changes in response. 5 Minute Preview
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
RFR.IC.2: : Identify key features of conic sections (foci, directrix, radii, axes, asymptotes, center) graphically and algebraically.
Circles
Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview
Ellipses
Compare the equation of an ellipse to its graph. Vary the terms of the equation of the ellipse and examine how the graph changes in response. Drag the vertices and foci, explore their Pythagorean relationship, and discover the string property. 5 Minute Preview
Hyperbolas
Compare the equation of a hyperbola to its graph. Vary the terms of the equation of the hyperbola. Examine how the graph of the hyperbola and its asymptotes changes in response. 5 Minute Preview
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
RFR.IC.3: : Sketch a graph of a conic section using its key features.
Circles
Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview
Ellipses
Compare the equation of an ellipse to its graph. Vary the terms of the equation of the ellipse and examine how the graph changes in response. Drag the vertices and foci, explore their Pythagorean relationship, and discover the string property. 5 Minute Preview
Hyperbolas
Compare the equation of a hyperbola to its graph. Vary the terms of the equation of the hyperbola. Examine how the graph of the hyperbola and its asymptotes changes in response. 5 Minute Preview
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
RFR.IC.4: : Use the key features of a conic section to write its equation.
Circles
Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview
Ellipses
Compare the equation of an ellipse to its graph. Vary the terms of the equation of the ellipse and examine how the graph changes in response. Drag the vertices and foci, explore their Pythagorean relationship, and discover the string property. 5 Minute Preview
Hyperbolas
Compare the equation of a hyperbola to its graph. Vary the terms of the equation of the hyperbola. Examine how the graph of the hyperbola and its asymptotes changes in response. 5 Minute Preview
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
RFR.IC.5: : Given a quadratic equation of the form ax² + by² + cx + dy + e = 0, determine if the equation is a circle, ellipse, parabola, or hyperbola.
Circles
Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview
Ellipses
Compare the equation of an ellipse to its graph. Vary the terms of the equation of the ellipse and examine how the graph changes in response. Drag the vertices and foci, explore their Pythagorean relationship, and discover the string property. 5 Minute Preview
RFR.IC.a: : Students should have opportunities to… Explore conics as loci of points satisfying stipulated conditions.
Circles
Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview
Ellipses
Compare the equation of an ellipse to its graph. Vary the terms of the equation of the ellipse and examine how the graph changes in response. Drag the vertices and foci, explore their Pythagorean relationship, and discover the string property. 5 Minute Preview
Hyperbolas
Compare the equation of a hyperbola to its graph. Vary the terms of the equation of the hyperbola. Examine how the graph of the hyperbola and its asymptotes changes in response. 5 Minute Preview
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
RFR.IC.b: : Students should have opportunities to… Explore conic sections with technology and manipulatives.
Circles
Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview
Ellipses
Compare the equation of an ellipse to its graph. Vary the terms of the equation of the ellipse and examine how the graph changes in response. Drag the vertices and foci, explore their Pythagorean relationship, and discover the string property. 5 Minute Preview
Hyperbolas
Compare the equation of a hyperbola to its graph. Vary the terms of the equation of the hyperbola. Examine how the graph of the hyperbola and its asymptotes changes in response. 5 Minute Preview
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
RFR.IC.c: : Students should have opportunities to… Connect the geometric and algebraic relationships of conics.
Circles
Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview
Ellipses
Compare the equation of an ellipse to its graph. Vary the terms of the equation of the ellipse and examine how the graph changes in response. Drag the vertices and foci, explore their Pythagorean relationship, and discover the string property. 5 Minute Preview
Hyperbolas
Compare the equation of a hyperbola to its graph. Vary the terms of the equation of the hyperbola. Examine how the graph of the hyperbola and its asymptotes changes in response. 5 Minute Preview
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
RFR.IC.d: : Students should have opportunities to… Use the method of completing the square to put the equation of the conic section into standard form.
Circles
Compare the graph of a circle with its equation. Vary the terms in the equation and explore how the circle is translated and scaled in response. 5 Minute Preview
RFR.ISS: : Interpreting Sequences and Series
RFR.ISS.1: : Model real-world situations involving sequences or series using recursive and/or explicit definitions.
Arithmetic Sequences
Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview
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
RFR.ISS.2: : Use covariational reasoning to describe sequences and series.
Arithmetic Sequences
Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview
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
Geometric Sequences
Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview
RFR.ISS.a: : Students should have opportunities to… Become fluent in working with arithmetic and geometric sequences and series.
Arithmetic Sequences
Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview
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
Geometric Sequences
Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview
RFR.ISS.b: : Students should have opportunities to… Explore several types of sequences, including but not limited to Fibonacci, telescoping, harmonic, alternating.
Arithmetic Sequences
Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview
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
RT: : Reasoning with Trigonometry
RT.ETT: : Extending Triangle Trigonometry
RFR.ETT.1: : Model real-world situations involving trigonometry.
Sine, Cosine, and Tangent Ratios
Reshape and resize a right triangle and examine how the sine of angle A, the cosine of angle A, and the tangent of angle A change. 5 Minute Preview
Sound Beats and Sine Waves
Listen to and see interference patterns produced by sound waves with similar frequencies. Test your ability to distinguish and match sounds as musicians do when they tune their instruments. Calculate the number of "sound beats" you will hear based on the frequency of each sound. [Note: Headphones are recommended for this Gizmo.] 5 Minute Preview
Translating and Scaling Sine and Cosine Functions
Experiment with the graph of a sine or cosine function. Explore how changing the values in the equation can translate or scale the graph of the function. 5 Minute Preview
RFR.ETT.4: : Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi - x, pi + x, and 2pi - x in terms of their values for x, where x is any real number.
Cosine Function
Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Sine Function
Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Tangent Function
Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview
RFR.ETT.5: : Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Cosine Function
Compare the graph of the cosine function with the graph of the angle on the unit circle. Drag a point along the cosine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Sine Function
Compare the graph of the sine function with the graph of the angle on the unit circle. Drag a point along the sine curve and see the corresponding angle on the unit circle. 5 Minute Preview
Tangent Function
Compare the graph of the tangent function with the graph of the angle on the unit circle. Drag a point along the tangent curve and see the corresponding angle on the unit circle. 5 Minute Preview
RT.ETT.a: : Students should have opportunities to… Go beyond using the mnemonic SohCahToa by interpreting the meaning of the trigonometric ratios as multiplicative comparisons of the appropriate sides of a right triangle. Situations should involve unknown sides and/or angles, as well as periodic functions and their inverses.
Sine, Cosine, and Tangent Ratios
Reshape and resize a right triangle and examine how the sine of angle A, the cosine of angle A, and the tangent of angle A change. 5 Minute Preview
RT.RTS: : Reasoning with Trigonometric Structure
RT.RTS.1: : Use the structure of a trigonometric expression to identify ways to rewrite it.
Simplifying Trigonometric Expressions
Choose the correct steps to simplify a trigonometric function. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview
RT.RTS.2: : Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Simplifying Trigonometric Expressions
Choose the correct steps to simplify a trigonometric function. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview
RT.RTS.a: : Students should have opportunities to… Recognize that a single identity can be manipulated into another identity.
Simplifying Trigonometric Expressions
Choose the correct steps to simplify a trigonometric function. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview
RT.RTS.b: : Students should have opportunities to… Apply the Pythagorean, sum, difference, double angle, and half angle formulas for sine and cosine to reveal and explain properties.
Simplifying Trigonometric Expressions
Choose the correct steps to simplify a trigonometric function. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview
Sum and Difference Identities for Sine and Cosine
Choose the correct steps to evaluate a trigonometric expression using sum and difference identities. Use step-by-step feedback to diagnose incorrect steps. 5 Minute Preview
RT.EPE: : Exploring Polar Equations
RT.EPE.a: : Students should have opportunities to… Convert points between polar and rectangular forms.
Points in the Complex Plane
Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview
RT.EPE.b: : Students should have opportunities to… Determine equivalent polar representations for a given point.
Points in the Complex Plane
Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview
RV: : Reasoning with Vectors
RV.EV: : Exploring Vectors
RV.EV.1: : Recognize vector quantities as having both magnitude and direction.
Adding Vectors
Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview
Vectors
Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview
RV.EV.2: : Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.
Adding Vectors
Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview
Vectors
Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview
RV.EV.3: : Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
Adding Vectors
Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview
RV.EV.4: : Solve problems involving velocity and other quantities that can be represented by vectors.
2D Collisions
Investigate elastic collisions in two dimensions using two frictionless pucks. The mass, velocity, and initial position of each puck can be modified to create a variety of scenarios. 5 Minute Preview
Adding Vectors
Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview
Golf Range
Try to get a hole in one by adjusting the velocity and launch angle of a golf ball. Explore the physics of projectile motion in a frictional or ideal setting. Horizontal and vertical velocity vectors can be displayed, as well as the path of the ball. The height of the golfer and the force of gravity are also adjustable. 5 Minute Preview
Vectors
Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview
RV.EV.5: : Add and subtract vectors, and multiply a vector by a scalar.
Adding Vectors
Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview
Vectors
Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview
RV.EV.a: : Students should have opportunities to… Distinguish vector quantities from scalar quantities. For example, distinguish the difference between velocity and speed.
2D Collisions
Investigate elastic collisions in two dimensions using two frictionless pucks. The mass, velocity, and initial position of each puck can be modified to create a variety of scenarios. 5 Minute Preview
Adding Vectors
Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview
Vectors
Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview
RV.EV.b: : Students should have opportunities to… Make sense of operations with vectors.
2D Collisions
Investigate elastic collisions in two dimensions using two frictionless pucks. The mass, velocity, and initial position of each puck can be modified to create a variety of scenarios. 5 Minute Preview
Adding Vectors
Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview
Vectors
Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview
RM: : Reasoning with Matrices
RM.UM: : Using Matrices
RM.UM.1: : Use matrices to represent and manipulate data.
Dilations
Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in
Solving Linear Systems (Matrices and Special Solutions)
Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview
Translations
Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview
RM.UM.2: : Use matrix operations to solve problems. Add, subtract, and multiply matrices of appropriate dimensions. Multiply matrices by scalars to produce new matrices.
Dilations
Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in
Solving Linear Systems (Matrices and Special Solutions)
Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview
Translations
Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview
RM.UM.3: : Find the inverse and determinant of a matrix.
Solving Linear Systems (Matrices and Special Solutions)
Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview
RM.UM.4: : Use matrices to solve systems of linear equations.
Solving Linear Systems (Matrices and Special Solutions)
Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview
RM.UM.a: : Students should have opportunities to… Explore the properties of matrices and their operations.
Dilations
Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in
Solving Linear Systems (Matrices and Special Solutions)
Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview
Translations
Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview
RM.UM.b: : Students should have opportunities to… Explain that the determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
Solving Linear Systems (Matrices and Special Solutions)
Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview
RM.UM.c: : Students should have opportunities to… Explore the roles of the zero matrix, identity matrix, inverse matrix, and the determinant of a matrix.
Solving Linear Systems (Matrices and Special Solutions)
Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview
RM.UM.d: : Students should have opportunities to… Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
Dilations
Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in
Translations
Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview
RM.UM.e: : Students should have opportunities to… Use the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers.
Translations
Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview
RM.UM.f: : Students should have opportunities to… Use matrices as a tool. Including but not limited to: producing new vectors from old vectors, creating transformations in the plane, calculating the area of geometric figures.
Dilations
Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in
Translations
Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview
RM.UM.g: : Students should have opportunities to… Explore matrices and solve problems with and without technology, as appropriate.
Dilations
Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in
Solving Linear Systems (Matrices and Special Solutions)
Explore systems of linear equations, and how many solutions a system can have. Express systems in matrix form. See how the determinant of the coefficient matrix reveals how many solutions a system of equations has. Also, use a draggable green point to see what it means for an (x, y) point to be a solution of an equation, or of a system of equations. 5 Minute Preview
Translations
Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview
PC.MP: : Standards for Mathematical Practices
PC.MP.1: : Make sense of problems and persevere in solving them.
Biconditional Statements
Make a biconditional statement from a given definition using word tiles. Use both symbolic form and standard English form. 5 Minute Preview
Conditional Statements
Make a conditional statement from a given fact using word tiles. Use both symbolic form and standard English form. 5 Minute Preview
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
Pattern Flip (Patterns)
In the Pattern Flip carnival game, you are shown a pattern of cards. The first cards are face-up so you can see the pattern, and the rest are face-down. Can you guess which animals are on the face-down cards? Use one of the preset patterns, or make your own custom pattern. Good luck! 5 Minute Preview
5.1.1: : Mathematically proficient students explain to themselves the meaning of a problem, look for entry points to begin work on the problem, and plan and choose a solution pathway. While engaging in productive struggle to solve a problem, they continually ask themselves, “Does this make sense?' to monitor and evaluate their progress and change course if necessary. Once they have a solution, they look back at the problem to determine if the solution is reasonable and accurate. Mathematically proficient students check their solutions to problems using different methods, approaches, or representations. They also compare and understand different representations of problems and different solution pathways, both their own and those of others.
Biconditional Statements
Make a biconditional statement from a given definition using word tiles. Use both symbolic form and standard English form. 5 Minute Preview
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
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
Linear Inequalities in Two Variables
Find the solution set to a linear inequality in two variables using the graph of the linear inequality. Vary the terms of the inequality and vary the inequality symbol. Examine how the boundary line and shaded region change in response. 5 Minute Preview
Modeling One-Step Equations
Solve a linear equation using a tile model. Use feedback to diagnose incorrect steps. 5 Minute Preview
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
Pattern Flip (Patterns)
In the Pattern Flip carnival game, you are shown a pattern of cards. The first cards are face-up so you can see the pattern, and the rest are face-down. Can you guess which animals are on the face-down cards? Use one of the preset patterns, or make your own custom pattern. Good luck! 5 Minute Preview
Polling: City
Poll residents in a large city to determine their response to a yes-or-no question. Estimate the actual percentage of yes votes in the whole city. Examine the results of many polls to help assess how reliable the results from a single poll are. See how the normal curve approximates a binomial distribution for large enough polls. 5 Minute Preview
Solving Equations on the Number Line
Solve an equation involving decimals using dynamic arrows on a number line. 5 Minute Preview
Using Algebraic Expressions
Translate algebraic expressions into English phrases, and translate English phrases into algebraic expressions. Read the expression or phrase and select word tiles or symbol tiles to form the corresponding phrase or expression. 5 Minute Preview
PC.MP.2: : Reason abstractly and quantitatively.
Conditional Statements
Make a conditional statement from a given fact using word tiles. Use both symbolic form and standard English form. 5 Minute Preview
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
PC.MP.3: : Construct viable arguments and critique the reasoning of others.
Biconditional Statements
Make a biconditional statement from a given definition using word tiles. Use both symbolic form and standard English form. 5 Minute Preview
5.3.1: : Mathematically proficient students construct mathematical arguments (explain the reasoning underlying a strategy, solution, or conjecture) using concrete, pictorial, or symbolic referents. Arguments may also rely on definitions, assumptions, previously established results, properties, or structures. Mathematically proficient students make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. Mathematically proficient students present their arguments in the form of representations, actions on those representations, and explanations in words (oral or written). Students critique others by affirming or questioning the reasoning of others. They can listen to or read the reasoning of others, decide whether it makes sense, ask questions to clarify or improve the reasoning, and validate or build on it. Mathematically proficient students can communicate their arguments, compare them to others, and reconsider their own arguments in response to the critiques of others.
Biconditional Statements
Make a biconditional statement from a given definition using word tiles. Use both symbolic form and standard English form. 5 Minute Preview
Conditional Statements
Make a conditional statement from a given fact using word tiles. Use both symbolic form and standard English form. 5 Minute Preview
PC.MP.4: : Model with mathematics.
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
PC.MP.5: : Use appropriate tools strategically.
Elapsed Time
Calculate the difference between the times given by two analog clocks. Rotate the hands of the clocks to change the time and see how the calculation changes. 5 Minute Preview
5.5.1: : Mathematically proficient students consider available tools when solving a mathematical problem. They choose tools that are relevant and useful to the problem at hand. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful; recognizing both the insight to be gained and their limitations. Students deepen their understanding of mathematical concepts when using tools to visualize, explore, compare, communicate, make and test predictions, and understand the thinking of others.
Segment and Angle Bisectors
Explore the special properties of a point that lies on the perpendicular bisector of a segment, and of a point that lies on an angle bisector. Manipulate the point, the segment, and the angle to see that these properties are always true. 5 Minute Preview
PC.MP.6: : Attend to precision.
Biconditional Statements
Make a biconditional statement from a given definition using word tiles. Use both symbolic form and standard English form. 5 Minute Preview
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
Using Algebraic Expressions
Translate algebraic expressions into English phrases, and translate English phrases into algebraic expressions. Read the expression or phrase and select word tiles or symbol tiles to form the corresponding phrase or expression. 5 Minute Preview
5.6.1: : Mathematically proficient students clearly communicate to others using appropriate mathematical terminology, and craft explanations that convey their reasoning. When making mathematical arguments about a solution, strategy, or conjecture, they describe mathematical relationships and connect their words clearly to their representations. Mathematically proficient students understand meanings of symbols used in mathematics, calculate accurately and efficiently, label quantities appropriately, and record their work clearly and concisely.
Arithmetic Sequences
Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview
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
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
Function Machines 2 (Functions, Tables, and Graphs)
Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. 5 Minute Preview
Geometric Sequences
Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview
Pattern Flip (Patterns)
In the Pattern Flip carnival game, you are shown a pattern of cards. The first cards are face-up so you can see the pattern, and the rest are face-down. Can you guess which animals are on the face-down cards? Use one of the preset patterns, or make your own custom pattern. Good luck! 5 Minute Preview
PC.MP.7: : Look for and make use of structure.
Pattern Flip (Patterns)
In the Pattern Flip carnival game, you are shown a pattern of cards. The first cards are face-up so you can see the pattern, and the rest are face-down. Can you guess which animals are on the face-down cards? Use one of the preset patterns, or make your own custom pattern. Good luck! 5 Minute Preview
5.7.1: : Mathematically proficient students use structure and patterns to assist in making connections among mathematical ideas or concepts when making sense of mathematics. Students recognize and apply general mathematical rules to complex situations. They are able to compose and decompose mathematical ideas and notations into familiar relationships. Mathematically proficient students manage their own progress, stepping back for an overview and shifting perspective when needed.
Arithmetic Sequences
Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview
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
Function Machines 2 (Functions, Tables, and Graphs)
Drop a number into a function machine, and see what number comes out! You can use one of the six pre-set function machines, or program your own function rule into one of the blank machines. Stack up to three function machines together. Input and output can be recorded in a table and on a graph. 5 Minute Preview
Geometric Sequences
Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview
Pattern Flip (Patterns)
In the Pattern Flip carnival game, you are shown a pattern of cards. The first cards are face-up so you can see the pattern, and the rest are face-down. Can you guess which animals are on the face-down cards? Use one of the preset patterns, or make your own custom pattern. Good luck! 5 Minute Preview
PC.MP.8: : Look for and express regularity in repeated reasoning.
Arithmetic Sequences
Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview
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
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
Geometric Sequences
Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview
Pattern Finder
Observe frogs jumping around on colored lily pads. Find, test, and reason about patterns you see in their jumping. 5 Minute Preview
Pattern Flip (Patterns)
In the Pattern Flip carnival game, you are shown a pattern of cards. The first cards are face-up so you can see the pattern, and the rest are face-down. Can you guess which animals are on the face-down cards? Use one of the preset patterns, or make your own custom pattern. Good luck! 5 Minute Preview
5.8.1: : Mathematically proficient students look for and describe regularities as they solve multiple related problems. They formulate conjectures about what they notice and communicate observations with precision. While solving problems, students maintain oversight of the process and continually evaluate the reasonableness of their results. This informs and strengthens their understanding of the structure of mathematics which leads to fluency.
Arithmetic Sequences
Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response. 5 Minute Preview
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
Geometric Sequences
Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas. 5 Minute Preview
Correlation last revised: 3/25/2021
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.
Student progress is automatically saved so that STEM Cases can be completed over multiple sessions.
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.
How Free Gizmos Work
Start teaching with 20-40 Free Gizmos. See the full list.
Access lesson materials for Free Gizmos including teacher guides, lesson plans, and more.
All other Gizmos are limited to a 5 Minute Preview and can only be used for 5 minutes a day.
Free Gizmos change each semester. The new collection will be available January 1 and July 1.
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