Illustrative Math Alignment: Grade 8 Unit 3

In this TI Nspire tutorial for the TI-Nspire CAS, the Graph Window is used to graph quadratic inequalities. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet.

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Video Transcripts

In this episode of Algebra Applications, two real-world explorations of inequalities are developed. Hybrid Cars. An examination of the equations and inequalities that involve miles per gallon (mpg) for city and highway traffic reveals important information about hybrid cars and those with gasoline-powered engines. Floods in Venice. Venice experiences a great deal of flooding, and with the expected rise of sea levels over the next century, this ancient city is in peril. Through a series of inequalities, students analyze the impact of flooding.

With the increasing demand worldwide for cars, the cost of gasoline continues to rise. The need for fuel-efficient cars makes hybrids a current favorite. An examination of the equations and inequalities that involve miles per gallon (mpg) for city and highway traffic reveals important information about hybrid cars and those with gasoline-powered engines. Students use the Graphs and Geometry features of the TI-Nspire.

The city of Venice is slowly sinking into the Adriatic Sea. So what does a city whose streets are full of water do about flooding? Venice experiences a great deal of flooding, and with the expected rise of sea levels over the next century, this ancient city is in peril. Through a series of inequalities, students analyze the impact of flooding, rising sea levels, and sinking have on this grand, ancient city. Students use the Lists and Spreadsheets and the Program Editor features of the TI-Nspire.

In this episode of Algebra Applications, students explore various scenarios that can be explained through the use of systems of equations. Such disparate phenomena as profit and loss, secret codes, and ballistic missile shields can be explored through systems of equations.

Profit and loss are the key measures in a business. A system of equations that includes an equation for income and one for expenses can be used to determine profit and loss. Students solve a system graphically.

Secret codes and encryption are ideal examples of a system of equations. In this activity, students encrypt and decrypt a message.

A ballistic missile shield allows you to shoot incoming missiles out of the sky. Mathematically, this is an example of a linear-quadratic system. Students graph such a system and find the points of intersection between a line and a parabola.

Used in just about any industry, inequalities, like equations, are fundamental building blocks of algebra. Written and hosted by internationally acclaimed mathematics educator Dr. Monica Neagoy, this video explores inequalities—concepts, properties, solutions, and notations— connects them to real-world contexts, and uses the TI-Nspire to make the algebra meaningful. The focus of this program is on linear inequalities in one and two variables. Concepts explored: Equations, inequalities.

In this Investigation we explore linear inequalities in one variable. This video is Segment 1 of a 4 segment series related to Algebra Nspirations: Inequalities. Segments 1 and 2 are grouped together.

In this Investigation we look at linear inequalities in two variables. This video is Segment 3 of a 4 segment series related to Algebra Nspirations: Inequalities. Segments 3 and 4 are grouped together.

Written and hosted by internationally acclaimed math educator Dr. Monica Neagoy, this video introduces students to systems of linear equations in two or three unknowns. To solve these systems, the host illustrates a variety of methods: four involve the TI-Nspire (spreadsheet, graphs and geometry, matrices and nSolve) and two are the classic algebraic methods known as substitution and elimination, also called the linear combinations method. The video ends with a summary of the three possible types of solutions. Concepts explored: equations, linear equations, linear systems.

In this Investigation we solve a linear system. This video is Segment 1 of a 4 segment series related to Algebra Nspirations: Solving Systems of Equations. Segments 1 and 2 are grouped together.

In this Investigation we use matrices and the elimination method to solve a linear system. This video is Segment 3 of a 4 segment series related to Algebra Nspirations: Solving Systems of Equations. Segments 3 and 4 are grouped together.

Video Tutorial: Algebra Tiles: Adding Integers Using Algebra Tiles. In this tutorial, review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to add integers using algebra tiles.

Description

Video Tutorial: Algebra Tiles: Dividing Integers Using Algebra Tiles. In this tutorial, review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to divide integers using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

Video Tutorial: Algebra Tiles: Dividing Integers Using Algebra Tiles. In this tutorial, review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to divide integers using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

Video Tutorial: Algebra Tiles: Modeling Equations Using Algebra Tiles. In this tutorial, review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to model equations using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

Video Tutorial: Algebra Tiles: Modeling Multi-Step Equations Using Algebra Tiles. In this tutorial, students review the basics of how to use algebra tiles to model equations. Then the video focuses on how to model multi-step equations using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

Video Tutorial: Algebra Tiles: Modeling Negative Integers Using Algebra Tiles. In this tutorial, review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to model negative integers using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

Video Tutorial: Algebra Tiles: Modeling One-Step Equations Using Algebra Tiles. In this tutorial, review the basics of how to use algebra tiles to model equations. Then the video focuses on how to model one-step equations using algebra tiles.

The following section reviews the basics of algebra tiles. Use this material as background material, and also as a supplement to the video series.

Description

Video Tutorial: Algebra Tiles: Modeling Positive Integers Using Algebra Tiles. In this tutorial, review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to model positive integers using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Video Tutorial: Algebra Tiles: Modeling Quadratic Equations Using Algebra Tiles. In this tutorial, review the basics of how to use algebra tiles to model equations. Then the video focuses on how to model quadratic equations using algebra tiles.

Description

Video Tutorial: Algebra Tiles: Dividing Integers Using Algebra Tiles. In this tutorial, review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to divide integers using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

Video Tutorial: Algebra Tiles: Modeling Variable Expressions Using Algebra Tiles. In this tutorial, students review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to model variable expressions using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

Video Tutorial: Algebra Tiles: Modeling Variables Using Algebra Tiles. In this tutorial, review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to model variables using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

Video Tutorial: Algebra Tiles: Modeling Zero Using Algebra Tiles. In this tutorial, students review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to model zero using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

The following section reviews the basics of algebra tiles. Use this material as background material, and also as a supplement to the video series.

Description

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

The following section reviews the basics of algebra tiles. Use this material as background material, and also as a supplement to the video series.

Video Tutorial: Algebra Tiles: Solving Multi-Step Equations Using Algebra Tiles. In this tutorial, students review how to model equations with algebra tiles. Then the video focuses on how to solve multi-step equations with algebra tiles.

Description

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

The following section reviews the basics of algebra tiles. Use this material as background material, and also as a supplement to the video series.

Video Tutorial: Algebra Tiles: Subtracting Integers Using Algebra Tiles. In this tutorial, review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to subtract integers using algebra tiles.

Description

Video Tutorial: Algebra Tiles: Dividing Integers Using Algebra Tiles. In this tutorial, review the basic definition of what algebra tiles are and how they are used. Then the video focuses on how to divide integers using algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

Description

Video Tutorial: Algebra Tiles: What Are Algebra Tiles? In this tutorial, learn the basics of what algebra tiles are and how they are used. Use this video segment when first introducing algebra tiles.

This is part of a collection of video tutorials the topic of Algebra Tiles.This series of videos describes what algebra tiles are and how they can be used to model numbers, operations, expressions, and equations. Use this series if you are incorporating the use of algebra tiles in your instruction.

In this TI Nspire tutorial, the Graph window is used to create a slider-based graph of a linear function in slope-intercept form. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet. .

In this TI Nspire tutorial, the Graph window is to find the slope of two points connected by a line. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet. .

In this TI Nspire tutorial for the TI-Nspire CAS, the Calculator Window is used to solve a linear system. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet. .

Related Resources

Video Transcripts

Topic

Equations

Definition

The "Not Equal To" symbol (≠) is used to indicate that two values are not equal.

Description

The "Not Equal To" symbol is crucial in mathematics as it denotes inequality between two expressions. This symbol is used in various mathematical contexts, such as solving inequalities, comparing numbers, and expressing conditions in algebraic equations. For example, in the inequality 𝑥 ≠ 5, it means that x can be any number except 5.

Topic

Equations

Definition

The Addition Property of Equality states that if you add the same value to both sides of an equation, the equality remains true.

Description

The Addition Property of Equality is a fundamental principle in algebra. It asserts that for any real numbers a, b, and c, if a = b, then a + c = b + c. This property is used to solve equations and maintain balance. For example, to solve x − 3 = 7, you add 3 to both sides to get x = 10.

Topic

Equations

Definition

An algebraic equation is a mathematical statement that shows the equality of two algebraic expressions. It's also another way of referring to a polynomial equation.

Description

Algebraic equations are central to algebra and involve variables, constants, and arithmetic operations. They are used to represent relationships and solve problems. For instance, the equation 2x + 3 = 7 can be solved to find x. Algebraic equations come in various forms, including linear, quadratic, and polynomial equations.

Topic

Equations

Definition

An algebraic expression is a combination of variables, constants, and arithmetic operations, without an equality sign.

Description

Algebraic expressions are fundamental components of algebra. They represent quantities and relationships without asserting equality. Examples include 3x + 4 and 5y − 2. Unlike equations, expressions cannot be solved but can be simplified or evaluated for given variable values.

Topic

Equations

Definition

Assigning values to variables involves giving specific values to variables in an equation or expression.

Description

Assigning values to variables is a fundamental process in algebra. It involves substituting variables with specific numbers to evaluate expressions or solve equations. For example, in the equation 

y = 2x + 3

assigning x = 4 gives y = 11.

Topic

Equations

Definition

A conditional equation is true only for specific values of the variable(s).

Description

Conditional equations are equations that hold true only under certain conditions or for specific variable values. For example, the equation 

x2 = 4 

is true only when x = 2 or x = −2. These equations contrast with identities, which are true for all variable values.

Topic

Equations

Definition

A constant term is a term in an algebraic expression that does not contain any variables.

Description

Constant terms are fixed values in algebraic expressions and equations. They do not change because they lack variables. For example, in the expression 

3x + 4

the number 4 is a constant term. Constant terms are essential in forming and solving equations.

Topic

Equations

Definition

The Division Property of Equality states that if you divide both sides of an equation by the same nonzero value, the equality remains true.

Description

The Division Property of Equality is a key principle in algebra. It states that for any real numbers a, b, and c (where 𝑐 ≠ 0), if 

a = b, then a ÷ c​ = b ÷ c 

This property is used to solve equations by isolating variables. For example, to solve 

3x = 12

divide both sides by 3 to get x = 4.

Topic

Equations

Definition

Equality is a mathematical statement that asserts that two expressions are equal.

Description

Equality is a foundational concept in mathematics. It indicates that two expressions have the same value, represented by the symbol "=". For example, in the equation 2 + 3 = 5, both sides are equal. Equality is used to form equations and solve problems.

In real-world applications, equality is used in accounting, engineering, and data analysis to ensure balance and accuracy. Understanding equality helps students develop logical reasoning and problem-solving skills.

Topic

Equations

Definition

An equation is a mathematical statement that asserts the equality of two expressions.

Description

Equations are central to mathematics, representing relationships between quantities. They consist of two expressions separated by an equals sign. For example, 

2x + 3 = 7

is an equation that can be solved to find x. Equations can be linear, quadratic, or polynomial, among others.

Topic

Equations

Definition

Equations in one variable involve a single variable and can be solved to find its value.

Description

Equations in one variable are fundamental in algebra. They typically take the form of ax + b = 0, where x is the variable. Solving these equations involves isolating the variable to determine its value. For example, solving 

2x + 3 = 7 

yields 

x = 2

Topic

Equations

Definition

Equations in two variables involve two variables and describe a relationship between them.

Description

Equations in two variables are essential in algebra and coordinate geometry. They typically take the form of 

ax + by = c

and represent lines in a coordinate plane. For example, the equation 

2x + 3y = 6

can be graphed as a line.

Topic

Equations

Definition

Equivalent equations are equations that have the same solutions.

Description

Equivalent equations are a key concept in algebra. They may look different but yield the same solutions. For instance, 

2x + 3 = 7

and 

4x + 6 = 14

are equivalent because both have the solution x = 2. Transformations such as addition, subtraction, multiplication, or division can produce equivalent equations.

Topic

Equations

Definition

A false equation is an equation that is not true for any value of the variable(s).

Description

False equations are equations that do not hold true for any value of the variable(s). For example, the equation 

x + 2 = x + 3

is false because there is no value of x that makes both sides equal. Identifying false equations is important in verifying the validity of mathematical statements.

Topic

Equations

Definition

An identity equation is true for all values of the variable(s).

Description

Identity equations are equations that hold true for all values of the variable(s). For example, the equation 

2(x + 1) = 2x + 2 

is an identity because it is true for any value of x. These equations are used to express mathematical identities and properties. In the equation above, the identity results from the use of the distributive property.

Topic

Equations

Definition

An inequation is a mathematical statement that shows the inequality between two expressions.

Description

Inequations, or inequalities, are statements that compare two expressions using inequality symbols such as >, <, ≥, and ≤. For example, 

x + 3 > 5 

indicates that 𝑥 x must be greater than 2. Inequations are used to represent constraints and conditions in mathematical models. Inequations sometimes involve the inequality symbol: ≠.