Card Towers and Linear Equations

I like to use alternative ways of assessing students' understanding.  A couple of years ago I used card houses as a way to teach algebraic expressions to my algebra and pre-algebra students.  This year, I designed a lesson where students built card towers, gathered data, then created a poster describing the method of building, the dependent and independent variables, the linear equation resulting, and a graph of their results.  Students used their equations to predict the height from the floor of a 10-story card tower, a 25-story card tower, and 131-story card tower, which is the height of the world record tower to date. Each pair of students presented their findings in poster format for their grade.  Oh, and I added a prize for the tallest free standing tower.  We began the lesson with a short video showing the record holder for the tallest card tower.  Our efforts were hampered by the new decks of cards I had purchased which were slippery.  However, that challenge spurred the students on to more innovative tower building strategies, as you can see from the photos here.  I did not get a photo of an interesting strategy that involved a series of boxes made from cards, stacked on one another.  The winning tower, shown above, was three stories tall, 27 cm.  They won their choice of sports-themed hacky sacks, a popular prize.  This year, I teach my eighth graders twice on Fridays so a longer plan such as this fits the bill.  I gave the students 30 minutes to build their towers, and used an online timer.  The lesson was successful, and produced great results. 

And, we had a lot of comic relief as the timer ticked down and the slippery cards slipped.  It was very interesting, and all were engaged. 

Jeopardy in the Math Class - Shift the rules for total engagement

I use Jeopardy frequently in my algebra and pre-algebra math classes.  Jeopardy Labs is a terrific and free source of jeopardy templates and pre-made games for a variety of subjects.  The interface is easy to access and search for very specific topics.  Above is an example of one of the games when I searched for "slope intercept form". The search results offer a clear list by relevance to your search criteria, and the list offers a "preview" option.  As you scroll down

the list, the answer previews display on the right so you can quickly choose the most accurate questioning for your students.  This makes it easy to fine tune your instruction class by class.
I have adapted the rules of play to keep all students engaged as follows:

The students are arranged in teams of three or four. Each student in the class has a whiteboard and marker.  Teams take turns choosing questions, which eliminates the wrangling over who had a hand up first, etc.  When a team chooses their question, ALL students in the room are doing the work.  I provide a time limit on the question.  To get it right, each person at the table must show the correct answer and the correct work on their white board.  If not, I offer an option to steal as follows:  I write a number between 1 and 20 on a notepad, and let all teams wanting to steal guess the number.  They guess in clockwise order starting with the team after the team that missed the question.  Closest to the number gets the steal opportunity.  When it is time to steal, the stealers must show their answers right away, maintaining the incentive for all students to work on all questions.  

 These rule adaptations keep all students engaged, eliminate the claims of "that's not fair", and prevent the game from being dominated by the advanced students in the class.

  I also use Jeopardy as a review tool.  In this case, each student gets a laptop.  I share a link to the Jeopardy game.  Students work individually, and need to answer 15 questions of their choice.  Again, they must show their work.  This is a great way to differentiate instruction.  I can circulate through the room and work with students who are struggling with some of the content. 
 

Fraction Flags - Creative Thinking

My students have graph paper journals I gave them this year.  Recently I asked them to design a 300 square flag, where they made a pattern using 1/3 of the flag, 1/5 of the flag, and 1/6 of the flag and the rest of the flag as patterns. I suggested they lay out their flag patterns in something more sophisticated than rectangles.  We got some interesting results.  My goal is to let students be creative while thinking, and for them to be successful.  Allowing space for thinking, and for open-ended questions leaves room for success for all my students, even those who find math a struggle.

Three Fourths How? Good thinking!!!!

This year I gave each of my students a composition book filled with graph paper.  They are kept in bins in the classroom, and at least weekly, we use them for some interesting critical thinking problems.  The first I gave them was to depict the amount 3/4 in as many ways as they could... using words, numbers, pictures, music, design, whatever they could come up with.  I told them I expected at least five ideas per student.  We got great stuff!!!  3/4 Time in music, 3 quarters taped to the page, 3/4 pizza, 3/4 as equivalent fractions... We shared in groups, and then with the whole class.  I then challenged the students to bring in another creative way to show 3/4 the next day as extra credit on the first test.
Here are some of the interesting ideas I received.

"Three Fourth Awakens" with 3/4 Darth Vader's face

 

"3/4 a Bearded Dragon's Shed Skin"

"3/4 the way there"

"3/4 the Cranes are Green"

"3/4 the Gloves are Pink"

Card Houses, Patterns, and Algebraic Expressions

 My pre-algebra and algebra classes are learning about algebraic expressions, also called variable expressions.  It is important for them to be able to observe patterns, to identify the constant and changing parts of those patterns and to write variable expressions that represent the patterns. Yesterday we built card houses with that intent.  I demonstrated and off they went.  With each layer, they had to measure how high the top was from the floor. They quickly found they didn't have to measure from the floor each time, because the height of the table didn't change. The house of cards were approximately 9cm tall per story, so each additional story added 9cm to the total.  We built a table of the data, and decided what would happen with a card house any number of stories high. We made the table below, then the students told me how far from the floor a card house of 10 stories would be, 15 stories would be.  They've got this... finding patterns, writing expressions, making predictions.  And, they had a blast.

Happy Pi Day - Dominos, Measurement, and Differentiation

   

So today is Pi Day, as in 3.14.  My math students think about pie, and an infinitely long decimal, and sometimes stories of a classmate long ago who could reel off the first 25 digits, or the first 100 digits.  However, I want the students to come away with an understanding that Pi is actually the ratio of every (yes, every) circle's circumference to its diagonal. Pi is an irrational number, meaning it cannot be written as a fraction with integer numerator and non-zero integer denominator.  We didn't have a chance to organize a pie-a-palooza for today, so I brought in round cookies (okay, a stretch to think of them as mini-pies, but they certainly are desserts and they are mostly round). 
  

 

 I intended to have my students watch my favorite Pi Day video  (you've got to check this out!!), especially because after its mesmerizing 3 minute 14 seconds (hey, wait a minute....) it explains the math in a very accessible way.  Then they would measure the cookies, measuring circumference with a string, and diameter and finding the ratio, and going from there.  Thanks to my colleague for sharing her idea, which was a table for measuring lots of round objects.  I borrowed it, wrote "cookie" in the first row, stacked a variety of round objects on the table,  and added some differentiation for my different classes as follows:  

My seventh graders watched the video and measured, calculating ratios.  We will discuss their results tomorrow.
One eighth grade class watched the video, measured, then built a scatter plot of their results with the data and drew a line of fit. I will show them how to find the line's equation tomorrow.
The other eighth grade class did all of the above and generated equations for the line of fit, which we wrote on the board.  Tomorrow we will compare equations, thinking the slope of each should be near 3.14.  Hmmmm. 

So, I spent Pi Day without eating a bite of pie, which is fine with me.  My students came away from Pi Day having eaten a well-measured cookie, understanding that Pi is irrational, and that it is the ratio of circumference to diameter of each and every circle.  Everyone was engaged! It was a very good day.



                                           

"Hall Math" or "How to Accomplish Review, Collaboration, Movement and Confidence with a Few Index Cards"

My high school algebra class is working with linear equations.  There is a lot of vocabulary.  Much of the vocabulary sounds vaguely similar so it is easy for the students to feel "mushy" about the whole unit.  In my book, "mushy" is not good, and leads to "mushier" as we move forward into inequalities, linear systems, and eventually quadratic equations.  Ideally, students will have facility with equation solving tools, with which they can manipulate linear equations into various forms.  And, they can see that certain forms of linear equations lend themselves to different applications.  We aren't there yet, but closer.  As I write this, I realize I need to design a lesson where they read problems and identify which form of the linear equation would be more applicable.  

This Algebra I class is particularly fidgety after lunch, so I proposed "hall math" to them.  The eight questions above were taped in the hall, within view of my room.  I asked them how they should comport themselves in the hall, and we reviewed.  Then I asked how many times I should have to prompt them to keep the noise down before I called them all back in.  They suggested five! (not surprised)  I told them that they would get one warning before I rounded them up, but I was sure they could handle it, and this was a trial run to see if we could have more hall math in the future... And, I asked them to pair up as follows:  If you are comfortable with the material, find someone less comfortable, and vice versa.  I gave them each clipboards with an eight-graph sheet attached and off they went.  They were fantastic!  

Last year, I attended the National Council of Teachers of Math conference which was held in Boston.   May I say, what a treat to be among hundreds of people who are enthused about mathematics?  I had sort of forgotten that many people continue to believe that Math is a mystery, or it became a mystery to them at some point in school and it moved forward without them.  That's a topic for another day. But, one of the sessions I went to was about incorporating some social conscience into math class.  It's easy to start; rather than having Roxanne contemplate which deal is better at the health club, you can formulate problems that involve giving, or that highlight some local or global concerns.

Here's an example of slope-intercept form in action:
 
Alex has set a goal of donating 76 books to the Boys and Girls club. He started by using part of his birthday money to buy 10 books.  Each week he uses some of his allowance to choose 4 books from the sale rack at the local bookstore. 

a. Write an equation representing Alex's progress toward his goal as a function of how many weeks have passed. 


b. Graph the equation.

c. How many books will Alex have left to donate after 6 weeks have passed?


Thanks for reading along... I'm excited about hall math... "Get thee to a laminator".