As an extension, I challenged some students to make a spreadsheet (with formulas) that would write a function for any linear function values they put into their table.
I had a couple of boys excited about the task and it was great to see some of their misconceptions drawn out through the activity.
Misconception #1: Rate of change is always just the difference in output values
Their initial table had x values increasing by 1. When they finally got a working function I congratulated them...and then asked, "What if the x's increased by 2 and didn't start at 1?". I even changed their table values and they realized their function now didn't work in this new situation. Motivated by a new layer of challenge they went back to work. Misconception #2: Zero term is always the term before the first term in the table
Prior to the experience they thought the zero term (connected to arithmetic sequence concept we'd been doing) was the term before the values represented in the table. For the table above that may have been for them the x = 4 or even x = 3 value rather than thinking of when x = 0.
Being nerdy and excited about spreadsheets myself I completed the challenge as well and made it a fun visual function machine. Here's the link to mine.
Today as an extension activity for students that have already mastered simplifying square root expressions, I offered students a chance to complete a symbolic example using my #codemaths structure.
Here was the minimal prompt given to students in a unit google document:
I gave a short explanation of how the variable "a" is randomly generated and when they type [a] it will output that value instead of the letter "a". (I recently switched from <a> to [a] to allow for simple html formatting to be used in code output. <b></b> for bold, etc.)
Students only needed to "fill in the gap" of the potential worked out solution from their perspective on how they would solve it--filling in enough detail that would help a fictional struggling student viewing it as a potential worked out solution.
The existing code in the worked out section looks like this:
And will print out:
For the sake of not giving away the solution to the problem, I will just share screenshots of the student output from the #codemaths worked out solutions they wrote and emailed to me:
"Remember to give credit to the makers!!!"--Melissa and Katelyn
"Here mr. schwen this is my super awesome code"--Ken
Pretty fun application for the kids and little time is spent explaining writing code because of the simplified process and kids are thinking more deeply about the mathematics embedded in an activity like this at a pretty highly symbolic level.
One student, wide-eyed and excited, said to me after getting his to work that "This is really fun!" I like too that there is a trial and error aspect to this as they refine and revise what they think will output and what actually does show up when they test it out.
In all honesty, not all my students are doing this but for the kids that are ready for it, it turned out to be a great discussion and application of where we had been in our mathematics and a good preview as to how math can be used in an increasingly more coded world.
I'm early on in developing this activity for my class and who knows--I may even be thinking about it incorrectly but what I do know is this game is both addicting and very relevant to my students right now. What I'm considering posing to them as a question in our exponent unit is:
"What is the best and worst case scenario of the amount of time I would need to 'beat' the game and get to the 2048 tile if I take 5 seconds with each move?"
If you don't know the game you can check out this video but I'll warn you--while you are playing this game life happens around you without you being aware of anything going on (very addicting).
During the game you continue to get new 2 and 4 tiles that you must push together to try and build up to a tile of value 2048 (211). In a worst case scenario you would continue to get all 2's as new tiles or best case scenario all 4 tiles (neither happens but it sets up a range of what could).
Here are some scenarios that could give you a range of values for acquiring the 8 tile and the 16 tile:
What I like about the idea of this problem is there is real motivation in wanting to get to an equation but the student will need to be sure that the range of moves can all be generated. I haven't presented it to students yet so more may be following on how that goes.
I will leave the 4 tile graphic out to allow students the opportunity to consider it on their own and maybe I won't even present them with these scenarios early on in the problem either. Comments or suggestions welcome as always in the comments below. If there is a better example of this somewhere else too please let me know!
Today in class I anticipated I'd have a range of abilities with what we are doing in wrapping up a multi-day lesson. I have become fully dependent on the half class set of chromebooks on days like today. I would even say that I am happy that I do not have a 1:1 ratio in my classroom because of the conversations rooted in the need to share and discuss their thinking.
Regular access to classroom technology is crucial for this day to happen in the way that it does.
We have been working toward solving systems of equations in algebra, the last 2-3 days have been spent on this algebra-like investigation.
As part of the review day, students were checking graphical solutions to a system of equations on this worksheet ("websheet"?) from http://www.mathematicsvisionproject.org. I asked students to do their algebra check on paper but I wanted them to graph (mostly) on Desmos (yay!) and at least once on a graphing calculator (...for future college board tests...meh).
On the second page of the MVP math sheet, #5 says:
"A theater wants to take in at least $2000 for a certain matinee. Children’s tickets cost $5 each and adult tickets cost $10 each. The theater can seat up to 350 people. Find five combinations of children and adult tickets that will make their goal."
I challenged the class to also come up with some equations that represented the situation and to think about what could be replaced by variables. I did not expect everyone at this point in our learning cycle to do so but was very pleased with some results I will share below. Other students in the class that weren't ready just found the 5 possible combinations, still laying the groundwork for next class when we discuss the problem as a group. Students could also get additional practice on this web app if they felt they needed it.
I was most pleased with a group that not only went to desmos, I was able to start talking to them about the inequalities they had written, which regions created contained points that were considered solutions and if they needed to add any (constraints) to better define the solution region. At the end of class they were excited and were going to continue working on it tonight even though they did not have to. (Here is what they made as an extension to the word problem shown above.)
Thank you Desmos for making free tools like this for math teachers. I really like that students can graph equations in standard form (not just in the form y=mx+b). It helped in building understanding for these students as they were looking at the solution area of their graph. On a related note, you should also check out their classroom activities page.
A symbolic extension to our current learning in class
I presented this as a challenge for my students today that were ready to move on from the review that others were still doing in class. All students worked on the initial question and a handful moved on to try to program the solution in #codemaths. I was happy to see that a couple of my female students were working on this and I talked to them about the movement to try and get more girls into programming in what is typically seen as a male career path--they thought that was pretty cool.
Here is the example the class worked on and the skills I went over to prepare students for what they would need to do in order to connect the Javascript to the mathematics they were applying:
What is the solution to ax+b≥c where a, b and c are any value? Are there any circumstances that cause your solution to be different?
Javascript skills needed: using the eval() function and joining a string with variable expressions.
Joining strings with variable values:
“This sentence includes the value “+b;
displays as
“This sentence includes the value 3”
when b = 3.
Using the eval() function:
You can do math using the eval() function as well:
“x>”+eval(a*b-c)
would display as
“x>-1”
when a=1, b=2 and c=3.
In this #codemaths challenge, you will want to change the variable ‘solved’ in the constraints section to output a correct solution for any inequality that may result for random variables a, b and c that is programmed in the question section.
Notice in the first line of code
if(a==0){a = 3;}
that a will be changed to 3 if it is ever equal to 0.
In starting a new unit today I wanted to give context to students for writing inequalities. In our algebra class students have not yet learned the triangle inequality theorem which states that:
The sum of any two sides in a triangle must be greater than the third side.
Seems like a great opportunity for some 'ruler math'. I kept the rulers on my desk until after the bell because spring break starts after today and I know how middle schoolers can we with unstructured time and these tools...
In anticipating the pacing to be different for each student I made the activity as a Google presentation and provided students the link, instructing them to do their work on paper and answer completely any questions asked of them in the activity--noting that I would be breaking in periodically to the whole class and facilitating some discussion.
My favorite part was #4 of the first drawing where many of them grew frustrated with not being able to construct a 1 cm, 2 cm, 5 cm triangle--thinking it was something they were doing wrong. Kind of a fun way to watch them squirm and I know I'm not the first math teacher to have ever done this.
I had some awesome conversations with students while they were working. One student changed what he said mid-sentence. He went from saying
"The two lengths need to be at least the length of the third...The two lengths need to be more than the third length..."
I asked him why he changed his wording and he explained the difference between the two correctly. Pretty cool--I made sure to highlight that to the class in our whole group time. This kind of mathematics is so much more fun to teach than the traditional "do this, copy me, don't think" style. I overheard the same student above before leaving say to his neighbor
"Math is my hardest class because you can't just memorize stuff and copy it down. You actually have to understand the concepts to do well."
I told him I will take that as a compliment.
I am not expecting all students to go here but I am hoping that some make some connections to they pythagorean theorem and how it relates to acute and obtuse triangles in the last slide.
I've already made some revisions to the activity since teaching it one period and I wouldn't be surprised if it changed a little more as the day goes on. As always I'm open to other ideas or revisions to make it a better experience for my students.
What is important? What message do I want to convey?
A year back I found myself presenting on some assessment tools I've made. As I looked at the room full of people I thought to myself,
"Is this who I am? Have I become the guy that's excited about testing tools?!"
I wanted the answer to be "No!" but my presentation topic and blog posts would probably say otherwise.
From that day on I have decided two things:
1) I'm really tired of presenting on assessment technology at conferences
2) I want to invest more time in content development and not content assessment for my students
In recognizing that I've been writing a lot less this year, I've decided I need to update The Internet on what I'm really about. I could either say what I'm about or I could give some examples of what I'm trying to do so here we go...
Meyer-ish activities
The more I follow and read stuff from Dan Meyer, the more I'm influenced as a math teacher. I find myself going to YouTube and looking for ideas, capturing way more math with my iPhone (there should be an educator tax credit for math teachers that have iPhones) and scouring Pinterest for the perfect picture for our next activity.
A recent one I've found is this video where these guys flood a bathroom in a house. The video is actually two years old which on YouTube is like an antique.
I have embedded it into a presentation and plan to use it today to introduce our volume unit. In one picture I layered a before and after image to actually help in finding the final height of the water before it crashes through the floor. On the last slide is a linked student handout.
What I love is that I was able to pull out some realistic answers based on some reference points in the video itself and with the ideas of bathrooms, flooding and ultimate destruction of a house I'm thinking it will be a hit with the 8th graders.
Extension projects for students
I've been continuing on with the idea of developing new ideas in class first with my students. As often as possible we are using physical tools, drawings, constructions and media to scaffold our learning. I do not introduce new things to students when they are away from me. When video is used, it is to review or reinforce what has taken place in the classroom. Students are reporting often that our activity in class replaced the need for the video altogether and that's great.
As an option for demonstrating understanding students can complete a unit project that is a deeper application of our learning. I love seeing the student creativity and individual choice as these are being completed. I have one student who has told me that he is now doing them "not because I have to but because they are just fun!". How cool is that? For more on that and an example you can check out this post.
In another example of a project students created a powerpoint based off of a flowchart they made. The flowchart consisted of "Yes" and "No" questions that would identify special quadrilaterals by their properties. In the end of each path, the quadrilateral present in the user's mind should be identified correctly. In transitioning to the powerpoint, students made links for a "Yes" and "No" response to individual slides that would be next in identification of the shape.
What I loved about this project is students are required to think as programmers as they design and test their product to make sure it works without the need at this point to learn any computer code.
Still working on it
I hope that I don't ever come across like I've figured it out. Each year that I teach I feel like I'm starting all over or realizing there is a much better way of doing something in class. What I do find myself thinking more this year is that what we are doing in class seems to jive more with what I hear research is telling us to do--differentiate, provide choice for students, allow the students to develop their understanding, require them to do the reasoning.
Organic development of definitions
Today felt like one of those lessons that went way better than I could have expected. We used a Venn Diagram comparing two shapes at a time to come up with similarities and differences between all squares, rectangles and rhombus. Of course there were the kids that spouted "a square is always a rectangle but a rectangle is not always a square" but they really had to then think about how that applied in the diagram. This was a good natural stopping point for creating our class definition of these three shapes. The conversations were passionate and on topic and it was fun to see their understanding of these shapes deepen as time went on. To conclude this portion of the lesson I had the circles represent rhombus and rectangles and asked what they had in common, trying to lead the conversation toward 'squares'.
Flow charts and modifications
Next we moved to creating a flow chart, given a parallelogram, that would help us identify each of these three shapes. I had an example set in which they had to name the shape that applied and then we started to mix it up. I gave them a shape and then had them determine the property questions and answers to choose that would get us there. Another modification was to give them the answers (yes then no for example) and they had to determine what questions would lead us to the given shape. Through this my classes were able to determine that having perpendicular diagonals is just another way to identify equilateral parallelograms and congruent diagonals is an alternate way to check for equiangular parallelograms. Here is a Google Drawing of the final flow chart activity we did.
Introducing programming in math
A secondary reason for the flow chart portion of the lesson was to introduce a programming extension I'm hoping some will choose. Without actually having to do code I want students to create a stack of PowerPoint cards with yes or no questions that will correctly identify any quadrilateral by its properties. I was hoping to hook them with a taste of it in class and give them some familiarity before taking on the bigger challenge. As the year has gone on and in my own personal learning I believe that getting my math students more opportunities to think like programmers will help them learn valuable programming skills and a deeper understanding of the math they are applying within it.
