Many students struggle with writing equations for linear functions, even with only 2 parameters to fill in (slope and the y-intercept are parameters for the equation). This approach make a connection between the table and graph with the equation. The relevant, real life context helps students.
This post presents a scaffolded and meaning making approach to exponents that are 0 or negatives.
The slide show below presents all 4 pages.
The handout starts with an initiation to preview the prerequisites for what is presented in the lesson. It also introduces a chart that will be used for discovery.
Page 2 presents a discovery activity of following a pattern of dividing by 2 down to the 0 exponent. The concept of exponents is presented as the number of occurrences of the base. This leads to the idea of a 0 exponent indicating the base is no longer present, but there is still 1.
Similarly, on the 3rd page the pattern of dividing continues into negative exponents to show the resulting fractions. The negative exponents are then presented as reciprocals.
For terms with multiple factors (e.g., 5x vs just x) the students are presented steps to write the factors separately. This unpacks the reason why the negative exponent acts only on one of the factors (unless both are grouped with parentheses).
Functions are perhaps the most prevalent and important topic covered in secondary math, aside from maybe 1 variable linear equations. The concept of a mathematical function is challenging for many students. This post provides details about a meaning making approach to introducing functions.
The introduction is presented on a Google Jamboard, to allow for movement in the pairing of inputs and outputs. It starts with analogies pairing of items using a gumball machine and a Coke machine and proceeds incrementally towards the various representations. The functions are contrasted with examples of relationships that are not functions.
Slides of the Jamboard
Slides1 and 2 present the gumball and Coke machines. Students can move the items to see how a quarter can result in 2 different color gumballs while the Coke button results in only 1 output.
In slides 3 and 4, the use of an hourly wage introduces input and output with quantities. Slide 4 shows two different pay amounts for the same number of hours worked. This taps into prior knowledge.
The sequencing progresses through
Each includes an example and a non-example.
The last slide provides a sorting activity.
Access to Jamboard
Here is the link. To access the Jamboard, you need to make a copy.
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.
To help a student unpack the concept of a spending limit and the act of overspending, I created an in person shopping experience at Barnes and Noble. I purchased a gift card with a balance of $1 (image below). The student I was helping was tasked with purchasing an item that cost over $1. He had experience with ordering and paying on his own with enough money provided. This time he was in a position to overspend. I was ready with cash and stepped in. This allowed him to experience, firsthand, the overspending and budget situation.
Clearly, we must take into account the level of anxiety a student may have with such an activity before undertaking it.
Fractions are challenging. Multiplying fractions is really challenging! This post presents a Google Jamboard to introduce students to the concept of multiplication of fractions.
The artifact is chunked to incrementally move from multiplication of whole numbers to whole number and fraction to multiplication of fractions. The representation of multiplication as number of objects in a group times number of groups is the structure used throughout. Cookies on a plate is the context used to draw upon prior knowledge and make the idea more concrete.
This serves as an introduction. Each chunk can be followed by practice before moving on to the subsequent chunk.
The Jamboard starts with a representation of multiplication as groups of objects, first with the number of objects in a group and the number of groups. This is presented first as cookies per person to connect to prior knowledge. Then presented per plate as the plate is subsequently used to model the fractions.
First, whole number times a fraction is presented. This allows for a connection to prior knowledge and introduces fractions in this representation. There are still 6 cookies per group, but now there is only 1/2 a group.
The students can move the cookies onto the plate to see the group of objects. Then they can cut the group in half.
To help make sense of the fractions used in the multiplication of two fractions, the fractional parts of the cookies are presented first.
For multiplication of fractions, the process is the same. There is 1/4 of a cookie in each group, then there is 1/2 a group. As was done previously, 1/2 the group is removed. Conceptually, you can explain to the students that they have 1/4 of a cookie and they split it with a friend.
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.
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.
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.
Shopping is surprisingly more complex that we realize for many students who are working on life skills math. Staying within a spending limit is one issue. The concept of a running balance is another. In this post, I detail an activity in which a gift card in real life and ones in simulations are used in shopping activities. The purpose is to engage students with spending limits and balances.
Real Life Activity
I provided a student with a $10 gift card to buy a hot chocolate at Barnes and Noble. This experience allowed the student to order, pay, and monitor the balance. At some point the card will not have a balance to cover another drink. Not only does this provide real life experience, it provides an anchor for other instructional activities.
The student identified the price, the tax, the total, and the balance. Then he computed the balance to see for himself how this works.
Simulations in Instructional Settings
In our instructional setting, the student is provided an image of a gift card on a Google Slide and is prompted to buy 1 item at a time. This allows for immediate computation and tracking of a running balance.
In the image below, you can see a prompt for the student reflect on how the balance is computed by referring to the in person experience. I have the student compute on Google Calculator to allow me to record the work.
The running balance is recorded on the Google Calculator image ($18.75 below) and the next item is purchased with a new running balance computed.
I am in a position to conduct in person instruction at other settings. Obviously, most teachers do not have this opportunity. My recommendation is to collaborate with a parent to facilitate such activities. For example, if the family is going out to eat, the student can be provided a gift card and allow him or her to choose an item and pay independently. Another possibility is that the parent provides a student an e-gift card to spend during class with the teacher.
Exponents and Basic exponent rules are challenging. The Power Rules add another layer of challenge. This post outlines an instructional approach. The original problem is decomposed and then recomposed to show how the underlying concepts of the Power and Power of a Product Rules.
The Jamboard is configured in similar fashion as the Jamboard used for the Product and Quotient Rules. The exponential terms and variables are moveable parts. The background is a scaffolded to guide the decomposition. Here is a FB Reel and a YouTube video showing how it works. NOTE: I decompose the expression down to individual X values in lieu of using the Product Rule because I want them to see how many Xs there are. Also, the Product would be relatively new to them, I wanted to reduce the task demand placed on the working memory.
Here is a link to the Google Jamboard. To get access, you must make a copy.