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Large Box
Grocery Bag
Marker
2 Small Plastic Cups
Scissors
Tape
5 Toilet Paper Rolls
Dice

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Construction of the function machine:

When creating the function machine, I choose to use materials I had available at my house. I quickly decided that I wanted something practical and simple. My students would not care about aesthetic quality or "wow" factors, only that the device worked and the activity I developed was engaging.

I started by choosing a box that was large enough in size to be used in a whole group setting, but small enough that students could handle it appropriately. I then cut a grocery bag to the appropriate size and added necessary graphics on the front, securing the paper with tape. Next, I traced two holes in the top and side of the design to determine where I needed to cut the box to allow the cups to fit. Then, I opened the box and created a slope using toilet paper rolls that connected to the two holes in the box. I secured the slope and cups with tape and practiced rolling a dice from one cup to the other, making slight modifications until the process was consistently successful.

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Process
Function
Rule
Input
Output

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-What strategies can you use to help you solve this?
-How can you check your work?
-How can you represent this a different way/.

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The explicit rule in a function, or change, is the component that needs to be analyzed by students (Van de Walle, 2016). This is why I chose that specific vocabulary on my function machine. It shows the "in" segment where a number is inputted, as well as an "out" area where the number is outputted. The arrow in between intentionally tells students there is a change occurring and prompts them with the question, "What is the rule?". As I introduced the function machine, I would go over the key vocabulary words mentioned to the left. Students would lead a guided discussion about these words, their prior knowledge, and questions they have. I will then explain the purpose of the function machine: Students will roll dice to receive a specific number. They will take this number and record it on their two column chart I have provided. I will demonstrate with a basic input with a dice that reads "1". I will then roll the dice through my device and determine where it lands. I will then record the output in the second part of my two column chart. For demonstration purposes, I will pretend the dice lands on "2". At the bottom of each graph, there is a line that reads "rule:" Here, children need to create an algebraic equation for their rule before filling in five other possibilities in their chart that follow the same rule. To explain this further, let's look back at my example with an input of "1" and an output of "2". Students could say the rule is "add one". Algebraically this may look like: X = Y + 1 (X = Output, Y = Input). Students would then fill in the next 5 spaces in their chart will other X and Y values that follow this rule. (X = 4, Y = 3; X = 20, Y = 19, etc).

However, this is not the only rule that could be constructed between these two numbers. A child could say the rule is: X = Y x 2 (X = Output, Y = Input). Some other possible values for these variables could be X = 4, Y = 2, X = 10, Y = 5, etc. I designed this activity to be naturally differentiated. What I mean by this is, I would ideally have a function machine for each small group of students. Although they would receive the same input and output numbers, there are countless ways to construct an algebraic equation that holds true for their digits. Students who struggle in mathematics more may initially see the difference between the numbers and create an addition or subtraction problem. Students with higher ability levels may challenge themselves to use multiplication, division, exponents, and even fractions. When presenting the assignment I would ask students who completed one rule for the given numbers to explore other options and record those as well. This means that students would not finish the assignment and be left without a task, students who struggle would be given ample time to find success with this activity, and students who need to be challenged would be given that extra opportunity. During this process, I would be walking around to monitor learning and make further accommodations where necessary.

To push students even further in their thinking, I could incorporate the Semantic Feature Analysis Task recommended by Gay (2002). For struggling students, I could give them a completed grid with multiple equations that hold true for our demonstration input and output. The columns could have more simplified options such as :uses multiplication, uses addition, positive slope, negative slope, etc. Students would simply have to consider each equation given and determine which qualities it possesses. For students who are doing well with the activity, they could have to construct their own grid, complete with equations and attributes.

This activity includes multiple components that engage students and develop deep understanding of algebraic concepts, even in elementary school. They are analyzing functions, finding patterns, generating data, and constructing expressions using variables. Students need this meaningful exposure to variable use in expressions now to help them with future mathematical practices (Van de Walle, 2016). The authenticity and possibilities with this activity make it applicable to all learners. I would be formatively assessing students throughout the lesson and end with an exit ticket that requires students to state the rule of a given input and output, and also explain their thinking.

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Gay, A. S., & Keith, C. J. (2002). Take time for action: Reasoning about linear equations. Reasoning about linear equations. Mathematics Teaching in the Middle School, 8(3), 146-148.

Van De Walle, J. A., Karp, K. A., & Bay-Williams, J. M. (2016).  Elementary and middle school mathematics: Teaching developmentally.  Upper Saddle River, NJ: Pearson.