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.

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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.

Where’s the Algebra?

You won’t know this, but this is my 2nd draft of this post.  (I even changed the title.)  I went running half way through what I was trying to write and everything that I really wanted to say came to me so clearly. I should have stopped and recorded the words that were so fluidly coming to mind, but I didn’t.

It made me wonder, however, about why the words that I was struggling to string together for the original post came to me so easily while out running.  I decided that my mind was free of the clutter of trying to say the right things and in the right way–allowing me the ability get at the heart of what I wanted to communicate.

This–I decided–is the reason that balance problems are a great way for students to interact with the reasoning of equation solving.  They are free of the clutter of the procedures and notation and get at what is really going on.

Yesterday, I read Michael Pershan’s post about the use of balance problems (which I consider a sort of puzzle) as part of a series on solving equations.

He is able to so clearly articulate all of the reasons that I believe that these sorts of tasks are a valuable tool in reinforcing the thinking of solving equations.  (I would read his post on this because he is really good at thinking through these sorts of things.)

I used this “balance reasoning” that he referred to in order to make the connection between what students know how to do naturally and the skill that I wanted them to walk away from 8th grade being fluent in–solving equations.

8th graders have done a ton of work solving equations, but still, many are not fluent.  I used these puzzles during warm-ups without expecting any sort of equation writing/solving.  Over the course of time…I wanted the students to begin to ask why we were doing them when they were so easy.  I used the balance problems from EDCbut I also made great use of some of the resources found here.

One resource I used from this site is Mystery Numbers.  Students were able to use guess and check, along with “balance reasoning”, to find the correct value.

 

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The “mystery” of these problems can be ramped up rather easily and foster the need for a more sophisticated strategy.  I began having the students simply write the equation that could be represented by the mystery number equation.  The students had been wrapped up in the fact that these were at once so easy, but now not so much.  I used the mystery number problems written side by side with its algebraic representation in order to show the students how the “balance reasoning” related to the algebraic manipulation.

Students seemed to invest in this process because it was connected to something that they knew they could do.  I talked with them a lot about how math is all about learning more and more sophisticated ways to represent problems–even ones that they could do with an elementary level of understanding.  I believe it was the connection to a task that they found doable-that allowed them to find the algebra less intimidating.

Additionally, I used problems like this from Sarah Rubin.

I’m very interested in following the rest of the train of thought over at Problem Problems