How to address violence in the news with your students.
Teachers can use GeoGebra to help make math more meaningful and visual for students. Teachers can quickly build digital worksheets that include simulations already created on GeoGebra. The tools also allow students to manipulate math concepts in one format and see them in another (such as how a 3D shape sits on a 2D plane, or how the algebraic function of a plane and a sphere changes as we change points on either -- or both). It helps students to make connections between different areas of math and how they relate to one another.
However, make sure to familiarize yourself with the extensive tutorials ahead of time in order to become an expert with the program and its capabilities before attempting to implement it in the classroom. Failure to do this could lead to some frustration for both teachers and students. Luckily, GeoGebra has made some great tutorials, and there's an extensive forum of questions and detailed definitions for each function. Each applet also has a number of built-in functions that provide hints and prompts to allow students to "play around" and see how things connect with one another. The downloaded programs have built-in help icons that take users right to the appropriate help section of the website when needed.Continue reading Show less
GeoGebra is a free software program that lets kids create mathematical constructions -- and models where they can drag objects and adjust parameters -- to explore algebra and geometry simultaneously (along with other math fields). It's browser-based and also has downloadable applets for computer and mobile devices. GeoGebra offers kids and teachers the option to use existing math explorations or build their own, and there are a lot of tools available to do this. The existing pool of explorations is vast, so it covers most high school Common Core math expectations, especially those involving graphing or geometry. Teachers can make their own interactive worksheets that include simulations, videos, text, multiple-choice questions, and more. Some of the preexisting worksheets are interactive, while others simply show a concept.
Teachers can create class groups to quickly share activities with kids. Alternatively, tutorials provide options to help teachers incorporate GeoGebra into their existing class site or learning management system (LMS).
GeoGebra gives kids a way to access math that moves beyond straightforward pencil-and-paper computations. Traditional methods of performing constructions with a compass and a ruler can be time-consuming and frustrating for students. GeoGebra makes it quick, easy, and fun as long as there are clear directions. Premade resources available on the site are built and shared by anyone who wants to be an author, so quality varies greatly; some don't provide specific directions for students, and others are basic illustrations of a simple topic with no interactivity. But teachers can create their own interactive worksheets that include the simulations built by someone else, along with their own helpful directions or context.
GeoGebra is a powerful example of flexibility paired with community. Teachers can make whatever they need for their students. But they don't have to start from scratch: They can build off of the work of others. This means that math simulations are being built every day by teachers all over the world. Teaching the difference between exponential and logistic models? One teacher shared an activity on the growth of Facebook. Need an analog clock with hands you can manipulate? GeoGebra has one of those, too!
Key Standards Supported
Arithmetic With Polynomials And Rational Expressions
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Find inverse functions.
Prove that all circles are similar.
Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
(+) Construct a tangent line from a point outside a given circle to the circle.
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Conditional Probability And The Rules Of Probability
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Expressing Geometric Properties With Equations
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Derive the equation of a parabola given a focus and directrix.
(+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1
Geometric Measurement And Dimension
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
(+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Identify the shapes of two-dimensional cross-sections of three- dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Interpreting Categorical And Quantitative Data
Represent data with plots on the real number line (dot plots, histograms, and box plots).
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
Compute (using technology) and interpret the correlation coefficient of a linear fit.
Distinguish between correlation and causation.
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Linear, Quadratic, And Exponential Models
Distinguish between situations that can be modeled with linear functions and with exponential functions.
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
For exponential models, express as a logarithm the solution to abct =dwherea,c,anddarenumbersandthebasebis2,10,ore; evaluate the logarithm using technology.
Measurement And Data
Tell and write time in hours and half-hours using analog and digital clocks.
Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
Reasoning With Equations And Inequalities
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
(+) Represent a system of linear equations as a single matrix equation in a vector variable.
(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Graph the solutions to a linear inequality in two variables as a half- plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
Similarity, Right Triangles, And Trigonometry
Verify experimentally the properties of dilations given by a center and a scale factor:
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
(+) Prove the Laws of Sines and Cosines and use them to solve problems.
(+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios.
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Vector And Matrix Quantities
(+) Add and subtract vectors.
(+) Multiply a vector by a scalar.
(+) 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).
(+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
(+) Add, subtract, and multiply matrices of appropriate dimensions.
(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
(+) 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.
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
(+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.