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.
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.
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.
Dr. Po-Shen Loh shared a possible new method for factoring a quadratic. This post provides a layman’s attempt to share the steps that teachers may find intriguing and possibly useful, especially for complex roots.
The premise of Loh’s method is that in lieu of considering two factors separately, you can focus on the following:
Average of the two factors which is the coefficient of x (linear coefficient) divided by 2. In the case below, that is 8/2 = 4.
The common distance of each factor from the average, d. In the case below, the factors are converted into the expressions 4-d and 4+d because both are d units away from 4.
This results in a single unknown, the distance d.
The aforementioned expressions with d replace the factors. Now we have an easy quadratic equation to solve using square root. Once d is determined, the factors of 12 are now known and we are on our way.
This method works for complex| factors. This makes Loh’s method less time intensive than using the Quadratic Formula, and there is no formula to memorize.
Word problems are challenging for many students. Writing a system of equations to model a word problem has unique challenges. This post provides details about a scaffolded handout with color coding can unpack the process for generating the appropriate system of equations.
Unpacking the Word Problem
A mistake I have witnessed over the years is students mistakenly using given values for both equations. In the problem below, students are far more likely to generate the equation for the yellow part: 2x + 3y = 24. The challenge is that the blue part has only 1 number so students will often write 2x + 3y = 10, using the dollar quantities a second time.
By highlighting the two parts of the word problem with given values, the students can match parts of the word problem with respective equations. The scaffolding separates the parts, and the color allows for matching.
The rest of the problem is prior knowledge with the students using one of the methods for solving. The scaffolding continues to lower the task demand by reducing the need to remember all the steps. This allows them to focus more bandwidth on the new steps.
Solving the variable on both sides is the Cerberus of 1 variable linear equations. It has multiple steps, simplifying expressions, and eliminating a variable expression. The later is a new step, added on to all the other steps. This post describes a scaffolded handout to guide students through the mental steps and the written steps.
The process starts with mental steps of identifying the two variable terms. This directs students to focus on identifying that the equation has a variable on both sides which in turn leads them to understand the algorithm they will follow. The circling focuses attention on the operations. Then the students choose which variable term to eliminate and identify the inverse operation. The written steps are then guided.
Choosing the Variable Term
As I did in the scaffolded handout for solving a 2-step equation, I have students solve the equation two different ways. This time by eliminating each variable term respectively. This allows them to see for themselves which term may provide the path of lesser resistance.
Solving 2-step Equations is an escalation in terms of task demand. Starting in elementary school, they are exposed to math sentences in which there is a single operation. Now they are asked to choose a number to eliminate. This post provides details about a scaffolded handout. It addresses the choice of number to eliminate and how one choice may provide the path of least resistance.
Similar to the handout for solving 1-step equations, this scaffolded handout engages the students with the mental steps in addition to the steps they write out when they show their work. A key concept is identifying that there are two numbers to eliminate and then eliminating them, one at a time.
Choosing the Number
In lieu of teaching the method of doing PEMDAS or order of operations in reverse, I focus on presenting both the respective steps for eliminating either number in the expression with the variable. I believe it strengthens their understanding of the steps for solving and it shows them a reason to choose one route over the other – but they are free to choose! Here is a FB Reel and Youtube video addressing this.
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.
I introduce key characteristics with parabolas and use the analogy of a rollercoaster. Riding once (and never a again) the Superman rollercoaster at Six Flags New England got me thinking about this. At one point the roller coaster hits ground level (a zero) and then goes underground (negative y values).
Here is a handout I use for the introduction. Here are images showing how I use the handouts. The table helps students visualize the x-values and y-values when looking at the graph. The rollercoaster provides extra context for the various characteristics, e.g., increasing means the rollercoaster is going up. Note the scaffolding by adding context clues for each characteristics.
I start with max height of the rollercoaster. I highlight the actual graph first, then the y-values in the table. Then point out we are looking for the x-values for what we highlighted.
The issue of highlighting the vertex for the increasing and decreasing values would be addressed when writing the interval. The idea of the rollercoaster at the tip top provides context to develop the concept.
Similarly, I start with the zeroes. Again, highlight the graph, then the y-values, then the x-values.
The issue of highlighting the zeroes for the positive and negative values would be addressed when writing the interval. The idea of the rollercoaster at the ground level provides context to develop the concept.
I has been effective to have students highlight the parts of the respective axis when discussing the domain and range (not discussed yet). A common challenge is understanding that the x-values continue to the right or left when it appears they simply go down. To address this, I use a very wide parabola to show more lateral movement.
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.
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.
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 are introduced in similar fashion, including with 0 toppings for the intercept.
Here is a scaffolded handout. The first problem has additional scaffolding to convert from addition to subtraction and vise versa. The first problem does not require the transformation of the operations but the second one does.