These short clips are the same videos many CMP3 teachers use to review key math concepts when they prepare to teach CMP3 lessons. Use these helpful videos to bolster your math knowledge as you help your student with homework. The video clips highlight important mathematical ideas and concepts in each CMP3 Unit in an easy-to-understand, visual way.
Rectangles With Area 12: Graphing Factor Pairs
The Grade 6 Unit Prime Time covered the relationship between factor pairs of a number and rectangles with area equal to the number. By superimposing the factor-pair rectangles for a number on top of each other, students can see the symmetry of the factor pairs. This video shows the inverse variation relationship that results when you graph those factor pairs as coordinates.
Inverse variation refers to a nonlinear relationship in which the product of two variables is constant. In the context of these rectangles, the two variables are length (I) and width (w). Their area, or the product of the variables l and w, is constant.
An inverse variation can be represented by an equation of the form y = k/x, or xy = k, where k is a constant. In an inverse variation, the values of one variable decrease as the values of the other variable increase.
The Grade 6 Unit Prime Time covered the relationship between factor pairs of a number and rectangles with area equal to the number. By superimposing the factor-pair rectangles for a number on top of each other, students can see the symmetry of the factor pairs. This video shows the inverse variation relationship that results when you graph those factor pairs as coordinates.
Inverse variation refers to a nonlinear relationship in which the product of two variables is constant. In the context of these rectangles, the two variables are length (I) and width (w). Their area, or the product of the variables l and w, is constant.
An inverse variation can be represented by an equation of the form y = k/x, or xy = k, where k is a constant. In an inverse variation, the values of one variable decrease as the values of the other variable increase.
Identifying Correlation CoefficientsThe correlation coefficient is a number between 1 and –1 that tells how closely a pattern of data points fits a straight line. This video shows the correlation coefficient of various sets of data points.
Connecting Equations, Tables, and GraphsThere are many ways to represent relationships between two quantitative variables, such as equations, tables, and graphs. This video illustrates the relationships among these representations. It is important for students to move freely among the various representations. By the end of the Unit, students should feel very comfortable with tables and graphs and with some simple equations. Students should also have an appreciation of the advantages and disadvantages of each representation.
Looking for Pythagoras
Proving the Pythagorean TheoremThe Pythagorean Theorem states that, if the lengths of the legs of a right triangle are a and b, and the length of the hypotenuse is c, then a^{2} + b^{2} = c^{2}. This video shows a geometric proof of the Pythagorean Theorem.
Growing, Growing, Growing
Exponential and Linear FunctionsWhen making tables, students usually generate each value in the table by working with the previous value. Either they add a constant to the previous value (in the case of linear relationships) or they multiply the previous value by a constant (in the case of exponential relationships). This process of generating a value from a previous value is called recursion, or iteration. It is important to distinguish between a constant growth factor (multiplicative), as illustrated in an exponential function, and the constant additive pattern in linear functions. This video illustrates successive iterations for linear and exponential growth.
Patterns in the Table of PowersIn this Unit, students begin to develop understanding of the rules of exponents by examining patterns in the table of powers for the first 10 whole numbers. This video describes some of these patterns.
Frogs, Fleas, and Painted Cubes
The Distributive Property and Equivalent Quadratic ExpressionsWhen a quadratic expression is written in factored form, its factors are often binomial expressions, or simply binomials. A binomial is an expression with two terms. In this Unit, students do a bit of factoring and multiplying binomials to find equivalent expressions. This video shows how you can think of the area of a rectangle divided into four smaller rectangles as the product of two linear expressions, the result of multiplying the width by the length. This produces the factored form of a quadratic expression. You can also think of the area as the sum of the areas of the subparts of the rectangle. This generates the expanded form of a quadratic expression.
Other Contexts for Quadratic FunctionsThe height of a jumping frog can be represented by the quadratic equation y = −16t^{2} + 12t + 0.2.
This video describes the graph of the height of a frog over time. The frog is 0.2 feet (or 2.4 inches) tall, so his initial height, or h_{0}, is 0.2. When the frog begins his jump, he takes off at a velocity of 12 feet per second, so v_{0} is 12. As the frog moves through the air, gravity slows his speed down, until he pauses for an instant at his maximum height. Then, gravity causes the frog to accelerate as he falls back toward the ground.
Butterflies, Pinwheels, and Wallpaper
Reflection in Parallel LinesIn very informal ways, students explore combinations of transformations. In a few instances in the ACE Extensions, students are asked to describe a single transformation that will give the same result as a given combination. This video shows how reflecting a figure in a line and then reflecting the image in a parallel line has the same result as translating the figure in a direction perpendicular to the reflection lines for a distance equal to twice the distance between the lines.
Reflection in Intersecting LinesThis video shows how reflecting a figure in a line and then reflecting the image in an intersecting line has the same result as rotating the original figure about the intersection point of the lines by an angle equal to twice the angle formed by the reflection lines. Notice that reflecting the triangle ABC in line 1 and then reflecting the image A′B′C′ in line 2 does NOT give the same result as reflecting triangle ABC in line 2 first and then reflecting the image in line 1.
It's In the System
Three Types of Solutions of a System of Linear EquationsThere are three possible outcomes for a system of linear equations: one unique solution, infinitely many solutions, and no solution. This video shows an example of each type of outcome.
Function Junction
Relating Vertex Form and TransformationsSo far in their study of quadratic expressions and functions, students have learned how both standard trinomial forms (for example, x^{2} + 5x + 6) and factored forms (for example, (x + 2)(x + 3)) reveal important information about the related functions and their graphs. This video shows how transformations of the basic quadratic y = x^{2} give clues to the form and location of the graph for quadratic functions expressed in what is called vertex form: y = a(x – h)^{2} + k.
Completing the Square, Part 1While the vertex form of a quadratic expression is very useful, once obtained, it is not a trivial task to take any standard trinomial form and transform it to equivalent vertex form. The process is called completing the square, because the tricky part is generating the (x – h)^{2} perfect square term. This video gives a geometrical interpretation of a perfect square trinomial.
Completing the Square, Part 2In most instances, it is not possible to write a quadratic expression as the product of two identical linear factors. For the cases that lead to two different linear factors, it is still possible to write the given quadratic in equivalent vertex form and reap the benefits of that expressive form. This video shows one such case.