Mini-Project on Translations, Reflections, and Rotations

I’ve been incorporating more art into math this year. This project continues this theme. I had the students draw a pre-image, and then accurately translate it, reflect it, and rotate it into images. They could do these transformations in any order; they just need to document all the transformation rules and label the vertices correctly. Here’s some student work:

Rubric for Transformation Project

Geometry – Construction Projects

It’s that time of year again…when parents and administration tour the school and my room to see interesting, creative, and, hopefully, relevant projects completed by my students. I try to pick a project that by its nature will be unique for each student. This way it will make the room more fun to explore, and the students can learn from each other’s work.

This year in my Algebra 1 class we are currently doing geometry : constructions, transformations, and congruency. This is because the students are going into an integrated common core curriculum in high school and I am preparing them to skip freshman Integrated 1.  Anyway, for their open house project, I gave each student a unique design (given design) that they had to reproduce using constructions (only a compass and straightedge). I got the idea and designs from this pdf , Pretty Pictures for Construction, after searching the web for such projects (thanks Mr. Baroody).  Go to the end of the file to find the designs to give to the students. On a poster they needed to 1) write out the steps they used to construct it,  2) show the finished construction with the construction marks, and 3) create a clean finished copy without marks. They also were required to do this again with their own design. Here’s my one page set of Construction Project for the students. The students loved doing this project; they worked on it inside (2 days) and outside of class (a lot). Some used this online construction tool to help them figure out the construction method; this tool is super handy once you get the hang of it.

Here’s some of my students’ work (for this first one a particularly artistic student put a paintbrush in her compass to construct the eye):

 

Intervention through Practice and Puzzles

I teach a 7th grade intervention class, and have been for several years. In addition to working to fill gaps in fundamental skills, I try to increase the students’ interest in math and to build their perseverance in problem solving.

To help fill the skill gap I use:

• White Board Practice: About twice a week, the students use small white boards to work out problems I put on the the front board. I only put two problems on the board at a time; walk around to check their work; go over the two problems with their input; then I put two similar problems on the board if the majority did the first set incorrectly, or I take it up a notch or switch to a different skill. I find that this practice with immediate feedback is one of the effective tools for skill improvement.  Update: An English teacher at my school regularly uses gaming strategies to invigorate practice via worksheets. Check out her post to learn how to gamify your worksheets: Worksheets=Complete and Utter Engagement.

• Playing Zonk:  Zonk is a fun, group-work game that I discovered my first year teaching, eight years ago. You can use it to review any topic. Basically it requires a pocket chart where you place 25 cards; each card has a picture on one side (mine has a bulldog, the mascot from my previous school) and on the other side are points or the word “ZONK”. I put a problem on the board, and the groups work together to solve it. The groups try to get consensus on the correct answer, helping each other were needed.  If group 1 gets the answer correct, they get to pull a card after we have reviewed the problem. If not, I go to team 2, and so on. The part the kids especially love is picking the card. If they get a ZONK, their turn is over and they didn’t earn any points. If they get a number card, they can keep those points, or risk them by picking another card; however, if they pull a ZONK, they lose all the points they just earned (points they earned from earlier rounds are unaffected).  The students never seem to grow tired of playing this game, and some serious review and student-to-student tutoring goes on during the game.

• www.buzzmath.com and  www.SumDog.com: I use buzzmath for  grade-level practice of topics and sumdog for general practice of basic operations.

Perseverance Skills:

• Almost everyday we start the class off by playing the daily online SET puzzle.   The first time the students play it, it takes about 20 minutes to solve, but after a few weeks we solve it under 3 minutes. We record all our times on the board. Our current record is 1 minute, 33 seconds. How do we play it as a class? I tell them that each column is labeled A through D, and each row is labeled 1 through 3. Students communicate their set using ordered pairs like (B, 3), (D, 2), (A, 1).  I think this game has taught them that working together can be more efficient than working alone, and from a math standpoint, there’s a lot of analysis, proving, and comparing/contrasting that goes into creating each set.
• Tower of Hanoi:  Another puzzle that pushes the students to improve and persevere. I eventually connect the game to math as we work out how to come up with a solid method for achieving the least number of moves, and predicting what the least number of moves is required per puzzle.

Student solving SET by himself, learning it takes a lot longer compared to when we solve it as a class.

My students get creative with Desmos

Last week I introduced my students to Desmos. After doing a short lesson to show off the power and “what-if” capabilities of desmos, I pretty much let them loose on creating equation-driven art. My main objective here was for students to get a deeper understanding of linear equations, and specifically more experience with restricting the domain and/or range of a linear equation. So I showed the students how to restrict the domain and range, and my only requirement of the art is that they include at least one instance of a restricted domain/range.  Very soon the kids wanted to know how to create and manipulate parabolas, circles, sine waves, etc.  The students absolutely loved this activity; I was hearing comments like “Desmos is the best app I ever”, and “oh, now I see how that works”.  I gave them two class periods to work on their art, and told them they can complete it at home and save it to the  google folder they have shared with me. Here’s some of the artwork I received back. This first one, the Hunger Games’ Mockingjay uses 40 equations and the student worked on it for over 3 hours.

To see Mockingjay in all its glory in Desmos:  https://www.desmos.com/calculator/ihxor3kijv

To see butterfly in Desmos: https://www.desmos.com/calculator/zwmkq6gxx5

To see Birds by the Sea in Desmos: https://www.desmos.com/calculator/poq9oyptce

To see star in circles in Desmos: https://www.desmos.com/calculator/tweud3vjdw

Number Talks: Great way to start the day

I’ve been doing a lot of professional development lately in preparation for common core. Everything from monthly meetings at the district office to a week with Silicon Valley Math Initiative this past summer. Also I am taking a Stanford MOOC called: Constructive Classroom Conversations, where I am waking up to the fact that my students need to learn how to talk to each other, and this class is providing me tools to teach them and ways to measure success (I’ll do a post on this later).

A common thread to all these PD sessions is for students to articulate their thinking and to listen to other students’ thinking in order to expand their own.  I’ve incorporated Number Talks into the start of my classes (especially on Mondays since I’m not checking homework on Mondays). My students and I like these because most feel comfortable participating because there’s no one right answer, and there’s lots of aha moments.

Number Talks is when you pose a math problem, math dilemma, or other math visual to the students, and they are challenged to think about a strategy for solving the problem mentally or to form an analysis on the problem.  Next you have them share their thought-process with the class, often after they have shared it with their partner. Also you get feedback on students’ progress by having them do a thumbs-up close to their chest (way better than raising their hand since students aren’t pressured).

These are the topics/sources I’ve been using for my Number Talks:

• Home brewed prblems relevant to my students:  19(21) ,  √1089
• Graphing Stories
• Visual Patterns
• San Diego Unified Routine Banks
• Mental Math for Junior High

If  Number Talks are new to you, I highly recommend learning about them and introducing them into your classroom.  There’s lots of resources about them on the internet, and even youtube videos.

Halloween: Scary Graphing Stories

This was a hoot! A couple of weeks my algebra students and I were going over graphing relationships. I had a story about Bob driving a car to demonstrate a speed .vs. time continuous graph. Later we used the example of a squirrel gathering acorns to demonstrate a discrete graph. The students started mashing the stories together to create a story and graph of Bob chasing the squirrel….it didn’t end well.

Anyway, scary graphing stories were born. So today for Halloween, I created this Scary Graphing Stories worksheet. After the students created the stories, we shared out. Here’s a small sampling of student work:

My students love jigsaw math puzzles

My 7th grade math intervention recently sharpened their fraction skills using this jigsaw puzzle from the wonderful nrich website; a website that I just learned about from the webinar on Problem-based Lessons and Tasks. My students loved solving the puzzle while working out the problems on their whiteboards. They didn’t finish the day they started, so unfortunately they had to start over the next day, but they were very eager to do so. They learned that  the same task is accomplished much faster the second time around 🙂 .  I am sold on these types of puzzles; my 8th grade algebra kids loved their simplifying algebraic expression puzzle so much also.

Pythagorean Triplets

A couple of years ago when I was teaching the Pythagorean Theorem to my Advanced 7th grade class, I gave the students a “formula” for creating Pythagorean Triplets (a,b, c values that form right triangles). They went mad with it. I had students creating lists of triplets into the the hundreds, just for fun!

Here’s how you do it:

If a ≥ 3 and a is odd, then b = (a²-1)/2, and c = b + 1

a=3  b= (3²-1)/2=4   c=4+1=5

a=5  b=(5²-1)/2=12  c=12+1=13

If a ≥ 6 and a is even, then b=(a/2)² – 1  and c = b + 2

a=6  b= (6/2)² -1=8    c = 8 + 2 = 10

a=8  b=(8/2)² – 1=15  c = 15 + 2 = 17

There are other variations on these formulas to finding the triplets, but I like how these work.  Also note, that if a Pythagorean triplet has an odd number in it, there has to be two odd numbers. The triplets always have either all even numbers, or  2 odd numbers  and an even number.

Simplifying Algebraic Expressions Puzzle

This week my Algebra 1 classes studied and practiced simplifying algebraic expressions.  To cap off the learning, we did the puzzle that I lifted from Middle School Math Madness Blog.

However, instead of using Tarsia, which looks like  great software but not available on the Mac yet, I just created the puzzle the old fashion way.  Anyway, the results were amazing. Total engagement by the students. The puzzle was hard enough to be really challenging, but achievable at the same time. About a quarter of the students finished it, and many more were very close. They were eager to see a picture of the finished puzzle the next day. Here’s some pics:

BTW, I love taking pictures of the kids in action. And they really enjoy seeing pictures of themselves the next day as we summarize the activity. Such hams!

Should I Evaluate or Simplify?

I’ve been doing some searching in textbooks and the internet, and there’s some ambiguity around the mathematical verbs: evaluate and simplify. I wish we could differentiate these better because I think a clearer, more consistent, distinction between the two words would make it easier on the students. They would have a better clue as to the objective of the request.

Many textbooks and teachers only use evaluate when they have an algebraic expression with known variable values.

Ex)  Evaluate  3x + 4 for x = 12

Once the variable is substituted in, then the student is to simplify the numerical expression. And then they use simplify for any other situation regarding expressions.  They use simplify for both numerical and algebraic expressions.

Simplify:  3(12) + 4          or     Simplify : 2(3x + 4) + x

36 + 4                                         6x + 8 + x

40                                               7x + 8

However, other sources define evaluate to mean finding the value of a numerical or algebraic expressions. If a single numerical value is sure to be found for an expression, then evaluate would be appropriate.  Both these examples would use the verb evaluate.

Evaluate:  3x + 4 for x = 12        or   Evaluate:  3(12) + 4

40                                                   40

Then when do you use simplify? Simplify is reserved for algebraic expressions whose variable values are unknown. Algebraic expressions that can potentially be compacted to a smaller number of terms by mathematical operations and/or combining like terms.  So in this case, simplify truly means to make the expression simpler in form even though it may not result in a single numerical value.

Simplify: 2(3x + 4) + x           or       Simplify: 2(3x + 4) – 6x

6x + 8 + x                                      6x + 8 – 6x

7x + 8                                               8

I really like this latter definition.  But mostly I would like the math community to agree on a consistent definition. What do you think?