Let's Make A Deal!

 This week's game was also based on a television game show: Let's Make a Deal!  It is a review game that covers unit rate and leads into constant of proportionality. 

Students are given two choices (which eventually evolve to three or four choices) for which is the better deal.  They then use unit rate strategies to calculate which is the better deal based solely on math. This is an important filter. We actively discuss that in real life people think about other factors besides cost - taste, quality, and so on - but for our purposes, the only factor is the cost.

Students then calculate the better unit rate and explain in a sentence, using my model if they are having trouble wording an answer. 

I then wander the room and check work.  Students know I've picked a random student and that student needs to have their work shown as well as a sentence explaining which is the better deal including the unit rate.  

If students are correct, that is if the secretly chosen student is correct, they get the point. If not, teachers get the point. 

Generally students write in cost per item ($3 per pound, $2.29 per gallon.  When they calculate the cookies, they get 0.23, which they generally write as 23 cents per ounce for Oreos. When students calculate the problem above, they get 1.69/250 = .00676.  Students are often confused how to write numbers correctly when values get more precise than two decimal places. How does it get represented? It is a great question since we never (rarely) use this many decimals for money in real life.

What does .00676 mean?  Well, .01 is one cent, so this value is less than a penny per napkin... that seems to make sense; napkins aren't that expensive... so that means it is part of a penny per napkin. So we can write it as $0.00676 per napkin or we can round and say six-one thousandths of a dollar, or six-tenths of a penny per napkin. What we wouldn't write is 6.76 cents per napkin. We then establish a class agreement that the cent sign will no longer be used, and everything will be written in terms of dollars since that is the calculation result we would get. 

I show students this real image and ask, "If you were buying things from this toy bin, how many toys can you get for $5?"  Most answer 5 toys, since the cost is 99 cents each, which is about $1.  We then look at the photo in more detail and notice that it doesn't say 99 cents, but .99 cents.  This is read ninety-nine hundredths of a cent. This means each toy is less than a penny! 

We redo the math: $5 broken into $0.99 each would be 5 parts with a little left over.  But $5 broken into part of a cent each... $5 / .0099 = 505 toys!  What a bargain!  

Not sure who would need 2600 bananas for $5... but if you are looking...

Welcome to ... The Price is Right

Good morning, readers! Currently our eighth grade class is working on estimating the value of non-perfect square roots and placing them on a number line.  This is a difficult concept for many of our students as we are introducing them to the idea of irrational numbers. Additionally, many students have the misconception that the square root means "divide by two." 

To help with this, we play a game based on a VERY popular game show - The Price is Right. 

Here is how we play.  Students are given an imperfect square (such as 40) and asked to estimate the square root.  We have a double-number line tool card which was developed by a teacher at my school to help students visualize where they are on the integer number line. Using this card, they identify the two perfect squares that the imperfect square is between. They then decide which perfect square is closer to the imperfect square.  The initial goal is for them to identify if the imperfect square is larger or smaller than the half mark.

For example, if they had the square root of 40, students would identify the square root of 36 and 49, decide that they are closer to the square root of 36, which means the value is larger than 6 but less than 6.5.  We would get estimates that range from 6.1 to 6.4. 

Once we have done a few practice problems, the game begins!

I randomly call four students to the front with their tool cards. I play the music as they 'come on down' and then display the 'fabulous new prize'

It is the BEAUTIFUL square root of 10!

The four contestants then start their work while the 'studio audience' (other students) do as well. After 90 seconds or so, I have the students in the audience start calling out bids, much like the actual game show. Contestants up front can listen to the suggestions or stick with their answer. 

Each contestant shares their answer with me and I write their estimate on the board.  I then reveal the correct answer, usually to three decimal points.  Students then have to decide who is the winner - the person that is closest without going over. 

We do this for a number of rounds and the winner of each round gets to go into the showcase showdown at the end of class.  This determines our daily winner.

It is amazing how good students get at estimating values. At first they start with one decimal point but after some time, practice, and friendly competition, they start going out to two and three decimal points. 

This strategy and game has really increased our students accuracy as well as confidence. They never use a calculator during the entire lesson and are able to estimate the square root of imperfect squares with amazing precision!  The results of the final round are below: 

After the game, we do return back to the learning target - estimating the square root of non-perfect squares. It is important to return to this, because student "H" in the image above didn't win as they had an answer that was 'over', but all four answers met the target of the lesson.  It was a good day!

#MarburnCon21

This week got to experience the wonder of #MarburnCon21.  This virtual conference hosted by my school, Marburn Academy, focused on the theme of "Closing the Gap."  It brought together researchers from across the country as well as England and Australia to share the most recent best practices, techniques, and strategies to help students that learn differently or in non-traditional ways. 

What I really love about conferences is being able to take information from these experts and filter them into two buckets - what can I do to change my own classroom in the long term and what can I get from these presentations that I can bring back to the classroom on Monday.  MarburnCon21 gave me a good opportunity to reflect on my current teachings.

Day One

The Keynote speaker, Dr. Carl Hendrick, shared so many good thoughts. He reminded us that memories are built around schema - that words in context could have different meaning and memory depending on this schema. Vocabulary is built based on this schema and novices need lots of practice as well as explicit practice to build memories - much more so than people that have previous connections. 

Dr. Hendrick also talked about authenticity in four parts: Expertise in the subject, passion, unicity (the purposeful linking of the teacher's experience with students), and building rapport. 

One takeaway from Dr. Hendrick for Monday connected to feedback. The purpose of feedback is to improve the student, not the assignment. Improving the one assignment will not transfer, but improving the student has a stronger chance of transferring to the next activity. 

Another speaker that left an imprint on me was Dr. Sarah Powell. Dr. Powell spoke on helping students with word problems - a very important part of my everyday teaching world. She opened up with ineffective word problem practice.  When I say ineffective, I mean based on long term studies, not based in opinion. 

Dr. Powell shared that attaching keywords to operations in word problems is not an effective practice. For instance, teaching students if they see "all together" they should always add. What research has shown is that students will scan for numbers, key words, and then use the clues to solve the problem without reading the context.  She shared that in norm-referenced tests, key words led to correct answers between 25 and 50 percent of the time. With multi-step word problems (the type seen in late elementary to middle school) this accuracy dropped to less than 10%.  

Instead, we have to teach our students attack strategies and help build their schema to recognize how problems should be solved. Dr. Powell also shared a great visual for students in a round table discussion after her presentation.  

Day Two

Day two continued first with Dr. Erica Lembke sharing her knowledge of data based learning decisions. Much of her presentation fell into the bucket of long-term thoughts and how I could revise my progress monitoring on various skills and, more importantly, have students track their own growth over time. 

Her presentation was followed by an amazing language acquisition presentation by Dr. Pamela Snow. First, her dedication to helping educators was shown off by the fact she is located in Victoria, Australia, meaning she was presenting to us at 2:00 am local time.  Amazing.

Dr. Snow led us down a conversation of components of language including form, use, and content. She went into biological vs. non-biological forms of language, the explosion of vocabulary between ages five and eight, and the different tiers of vocabulary from everyday oral vocabulary (expression of feelings or needs) to general contextual reading (knowing words for comprehension), to lexicon in specific classes (hypotenuse, quotient.)  

I finished my MarburnCon21 experience with information from Dr. Amber Ray as well as Dr. Elizabeth Hughes. Both of them shared many ideas that I plan to incorporate into my lessons over the next few weeks. Dr. Ray spoke on SRSD as an approach to writing, while Dr. Hughes talked about the importance of precision of vocabulary in the classroom. Dr. Hughes had many connections back to Dr. Snow as well as Dr. Hendrick, and I reflected on how as a math department we have spent the past couple of years focusing on our vertical alignment of vocabulary.  It is a great feeling now when students join us in seventh grade and say, "I'm going to plus 5 and 9" and I hear a cry of "You don't plus numbers, you ADD them!"

Thank you to all of the organizers, presenters, and behind-the-scenes members that made this event possible! 

Day One: Where Do I sit?

It was so wonderful getting back into the classroom this week for so many many reasons. One was that I got to use one of my favorite first day of school activities called silent line-up.

This activity starts by having students lining outside my classroom. From there I told them directions are on the board and once they enter the room they may not talk or write anything.  As they come in they see the desks with numbered post-its and this screen:

I usually include "no mouthing words" but with masks, this wasn't necessary.

It is always interesting watching their reactions. They first start by staring at each other. and then staring some more.  This year I actually had a student go straight to the first desk and sit down while another sat in the last desk in the room. Bold moves, I exclaim.

Eventually they find some systems that kind of works. This class saw a calendar on the wall and a periodic table. Some pointed to the numbers on the calendar while others pointed to various elements.  Other students had no idea what they were trying to communicate.  

I was born in Rubidium

After 10 minutes I gave the student in the last desk a voice so she could talk but nobody else could.  It didn't help matters much. 

Five more minutes pass and after 15-minutes all the students were seated and student #7 said, very meekly, "We are ready for greatness." I asked them to give me a thumbs up if they felt they were in the right seat. Not many did.  We went around the room and shared birthdays. The first student called out January 5th, the next was later in January.  The third student called out a birthday in August.  Moving along other students realize there are many people out of place.  

I tell them they'll have 5 minutes to figure out their correct desk and will allow them to talk thsi time.  A couple minutes later all students are back in their seats and we go around again.  Turns out we have a correct order this time and also learn that two students in this class have the same birthday! (A math lesson for another day, perhaps.)

When we were finished, I processed through the activity. I asked them how easy the activity was. Were they frustrated? Did they want to quit? Was it getting easier or more frustrating as the class went on?  We then connected it to their educational experiences, not just in math.  When they are stuck, what is their strategy? Do they sit quietly and not ask for help? Do they build their frustration? Do you wait for the one person in the class to talk?  How much faster and easier did this problem get solved once you were able to all communicate with each other?

Eventually we talk about how school can be difficult at times, but by staying silent and not asking for help it will only increase the frustration and difficulty. Many times the rest of that week a student would need help (from I don't understand the material down to I don't' have a pencil.) Usually they spoke up, but when they didn't I reminded them of this activity. 

It also benefits me in the classroom as well. I see who the natural leaders are, or who students perceive as the leaders. Which students gravitated towards each other? Which ones tried to be a tribe of one? Which tried to just be invisible?  I learned of friend groups and potential discipline issues before anyone said a single word in my class. It is a powerful fifteen minutes.

I hope that all of you are having an amazing return-to-school experience. Stay safe!

Reflections After Hitting a Milestone

Recently a fellow educator on twitter, MorganLeming3, replied to a rather fun post from Nicholas Ferroni: How long have you been in education? (only using references and not the number)

My students have this insane curiosity about how old I am. Ranges go from 30 (so kind!) to 63 (so... ummm.... unexpected?)  They try to use context clues (the age of my kids, the fact that I've been teaching longer than any student in the school has been alive...)   I run lots, so I can't be THAT old, but I have a son in college, so I have to be at least 50, but I have a daughter that is only a year older than them so then I have to be about their parents age. 

So generally I give references like that to the question posed in the twitter post.  Realizing some facts, I decided to see how long I was a teacher.

What is crazy... what I found amazing... is from the day I was born to the day I first entered my first classroom after signing my first contract is LESS THAN the number of days from that day to today. 

I have literally been a teacher for longer than I have NOT been a teacher. 

This statistic amazed me. I have literally been a teacher over half of my life. 

And that has led to lots of reflection.  I started teaching back in the late 90s - before standardized testing, before common core, before Google was a publicly traded company. I started teaching before students had cell phones or social media.  I started teaching before you could get a new overhead projector with free two-day delivery... and back when we said overhead projector, we meant this:

Say hi to Elmo!

It is amazing both how much education has changed, and how it has stayed the same over these 20+ years. We have made so much growth in understanding how the pre-adolescent and adolescent brain encodes information. We have made so much progress on understanding various learning difficulties and how to best serve students that learn in non-traditional ways. 

There are so many changes that have happened within the classroom, both with this new knowledge, and also to keep up with the latest trends. I have been teaching a while, but looking back at how much technology has changed in the past two decades, it is amazing that there is anything that looks the same anymore. What do students still sit in rows and columns? (pre-covid, that is.) Why do students still give study guides the day before tests? Why do we still punish students with lower grades for late work or for retakes? Why do we still reward students with extra points on assignments that don't connect to learning targets (one box of tissues = 5 bonus points!) Why do we still average in zeroes to final grades? Why do we still depend so heavily on worksheets (PS - if you do, I highly recommend the book Boredom Busters by Katie Powell) 

Education has so much room for growth still and I hope to witness, and even be a voice in that change, over the next part of my career. 

When Will I Ever Use This In "Real-Life".....

 The most common question I think many teachers receive is "When will I ever use this in real life???" I get this question from students, from my own kids, my wife (when have I ever used...)

First, I always find it funny that math is generally the main target for this question. The one subject that arguably is needed for any profession.  So let's look at math skills and where they are used in real life.

"When will I ever need to know that a^2 + b^2 = c^2?" 

Let's look at the skills needed to solve this problem:

1) Following Steps in Order.

In order to solve this problem you have to follow steps in the correct order. Assuming you are solving for a leg, you need to square then subtract then take the square root. If you are solving for the hypotenuse you need to square then add then take the square root. If you don't follow the steps in order, you end up with an incorrect answer. 

Now, reflect on the times in your profession or in your life you had to follow steps in the correct order to complete a task. Recently I was tasked with putting together a squat rack with a lat pull-down attachment. There were over 20 steps and hundreds of different pieces of hardware. Following the steps in the correct order was key to putting it together. 

2) Attention to Detail

Looking at the Pythagorean problem, you need to know if you are solving for a leg or a hypotenuse. You need to label your values correctly and put them in the correct part of the equation. You have to notice if you are adding or subtracting a constant.  25 + 36 = c^2  is a different problem than 25 + b^2 = 36. Students get training on noticing small details and the importance of those details. How many times have you heard someone say, "I only forgot the negative sign, why did it get marked wrong?" That one symbol is the difference between MAKING $40 and LOSING $40.  

Obviously when putting together the squat rack, this skill was vital. There were bolts of different sizes, including a 72mm bolt and a 76 mm bolt. These are almost identical in size, but different enough that putting the wrong one in the wrong place would make the equipment non-functional. 

3) Building Social Skills (especially to ask for help!) 

The math classroom is the prime place for students to practice social skills and language to ask for help - not only from an adult but also from peers. Students solving for the missing value of a right triangle can check in with each other, practicing good social questions such as, "What did you get for the answer - I got 15."  They can work on respectful dialogue when answers do not match "Oh, I got 9." From there they can use respectful, responsive language to determine who made a mistake. 

This language is a skill and needs to be taught. Students do not inherently know how to handle conflict, especially if they are the one that made an error. 

You can bet I was asking for help when I needed to attach the two vertical sides of the cage.

4) Using Resources

Sometimes you don't have the answer in your brain. Sometimes you need to use a resources. In the math classroom this can be a tool card, a notebook, a digital reference... each classroom has their own system. I can't think of a single profession where this skill isn't valued. Doctors have scores of medical text, mechanics have diagrams and reference cards for various vehicles. No profession requires you to know all of the answers to all of the questions off the top of your head. In teaching, like other professions, our greatest resource is each other - getting ideas on how to help students from other teachers that have had similar situations is vital, which connects right back to having the social skills to talk to others.

So many skills in one skill!

This is just a small selection of thoughts that came to mind as I have thought about this question. I've left off other skills such as patience and perseverance.  Math is more than a list of algorithms. The math classroom should be a fluid, open classroom with dialogue and discourse. Students should be engaged in discussion and asked to defend their answers. Incorrect answers and appropriate-levels of struggle should occur. Those skills translate to the real world in many important ways. 

#FunWithMath - Part 2, Minimum wage vs. Housing

  There has been lots of talk about raising the minimum wage to $15.  I have done a number of calculations about this - not to prove or disprove any one side, but just to show the math behind the numbers.  This will be a multi-part series with the end-goal of education - There will be math... and math always evokes emotion.  I'm curious what questions come up. Please let me know.

This week's blog focuses on housing.  Since both rent and housing prices vary so much nationwide, there is not a general 'national average', so instead I'll focus on a few regions of the country along with the minimum wage in that region.  For instance, federal minimum wage is $7.50, but Ohio has a state minimum wage of $8.55 per hour.

According to many sources, an individual should not spend more than 30% of their income on rent and utilities.  How does that math translate to minimum wage?  

Someone living in Ohio making minimum wage would earn about $1348 per month for 40 hours per week. This is after about 9% withholding for taxes and social security (Here is a calculator, though I used excel so numbers may not match exactly.) If we take 30% of $1348, we get about $400 per month for rent and utilities.  According to RentCafe, this price point doesn't exist, and only 20% of apartments will come in under $700 per month.  At $700, someone would be spending over 50% of their income on rent, or nearly twice the recommended amount.

These numbers held pretty true in other locations I tested.  In Rochester, NY the average price of an apartment was about $1090 per month. With minimum wage in New York at $12.50 per hour, someone would spend 55% of their post-tax income on housing.  One of the worst ratios I found was Jackson, Mississippi.  Minimum wage in Mississippi is $7.25 and the average apartment cost $835 which meant someone working minimum wage for 40 hours a week would spend 73% of their post-tax income on rent. It also means that if two people shared a one-bedroom apartment, they'd still spend nearly 40% of their combined income on rent. And that is assuming both can find jobs that provide a guaranteed 40-hours per week..

Projected yearly earnings for $15 minimum wage

Yes, boroughs of New York were worse, with someone from Brooklyn spending over 100% of their $15 per hour income on rent for a 650 square foot apartment, while someone in the Bronx would spend 70% for a similarly sized apartment. 

What is notable is that even if minimum wage was set to $15 per hour tomorrow (clearly not a plan) these ratios are still relatively high.  Someone would earn $2231 per month (post tax) working at a $15 minimum wage. The percentage spent on rent in Rochester, Columbus, and Jackson after the wage hike would be 33% for Columbus, 37% for Jackson, and 48% for Rochester.  

Again - will increasing the minimum raise solve these issues?  I don't know. I'm not a finance expert nor do I pretend to be one.  I can only speak in the here and now which tells me that living expenses far exceed what someone earning minimum wage could afford.

#FunWithMath part one - Minimum Wage Vs. College Tuition

 There has been lots of talk about raising the minimum wage to $15.  I have done a number of calculations about this - not to prove or disprove any one side, but just to show the math behind the numbers.  This will be a multi-part series with the end-goal of education - There will be math and math always evokes emotion.  I'm curious what questions come up. Please let me know .

My son is about to graduate college. He is at a two-year private trade school. I knew the cost of secondary education had gone up since I last attended, but it wasn't until I did the math that I realized how much it went up.  What baffled me was the rate that it went up when compared to the federal minimum wage.  Here is a chart of what I mean:

Let me interpret some of this for you.  The first three columns are data sets - year, the federal minimum wage, and the average cost of a 4-year public college per year.  I used values published by USA Today, but also found comparable numbers on other sites. 

That fourth column is the eye-opening one for me. That is how many hours a student would have to work to pay off a yearly tuition bill.  It is then broken into that student working either 40 or 50 weeks of the year. 

What this means: Back in 1980 a college student would have had to work 20 hours per week for 40 weeks to pay off their college tuition bill.  By 1990 it was 33 hours. By the year 2000 it was 41 hours.  Today it is 77 hours.... yes...  a student would have to work 77 hours PER WEEK for 40 weeks, or 62 hours PER WEEK for 50 weeks, to pay off their college tuition that year.  This would be on top of going to college classes, completing homework and labs, and other educational duties. 

What happens if we increase minimum wage to $15 per hour? The numbers of hours needed to work drops to 30 hours per week if the student works 50 weeks a year, putting us back to Generation X levels of the mid-1990s, but nowhere near the numbers for the Baby Boomer years.  For that to happen, minimum wage would have to be around $25 per hour. 

2020a with a $15 minimum wage, and 2020b with a $25 minimum wage

'A' gets the numbers back to 1993 levels while 'B' gets to the 70s.

It is physically impossible for college-aged students to pay for tuition with just "a good ole hard work ethic."  There are not enough hours in a year to do it.  This is a reason student loan debt is as high as it is.  Right now citizens in the US have $1,570,000,000,000 (that's trillion) in student loan debt, about $35,000 per borrower. And remember, this doesn't include anything other than tuition: books, meals, dorms, other fees - none of those are included in these calculations.  

I'm not saying raising the minimum wage would solve any of these problems - but clearly this is not a sustainable model for higher education in the country. 

2021 One Word

 With 2020 coming to an end, lots of emotions come flooding to my mind, body, and soul.  It was a wild year for everyone with events and outcomes nobody could have predicted. My world, along with so many other people, was turned upside down and inside out. 

So as I was reflecting on my 2021 word, one came to mind rather quickly. 

Recenter (should I hyphenate that? Maybe? It is my word, so I'm going with it 😋)

I need to recenter my world.  Last year norms were shifted, relationships were impacted, digital time exploded exponentially... I, like so many others, was trying to reinvent normal without knowing what normal was. It led me to having more anxious days and nights, lower self-esteem and self-confidence, and an overall feeling of helplessness.

That isn't who I am.  So in 2021 I will recenter my world.

Professionally

As an educator I have never felt more lost, scattered, and yet supported than this year. I'm blessed to be part of a wonderful professional community, both in my school placement as well as in the twitterverse. 

My goal here is to recenter my support to others. I'm the math department chair / coach at my school. Over break I created some ideas to share with my colleagues and will offer them weekly as ways for students to do quick (5 to 10 minute) reviews of previously learned concepts so students continue to feel successful with current and previously presented material.  

As many of you know I have a first-year co-teacher this year. It has been a wonderful experience. I really enjoy the mentor role and also get excited to learn from her as well. I'm truly looking forward to the second half of the year and hope to continue to guide and push her in her rookie year!

I also know I need to blog more. I pressure myself into a blog-post-a-week schedule.  This, I've learned, is not reasonable and leads me to feeling helpless about being successful. Once-a-month, however, is well within reason. One lesson, one activity, one experience that I had as a teacher that I can share with the world. And have the courage to know that it may help another educator... 

Personally

I spent lots of time running. LOTS of time - more miles in 2020 than ever before. 2020 was supposed to be a banner year for me for events. I was scheduled to run my first 50 mile / 50k event(81 miles total) and my first 100 miler.  Unfortunately both events were cancelled... no. not cancelled. postponed. They are both scheduled to happen this year. However I fell off the training plans when I learned they were not going to occur. 

This year I am training for those two events even if the actual events do not occur. In June I'll run my 50/50. In October I'll go for my first 100 miler. 

Mentally / Emotionally

I know this is where I struggled most in 2020. I also know there are things I can do to recenter. I set a goal to read 30 books this year. Last year I read about that many, but most were during the lock down when I was literally not able to do much else. I also read a number of books that were re-reads for me. I'm going to dedicate time most evenings to getting off the screens, away from the apps, and getting back into a book. 

I have a couple of books lined up, but if you have any good book recommendations, please comment them below!

I've also set a goal to create a couple of playlists on spotify to help me when I'm having those tough mental moments. Music truly helps my emotional balance, and the right songs can help sway me back to my center.  

I'd also love to get into yoga, meditation, or other mindfulness activities. I've never been successful with these in the past, so if you have any suggestions, I'd appreciate it.

As we all embark on 2021, I want to wish all of you a healthy, happy, and safe new year.  Look for the blessings where they are, find hope and joy in each day, and please reach out if you need anything!