# Description

In this module students learn the basics of integers, with plenty of real-world applications to ground their knowledge. Two short videos introduce the topic of integers and how to graph integers on the number line. A Quizlet activity allows students to practice the skill of identify positives, negatives, or zero. Four formative assessment items test a student's understanding of integers and graphing integers on a number line.  allow students to test their understanding of these topics.

For the problem solving activity, students learn about super-cooling electronic circuits for certain high-tech applications. Students use their knowledge of integers to arrange temperature values on a number line.

This module can be used as an introductory lesson on integers for pre-algebra or algebra. The content aligns with the grade 6 Common Core State Standards, but this module can be used as a refresher for higher grades.

This module can be assigned to individual students or groups of students. Students should be able to complete this lesson in 20 minutes.

• Algebra

#### Learning Objectives

• Define integers
• Use integers for measurement
• Graph integers on a number line

#### Prerequisite Skills

• Comparing and Ordering Whole Numbers
• Graphing Numbers on a Number Line
Common Core Standards CCSS.MATH.CONTENT.6.NS.C.6 20 mins 6th - 7th Grade

# Description

In this lesson students will use their basic understanding of circles to learn how circular structures are built. The example shown is that of the Roman Colosseum. Students will construct an oval shape from circular arcs to simulate the elliptical shape of the Roman Colosseum.

For this lesson make sure that students have ready access to a compass, ruler, grid paper (graph paper with x-y axes marked is preferred), and pencil with an eraser. The first construction has students creating a teardrop shape from circular arcs that have overlapping points of tangency. This first construction sets up the more elaborate second construction.

Students then look at a real-world application of using circular arcs to approximate the elliptical shape of the Roman Colosseum in a highly engaging video. Students analyze the architecture of the Roman Colosseum and are then shown how to build a scale model of the interior of  the Colosseum.

This lesson can be assigned to individual students or teams of students. The lesson can be completed in about 20 to 25 minutes.

#### Math Concepts

• Geometric Constructions
• Circles

#### Learning Objectives

• Define the terms circular arc and tangent
• How to construct circular shapes using circular arcs
• How to analyze the architecture of circular structures

#### Prerequisite Skills

• Basic knowledge of circles
• How to use a compass
Common Core Standards CCSS.MATH.CONTENT.HSG.C.A.4, CCSS.MATH.CONTENT.HSG.MG.A.3 20 mins 8th - 10th Grade

# Description

In this module, students explore algebraic expressions to model different quantities. They look at expressions that involve addition, subtraction, and multiplication. Then they look at real world data from the Star Wars movies since the Disney acquisition of the franchise. Students analyze whether the purchase of the Star Wars franchise has been profitable for Disney.

For this lesson make sure that students are familiar with the definitions of variables, unknowns, and constants. Review definitions are provided.

Students will learn about modeling and evaluating algebraic expressions. In particular students will look at an expression of the form $px-C$, where p is the price and C is the cost of putting on an event (concert, movie).

Students then look at a real-world application of the Disney purchase of the Star Wars franchise. Students analyze box office data and arrive at an algebraic expression using this data set.

This lesson addresses Common Core standards from grades 6 and 7. This lesson can be assigned to individual students or teams of students. The lesson can be completed in about 20 to 25 minutes.

#### Math Concepts

• Expressions
• Equations
• Algebra

#### Learning Objectives

• How to model and evaluate algebraic expressions
• Using algebraic expression to mode real word situations
• Analyze real world data

#### Prerequisite Skills

• Knowledge of variables, constants, and unknowns
• Rounding numbers
Common Core Standards CCSS.MATH.CONTENT.6.EE.A.2, CCSS.MATH.CONTENT.7.EE.B.3 20 mins 6th - 9th Grade

# Description

In this module model students will learn how to use algebra tiles to model positive numbers, negative numbers, and zero. For this module make sure students have access to set of algebra tiles to use to model integers. Students will then extend their knowledge of integers to learn about matter and anti-matter.

In this module we use red and yellow tiles. The yellow tiles are used to model positive integers, and the red tiles are used to model negative integers. Three short videos describe how to use these tiles to model positives, negatives, and zeros. This module does not address number operations.

As an extension activity students apply their knowledge of zero pairs to the context of matter and anti-matter. A short video describes the concept.

Throughout the lesson various assessment items are included, which are then scored. A teacher's guide includes a detailed description of all components of this lesson.

This lesson addresses the Grade 6 Common Core Standards but it can also be used at any point where algebra tiles are introduced.

This lesson can be assigned to individual students or teams of students. The lesson can be completed in about 20 to 25 minutes.

#### Math Concepts

• Number Concepts
• Problem Solving

#### Learning Objectives

• Define integers
• Model positive numbers, negative numbers, and zero with algebra tiles
• Solve problems using algebra tiles

#### Prerequisite Skills

• Understand whole numbers
Common Core Standards CCSS.Math.CONTENT.7.NS.A.1 20 mins 6th - 7th Grade

# Description

In this module students develop their skills at translating version expressions into mathematical ones. This is a crucial skill that often doesn't get enough attention. A student's ability to successfully translate words into mathematical expressions and equations puts that student on a path to successfully solving more complicated problems.

In this module students focus on subtraction expressions. This includes numerical and algebraic expressions. Students progress through these form:

• a - b, for integers a and b
• x - a, for integer a
• ax - b, for integers a and b

As an extension of their work with translating words into mathematical expressions, students explore how the Inca used Quipus to identify numbers. This becomes an opportunity to translate the visual symbols from the Quipu into numbers in the proper place value. They also translate visual symbols into verbal mathematical expressions.

#### Math Concepts

• Expressions
• Number Concepts
• Problem Solving

#### Learning Objectives

• Translate verbal expressions into algebraic expressions
• Solve problems using algebraic expressions

#### Prerequisite Skills

• Familiarity with integers
• Familiarity with algebraic expressions
• Place value
Common Core Standards CCSS.MATH.CONTENT.6.EE.A.2.A, CCSS.MATH.CONTENT.6.EE.B.6 20 mins 6th - 7th Grade

# Description

In this module students apply their knowledge of linear functions to the context of distance vs. time graphs. They look at data sets, graphs of coordinates, and algebraic representations of distance vs. time functions. Then students go on a field trip to a Nascar race to see how timing at the pit stop has an impact on distance vs. time data.

Students begin by analyzing linear function data by identifying the common difference. The goal is to see the relationship among the common difference and the slope of the linear function.

Students use the Desmos graphing calculator to explore graphs of data and equations. Both types of linear functions are explored:

• ax
• mx b

The goal here is to connect the slope of the distance vs. time graph to the speed of the car. Students then explore the graphs of cars with different speeds and initial distances.

Students then watch a video about Nascar pit crews and learn about the rapid-response pit crews who change the tires on the race cars and how the timing of this affects the distance vs. time graph of the cars. Students use the information in the video to analyze linear function models for different pit crew times.

Note: Be sure students are familiar with the concept of slope and the basic definition of a linear function.

Throughout the lesson various assessment items are included, which are then scored. A teacher's guide includes a detailed description of all components of this lesson.

This lesson addresses the Grade 8 Common Core Standards but it can also be used in grades 9 and 10 for review purposes.

This lesson can be assigned to individual students or teams of students. The lesson can be completed in about 20 to 25 minutes.

#### Math Concepts

• Functions
• Problem Solving
• Equations

#### Learning Objectives

• Model distance vs. time linear function data
• Write linear functions in slope intercept form
• Graph linear functions
• Solve problems involving speed

#### Prerequisite Skills

• Understanding of linear functions
• Familiarity with the slope formula
Common Core Standards CCSS.MATH.CONTENT.8.F.A.2, CCSS.MATH.CONTENT.8.F.A.3, CCSS.MATH.CONTENT.8.F.B.4 20 mins 8th - 10th Grade

# Description

In this lesson students learn how to use the slope formula to calculate steepness. In particular, students learn how to calculate steepness in the context of cycling. Cyclists use a measure called grade to calculate the steepness of a hill or mountain. Students apply their knowledge of slope to the concept of grade.

Student learn to use the slope formula and then apply it to the context of grade. Several instructional videos provide the background on using the slope formula. Examples of using the slope formula are then provided.

Assessments include two drag-and-drop activities that call on students to carefully analyze the use of the slope formula.

• Slope
• Algebra

#### Learning Objectives

• Calculate slope using the slope formula
• Express slope as a percent
• Apply slope to the context of cycling

#### Prerequisite Skills

• Graphing coordinates
• Using rational numbers
Common Core Standards CCSS.MATH.CONTENT.8.F.B.4, CCSS.MATH.CONTENT.7.NS.A.3 20 mins 6th - 8th Grade

# Description

In this module, students explore a physics-based application of linear functions: Hooke's Law. By exploring the properties of springs, a simple linear model is developed. Students then explore applications of Hooke's Law, from weight scales to bungee cords.

Students investigate the properties of springs and identify two variables: the displacement of the spring (extension or compression) and the amount of force involved. From this students identify the independent variable and dependent variable. A graphing calculator activity (using the Desmos graphing tool) allows students to explore the value of k in the function F = kx.

#### Math Concepts

• Functions
• Problem Solving

#### Learning Objectives

• Create a linear function model
• Test the model for accuracy
• Graph a linear function

#### Prerequisite Skills

• Understands the basics of linear functions
• Graphing on the coordinate plane
Common Core Standards CCSS.MATH.CONTENT.8.F.A.1, CCSS.MATH.CONTENT.8.F.A.3, CCSS.MATH.CONTENT.8.F.B.4 20 mins 6th - 8th Grade

# Description

When a rocket is launched into space, it starts from rest and within minutes reaches speeds of tens of thousands kilometers per hour. In other words, the rocket accelerates.

In this module, students apply their knowledge of linear functions to explore the speed vs. time function. In the process they learn about acceleration, as well as the properties of this linear function.

Students first explore the equation for calculating acceleration. Then they use that to develop the speed vs. time linear function.

This module can be completed in about 20 minutes. Make sure that students understand the basics of linear functions in slope-intercept form.

#### Math Concepts

• Functions
• Problem Solving

#### Learning Objectives

• Create a linear function model for speed vs. acceleration
• Test the model for accuracy
• Graph a linear function model on a Cartesian coordinate system

#### Prerequisite Skills

• Understands the basics of linear functions
Common Core Standards CCSS.MATH.CONTENT.8.F.B.4, CCSS.MATH.CONTENT.8.F.B.5, CCSS.MATH.CONTENT.8.F.A.3 20 mins 6th - 8th Grade

# Description

As the size of a circle changes, so does the size of the diameter and that of the circumference. In fact, there is a linear relationship between these two measures. This relationship can be modeled with a linear function. In this module students will study this linear function and examine its properties, including the fact that the slope of this function is π itself.

This is a hands-on module in which students will measure the diameters and circumferences of a number of different containers. This data gathering will lead to graphing the data. From that students develop a linear model using the Desmos graphing tool. Students will see that the relationship between circumference and diameter has to do with π. In fact, the slope of the linear function is π itself.

For the hands-on part of the lesson, make sure you have all the materials: Different size cylindrical containers (bottles, cups, etc.), string, marker, and a ruler (preferably a caliper). Collect all the student data and use the embedded Desmos graphing tool to graph the data and explore the linear function.

The module concludes with an overview video about the number π.

• Functions
• Geometry
• Circles

#### Learning Objectives

• Create a linear function model for circumference vs. diameter
• Graph the function and interpret slope, y-intercept, domain, and range

#### Prerequisite Skills

• Understands the basics of linear functions
Common Core Standards CCSS.MATH.CONTENT.8.F.A.3, CCSS.MATH.CONTENT.8.F.B.5 20 mins 8th - 10th Grade

# Description

What do drones have to do with domain and range? In this module, students learn about an emerging high-tech delivery system and use that as a vehicle for learning about a function's domain and range. Students will graph data from a table and explore domain and range, and then they graph a continuous function. This highly engaging module will give students a solid understanding of algebraic functions, specifically domain and range.

In this module students will learn the following concepts:

• Finding the domain and range for a set of coordinates that define a function
• Finding the domain and range of a continuous function that doesn’t extend to infinity
• Finding the domain and range of a continuous function that extends to infinity

#### Math Concepts

• Functions
• Graphing Numbers

#### Learning Objectives

• Finding the domain and range for a set of coordinates that define a function
• Finding the domain and range of a continuous function that doesn’t extend to infinity
• Finding the domain and range of a continuous function that extends to infinity

#### Prerequisite Skills

• Understands the basics of linear functions
Common Core Standards CCSS.MATH.CONTENT.8.F.A.1, CCSS.MATH.CONTENT.8.F.A.3 20 mins 8th - 10th Grade

# Description

In this module students explore indirect measurement by seeing how simple angle measure, height measurements, and tangent ratios can be used to calculate distances. The context of castles provides a historically relevant military purpose for the tallness of castles.

This module explores Himeji Castle in Japan, as well as other castles. Some of the concepts explored in this module include:

• Angle measurements and alternate interior angles of parallel lines
• Tangent ratios
• Linear functions, in particular the functional relationship between a castle's height and its line of sight for a given angle

This module provides a nice blend of algebra and geometry topics and can be used in an algebra unit on linear functions, a geometry unit on tangent ratios, or even a pre-calculus lesson on tangent ratios and functions.

#### Math Concepts

• Functions
• Ratios and Rates
• Triangles
• Geometry

#### Learning Objectives

• Define trig ratios
• Solve problems involving trig ratios
• Use linear functions to model real-world phenomena

#### Prerequisite Skills

• The concept of a ratio
• Basic properties of right triangles
• Basic understanding of a functional relationship between two variables
Common Core Standards CCSS.MATH.CONTENT.6.RP.A.3, CCSS.MATH.CONTENT.8.F.B.4, CCSS.MATH.CONTENT.7.G.B.5 30 mins 7th - 10th Grade

25. \$3.99

# Description

Have you noticed how wrinkled an elephant's skin is? What purpose does it serve and what does math have to do with explaining this phenomenon? Well, the explanation for an elephant's wrinkled skin is almost entirely a math story.

In this module students explore rational expressions and functions in the context of the ratio of surface area and volume for various three-dimensional figures. Such figures can be used to model the basic shapes of animals.

This ratio reveals a lot about how an animal is able to retain heat or lose it rapidly, depending on the animal's habitat. The geometry of heat transfer also has applications in architecture and design.

• Calculate the ratio of surface area and volume
• Graph rational functions
• Solve real-world problems using rational functions

Make sure your students know the basics of rational numbers and functions.

#### Math Concepts

• Functions
• Ratios and Rates
• Geometry
• Algebra

#### Learning Objectives

• Calculate the ratio of surface area and volume
• Graph rational functions
• Solve real-world problems using rational functions

#### Prerequisite Skills

• Understanding of rational numbers
• Understanding of the basic properties of functions
• Understanding of area and volume formulas for simple geometric shapes
Common Core Standards CCSS.MATH.CONTENT.HSF.BF.A.1.B, CCSS.MATH.CONTENT.7.NS.A.3, CCSS.MATH.CONTENT.HSG.MG.A.1 30 mins 8th - 11th Grade