# Barbie Bungee Implementation

I think that there are a million and one ways you can do almost anything.  Dan Meyer tweeted this out the other day…

I was quick to comment.

I think it’s important to note that I work most closely with 8th grade (on grade level) students.

After seeing the subsequent comments…I wanted to add that IMPLEMENTATION can be more than just here’s your worksheet, “get on with it.”

I like to lead up with this…

Can you guess what their answer is???

Then I show them this…

Of course this changes their answer.

Then I move onto this…

And I have students come up to put a point…usually around a y-value of about 10 or so.

Then, I click to reveal additional points.

The point is…I make the point that collecting multiple pieces of data helps to make better predictions.  I also ask the question about what mathematical models can we create to help make predictions in math.

• Graphs
• Tables
• Equations

Just what I want.

Then I show one of the many videos that you can find on YouTube.

The last thing I do is pass out a worksheet.

# We Need More Drawing in Math Class

I worked with a 6th grade student in our In School Intervention classroom today.  I do this every day.  A boy had been assigned a set of pages from a workbook and had begun the work on his own. From what he had already completed, I could tell that this student had a strength in math. He was unsure of what he had done, so he was looking for validation from me.  We looked at this next problem together:

He said, “so you gotta subtract right?” I said, “yes you do, but that’s if you’re looking for the answer. This is asking you to write an equation.”

He began to write y = x – 14.

I talked to him about variables, and how many unknowns were in this problem and he could tell that there was only one.

I asked him to draw a picture of the situation.  Here is what he drew.

HMMMM….(maybe they’ve recently worked on area and perimeter???)

I then asked him to reread the problem and prompted him to draw the house and the airport.

He drew the line between the house and the airport, and I had to do some questioning to get him to realize it represented the distance of 29 miles.  We reread the problem and I asked him about the 14 and how he would represent that on the drawing.

He thought to himself for a moment and then began drawing the tick marks.  I could tell that he was counting them.  He labeled the 14th tick mark and the 29th, and then circled the 14.

I drew the bracket and asked him what that would represent.  He said with a questioning tone, “the remaining distance?” Thinking for a minute, he then said, “oh I know…would it be 14 + x = 29?”

We did one more problem together that we represented with a drawing, and then I left him to be independent in the rest of his work.

He came over to show me one of his drawings and the equation he had written:

(In a previous part of the problem they were told there are 150 seats in coach seating.)

We high-fived!

Teaching students how to create math drawings is a valuable sense-making tool that can be over looked in math class.  I know numerous reading language arts teachers that ask students to draw pictures to portray what’s going on in a story.  We need more of that in math class.

# Noticing Subtleties…Reflection

So I wrote this post yesterday, and today we tried it out.

The purpose was to give students an opportunity to notice the subtleties in the language associated with three different mathematical scenarios they may need to represent.

We read each scenario out loud and then asked the students to compare and contrast each problem type with their table partner.

Some students began by discussing the similarities and differences of the contexts…

Others began by making a list of what they noticed on the back…You can see that this student paid attention to more of the mathy parts~understanding what was meant by a one-variable versus a two variable equation.

And then you’ll see below where students were able to make sense of each scenario and the math required.  However, the first student used an equation in two variables for the first scenario and created a table to find the solution.

We didn’t get to the whole class conversation part of this lesson…I want to talk about each problem type and how to recognize the differences.  Notice that two of the students above wrote the equation for the two variable scenario, but the third student created a table.  I think we need to talk about why that is.  Also, I think I may want to do three more scenarios that would produce equations in standard form, to see if they would recognize the differences then.

I definitely think that this was a useful exercise and would do it again.

One little shout out…one student pulled out their phone because they wanted to check out the equations for the third scenario on their Desmos app!!!

# Vacas y Pollos ~ Best Day Ever

Yesterday I was talking to our Algebra teacher.  She had asked the Spanish teacher in the building to translate the  problem below so that she could pose it to her students that take both Algebra and Spanish.

Our population of students that are emerging bilingual has grown significantly over the past several years.  The grade 8 students are also working on systems of equations and I had the perfect teacher in mind to pose this problem (written in Spanish) to his students as well.

His last class has the greatest number of Spanish speaking students and I wanted to see the look on their faces when he posted this problem on the board…

Priceless. I watched the biggest smile appear on their faces! The look of joy of what they knew. The boy that participates the least read the problem out loud so quickly and proudly…it was AWESOME! The rest of the students were in awe. They tried to figure out pieces of the problem. But, the students that usually have to spend time using Google translate to figure out what was going on, were the first ones to get to dig in this time.

One girl then said…”wow…they have to do this all the time.”

Yep.

Best day ever.

I was inspired to coordinate the Intentional Talk book study this summer by a conversation between Tracy Zager and Dylan Kane.

The Intentional Talk book study began in June with various people taking the lead–including the authors of the book Elham Kazemi and Allison Hintz.

This week, Allison Hintz posted a question that lead to a conversation about effective vs efficient strategies.  I tried to capture my current wonderings in this 140 character tweet:

At the end of this past school year I came across this task.

I worked with on-level 8th grade students this year, but as the school Instructional Resource Teacher, I spoke with our 7th Accelerated teachers and Algebra 1 teacher to see if they would be on board with giving this task to each of these cohorts.  All of the students had been taught systems of equations this year–but notice how unstructured and simply worded this task is.  I wondered what sort of strategies the students (at all of these various levels) would apply towards getting a solution for this task.  We used it to gauge the sort of student that would be appropriate for a 45 minute algebra class vs a 90 minute algebra class.

The range of approaches was very interesting to me.

The following examples are from the 8th grade students that I worked with…I organized these student responses in the sort of continuum that occurred.  (Interestingly, the 7th accelerated and the Algebra 1 students didn’t have too dissimilar a range of solution strategies.)

Students that chose graphing…

Student 1:  extended the lines but without precision or using any tools

Student 2:  asked for a ruler in order to extend the graph in a more precise fashion

Student 3:  asked for graph paper and created a graph using discrete points

Students that chose tables…

Student 4:  created two separate tables beginning at x = 0 and found the values that represented the intersection point

Student 5:  created a single table beginning at x = 10.  This student did not pull the correct information from graph to begin the table.

Students that wrote a one-variable equation…

Student 6:  wrote each linear equation and then used the substitution method to find the solution.  This student did have difficulty at the end because she struggled with what to do with (1/2)x = 8.  She eventually made sense of that part.

Student 7:  was able to use the substitution method to write the one-variable equation as well.  This student is not your typical top student and she was able to easily handle .5x=8.  I asked her if she used the calculator (which I had allowed) because she didn’t get stuck like a few other of the “smart” kids.  She said that she knew it took two groups of .5 to make 1 so she multiplied 8 by 2 and got 16.  It was amazing to me how it was so easy for her to flip to ratio reasoning when that last bit was a challenge for some of the “top” students.

This all leads me back to my original wondering…graphing, tables, equations were all effective methods (for some).  But, I am considering an efficient method to be one that utilizes grade level understandings and the goal should be to help students work through this continuum.

There has been a lot of discussion regarding the meaning of efficient.  Does this mean fast?  I think you are probably only able to use an efficient strategy if you have multiple strategies to choose from.  Otherwise, your ONE strategy is the effective/efficient strategy.

It’s important to give students these sort of unstructured tasks and let them figure out what they’d do on their own.  I would definitely use this task again and then use one of the targeted discussion strategies from Intentional Talk.

# A Pythagorean Stations Lesson

I’m so glad that I’ve joined the group trying to blog weekly…although I’ve not been very good at achieving the goal.  Teaching an undergrad class (Math for Elementary Teachers) came up suddenly and it’s completely absorbed my free time. This week’s prompt is October Goals (you can also share a weekly summary).  Check out @luvbcd blog here and @druinok blog here for posts that are a part of this challenge. I wanted to share a lesson that I wrote for 8th grade teachers to use when beginning a unit on Pythagorean Theorem.  My goal for October will be to work on writing other lessons with a similar degree of detail. When I was still in the classroom…all of the detail of my lessons lived in my head.  I’m really working on how to articulate more clearly/explicitly the details of a lesson so that someone else can implement with ease. When researching lessons to introduce Pythagorean Theorem…I considered:

I then did some Internet research and came across this idea as a way to launch into an investigation: I liked this as a way to keep students coming back to the purpose of their investigation…which offer would you choose? I decided I wanted students to experience multiple ways of looking at the relationship between the areas of the squares on the sides of the right triangle…so I developed three different stations.  The Google Drive folder with the materials for this Discover Pythagorean Stations is here.  You should make a copy of the materials so that you can edit and make your own.

Here is what I did at each of the three stations:

Some think abouts for using this activity:

• For this activity, I would definitely get another set of hands in the room (I was the second set of hands in my building).  We ran 2 groups of each station…so that you can split your kids up into 6 groups.
• For the NLVM site…make sure the computers you get are up to date with Java.  You need to do this prior to doing the activity.  (For groups of 3-4 we used only 1 or 2 computers at each of these stations–kids worked together).
• Kids will need the most help on the computer station and the tangram station.  Here is the answer key for the tangram station (if kids struggled…I would get them started with the big red piece location).
• For the recording document…I tried to get them to consistently draw a sketch of what they saw with any quantities that were important to figuring out which students chose wisely re: the gold.
• I kept coming back to what we were trying to figure out.  Who made the best decision…two smaller squares of gold or the one large square of gold.  Kids figured it out after the first station…and their a-has were priceless.  I still think it was useful to work through each of the three different ways to look at it.  So at the end, I asked them to reflect on what they learned and which station was most effective for their learning.

Here is student work from the recording document:

Some Pythagorean Problems to solve:

If you haven’t already…Check out The Centre for Education in Mathematics and Computing and their Problems of the Week.

I came across this Problem of the Week below in their book of problems and solutions grades 7/8 for the years 2012-2013.

I modified the task to make it appropriate for the new year 2014.

I planned to use this with a group grade level 8th graders.  I thought it looked like a perfect problem to teach the problem solving strategy of starting with a simpler problem.

Just I had imagined, when I presented them with the problem, they immediately shunned it and put it off by saying they had no idea.  They wanted to try and put it into the calculator (which offers up its own set of learning opportunities).

I asked, “Well, what DO you know?”  Of course they started with $5^3$ is 125.  So, I said well let’s write that down.

Then I asked, “Well, what ELSE do you you know?”  And someone shared $5^2$ is 25.

At this point, we talked about how we organize information in math.

We continued our table, and many students began to recognize the pattern.  I needed to use guiding questions for some to be able to communicate just how the pattern worked.

I think this is a very simple, yet powerful, problem for explicitly teaching a very specific problem solving strategy.

End note:  If you haven’t already read this book…YOU MUST!  It validated many things that I already believed…but gives VERY useful and practical strategies for being explicit about how you teach problem solving.

I’d like to put together a set of tasks that all make use of this problem solving strategy so that students have the opportunity to apply it themselves.

I’d love to hear about how you teach problem solving!