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.
Parallel Lines and a Transversal
Many important geometric structures make use of parallel lines, so it is useful to know how to check whether two given lines are parallel and how to construct parallel lines. The key principle in both tasks is the relationship between parallel lines and any third line that intersects them. The line that intersects two parallel lines is called a transversal. As a transversal intersects two parallel lines, it creates several angles. This video shows the special relationships among these angles.
Shapes and Designs
Interior Angles of a PolygonOne of the most important theorems in all of Euclidean geometry states that the sum of the angles of any triangle is always equal to a straight angle, or 180°. Based on that property of triangles, students can then reason to more general results about the angle sum of quadrilaterals and other polygons by showing how those figures can be decomposed into triangles. This video shows how to find the sum of the interior angles of a polygon with n sides.
Parallel Lines and a TransversalMany important geometric structures make use of parallel lines, so it is useful to know how to check whether two given lines are parallel and how to construct parallel lines. The key principle in both tasks is the relationship between parallel lines and any third line that intersects them. The line that intersects two parallel lines is called a transversal. As a transversal intersects two parallel lines, it creates several angles. This video shows the special relationships among these angles.
Accentuate the Negative
Subtracting With a Chip Board ModelColored chips can be used to develop a strategy for adding and subtracting integers. Using this model requires an understanding of opposites. The chip model uses one color of chips (black) to represent positive integers and another color (red) to represent negative integers. Because each chip represents 1 unit, either positive or negative, a red chip and a black chip are thought of as opposites. Combining two opposite chips makes zero.
Stretching and Shrinking
Using a Coordinate Grid to Draw Similar and Congruent FiguresIn general, algebraic rules of the form (nx, ny), where n is the scale factor, are called similarity transformations, because they will transform a figure in the plane into a similar figure in the plane. This video shows how to transform a figure on a coordinate grid.
Angle-Angle Similarity PostulateFor most polygons, when considering similarity, you must make sure that the lengths of corresponding sides increase by the same scale factor and that corresponding angle measures are equal. For triangles, you need to check only one of these characteristics to determine if two triangles are similar. This fact is called the Angle-Angle-Angle Similarity Postulate. This video shows a graphic explanation of the postulate.
Moving Straight Ahead
Solving a Linear EquationThe key to solving equations symbolically is understanding equality. In this Unit, students explore the properties of equality informally by examining metaphorical equations that involve $1 gold coins and mystery pouches of coins. We assume that all pouches in an equation have the same number of coins and that both sides of the equality sign have the same number of coins. This video shows how to solve an equation that consists of gold coins and pouches.
Representing Solutions of InequalitiesIn Problem 3.5, students solve linear inequalities in the context of Fabian’s Bakery. The bakery’s expenses E for making and selling n cakes can be represented by the equation E = 825 + 3.5n. The bakery’s income I for selling n cakes is represented by the equation I = 8.20n. For simplicity, students assume that the bakery sells all the cakes it makes.
In Question E, students are asked to find the number of cakes for which the bakery’s expenses are less than $2,400 and the number of cakes for which the bakery’s income is greater than $2,400. This requires students to use their knowledge of inequalities from Accentuate the Negative to find the solution set to the inequality statements
825 + 3.25n < 2,400 and 8.20n > 2,400.
This video shows how you can graph the associated equations and then find the values of n that satisfy the inequalities.
Two Methods for Solving a Linear EquationTwo arithmetic expressions are equivalent if they have the same numerical value. For example, 7 – 8 is equivalent to 7 + (–8). Two algebraic expressions are equivalent if they have the same numerical value regardless of the values of the variables involved. For example, a – b is equivalent to a + (–b).
Equivalent expressions were first introduced in Grade 6 using the Distributive Property—first as equivalent numerical expressions in Prime Time and then as equivalent algebraic expressions in Variables and Patterns. For example, the product of the two factors 3 and (5 + 8) can be written as the sum of the two terms 3(5) and 3(8). That is, 3(5 + 8) = 3(5) + 3(8). Similarly, 3(x + 2) = 3(x) + 3(2).
This video shows how equivalent expressions arise in this Unit as students solve equations involving pouches filled with gold coins.
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.
What Do You Expect?
Important Aspects of ProbabilityExperimental probability is probability that is determined through experimentation. Theoretical probability is probability obtained by analyzing a situation. One example of a theoretical probability that students study in this Unit involves the roll of a number cube. This video illustrates some important aspects of the concept of probability using the action of rolling a number cube.
Filling and Wrapping
Covering a Circle With Radius SquaresIn Investigation 3, students find the area of a circle by counting the number of unit squares needed to exactly cover the circle. Students may find interesting ways to do this. Some might divide the circle into several congruent sectors, count the squares needed to cover one of the sectors, and then multiply by the number of sectors. Some might enclose the circle inside a square, and then subtract the unit squares and parts of unit squares that lie outside the circle. The standard formula for the area of a circle is developed by finding the number of squares, whose side lengths are equal to the radius, that cover the circle. Again, students will use a variety of interesting strategies to do this. This video shows one possible method.
Samples and Populations
Using MAD to Analyze VariabilityVariability is found in any set of data. Not all math test scores are the same, not all students watch the same number of hours of TV, and not all basketball players are the same heights. When students compare two sets of data, they should expect the distributions to be different. They need a way, however, to decide whether the differences are expected due to natural variability or to meaningful differences between samples. This video shows how to use means and mean absolute deviations (MADs) to compare two data distributions.
Generating a Random SampleA census takes information from the entire population. Generally, conducting a census is not possible or reasonable because of factors such as cost and the size of the population. Instead, sampling is used to gain information about a whole population by analyzing only a part of it. To ensure that samples are representative, or fair, statisticians try to use random sampling plans. Each person or object in the population needs to have an equally likely chance to be included as part of a sample. Random samples are not always representative of a population. All types of samples vary. Random samples, however, have two important characteristics:
Random samples are free from bias.
Random samples of a large enough size generally give good predictions about the populations from which they are drawn.
Students study a number of strategies for selecting samples at random in this Unit, including those shown in this video.