Day 163: The Next Three Terms

Algebra and geometry both had concept tests today. I spent half the period reviewing a few concepts for a school wide diagnostic test they are taking on Monday. After they finished the test I gave ’em this pattern.

Numbers

A lot of students were randomly guessing what the next three terms would be. In response, I asked them what the pattern was. They are still used to yes/no answers to questions, which doesn’t encourage any critical sort of thinking.

This is a fun problem because each term is independent of the rest. Students are wired to try and find a common difference/ratio and when that doesn’t work they look for extended versions of those patterns like +1, +0, +2, +0, +3, ect.

Day 122: Visual Pattern

Today I want to share a fun visual pattern we went over in class.

V.P.

It is fun to see how far my classes have progressed since the beginning of the year. They struggled with this but after a bit of time they were able to come up with some quadratic rules.

V.P.1

V.P.2

There were a ton of different creative ways students saw this. It was really fun to hear students talk about moving different pieces to create shapes they were more familiar with.

If you are interested in how we do visual patterns, check out this post.

Day 95: A bit of everything

Over the last few weeks I have been using openers as more of a tool for review than anything. This time of the year is tough and I get caught up in all the madness of deadlines and content expectations.

I pulled myself together and gave students this opener today:

Day95.1Here is a list of the estimates

Day95

All of these make sense to me. What I get crazy for is watching the answer. If you haven’t clicked on the link above here is another opportunity.

The spoon crushed all hopes and dreams across the board in my 5 classes. Watching student’s reactions to the video was the highlight of my week.

What is interesting to me was how much student engagement increased since I started getting back in the swing of openers from visual patterns and estimate180. I believe students come into class expecting an 50 minute chunk of bookwork/lecture, which isn’t fun for anyone. But lowering the entry point of math class by giving students a low-risk opportunity has had a huge impact on my classroom even this far into the year.

Day 83: Visual Pattern

A lot of interesting things went on today; however this turn of events is near the top.

I gave my students this visual pattern today:

Day83

I gave em about two minutes to work on it. The conversations started out painfully slow. After 5 minutes this is what students had come up with

Day83.1

I gave them even more time in silence… something awesome happened a few minutes later.

Day83.2

S1: “If we X out the column on the far left column each time it makes a backwards L”

S2: “I see three squares there…”

T: “Are all the squares the same for each step?”

S2: “The three of them are”

S1: “The rule I see is 3(step) + step”

Day83.3

T: “If we call x the step we have 3x + x, does this work for the first three steps?”

S3: “No, the rule is 3x^2 + x”

Day83.4

T: “The rule works, does it make sense if we look at the actual pattern?”

Day83.5

The students came up with this explanation the rule completely intertwined with the pattern; this was amazing to be part of. I didn’t have to prompt the students with anything other than silence. They did the rest.

Day 32: HWD Visual Patterns

My school has 3 days a year that are used as “attendance incentive days”. Basically, if a student has less than 2 absences and maintains higher then a D in all their classes they are not required to attend school these days. Today was one of those days. It was nice to have time to work with students who were behind and fill in a few conceptual holes in the process.

I have been wanting to write about How We Do Visual Patterns, so here it is. (I cycle through several openers each week; Visual Patterns, Number Talks, Mental Math, Would You Rather, and Estimation)

This process has been evolving since the beginning of the year. I started off with something very similar to Fawn’s first hour found here. And I want to lead you through where things started and where they are right now.

For the first couple weeks, every pattern was about finding the squares/circles/shapes in a particular step;

VP.6

At this point, students were too focused on numbers. They wrote out tables and looked for patterns within the numbers.

VP.7

In the example above, most students overlooked the shape and saw a +3, +5, +7 pattern, which is fine, but I wanted to shift their thinking towards writing rules.

So I started asking for an equation. I went through about 2 visual patterns asking: “Find an equation to solve for the number of ________ in any step”.

This was WAY too complicated, so I changed the question:

VP.8

Things started falling into place at this point., Almost every student can now contribute to our discussions; a simple rewording drastically lowered the entry point for all students.

VP.9

Now instead of throwing around variables students write it as they see things, which in my opinion is awesome. The picture above gives an idea of how students create a rule:

S1 (Blue):From step 1 to step 2 you take the original pattern and add 4″

S2 (Red): “I did almost the same thing but called the pattern before the one we want to find x”

S3: (Green): “It also works if you take the step, multiply it by 4, and add after”

S4: (Green): “I did the same thing but called the step we want to find x” 

This was all facilitated by the students; no joke.

So, right now here is how we do visual patterns.

  1. The bell rings and I let students know we are opening with a Visual Pattern.
  2. I pull up a stopwatch in front of students and tell ’em how long they have (2-3 minutes).
  3. I remind students that it is OK if they don’t reach a rule, we are more interested in the process of finding one. It is also OK if the rule doesn’t work.
  4. I put something like this up on the board and start the timer.VP.5
  5. By themselves, students create a rule; no talking during this time.
  6. The timer goes off and I pull up our seating chart; I have the option to randomly choose a student. I do this in front of them.
  7. Whoever is called on shares where they are at in the process of writing a rule; it doesn’t have to be anything fancy. Sometimes the share what they see changing, sometimes they have a rule, they used to try and opt out; that didn’t fly.
  8. After 3 people have been randomly chosen I open things up for discussion; sometimes it ends there, other times it goes on for 10+ minutes.

Here’s the deal: I don’t help students along or provide a “correct” rule at the end. Things have gone this way since they very first Visual Pattern. Sure there are always times where I want to step in and some days we don’t get very far into creating a rule.But, I believe that students deserve opportunities to engage in discussions and share their thinking without being critiqued.

So much of what happens in math classrooms is about the teacher; we provide correct answers, solution methods, assignments, and assessments. The focus needs to shift towards the students and their perspectives. Taking 10 minutes out of one period, once a week to provide students with this opportunity has directly impacted what my students get out of the other 240 minutes each week and created an amazing culture of patient problem solving in my classes.

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Here are a couple resources for Visual Patterns

Visualpatterns.org

OUSD Instructional Toolkit

Day 11: HWD (How We Do) – Homework

In Geometry we played around with formulas and put some visual patterns into the mix as well. However, today we calculated perimeter instead.

day12.3

In Algebra we wrapped up Crossing The River and talked about the equation 25 = 4A + 1. I spent some class time going through a few examples of one and two step equations introducing some vocab along the way.

After we fished up I gave students some practice problems for the last 20 minutes of class.

I want to use today to write about how homework/classwork goes in my classes.

I have students who come from 6 or 7 different schools so there is a wide range of abilities. I don’t believe that giving ever student a single assignment  of 2 – 50 evens reaches students the best way. So here is how classwork/homework goes.

day12Two different assignments for two different ability levels. Students pick one and get to work. I explain that the /homework part of this only comes into play if they choose not to use their time in class productively. I do not ask students to take the problems they didn’t complete home and work on them there. I am more interested in the quality of student’s work over the quantity.

With two minutes left in class I throw up something along these lines:

day12.2

Student pick one of the two questions and bring it in complete the next day.

Day 9: Pairs of Angles

Today in Algebra we went over multiplying and dividing fractions. I followed along with Fawn, the day went OK. We used rectangles to model addition, subtraction, multiplication, AND division and I believe students were overwhelmed with using the same representation for all four different operations.

Next year I will probably use rectangles only for addition/division. It was also difficult for students to shift from the rectangle representation of division to the algorithm. I explained to them that the goal for today and yesterday was to look at how fraction operations worked rather than just memorizing algorithms.

In Geometry we started with a Visual Pattern.

day9

Student did an awesome job on this, it was really interesting to see which students counted the upside down triangles and the reactions when other students listen to a method they had not thought of. I have been doing visual patterns around once a week and LOVE them.

Students took a few notes on Adjacent Angles and Linear Pairs and worked through Pairs of Angles from Math Teacher Mambo. This was a great way for students to practice identifying and talking about different types of angles.I showed students a short proof on vertical angle congruencey and they went back through the worksheet and starred which angles were vertical.