CCSS.Math.Content.HSN-Q.A: : Reason quantitatively and use units to solve problems.
CCSS.Math.Content.HSN-Q.A.1: : Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Density
Measure the mass and volume of a variety of objects, then place them into a beaker of liquid to see if they float or sink. Learn to predict whether objects will float or sink in water based on their mass and volume. Compare how objects float or sink in a variety of liquids, including gasoline, oil, seawater, and corn syrup.
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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.
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Create a graph of a runner's position versus time and watch the runner complete a 40-meter 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.
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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.
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Create a graph of a runner's position versus time and watch the runner run a 40-meter 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.
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Explore the energy used by many household appliances, such as television sets, hair dryers, lights, computers, etc. Make estimates for how long each item is used on a daily basis to get an estimate for the total power consumed during a day, a week, a month, and a year, and how that relates to consumer costs and environmental impact.
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CCSS.Math.Content.HSN-Q.A.3: : Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Unit Conversions 2 - Scientific Notation and Significant Digits
Use the Unit Conversions Gizmo to explore the concepts of scientific notation and significant digits. Convert numbers to and from scientific notation. Determine the number of significant digits in a measured value and in a calculation.
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CCSS.Math.Content.HSN-CN: : The Complex Number System
CCSS.Math.Content.HSN-CN.A: : Perform arithmetic operations with complex numbers.
CCSS.Math.Content.HSN-CN.A.1: : Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.
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.
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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.
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CCSS.Math.Content.HSN-CN.A.2: : Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
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.
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CCSS.Math.Content.HSN-CN.A.3: : Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
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.
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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.
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CCSS.Math.Content.HSN-CN.B: : Represent complex numbers and their operations on the complex plane.
CCSS.Math.Content.HSN-CN.B.4: : Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
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.
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CCSS.Math.Content.HSN-CN.B.5: : Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
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.
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CCSS.Math.Content.HSN-CN.B.6: : Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
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.
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CCSS.Math.Content.HSN-CN.C: : Use complex numbers in polynomial identities and equations.
CCSS.Math.Content.HSN-CN.C.7: : Solve quadratic equations with real coefficients that have complex solutions.
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.
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CCSS.Math.Content.HSN-CN.C.8: : Extend polynomial identities to the complex numbers.
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.
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CCSS.Math.Content.HSN-VM: : Vector and Matrix Quantities
CCSS.Math.Content.HSN-VM.A: : Represent and model with vector quantities.
CCSS.Math.Content.HSN-VM.A.1: : Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
Adding Vectors
Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation.
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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.
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CCSS.Math.Content.HSN-VM.A.2: : 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.
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CCSS.Math.Content.HSN-VM.A.3: : 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.
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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.
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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.
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CCSS.Math.Content.HSN-VM.B: : Perform operations on vectors.
CCSS.Math.Content.HSN-VM.B.4: : Add and subtract vectors.
CCSS.Math.Content.HSN-VM.B.4.a: : Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
Adding Vectors
Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation.
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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.
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CCSS.Math.Content.HSN-VM.C: : Perform operations on matrices and use matrices in applications.
CCSS.Math.Content.HSN-VM.C.7: : Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
Dilations
Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in (x, y) form and in matrix form.
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CCSS.Math.Content.HSN-VM.C.10: : Understand that 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. 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.
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CCSS.Math.Content.HSN-VM.C.12: : Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
Translations
Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation.
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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.
