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…

barbie bungee one

Can you guess what their answer is???

Then I show them this…

barbie bungee two

Of course this changes their answer.

Then I move onto this…

barbie bungee three.PNG

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.

barbie bungee four

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.

Their answers:

  • 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.

 

 

Elementary Observations from a Secondary Math Teacher

Here is the interaction that I had with my 5th grade son yesterday:

Me: “What does 6 ÷ 2 mean?”

Son: “Six split into two equal groups.”

Me: “What else could 6 ÷ 2 mean?”

Son: [crickets]

Me: “…how many groups of two are in six… so…what does 6 ÷ ½ mean?”

Son: “…how many groups of one-half are in six…12.”

Me: “What is 6 ÷ ¼?”

Son: “…how many groups of one-fourth are in six…24.  I remember this every time you remind me…but I always forget.”


My son is very strong with fractions, and he’ll go straight to invert and multiply if I let him.  I often engage him in this same conversation…working on the understanding of dividing a whole number by a unit fraction because I know how difficult division of fractions can be for students.

My hypothesis is that it has to do with the two interpretations of division:

(a) How many groups?                      and                     (b) How many in each group?

***At this point…I will hope that elementary people will tell me I’m wrong if I’m wrong…

I began teaching Math for Teachers at the local college (St. Mary’s College of Maryland~my alma mater) two years ago.  I use Sybilla Beckmann’s Math for Elementary Teachers fourth edition for this course.

The first activity for the division unit is this:

Write a simple word problem and make a math drawing that you could use to help children understand what 10 ÷ 2 means.

Each year I’ve taught this course EVERY single student wrote a how many in each group problem.  This is called the sharing model of division.

I think it’s less natural for students to consider the how many groups (or measurement model) interpretation of division.  But it seems that this is the model that makes the most sense for the division of fractions.

I think this less familiar interpretation of division can also impact a student’s ability to be successful with long division.

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A student needs to be able to think, “How many groups of 30 are in 1429??”


I think it’s important to be purposeful with the language we’re using with students, and how we are exposing them to different interpretations of division specifically.  Secondary math teachers can learn A LOT by digging into the elementary material.

Learning from a 5th Grade Math Team

When my oldest child entered kindergarten I wanted a way to volunteer my time at the school, so I began coaching the 5th grade math team.  I saw this as an opportunity to better understand the math that elementary students bring with them to middle school. The elementary school that my children attend is a feeder school to my middle school.

I used the set of resources provided by the school system to train my mathletes that year. One problem I tasked them with was titled Kicking Tees below:

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I didn’t attempt this problem before giving it to the students that day.  I watched as they solved it…easily handling #1, skipping #2, and then answering #3.  In my mind I thought, “Wow! I’d create an equation to solve #2…how would a 5th grader figure this out?”

I can’t be too dissimilar from other secondary certified teachers, where an algebraic approach is the first that comes to mind.  It took me a few moments to think about using a table, or simply guessing and testing given the boundaries offered by the answers to #1 and #3.

This was a revelation for me though…That students might have skills to approach problems in which I’d use a more sophisticated method.

I was curious about what a 7th grade Algebra class in the midst of learning about systems of equations might do with this problem.

They did the EXACT same thing that the 5th graders did~they skipped #2!

This began my thinking about the intersection between the teaching of content, skill, and strategy.  And the connection between elementary math and algebraic thinking.

I used this problem solving experiment to talk with the 7th grade students about math learning and about the connection between arithmetic and algebra.

For teachers, this highlights the importance of horizontal content knowledge

“a kind of mathematical ‘peripheral vision’ needed in teaching, a view of the larger mathematical landscape that teaching requires”

“According to Ball and Bass (2009), HCK is an awareness of where and how the mathematics being taught fits into the structures and hierarchies of shared collective mathematical knowledge. This awareness serves both to engage students and to provide meaning to the present mathematical experience”

Further, this article reports

“The teachers in our study seemed to be more concerned about the mathematical content at the level they were teaching than the broader (more advanced) mathematical context—which can be referred to as the mathematical horizon”


It’s become incredibly important for me in the work that I do with teachers, to help them see how the math that they are teaching fits into the learning the students do across a mathematical spectrum.

…in addition to helping teachers see the importance of this “horizontal content knowledge” as a way to create instruction that engages, inspires, and makes math a meaningful, connected body of work.

 

Mosvold, R., & Fauskanger, J. (n.d.). Teachers’ Beliefs about Mathematical Horizon Content Knowledge. Retrieved May 27, 2016, from http://www.cimt.plymouth.ac.uk/journal/mosvold2.pdf

 

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:

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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.

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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.Screen Shot 2016-05-17 at 5.56.08 PM

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:

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(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…noticing subtleties 1

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.

noticing subtleties 2

 

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!!!

 

Noticing Subtleties

This just happened…

I walked into an 8th grade classroom finishing up writing and solving systems of equations given word problems.  They began this learning last week.

I overheard the teacher and a student having a discussion about drawing pictures to represent the problem versus trying to write the equations.  This told me that the weekend was too big of a gap from the examples that they had done last Thursday (they had a sub on Friday).

Here is a problem from the set that they were working on:

motorcycles and cars

I briefly walked over to a different student and noticed that she only had one equation to represent this cars and motorcycles problem. She defined the variables correctly, but had written a single equation that mixed the information about the number of wheels and the total number of vehicles.  She hadn’t made sense of what her equation actually meant.

But, I thought that maybe it goes further than that.

I think the students need to see the difference between problems that require a one-variable equation, a two-variable equation, and a system of equations side by side. Here is what I came up with:

one var vs two var vs system

It’s nothing fancy or mind-blowing…I just want to see if this gives them a structure to look for when they are deciding how best to represent a given scenario.  I want them to see that there are two unknowns in the third case, and that they need to create an algebraic representation for Eli’s savings and for Lucas’ savings.

I’m wondering if we should ask them to notice and wonder???  Or simply ask…what is the same and what is different…

I’m also wondering if I should remove the headings from the table before we ask them to notice and wonder???

We’re going to try this tomorrow…so I’ll let you know how it goes.

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.

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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…

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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.