# Rational Exponents3: Open Middle Problem

Directions: Using any number between 1 and 9, fill in the boxes to create a true statement.  You may only use a number once.

# Diving Back In

Its the second day of classes, and I have done my usual upsetting of student’s beliefs of what Math Class is in School, even one in a Juvenile Center.  (Sorry I have taken a hiatus from the blog over the summer, but it was a much needed break and a lot of recharging)  Before I dive into blogging about things we learned this year, I want to reflect on a couple of the challenges that I continually face here so you understand some more of the challenges teachers in my situation face.

First- because of confidentiality, there are many things that tie my hands as an educator.  Although we can have some limited internet access, students continually find ways around firewalls and security to make contact with friends and family outside- which is a big No-No.  Because of this, I don’t have the luxury of implementing many of the inter-tech goodness that I loved at my old school (losing Desmos in this way is really killing me!).  I can’t video myself teaching, or students discussing things in class- and it’s very touchy for me even snapping pictures to post on this blog or Twitter.  This almost throws me into old-school methods for a new-school teacher, which can be a challenge of its own.  As such, I’ve had to be creative in what I do in order to keep things fresh in the classroom.  I DO have a SMARTBoard, so I can at least give them access to some of those things through my computer and access.

Second- the nature of having students in a Juvenile Center means that they have done some things that are socially not acceptable.  Many times the consequence of that is Separation- the inability to interact with a particular peer, or peers of the opposite sex.  Once again the start of this year, my students were on multiple separations.  I will have to wait a week (or until they earn their privileges back) before I can start instruction on group work. A result of this is that my room has not been set up in a work-station format instead of group-table format.  This is how my room now looks:

(or, it looked this way pre-service as I was cleaning from summer classes/setting up)

As many of you who know me, this is killing my normal teaching routine/style.  I try to keep my students working in groups and talking with each other about math as much as possible, so in an attempt for compromise between Center expectations and my own, I took two perimeter bookshelves from the room and placed them back to back in the center of my room.  That gave the tables room to be placed around the outside of my room and it set up a space I am loving so far.  I also teach LEGO Robotics so I need a workspace to run test tracks.  The two bookcases left a large crack between them, which doesn’t make a nice workspace for assignments or robots (I would be worried about what I would find at the end of the year), so I built a counter-top for it, one made out of white panelboard so I can use dry-erase markers on it.  I could have put in a work order for it, but the project only cost me \$65 and a couple hours of work to assemble/finish the wood.

I am really liking this space so far, I have yet to use my whiteboard in class.  I have the students gather around the center island workspace to work out problems and brainstorm.  Students are also really liking the ability to write on the surface, stand and work, as well as having a more casual atmosphere for discussions.  Talks are flowing more naturally, like they would if were were just standing around talking about what they were going to do later that night.

This group is pretty rough so far, there are many new faces and they are not used to my approach to math.  Many have fully bought into the facade of mathematics: if they are smart/stupid, fast/slow, good at math/suck at math.  These first few days have been different for them, a struggle against what they believe math is.  They are having problems with they expect I want (procedures and answers) from them versus what I am asking of them (to just play with Math!).  So far they have experienced WODB, Visual Patterns, Open Middle, Gemini Puzzles and Four 4’s.  Tomorrow I will hit them with the 10 challenge from #BecomingMath, Estimation 180 and Graphing Stories.

This first week is about breaking down those established expectations and getting these Students in a place where they feel comfortable playing with mathematics.

# Vectors- Open Middle Problem

Directions: Using the digits 1-9, each only once, fill in the blanks to make the following vector relationship true.

What vectors maximize ab?

What vectors minimize ab?

# Changing the way we ask questions…

Reading through a book today I saw this questions posed:

Make up an addition problem where 2, 3 and 4 are used somewhere in the problem or answer.

Ask that question of your students and see what types of responses you get.

Then twist the question slightly and ask it again (this is what I was thinking when I originally read it, so I wondered what type of varied response it would get and why).

Make up an addition problem where 2, 3 and 4 are used both in the problem and the answer.

What difference did it make?  Was the change significant for teaching?

I would like to know your experiences with this, thanks for sharing.

# Open Middle Problem- Open Number Sentence 2

Directions: Using the Integers 0 to 9, each only once, how many different ways can you fill in the blanks to make the statement true?

# What is “Rounding Up”?

So, Christopher (@trianglemancsd) posed this question to twitter over the weekend:

Now, this weekend I was out in the woods hunting deer, so I got this notification while trudging through a cedar swamp.  I quickly jotted down a reply that first hit me, put the phone in my pocket and trudged on, but of course that is never enough with Christopher…

DOH! Yes Christopher, it is.  I’m in the WOODS, leave me alone!  So when I came in for dinner I tried to construct a better response..

It’s not the most mathematical definition, so I was wondering what my students would say… This was awaiting them when they entered the room today:

I haven’t had a lot of time with the group I currently have to work on constructing good thinking responses so I didn’t know what I would get, but here is what I got.

The -0.13 Camp:

Overall, this group considered “rounding up” as only “making the digit bigger”, and had their mind blew when I asked how rounding up was making it negative 0.13?  There was a long moment of pause; ideas flashing across their features as they struggled with this concept.  Many became unsure of their answer.

The -0.12 Camp

Overall, this group was confused by the rounding “rules”.  Many explanations would not lead to correct rounding for positive numbers and these students need a quick refresher.  There was one student who understood what to do and took that into consideration when rounding, going to -0.12 because to round up was to make a bigger number.

The “Other” camp:

I am not going to post a picture of these responses, but these students had answers other than -0.12 and -0.13, and had major errors in their mathematical thinking about rounding or just guessed.

The result:

It was really had to discuss this question without imposing my idea of what the answer should be.  I had many students ask me what the correct answer was throughout the discussion.  I told them that was what we were trying to discover, and would not tell them my answer until they all agreed upon their way to round this number.  They were confused with this concept at first because it following their rules did not produce what they expected- but only when that was implicitly pointed out to them.  Many did the mechanical procedure for rounding and didn’t examine the number or it’s implications.

Once we finished our discussions, each group came to the same conclusion.  That while they want math to be consistent- this did not appear to be until you considered the concept of negative.  They initially wanted the procedure to be the same, by using the terminology of rounding up they wanted the number to be larger.  Then they moved into the number line and comparing the distance from specific numbers.  Since this was a half number- that caused a little more discussion about which way to go.  They decided to round it to -0.13 because it would remain consistent with their concept of rounding, but with reflection around 0.  Since a number would be rounded up in the positive, it would “round up to more negative”.

I challenge those of you who read this blog to introduce the question and discussion to your students, and blog about it.  There was a lot of great mathematical thinking that happened today.

# Food for thought…

So I found this blog and this article got me thinking about the current direction of testing.

# Kids Do Worse on Computers When Taking Tests

A growing number of studies conclude that students perform worse on tests when they take them online than when the questions are on paper.

A study published by MIT and conducted at the U.S. Military Academy found that the students who did not use computers scored significantly higher than those who did.

The researchers suggested that removing laptops and iPads from classes was the equivalent of improving the quality of teaching.

The study divided 726 undergraduates randomly into three groups in the 2014-15 and 2015-16 academic years. The control group’s classrooms were “technology-free,” meaning students were not allowed to use laptops or tablets at their desk. Another group was allowed to use computers and other devices, and the third group had restricted access to tablets.

“The results from our randomised experiment suggest that computer devices have a substantial negative effect on academic performance,” the researchers concluded, suggesting that the distraction of an electronic device complete with internet access outweighed their use for note-taking or research during lessons.

The research had an unusual twist: the students involved were studying at the West Point academy in the US, where cadets are ruthlessly ranked by exam results, meaning they were motivated to perform well and may have been more disciplined than typical undergraduates.

But even for the cream of the US army’s future crop, the lure of the digital world appears to have been too much, and exam performance after a full course of studying economics was lower among those in classes allowed to use devices.

“Our results indicate that students perform worse when personal computing technology is available. It is quite possible that these harmful effects could be magnified in settings outside of West Point,” the researchers concluded.

The Hechinger Report reported that writing online essays may contribute to a widening of the achievement gap.

The U.S. Department of Education launched a study of fourth graders using computers for writing compared to fourth graders using paper and pencil.

High-performing students did substantially better on the computer than with pencil and paper. But the opposite was true for average and low-performing students. They crafted better sentences using pencil and paper than they did using the computer. Low-income and black and Hispanic students tended to be in this latter category.

“(T)he use of the computer may have widened the writing achievement gap,” concluded the working paper, “Performance of fourth-grade students in the 2012 NAEP computer-based writing pilot assessment.” If so, that has big implications as test makers, with the support of the Department of Education, move forward with their goal of moving almost all students to computerized assessments, which are more efficient and cheaper to grade.
In the study, high-performing students — the top 20 percent of the test takers — produced an average of 179 words per assignment on the computer, three times the number of words that the bottom 20 percent produced. They also used spellcheck, backspace and other editing tools far more often. The researchers found that these high-performing students were more likely to have access to a computer and the Internet at home.

But these high achievers were in the minority. More than two-thirds of fourth-graders’ responses received scores in the bottom half of a 6-point scoring scale that rated grammar and writing quality. Overall, the average fourth-grader typed a total of 110 words per assignment, far less than the 159-word average on the 2010 paper test.

In looking for explanations for the disparity in performance, it seems likely that the high-performing students are more familiar with computers than low-performing students or even those in the middle.

But it is also likely, at least to me, that it is easier to read and re-read a passage when it is on paper than to read it online. Some young children may have difficulty scrolling up and down the page.

And there may be a difference in recall associated with the medium. That requires further study.

Let me confess that I have tried and failed to read books on a Kindle or similar device. It is easy to lose your place; it is hard to find it again. Maybe the difficulty is age-related; after all, I have only been using a computer for 32 years and began using it as an adult. Children who grow up in the digital age may not have the same visual problem that I have in reading large blocs of text. But it will take more studies to figure out when it is beneficial to use the computer and when it is not. Unfortunately policymakers have rushed into online instruction and online assessments on the assumption (untested) that there are no downsides. They do this, as the Hechinger Report says, because the computer makes it easier and cheaper to grade tests. Standardization has some benefits. But it also has drawbacks. We should be aware of both.

# #MTBoS30- Day 18

Write about a time when work felt real to you, necessary and satisfying. Paid or unpaid, professional or domestic, physical or mental.

Surprisingly this prompt could be quite lengthy and complex for me- so I will trim it down a bit and talk about the first time work (nonacademic) felt necessary and satisfying.

I grew up on a small family farm, there was plenty of work to go around.  We had cattle, chickens, pigs, ducks, geese, lambs and more.  We had many jobs to do for feeding our animals and upkeep of their shelters and grazing land.  When you are a 8 year old boy there isn’t a lot of that type of work that feels necessary or satisfying.  Dad knew this- he fought to get my brother or I out doing chores, until he became smarter than the average bear.

Our main livestock was cattle, and it takes a lot to maintain them.  We had to put up hay every summer- trying to get 2 crops in each summer.  We had to fence in pasture and maintain it.  We had to give them shots, take care of hooves, sew lacerations, assist with births.  When the calves come, you have to clean them and feed them- first with bottles and then pails.  Hopefully mom will take over the feeding duties- if not you have to take care of them 3 times a day, every day.  As an 8 year old, none of “activities” are fun, interesting, or satisfying.  They are just necessary.

Dad put a twist on that.  One fall he asked if I wanted to go with him to the cattle yard.  I said sure- getting away from home was a rare occurrence.  We went to this place that held a large metal shed, and corrals that seemed to stretch forever.  There was so much going on I couldn’t take it all in.  We found a spot in the auction hall and watched the sale of cattle.  This held my interest for about 10 whole minutes until I got bored and started keeping myself busy- counting rafters, examining how the shed was built, finding patterns in cowboy hats, etc.  This all was all washed away when I recognized the next bull that was led into the arena, it was ours.  I was absolutely mesmerized with the bidding process on our animal, and the next 29 after that.  People were paying us money for our animals, and paying well.  It left quite an impression on my young mind.  On the way home Dad told me “Next year you are going to own your first cow, you will take care of her and you will get to sell any of the calves she has.”

This blew my mind, I was going to get a cow and I got to sell the calves I raised for once- and get money.  This opened up a whole new world to me.

Later that fall we went to another cattle sale and I got to look for my new calf.  My calf.  All of a sudden I became an expert on cattle, I wandered the yard looking at calves.  I was judging them by appearance- how clean they were, how the acted in the corral, how big they were- I was such the eagle eye.  I finally picked one and we got her at a good price, her name was GiGi.

From then on, any work around the farm was necessary and satisfying- I knew the importance of keeping GiGi fed, sheltered and kept safe.  I no longer complained about chores and even took initiative to do them on my own.  That was my first experience of ownership, and hopefully I can help my students find theirs as my Dad did for me.

# #MTBoS30- Day 12

Perpetual cycle…

I work at a school that services a juvenile center, as such I get to see the product of the bad side of the modern family unit every day.  I have had many people ask me how I can teach here, and work with these kids.  The fact is: these kids need me, they need someone who is open to listening to them- no matter what they have done before.  The bad side: many of these kids don’t think they have the power to change their lives.

Case in point- there was a student here who lost hope, he figured that he would never be released from the center.  We talked about things, he seemed to understand and actually talked rationally about situations and what he could do to change the outcome.  The thing is- he never believed it.  One thing we always talk about is how running from your problems only delay the outcome, and typically the punishment becomes harder and longer because you try to avoid it.  He seemed to understand and talked a big game.  I thought he was actually making a change.

He ran today, and has not been seen.

# Polynomial Division- Open Middle Problem

Directions: Using the integers 1 to 9, using each number only once, fill in the boxes to create a polynomial and its factor.

What is the other factor?

What patterns do you notice?

Can you expand this to a polynomial of degree 4?