Intro to Graphing Linear Functions using Jamboard

Graphing linear functions may be the most important topic in Algebra 1. While proportional reasoning is a prelude to functions, this is the first formally identified function presented to them. The graphing leads to slope and intercepts, beyond the entry point for graphical representations to functions. This post presents an activity that can serve as the entry point for linear functions.

Overview

The activity is presented on a Google Jamboard, which provides manipulatives. It begins with a relevant context for students, money and being paid for a job. This allows them to engage the function using money. Before using numbers, they engage the work context through images. They are presented the table and then graph representations of the function before getting into the equation. Here is a FB Reel showing the movement of the images.

The Slides

There are 3 categories of slides. Here is a description of each.

  • Table and graph clocks for hours and dollar bill for the money.
    • They graph the whole hours first, then fractional hours (1/2 and 1/4) to see that there are points “squeezed in between each other. This leads to the idea of infinite number of points. In turn, this leads to the idea of the line are a visual means to present all the points. The points can be presented as solutions. Hence, the graph presents all the solutions for the function.
  • Table and graph with numbers on sticky notes that can be moved from the table to ordered pairs to positions on the coordinate plane.
  • The equation, with sticky notes to show numbers substituted in for the variables and then moved to ordered pairs with parentheses.

Here is a link to the Jamboard. You need to make a copy to access it.

Intro to Intercepts Using Taco Bell

This post provides details for a discovery activity for intercepts. Students shop for tacos and burritos at Taco Bell. They are to find combinations of number of tacos and number of burritos. This will include no tacos or no burritos. The activity is provided on a Google Doc.

Overview

Student prior knowledge is leveraged to provide a meaning making activity. The concept of intercepts is developed through 0 tacos or 0 burritos. Here are the steps for the activity.

  • The students are tasked with spending the entire balance of the $30 in gift cards on tacos and burritos.
  • They are to find all the combinations that are possible.
  • Students enter the combinations into a table found on a linked graph on Desmos. The image below shows the final product, upon completion.
  • Then they are tasked to identify the two combinations that standout in the table.
  • Finally, they unpack why these two combinations are unique in the graph.
  • At that point, the teacher can present the “mathy” term, intercepts.

Google Doc

Below are images of a Google Doc with the activity. You have to make a copy to edit it. Here is the link to the Demos graph with blank table and labeled axes. It is included in the Google Doc.

Word Problems with Unit Rates

Word problems are challenging. In middle school and high school, word problems for proportional relationships and linear functions are particularly challenging. This post shows an visualization approach to unpacking the unit rate component of such word problems.

Visualizing Unit Rate

Below is an excerpt from a handout used to introduce such word problems. It starts with the unit rate and unit quantity provided. This is more accessible for students and allows for a visual representation. This allows students to “see” the multiplication and make a connection to the word problem.

This is followed by situations in which the unit quantity is unknown. Students still draw a visual, but use the 3 dot symbol for etc. to indicate the unknown quantity.

Fading Visuals

The same problems are used but students now write the symbolic representation. First they write the expression with both factors provided, then with the unknown. The “?” emphasizes there is an unknown quantity before writing the variable.

The Handout

Here is a link to the handout. It is in WORD format to allow you to enter your own problems.

Intro to Slope of a Line

Slope is one of the most important topics in algebra but perhaps one of the most challenging topics. One issue is that students often think of the formula as opposed to the idea of slope. The images shown below are for a handout that unpacks the idea or concept of slope.

MathBootCamps

Handout

The images show sections of a handout. Slope is a measurement of a graphed line. It measures steepness and also indicates direction. The handout starts by drawing on prior knowledge of measurements for newborns.

Prior knowledge of steepness and up and downhill are invoked.

How to measure steepness is introduced through a focus on stairwells.

The measurement of the length and height of steps leads into the rise and run of a hill. I use “right” and “up” or “down” which focus on direction. The terms “rise” and “run” can be addressed subsequent to the introduction. This part of the handout is scaffolded to reduce task demand and focus on the concept.

Finally, the students are provided a line and apply the ratio procedure. Note: the slopes are not the same as in the prior section.

Jamboard

An accompanying Jamboard will eventually be shared in this space.

Introduction to Scatter Plots with Google Sheets

This post outlines an activity to introduce linear functions (or scatter plots). The students are tasked with shopping for a used car – a specific make and model. They go to Carmax.com to find mileage and price for 10 cars for sale. They have to find a make and model that has at least 10 cars and can change the search radius to include all locations of Carmax as necessary.

They enter the data for each car into a table on a Google Doc. They are not to include the “k” or the “$” or “,” for price. This allows easier transfer of data. I do not use 0s for the mileage as the slope is more meaningful per thousand miles. For example, -$104 per thousand miles vs $.104 per mile.

Before they graph, you can provide them a common set of data to guide them through a trial run. This way you can show them your graph of the data to allow them to verify that they did it correctly. The data sets shown below are linked at the bottom of this post. (This can be useful for introducing systems of equations as Mustangs typically have a higher intercept and a steeper slope, which allows for a cluster of dots from both in an intersection.)

They copy and paste the mileage and price into a Google Sheet and attempt to graph. You can provide a link to a YouTube video on graphing a scatter plot to free you up to help individuals. The title of graphs should have the variable(s) and the individuals under study. A subtitle can be included to show when data was collected or a data set was accessed. The variables should include units.

A next step would piggy back off of this activity with a Jamboard addressing mileage and price to help students interpret scatter plots.

Here are links to items used for this activity.

Intro Linear & Proportional Relationship

This post provides a handout to introduce linear functions and proportional relationships. Multiple instructional strategies are implemented. There is context that draws upon prior knowledge, scaffolding, and visual representations. Graph and table representations of the functions are used as an entry point, without the use of equations.

Overview

There is a difference between proportional relationships and linear functions, which is addressed in another post. Proportional relationships are a subset of linear functions. They can be used as an entry point by citing it as prior knowledge and then showing how they have a constant rate of change.

Proportional Relationship

The students complete the table with the images then the table with the variables. This leverages the context to help them make sense of the table and graph. You can follow up by asking them the total for 0. This allows you to highlight the intercept.

Linear Functions

Linear functions are introduced in similar fashion, including with 0 toppings for the intercept.

Handout

Here is a link to the handout.

Introduction to Intercepts – Mini-lesson with Scaffolded Section for Computing

Here is a link to the document, with images showing the notes. This is a mini-lesson with the following components.

  • A fill in the blank for writing the lesson objective.
  • A Do now which serves as an initiation to the lesson. The y-intercept can be discussed in the context of buying 0 slices of pizza and paying $1.
  • A notes section on what an intercept is.
  • Practice session on identifying intercepts in graphs and tables.
  • A scaffolded steps section on computing the ordered pair of the intercepts.

Matching Activity for Linear Function Representations

Here is a matching activity on a Google Slides file for various representations of a set of linear functions: verbal, symbolic (equation), graphical, and tabular (or data). The students use gallery view of the slides and sort them by function. Then they can change the background color with a different color for each function. This invokes their analytical skills to decipher key elements of the function and of each representation, for example they may identify the value of the y-intercept in the equation and find a graph with the same value.

Proportional Relationship (with k) vs Linear Function (with m)

There have been interesting discussions on various Facebook Teacher pages about proportional relationships vs linear functions. This mirrors discussion about the constant of proportionality vs slope vs unit rates.

The difference between the proportional relationships and slope is context and the ratios. The ratio for the former addresses the variables themselves. The ratio for the slope addresses the change in the variables. This arises from context. To flesh this out let’s use the pay as a function of hours, with $15 an hour for the hourly rate.

  • If we focus on the fact that to compute the total pay, we multiply by the hours worked by 15 we have a proportional relationship and the $15 an hour is a constant of proportionality.
  • If we focus on the fact that every increase of 1 hour results in an increase of $15 in our total pay, we have a linear function and the $15 an hour is a rate of change (or slope of the line). Because of the context, we have different constants (k vs m).

Algebra 1: Concepts and Skills

I had an interesting discussion through a Facebook post recently regarding concepts vs skills. I want to share some information I have gathered regarding this topic. I do so, because there were a substantial number of teachers advocating for skill based learning. I hope to initiate some meaningful discussion.

Below left is a photo of an information processing model presented in a graduate level course on learning I took at UCONN. A key element I want to highlight is that information is more effectively processed if the information is meaningful. A theory behind this is Gestalt Theory in which the brain want to make information meaningful or organize it, e.g., the closure model in which our brains complete the triangle in the middle of the circle portions.

The meaning underlying math skills originates in the concepts. Below are the definitions for both, with the concepts being the “how or why” underlying the skills which are the “what to do” part.

I am not arguing that skills are unimportant or that rote practice is wrong. My position is that the concepts should drive the process. Here is a cartoon I think highlights the challenges with students having only skill based knowledge for topics that have important underlying concepts. I witnessed this first hand as a college adjunct instructor and found that a substantial number of students only understood slope by its formula. I also see a substantial number of students receiving special ed services who are taught at a skill level only to allow for progress. Often this is challenging for them when they have working memory or processing issues.

I will summarize in my own words an interpretation an article I read on the definition of Math, which stated there is no singular definition. The following was a theme that appeared to emerge. Math is a set of quantitative related ideas that can help explain the phenomena and the world. The mathematical symbols are used to represent these ideas. There are different ways to represent these ideas, e.g., we represent functions with tables, graphs, and equations. Formal proofs in Western Civilization are not the same a those in the East. Computer based proofs are not fully accepted by many math experts.

Technology has provided amazing ways to represent mathematical ideas. The most genius approach I have encountered is Dragonbox. The image below shows their initial representation of an equation through their algebra app. It develops the concept and the skills simultaneously.

Below is a list of some algebra 1 topics and some of the associated concepts. These are largely derived from math sources but include some massaging by me. I am happy to hear the working definitions of others.