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.

Temperature is one of the most important measurements that we deal with on a daily basis. Weather, climate, food preparation, health, and other phenomena involve some type of temperature measure. The two most common units of temperature measure are Fahrenheit and Celsius.

There is a linear function that allows you to convert from one unit to another. In this module, you'll learn about this linear function. In fact, students will learn about this function and its inverse.

The module starts with an analysis of Celsius-to-Fahrenheit data. They look at the functional relationship between the variables and develop a linear model using the Desmos graphing calculator. They analyze the properties of this linear function and look at its graph.

Next, students analyze Fahrenheit-to-Celsius data. They also develop a linear function model using the Desmos graphing calculator.

Finally, they compare the graphs of the first function and its inverse to identify properties of functions and their inverses.

Cheetahs can accelerate up to 75 mph and can easily outpace a gazelle. But gazelles have adapted to keep cheetahs at bay long enough to tire them out. We can analyze this phenomenon mathematically through the use of some basic concepts involving functions.

In this highly engaging module students learn about functions, domain, range, and mathematical modeling. They will look at the following types of functions:

• Speed vs. distance
• Distance vs. time
• Displacement vs. time

These three functions are analyzed using function notation, and the domains and ranges are clearly defined. Students explore a mathematical model that shows whether a cheetah will catch the gazelle or if the gazelle escapes.

This module also uses the Desmos graphing calculator extensively.

To come.

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.

In this module students learn the properties of linear functions. They look at data sets, graphs of coordinates, and algebraic representations of functions. Then students go on a field trip to the US Mint to see how money is printed. From this they develop linear function models for calculating the number of bills printed, along with their dollar value.

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:

• y = ax
• y = mx + b

Students then watch a video about the US mint and how currency is created. Students use the information in the video to develop two linear function models: one for calculating the number of bills produced for every sheet of printed bills, plus another for calculating the dollar value of the printed bills.

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

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:

• y = ax
• y = 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. 

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.

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.

In this Algebra Application, students develop a linear mathematical model based on the maximum heart rate during exercise based on age. Using this model, students investigate heart rate for moderate and vigorous workouts. The math topics covered include:

• Mathematical modeling
• Linear functions
• Families of functions
• Computational thinking

The culminating activity is for students to create a simple program (using either a spreadsheet or a programming language like Python) to develop an exercise chart.

This is a great back-to-school activity for middle school or high school students. A relevant real-world application allows them to review math concepts.

Note: The download is a PDF file that includes the link to the GoogleDoc version.

In this Algebra Application, students study the direction between diameter and circumference of a circle. Through measurement and data gathering students analyze the line of best fit and explore ways of calculating pi. The math topics covered include:

• Mathematical modeling
• Linear functions
• Data gathering and analysis
• Ratios
• Direct variation

This is a great back-to-school activity for middle school or high school students. This is also a great crossover activity that ties algebra and geometry.

Note: The download is a PDF file that includes the link to the GoogleDoc version.

In this presentation we show how to convert a linear equation in Standard Form to a linear function in Slope Intercept Form. We go over the reason for such a conversion and applications that give rise to these equations. Note: The download is a PPT.