Illustrative Math-Media4Math Alignment

 

 

Illustrative Math Alignment: Grade 6 Unit 9

Putting it All Together

Lesson 4: How Do We Choose?

Use the following Media4Math resources with this Illustrative Math lesson.

Thumbnail Image Title Body Curriculum Topics
Math Example: Fraction Operations--Dividing Fractions: Example 15 Math Example: Fraction Operations--Dividing Fractions: Example 15 Dividing Fractions: Example 15

Topic

Fraction Operations

Description

This example illustrates the division of fractions by converting the operation into multiplication using the reciprocal of the divisor. The image provides a clear step-by-step guide, emphasizing the importance of reciprocal multiplication. The result is a mixed number. Skills involved include recognizing and applying reciprocal relationships, performing multiplication, and simplifying fractions. This example is crucial for understanding the fundamental principle that dividing by a fraction is equivalent to multiplying by its reciprocal, a key concept in fraction operations.

Fractions and Mixed Numbers
Math Clip Art--Geometry Concepts--Circle Illustrations--Unit Circle Math Clip Art--Geometry Concepts--Circle Illustrations--Unit Circle Math Clip Art--Geometry Concepts--Circle Illustrations--Unit Circle

This is part of a collection of math clip art images about selected topics in Geometry. Also look at our Geometry Basics math clip art collection.

Definition of a Circle and Applications of Coordinate Geometry
Math Clip Art--Vertical Angles--Labeled Math Clip Art--Angle Illustrations--Vertical Angles--Labeled Math Clip Art--Angle Illustrations--Vertical Angles--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Vertical Angles--Unlabeled Math Clip Art--Angle Illustrations--Vertical Angles--Unlabeled Math Clip Art--Angle Illustrations--Vertical Angles--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Acute Angle--Labeled Math Clip Art--Angle Illustrations--Acute Angle--Labeled Math Clip Art--Angle Illustrations--Acute Angle--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Acute Angle--Unlabeled Math Clip Art--Angle Illustrations--Acute Angle--Unlabeled Math Clip Art--Angle Illustrations--Acute Angle--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Adjacent Angles--Labeled Math Clip Art--Angle Illustrations--Adjacent Angles--Labeled Math Clip Art--Angle Illustrations--Adjacent Angles--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Adjacent Angles--Unlabeled Math Clip Art--Angle Illustrations--Adjacent Angles--Unlabeled Math Clip Art--Angle Illustrations--Adjacent Angles--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Angle Bisector--Labeled Math Clip Art--Angle Illustrations--Angle Bisector--Labeled Math Clip Art--Angle Illustrations--Angle Bisector--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Angle Bisector--Unlabeled Math Clip Art--Angle Illustrations--Angle Bisector--Unlabeled Math Clip Art--Angle Illustrations--Angle Bisector--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Angle--Labeled Math Clip Art--Angle Illustrations--Angle--Labeled Math Clip Art--Angle Illustrations--Angle--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Angle--Unlabeled Math Clip Art--Angle Illustrations--Angle--Unlabeled Math Clip Art--Angle Illustrations--Angle--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Math Clip Art--Circle Illustrations--Circle

Topic

Geometry Concepts

Description

This clip art illustrates a basic circle, which is a fundamental shape in geometry. A circle is defined as the set of all points in a plane that are equidistant from a central point. This image serves as a visual representation of this concept, showcasing the perfect symmetry and continuous curvature that characterize circles.

Definition of a Circle
Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered at Origin of Coordinate Grid Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered at Origin of Coordinate Grid Math Clip Art--Circle Illustrations--Circle Centered at Origin of Coordinate Grid

Topic

Geometry Concepts

Description

This clip art depicts a circle centered at the origin (0,0) of a coordinate grid. It illustrates the relationship between circles and the Cartesian coordinate system, a fundamental concept in analytic geometry. The image shows how a circle's position can be defined in relation to the x and y axes.

Definition of a Circle
Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered at Origin of Coordinate Grid--with Radius Labeled Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered at Origin of Coordinate Grid--with Radius Labeled Math Clip Art--Circle Illustrations--Circle Centered at Origin of Coordinate Grid--with Radius Labeled

Topic

Geometry Concepts

Description

This clip art shows a circle centered at the origin of a coordinate grid with its radius clearly labeled. This illustration is crucial for understanding the relationship between a circle's radius and its equation in the coordinate plane. The labeled radius helps visualize how the distance from the center to any point on the circle remains constant, which is the defining property of a circle.

Definition of a Circle
Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered in Q1 Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered in Q1 Math Clip Art--Circle Illustrations--Circle Centered in Q1

Topic

Geometry Concepts

Description

This clip art illustrates a circle centered in the first quadrant (Q1) of the coordinate plane. This representation is crucial for understanding how the position of a circle's center affects its equation and properties. In Q1, both x and y coordinates of the center are positive, which influences the circle's equation and its intersection with the axes.

Definition of a Circle
Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered in Q2 Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered in Q2 Math Clip Art--Circle Illustrations--Circle Centered in Q2

Topic

Geometry Concepts

Description

This clip art shows a circle centered in the second quadrant (Q2) of the coordinate plane. In Q2, the x-coordinate of the center is negative while the y-coordinate is positive. This positioning is crucial for understanding how the location of a circle's center affects its equation and its relationship with the coordinate axes.

Definition of a Circle
Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered in Q3 Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered in Q3 Math Clip Art--Circle Illustrations--Circle Centered in Q3

Topic

Geometry Concepts

Description

This clip art depicts a circle centered in the third quadrant (Q3) of the coordinate plane. In Q3, both the x and y coordinates of the center are negative. This positioning is essential for understanding how the location of a circle's center influences its equation and its interactions with the coordinate axes.

Definition of a Circle
Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered in Q4 Math Clip Art--Geometry Concepts--Circle Illustrations--Circle Centered in Q4

Topic

Geometry Concepts

Description

This clip art illustrates a circle centered in the fourth quadrant (Q4) of the coordinate plane. In Q4, the x-coordinate of the center is positive while the y-coordinate is negative. This positioning is crucial for understanding how the location of a circle's center affects its equation and its relationship with the coordinate axes.

Definition of a Circle
Math Clip Art--Complementary Angles--Labeled Math Clip Art--Angle Illustrations--Complementary Angles--Labeled Math Clip Art--Angle Illustrations--Complementary Angles--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Complementary Angles--Unlabeled Math Clip Art--Angle Illustrations--Complementary Angles--Unlabeled Math Clip Art--Angle Illustrations--Complementary Angles--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Exterior Angle of a Triangle--Labeled Math Clip Art--Angle Illustrations--Exterior Angle of a Triangle--Labeled Math Clip Art--Angle Illustrations--Exterior Angle of a Triangle--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Exterior Angle of a Triangle--Unlabeled Math Clip Art--Angle Illustrations--Exterior Angle of a Triangle--Unlabeled Math Clip Art--Angle Illustrations--Exterior Angle of a Triangle--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Obtuse Angle--Labeled Math Clip Art--Angle Illustrations--Obtuse Angle--Labeled Math Clip Art--Angle Illustrations--Obtuse Angle--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Obtuse Angle--Unlabeled Math Clip Art--Angle Illustrations--Obtuse Angle--Unlabeled Math Clip Art--Angle Illustrations--Obtuse Angle--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Function Concepts--Graphs of Functions and Relations--Rational Function Graph Math Clip Art--Function Concepts--Graphs of Functions and Relations--Rational Function Graph Math Clip Art--Function Concepts--Graphs of Functions and Relations--Rational Function Graph

Topic

Functions

Description

This clip art depicts a rational function graph. Rational functions are quotients of polynomials and can have a wide variety of shapes depending on the degrees of the numerator and denominator. They generally pass the vertical line test and are thus functions, though they may have discontinuities.

Rational Functions and Equations
Math Clip Art--Right Angle--Labeled Math Clip Art--Angle Illustrations--Right Angle--Labeled Math Clip Art--Angle Illustrations--Right Angle--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Right Angle--Unlabeled Math Clip Art--Angle Illustrations--Right Angle--Unlabeled Math Clip Art--Angle Illustrations--Right Angle--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Straight Angle--Labeled Math Clip Art--Angle Illustrations--Straight Angle--Labeled Math Clip Art--Angle Illustrations--Straight Angle--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Straight Angle--Unlabeled Math Clip Art--Angle Illustrations--Straight Angle--Unlabeled Math Clip Art--Angle Illustrations--Straight Angle--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Supplementary Angles--Labeled Math Clip Art--Angle Illustrations--Supplementary Angles--Labeled Math Clip Art--Angle Illustrations--Supplementary Angles--Labeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Math Clip Art--Supplementary Angles--Unlabeled Math Clip Art--Angle Illustrations--Supplementary Angles--Unlabeled Math Clip Art--Angle Illustrations--Supplementary Angles--Unlabeled

This is part of a collection of clip art images showing different types of angles. Each type of angle measure includes a labeled and unlabeled version.

Angles and Applications of Angles and Planes
Definition--Rationals and Radicals--Cube Root Definition--Rationals and Radicals--Cube Root Cube Root

Topic

Rationals and Radicals

Definition

The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Description

The cube root is a fundamental concept in Radical Numbers, Expressions, Equations, and Functions. It is the inverse operation of raising a number to the power of three. Understanding cube roots is crucial for solving equations involving cubic terms and for simplifying radical expressions.

Cube roots also appear in various real-world applications, such as calculating volumes and in certain physics equations. They are an integral part of higher-level mathematics, including algebra and calculus.

Radical Expressions
Definition--Rationals and Radicals--Extraneous Solution Definition--Rationals and Radicals--Extraneous Solution Extraneous Solution

Topic

Rationals and Radicals

Definition

An extraneous solution is a solution derived from an equation that is not a valid solution to the original equation.

Description

Extraneous solutions often arise in the context of Rational and Radical Equations. They are solutions that appear during the process of solving an equation but do not satisfy the original equation. This can happen when both sides of an equation are squared or when other operations introduce additional solutions.

Rational Functions and Equations
Definition--Rationals and Radicals--Graphs of Rational Functions Definition--Rationals and Radicals--Graphs of Rational Functions Graphs of Rational Functions

Topic

Rationals and Radicals

Definition

Graphs of rational functions are visual representations of equations involving rational expressions.

Description

Graphs of rational functions are fundamental in the study of Rational Numbers, Expressions, Equations, and Functions. They help in understanding the behavior of these functions, including their asymptotes, intercepts, and regions of increase and decrease.

Rational Functions and Equations
Definition--Rationals and Radicals--Inverse Variation Definition--Rationals and Radicals--Inverse Variation Inverse Variation

Topic

Rationals and Radicals

Definition

Inverse variation describes a relationship between two variables in which the product is a constant. When one variable increases, the other decreases proportionally.

Rational Functions and Equations
Definition--Rationals and Radicals--Irrational Number 2 Definition--Rationals and Radicals--Irrational Number 2 Irrational Number 2

Topic

Rationals and Radicals

Definition

An irrational number is a number that cannot be expressed as a ratio of two integers. Its decimal form is non-repeating and non-terminating.

Rational Expressions
Definition--Rationals and Radicals--nth Root Definition--Rationals and Radicals--nth Root nth Root

Topic

Rationals and Radicals

Definition

The nth root of a number is a value that, when raised to the power of n, gives the original number. It is denoted as n√a

Description

The nth Root is a fundamental concept in the study of Radical Numbers, Expressions, Equations, and Functions. It generalizes the idea of square roots and cube roots to any positive integer n. For example, the cube root of 8 is 2 because 

23=8

Radical Expressions
Definition--OrderOfOperations.jpg Definition--Order of Operations Definition--Order of Operations

Watch the following video on Order of Operations. (The transcript is included.)

Video Transcript

 

A numerical expression includes numbers and operation symbols, addition, subtraction, multiplication, and division.

Because addition is commutative, adding from left to right, or right to left, gives you the same result.

The expressions 2 + 3 and 3 + 2 give the same result. But this isn't the case with all operations.

Subtraction isn't commutative.

Numerical Expressions and Variable Expressions
Definition--Ratios, Proportions, and Percents Concepts--Percent Definition--Ratios, Proportions, and Percents Concepts--Percent Percent

Topic

Ratios, Proportions, and Percents

Definition

A percent is a ratio that compares a number to 100.

Description

Percentages are a fundamental concept in mathematics, representing a ratio out of 100. They are used in various applications, including finance, statistics, and everyday calculations such as discounts and interest rates.

For example, if you score 45 out of 50 on a test, your percentage score is 

(45/50) × 100 = 90%

Percents
Definition--Ratios, Proportions, and Percents Concepts--Proportional Definition--Ratios, Proportions, and Percents Concepts--Proportional Proportional

Topic

Ratios, Proportions, and Percents

Definition

Proportional refers to the relationship between two quantities where their ratio is constant.

Description

Proportional relationships are fundamental in mathematics and science, describing how one quantity changes in relation to another. This concept is used in various fields, including physics, economics, and engineering.

For example, if the speed of a car is proportional to the time it travels, doubling the time will double the distance covered. Understanding proportionality helps students solve complex problems and apply mathematical reasoning in real-world situations.

Proportions
Definition--Rationals and Radicals--Radical Expression Definition--Rationals and Radicals--Radical Expression Radical Expression

Topic

Rationals and Radicals

Definition

A radical expression is an expression that contains a radical symbol, which indicates the root of a number.

Description

Radical Expressions are a core component of Radical Numbers, Expressions, Equations, and Functions. These expressions involve roots, such as square roots, cube roots, or higher-order roots, and are denoted by the radical symbol (√). For example, the expression 

√16

 

Radical Expressions
Definition--Rationals and Radicals--Radical Function Definition--Rationals and Radicals--Radical Function Radical Function

Topic

Rationals and Radicals

Definition

A radical function is a function that contains a radical expression with the independent variable in the radicand.

Description

Radical Functions are a vital part of Radical Numbers, Expressions, Equations, and Functions. These functions involve radicals, such as square roots or cube roots, with the independent variable inside the radical. For example, the function

f(x)=√x

Radical Expressions
Definition--Rationals and Radicals--Radical Symbol Definition--Rationals and Radicals--Radical Symbol Radical Symbol

Topic

Rationals and Radicals

Definition

The radical symbol (√) is used to denote the root of a number, such as a square root or cube root.

Description

The Radical Symbol is a fundamental notation in the study of Radical Numbers, Expressions, Equations, and Functions. This symbol (√) indicates the root of a number, with the most common being the square root. For example, the expression 

√25

Radical Expressions
Definition--Rationals and Radicals--Radicand Definition--Rationals and Radicals--Radicand Radicand

Topic

Rationals and Radicals

Definition

The radicand is the number or expression inside the radical symbol that is being rooted.

Description

The Radicand is a key component in the study of Radical Numbers, Expressions, Equations, and Functions. It is the number or expression inside the radical symbol that is being rooted. For example, in the expression 

√49

Radical Expressions
Definition--Ratios, Proportions, and Percents Concepts--Rate Definition--Ratios, Proportions, and Percents Concepts--Rate Rate

Topic

Ratios, Proportions, and Percents

Definition

A rate is a ratio that compares two quantities with different units.

Description

Rates are used to compare different quantities, such as speed (miles per hour) or price (cost per item). Understanding rates is essential for interpreting data and making informed decisions in various contexts, such as travel and budgeting.

For instance, if a car travels 60 miles in 2 hours, the rate is 30 miles per hour. Learning about rates helps students analyze real-world situations and apply mathematical reasoning to everyday problems.

Ratios and Rates
Definition--Ratios, Proportions, and Percents Concepts--Ratio Definition--Ratios, Proportions, and Percents Concepts--Ratio Ratio

Topic

Ratios, Proportions, and Percents

Definition

A ratio is a comparison of two quantities by division.

Description

Ratios are used to express the relationship between two quantities, providing a way to compare different amounts. They are fundamental in various fields, including mathematics, science, and finance.

For example, the ratio of 4 to 5 can be written as 4:5 or 4/5. Understanding ratios helps students analyze data, solve problems, and make informed decisions in real-world situations.

Ratios and Rates
Definition--Rationals and Radicals--Rational Expressions Definition--Rationals and Radicals--Rational Expressions Rational Expressions

Topic

Rationals and Radicals

Definition

Rational expressions are fractions in which the numerator and/or the denominator are polynomials.

Description

Rational Expressions are a fundamental aspect of Rational Numbers, Expressions, Equations, and Functions. These expressions are fractions where the numerator and/or the denominator are polynomials. Simplifying rational expressions often involves factoring the polynomials and canceling common factors. For example, the rational expression 

x2−1x−1

can be simplified to x + 1, provided that 

x≠1

Rational Expressions
Definition--Rationals and Radicals--Rational Functions Definition--Rationals and Radicals--Rational Functions Rational Functions

Topic

Rationals and Radicals

Definition

Rational functions are functions that are the ratio of two polynomials.

Description

Rational Functions are a key concept in the study of Rational Numbers, Expressions, Equations, and Functions. These functions are the ratio of two polynomials, such as 

f(x)=P(x)Q(x)

where P(x) and Q(x) are polynomials. Understanding rational functions involves analyzing their behavior, including identifying asymptotes, intercepts, and discontinuities. For example, the function 

f(x)=1x

Rational Functions and Equations
Definition--Rationals and Radicals--Rational Numbers Definition--Rationals and Radicals--Rational Numbers Rational Numbers

Topic

Rationals and Radicals

Definition

Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not zero.

Rational Functions and Equations