What is the difference between 3x and x3? Many of my students in Algebra 1 (and Geometry and Algebra 2, if I’m being honest) struggle with the distinction, especially when it comes to adding three x’s together to get 3x versus multiplying three x’s together to get x3.
The final products. 4x is represented by 4 candles (two sets of “Love Candles”), and x4 is one candle with four lights.
In her article “Rethinking Technology & Creativity in the 21st Century: Crayons are the Future”, Punya Mishra (2012) cites Lako and Nunez (2000)as having “suggested that abstract mathematical concepts are grounded (through thinking in metaphor) to sensory-motor experiences based on perception and action in the physical environment.” It was in this spirit that I decided to try combining paper circuits and exponents. Could students come up with a way to represent monomials with a paper circuit? To do so would demonstrated a deeper understanding of the meaning of a term such as 4x than just doing problems over and over again.
In this activity, students will create two paper circuits: one representing a monomials such as x2, the other representing the same variable with the exponent now a coefficient, in this case, 2x. They can do this in any way that makes sense to them, provided that their mathematical reasoning is accurate and they can justify their representation. When projects are completed, students will explain their circuits to the class, then write a blog post about their representations for the class website, including pictures of their work. This is intended to be a tool for other students struggling with exponents.
Students often confuse or struggle with what to do with the exponents of monomials in multiplicative and additive situations. The purpose of this activity is primarily to clarify when the exponent of a variable expression changes (multiplication) and when it stays the same (addition). Secondarily, students will make the connection that raising numbers to a power increases their value very quickly, especially compared to multiplying by a coefficient.
3+3 = 6 but 33 = 27
Thirdly, students will learn to create a basic series and parallel circuit.
Algebra 1 students or any student learning exponent rules, or any student struggling with the product rule v. combining like terms. This activity was designed with at-risk, poor students who have low confidence in their abilities and a history of struggling with mathematical concepts.
Commonly students “learn” these rules by taking notes, making flashcards and sheer repetition of doing problems. Even with these methods, however, many students still do not recall what to do when presented with a problem like x4 + x4. Some will think x8 (incorrect), some 2x4 (correct), others x16 (incorrect) and some will just give up. This lesson attempts to hit the “dynamic equilibrium” refered to by Mishra & Koehler (2009) in their TPACK model. It combines content (exponent rules), technology (paper circuits) and pedagogy (we have to understand the exponent rules to represent them with the technology) to provide students with a new experience of material that has historically been difficult.
Dale Dougherty, founder of Makezine, says “Making creates evidence of learning” (2012) in the article “Learning by Making” at Slate.com. By allowing students to demonstrated their understanding of exponent rules in any circuit they can imagine, students will be forced to fully understand the rule, but allow for expression of deeper understanding. Those that just grasp the “rule” can make a circuit as valid as the students who understand the power of exponential growth and are able to incorporate that knowledge into their creation.
Students will start, persevere, perhaps struggle with, and finish a product over the course of three days. Gee and Fulton in “Digital Media and Learning: A Prospective Retrospective” claim that humans learn through well-designed experiences and that after lots of time, effort, and practice in different experiences, the mind finds patterns and associations (2013, p.1). In other words, to learn, we must struggle, persist and be patient. This task requires students to stick with one task over the course of three days–something this cohort is not used to doing. It requires most of them to learn something totally new (circuitry), and use that as a language to express what is hopefully by now a less-new concept (exponent rules).
Finally, I hope to empower my students. I want them to believe they can create anything they want to. Creating a product from scratch provides learners with a sense of empowerment and ownership, according to Melanie Kahl. In her blog post “Recasting Students and Teachers as Designers”, she says of students who create: “their sense of self efficacy and power was through the roof. They develop a new swagger, but also a tenacity that they didn’t have” (2012).
I have allotted three 80-minutes blocks for this, as I have never done an activity like this before and want to make sure students do not have to rush. I also expect akwardness and resistance the first day. A minimum of two 45 minute class periods is required just to make the circuits. It is useful to note in the timing that my classes are generally no more than 20 students. The basic lesson template is:
- Warm up- review exponent rules, 6 problems (10 min)
- Introduce paper circuits- make a fun one for practice (45 min)
- Share circuits (10 min)
- Clean up (10 min)
- Bell ringer – 4 problems on exponent rules (10 min)
- Introduce the task of representing 2 monomials with paper circuits & partner up (10 min)
- Each pair makes 2 circuits (45 min)
- Clean up (5 min)
- Formative assessment: give warm up from Day 1 again to see any improvement (10 min)
- Finish up time, if needed (15 min)
- Present circuits to the class (20 min)
- Write explanation of both circuits to be published on class web page (40 min)
How to Make a Paper Circuit that Represents 2x and x 2
- copper tape
- 3V batteries
- LED lights
- construction paper
- hole punch
To Create the Circuits
Click on the top left picture below to go through the steps to create a series and a parallel circuit.
Draw the difference between your monomials. I decided to make 4 candles- two sets with wicks tha meet (for 4x) and one with 4 wicks (for the exponential).
The longer stem of the LED is positive and must touch the copper wire that leads to the positive side of the 3V battery.
Bend the ends of the LED down so they will lie on top of the copper wire.
Push the bulb through the hole, leaving the stems on the back side.
Align the stems on the copper wire.
Tape the stems to the page and the wire. I have two LED’s in this circuit to represent x2.
Put the battery in the corner. Fold the paper over so that each side is touching one segment of the copper wire. This is the switch to turn your circuit on.
I actually have 2 LED’s in this circuit to represent the coefficient 2. Here’s the back side.
Here’s the front side, decorated!
For the parallel circuit, I started by punching holes where I wanted the wicks of my candles to be.
Then I laid the copper tape on either side of the holes to create my circuit.
Now all I need is 4 LED’s and a battery.
First the battery….
Then test the bulbs before the taping.
Pop the 4 bulbs through the holes, just like in the series circuit.
Remember that the positive prong of the LED has to touch the positive lane of copper tape.
If the lights are not constantly on, bend the prongs so that the are digging into the copper wire.
Side view of the bend in the prongs.
The final products. 4x is two “love candles” ❤
Next Steps – Share
- Once the circuits are made, students will present their creations to the class, explaining how their circuits represent the monomials they were given.
- They will then be asked to use Padlet tell how this experience was for them. Do they feel like they learned more doing a project like this? What were some things they didn’t like?
- This is an opportunity for the teacher and students to help guide future learning in the classroom together.
- Finally, students will post a written or audio explanation of their circuits on the class website, to be used as a tool for current or future students struggling with exponents.
- Assign partners for the students. This saves time, hurt feelings, and you can build in a scaffolding structure by either placing kids at the same level together or a stronger with a weaker mathematician, depending on your situation.
- Each student could be given just one monomial to make, then they have to find their “partner,” i.e. the person that makes 4x has to find x4.
- Students who get done early can document what is going on, taking pictures of the process of other students, or can assist those still working who may need help.
- Give out strips of copper a little at a time to force them to use what they have before they get more.
- Implement a “no throwing anything away” rule. This will drive home the point that everything can be repurposed. I found this out myself after a few mistakes with the copper tape. I just pulled it up, flipped it over and taped it down. The LED still worked beautifully!
My prototype went through several iterations, as did my “final” product. At every step of the way I find myself changing and tweaking and modifying, as I am sure will happen when I implement this in class.
From prototype to finished product, though either set could be a finished product.
I really struggled with finding a maker activity that could demonstrate knowledge of the content my students are expected to know. I discovered in the process, however, that it is possible to do, and that much of my resistance to the idea has to do with my own fear of letting go of control. What if the students don’t buy in? What if they don’t stick with it? What if they still don’t know exponents when we’re done? What if I have to start sleeping at school because it takes so much time to plan these things (I’m not saying I have rational fears)?
What I complete this project knowing is that I have nothing to loose, but my students have everything to gain. I owe it to them to try something new, knowing the old way doesn’t work.
I’ll be back with a full report…
Dougherty, D. (n.d.). Want To Improve Science Education? Let Kids Build Rockets and Robots Instead of Taking Standardized Tests.. Slate Magazine. Retrieved July 20, 2014, from http://www.slate.com/articles/technology/future_tense/2012/06/maker_faire_and_science_education_americ
Kahl, M. (2012, October 1). Recasting Teachers and Students as Designers. MindShift. Retrieved July 20, 2014, from http://blogs.kqed.org/mindshift/2012/10/recasting-teachers-and-students-as-designers/
Mishra, P., & Koehler, M. J. (2009). ERIC – Too Cool for School? No Way! Using the TPACK Framework: You Can Have Your Hot Tools and Teach with Them, Too, Learning & Leading with Technology, 2009-May. ERIC – Too Cool for School? No Way! Using the TPACK Framework: You Can Have Your Hot Tools and Teach with Them, Too, Learning & Leading with Technology, 2009-May. Retrieved July 20, 2014, from http://eric.ed.gov/?id=EJ839143
Mishra, P., & The Deep-Play Research Group (2012). Rethinking Technology & Creativity in the 21st Century: Crayons are the Future. TechTrends, 56(5), 13-16.