Probability Pi Day Carnival

It’s that time of year again!  A few years ago, one of my colleagues finished a probability unit by having a mini-carnival with student-created games.  The next year, the entire seventh-grade team took part, and now, the mini-carnival has turned into the capstone math event of the year – the Probability Pi Day Carnival.  Hundreds of students attend, and it is a terrific celebration of learning and math.

Students create their own carnival game, calculate the probability of winning, do an experiment, make some predictions, and pitch their game to the carnival committee.

I described the project to students by creating a video describing the requirements and going over an exemplar.

Youtube screen capture

 

The kids came up with some clever and creative games!  They’re currently working hard on calculating the probability of winning, both theoretically and experimentally.  Then they will need to predict how many prizes they need to give out, so the carnival committee knows what supplies are needed.  I’ll post more about the games they’re creating tomorrow.

Their final project will be a model of their game, and a media presentation describing their game and the math behind it.  It can be a document, presentation, video, or anything they wish.

The best part is the school carnival, which is on Pi Day.  It’s awesome to see hundreds of kids in the building, celebrating their learning in math.  Kids talk about their projects for years to come. The community supports us by donating prizes to give away, and we sell circular treats such as pizza, fruit, and cookies.

I will update more as we get closer!

 

 

Workshop Using Google Docs

I’m settling into a recipe for using Google Docs for my cooperative learning activities, and I wanted to share.  I am very pleased with myself for all but eliminating paper for those activities, and am hopeful I can reduce the paper usage even more.  Using Google Docs has also opened up interesting ways for the students to get engaged and collaborate, but as with anything, it introduces pitfalls too.

Pre-work:  Before class, I create the cooperative activity just as I normally would have made a worksheet.  I create it in Google Docs – sometimes as a document, sometimes as a spreadsheet, or whatever tool the activity calls for.  I share it with my entire class with “view-only” permissions.

A:  Log on and warm-up.  Our class netbooks take some time to fire up.  I give the kids their daily planner update / learning target / plan of the day, and then send them in small groups to get their netbooks and log on.  While the netbooks are doing their thing, kids answer warmup questions in their math journals and demo at the board.  We process the warmups.

B:  Opening and mini-lesson.  Kids are logged on at this point, and usually I have them lower the screen while we participate in the whole-class opening.  Here we pose some introductory questions that have to do with the lesson, and I model the thinking that we will be doing that day.

C:  Ready to work.  All activities are posted on my Blackboard website and coded for the date.  Inside the folder for today are these activities.  I bring them up on the board and label them.

My Blackboard links with instructions for the kids.

D:  Work time.  Kids click the link to open the Google Doc.  One group member shares the document with their table group.  A leader is selected and that student keeps the rest of the group on the same problem, together.  My main role during this time is to question and guide their thinking.  Often, this is encouraging them to not go with the quick answer, but to carefully break down the problem.  Sometimes, of course, I police the teamwork. 🙂
In this case, I posted a little interactive activity to help them with the work.  Part of the document had spinners, and I had little moving spinners they could test if they got stuck.

Students analyze probability situations and write about their findings in the Google Doc.

E:  Reflection and summary.  On my Blackboard site, I place a link to a Google Form.  The forms are very easy to create, and all of the form responses go into a spreadsheet for easy access!  The kids click File –> Share and copy the link that shows up.  This link is pasted into the Google Form along with their reflections and any summary responses.

Sample form for collecting reflections.

 

F:  Summarize and wrap-up.  We share any interesting answers and methods for solving the problems, and then log off the computers.  If it’s not the last class of the day, students stack the netbooks on a shelf instead of trying to cram them into the cart.  The last class of the day gets to put them into the cart.

G: My turn.  I read through the work I get from the groups.  I might decide to add feedback directly to the documents.  I let their work guide me in what we’re doing the next day.  I reflect on the mistakes and pitfalls and misconceptions and what could go better, and often there is quite a lot.

I’m still very, very new at doing cooperative learning in this way, so there are some problems that have come up, and also some incredible advantages.

Problems:  Breaches of etiquette in using information technology come up often.  When they started learning, I set (what I thought were) clear expectations, and then I reminded myself that they’re twelve and they think it’s funny to post “WAZZZZUP EVERY1” in the comments and highlight everything in red and add pictures of spiders.  I’ll be giving the kids feedback for quite some time on how to collaborate properly – and reminding them that the computers, and all information that passes through them, do not belong to the kids and must be treated professionally.  There are the usual problems of workload sharing, which are the same whether work is done on paper or electronically.

Advantages:  Everyone gets to see everyone’s thinking laid bare right there in the document. No straining over each other’s shoulders, and no “what’d you get? What’s the answer?”  Instead, the conversations turn to “how’d you get that?” and “It says explain. We should explain.” and “It’s not yellow, yellow. It’s yellow, green.”  I loved this.  The students also found it engaging and there really was less “policing” than usual.  It could be that we have settled in as a community of learners a little more, and that almost certainly played a part.  But the format made it much more possible for everyone to be part of the action.

 

Things to watch for:  The type of work has to be conducive to the format.  Hand computation isn’t a good fit.  Drill / practice is better suited for other media, not collaborative documents.  They have to be rich tasks that require a lot of thinking and writing and explaining.

You have to have a good opening to scaffold the thinking.  You need to leave a lot of time at the end for summary and clean-up.  I am still struggling with this, because I see the kids doing good work and I’m hesitant to stop them.  But 8 minutes isn’t enough time to wrap up, reflect, summarize, and put everything away.

I plan to continue cooperative learning this way whenever I possibly can.  I love it and go home with a clean conscience from my reduced paper usage.  🙂

 

Google Docs and Interest

I am finishing up a unit on ratio, rate, and percent.  I am finally (finally!!) starting to get into the groove of using my class netbooks.  The first time we used Google Docs, it was just a disaster.  The whole class period was spent finding files and troubleshooting login problems.  I stayed away from Google Docs for quite a long time after that.  I created this lesson with an alternate plan for students who couldn’t get Google Docs to work, and I’m glad I did.

Standards for this lesson:
7th Grade Common Core: Use proportional relationships to solve multistep ratio and percent problems (7.RP.3)
Personal Financial Literacy: Solve problems involving percent of a number, discounts, taxes, simple interest, percent increase, and percent decrease (PFL)

Although the standard only mentions simple interest, I think it’s worthwhile to have the discussion about simple vs. compound interest.  The goals of the lesson were:

Describe simple vs. compound interest
Calculate simple interest if given the principal amount and the rate
Discuss how interest is part of saving for goals

I gave the students some notes on vocabulary terms yesterday. We reviewed them briefly today.

Interest_Notes

I told them that today, I wanted them to calculate how much money someone would have if they used simple interest versus compound interest for their savings account.  I wanted them to be able to compare simple versus compound interest, and which one would save you more money.  I pointed students to my Blackboard website and said we’d be working on spreadsheets to do our calculations.  I asked them to discuss the pros and cons of Excel versus Google Docs.  The students were very opinionated about it!
– Google Docs is convenient because you can get your files from home or school
– You can use Excel even if the internet is down
– Google Docs auto-saves whereas you need to remember to save Excel
– You can share Google Docs with someone to get help

Most students favor Google Docs.  However, not all students were able to view the file I shared, and I think it has to do with the settings for viewing their Google Drive… but I couldn’t troubleshoot it then and there.  I told students who couldn’t see my file to please open Excel. We spent some time just getting the files to open.

I guided them through making the calculations for Simple Interest, where you have $5000 in the bank and get paid 4% interest every year. Every year, someone would earn 4% of $5000 and still have $5000 in the bank.  The money was paid out directly and not put back into the principal.  The students decided that after the sixth full year (seven interest payments), the person would have earned $1400 in interest and still have $5000 in the bank, for a total of $6400.

I started them on the calculations for Compound Interest.  Students calculated 4% of the balance, added the interest to the beginning balance, and then this became the balance for the next year.  Next year, the bank account had more money and would earn 4% on that larger amount, and then that interest got added in, and so on.  I let the students finish with help from their friends.  A few students got stuck and stayed stuck, however most were quite persistent and sought help from friends to get the math done.  Some were really savvy about sharing their documents to troubleshoot together.  I felt good, as this was a great use of technology in the classroom and I think we are getting good bang-for-the-buck from the netbooks.

Some students were able to start on the second interest scenario, but many did not get there.  We discussed the differences between simple and compound interest, and we noted that compound interest earned more money over the years.  This was very easy to tell from the spreadsheet. Doing the calculation for each year was powerful.

I regret that I did have some students that stayed stuck on the task.  I will be targeting them over the next few lessons.

I enjoyed the task and I think the kids learned a lot from it.  I am attaching the blank spreadsheet, the key spreadsheet, and just for fun, my ticket-out-the-door responses (without names).  What do you think?  Did they get the idea behind the lesson?  Clearly I have some more spiraling to do on this topic, but for the first time they’ve calculated an interest scenario, I was pleased with the progress.

InterestLessonBlank  : Blank spreadsheet in Excel format

InterestLessonKey :  Filled-out spreadsheet

Ticket_no_names : My ticket-out-the-door responses.  I worded the first question badly.. we did the “analyzing discounts” lesson yesterday and I meant to say “calculate percent increases and decreases”.  But students responded as best as they could.  I did this with a Google Form and it worked beautifully.  Aren’t some of their answers awesome?

 

 

Magic! Help me do it again.

I had a wonderful day today.  Classroom magic happened.  It is so rare and beautiful that I didn’t know what it was at first.  I processed with a colleague afterward, and he asked what I put in place that made the magic happen.  I had a tough time with this question.  I always think I put my best effort into a lesson, but something happened with it during period 7 today that made the whole classroom do mathematics together.  So here’s how it went down.  What do I need to be sure to do tomorrow to re-create magic?

We started the lesson yesterday.  We did whole-class questions on percent of a number and percent discounts, and had volunteers solve the problems in different ways on the board.  Kids demonstrated benchmark percents, multiplying by decimal equivalents, converting to fractions, and hybrid strategies.  I told the kids to get their netbooks and log on while I gave them instructions for the group work time.

In their groups of 3 or 4, I asked for one person to create a new Word document while the other group members downloaded the activity .pdf file.  Their learning target was to compute and analyze discounts to find the best deal.

I got the activity from the Financial Fitness for Life book that I picked up at a recent training.  I originally had in mind that the learning target would be to compute percent discounts, but I liked that this activity had different kinds of discounts, not just percents, so kids really had to read the word problems carefully and not just apply an algorithm to get the cost.  It also spiraled back to rates and unit pricing, which I liked.  I will attach one page of the activity so you can see what it’s like, but you should get the book.  It’s wonderful.

coupon_p3

While the kids were logging on to the netbooks and downloading the file, I asked for a little multitasking attention and modeled an example problem for them.  At MexiFiesta, I have two coupons:  One that is “buy three, get one free” and one that is for 15% off my total bill.  I planned to buy 3 burritos at $6.00 each and one kids’ meal at $3.50.  I thought out loud and analyzed the two discounts I would get, determining that the first coupon would save me $3.50 while the second coupon would save me $3.23, so the first coupon was a better deal.

I told them they would be seeing ten problems, each with 2 possible coupons to use.  They would have to analyze the discounts with their teammates and calculate the better deal.  The teammate with the Word document would capture their complete-sentence responses and submit the team’s answers and reflection.

They started, but most teams only got 1 or 2 problems in before the period ended.  So we continued the next day.

——————-

Day 2!

I asked the students to log on to their netbooks and open the Coupon activity or Word document, whichever they were working on yesterday.  While they logged on, they worked on warmup problems on computing percent of a number.  Randomly selected students showed their work on the board.

I pointed out one tricky problem in the problem set, in which students paid different prices for peanut butter but got different amounts of peanut butter.  I demonstrated how someone could organize the information: dollars, ounces, dollars, ounces… and suggested maybe unit rates would help, but they ought to consider whether dollars per ounce or ounces per dollar made more sense.

I told them to share their thinking with each other and reminded them to capture complete-sentence responses in the Word document.

And then magic happened!!!

I circulated, waiting for a group to need reminding to get on task.  Waiting for a group to say “I don’t get this problem.”  They never did.  The room was filled with the sound of debate and discussion and “I think I figured it out” and “My answer is different” and “How’d you get that” and “We got done 5 problems, let’s do the next one.”  They made mistakes, but they worked them out as a team instead of summoning me over.  After the first 10 minutes, a student said “We’re all doing math. This is weird. This isn’t like our class.”  EVERY kid was doing math and even doing hand computation just to prove they could.  No one asked to use the restroom or sharpen a pencil.  Anything I said would have been an interruption to their work.  There was noise in the room, but it was a busy working noise.  Amazing.

They were excited to share their answers and write their reflection statements.  They put the netbooks away neatly and got out their journals to capture vocabulary for the next lesson.  I was sad to see the class end when it did.

I think I organized the lesson the same for period 5 and also period 1, but it was in that last class of the day that every kid on the room decided to be a mathematician.  There are probably other factors I haven’t thought about.

Period 7 is smaller than my other classes, with 20 students instead of 30.
I really can’t think of any other factors that would have played a role. The demographics of each class are really similar as far as the mix of partially proficient, proficient, and advanced students.  And to be fair, the other classes did a good job.  I just had to do a little policing as well and in each class, there were one or two groups that had a tough time getting things going.  Not so in that last class.

Did I just get lucky, or what other factors make magic happen in a classroom?
 

 

Every Choice Has a Cost

I attended some personal financial literacy training last weekend, presented by the Colorado Council for Economic Education.  In Colorado, we’ve adopted the Common Core State Standards (CCSS) mostly, but with added standards for Personal Financial Literacy… which show up in our standards tagged with (PFL).  A couple of co-workers and I thought the training would be a good introduction to what we need to teach in the new standards, and we weren’t disappointed!

If there’s one enduring understanding I could take away from personal financial literacy, it’s that “every choice has a cost”.  The ability to analyze and make informed judgments on the costs of your choices could make a big impact on your future success.

In sixth grade, the standards include saving for goals, using percents to solve saving and investing problems, and explaining the difference between saving and investing.

In seventh grade, we move on to understanding taxes – computing taxes, describing the role of taxes in society, and demonstrating the impact of taxes on your income and spending.  In addition, students should be able to compute interest and use unit rates to make purchasing decisions.

In eighth grade, students should understand how credit and debt impact their life goals, and describe the components of credit history.

There is a lot there, and we don’t tend to cover financial literacy well now.  However, it helped me to think of the big understanding, that the thread woven through many units of study must be helping students to analyze the cost of choices and make good judgments based on that analysis.

I want to put in a plug for the Council for Economic Education and the Financial Fitness for Life books.  The books have kid-friendly resources that cover finance problems, and put them in context of analyzing choices.  You can view some activities on the companion web site for the middle school books – click on a lesson and get to some information about the lesson and web resources.

I’m working on adding this analysis to the conversation as we work on our units on percents, probability, and linear algebra.

 

Rate Activities

claimtoken-510dd5ae75b07

We are working in a unit on rates and ratios, and I wanted students to be able to understand different applications of rates, such as area.  In one workshop, I introduced some warmup questions and then a mini-lesson on unit rates, followed by some work time on some more complicated area questions.  These are just old CSAP/TCAP released items, but they’re great problems that can be solved a number of different ways.  The kids worked hard and came up with some clever solutions.  Some pitfalls came up – some obvious, others not so.  I’ll put a link to the whole activity below.  It was exported from the SMART board software.

Area and Unit Rates
Misconceptions & pitfalls:

Moving right into calculating the time to plow the field, without calculating the “square feet per minute” rate first.
Calculating “feet per minute”, or the length of the field that can be plowed in one minute, instead of area.
Dropping zeroes when multiplying, making the field an order of magnitude smaller.
Not converting minutes to hours, or converting incorrectly (for example, 240 minutes equals 2 hours, 40 minutes)

I really encouraged students to think about the reasonableness of their answers, knowing any of the above mistakes would give them answers that should raise flags.  All in all, it was a good activity and I was pleased with how well the students worked together and struggled with the problems.  Some groups moved on to the food court problem, but many groups did not get there yet, and we haven’t summarized that part.

——————

We have 3 teachers on the seventh-grade team at our school, so we often have flexible grouping days.  We use these days to split the kids into “support”, “target”, and “enrichment” groups.  We work with the kids on remediation of some concepts they are not secure with, or extension and depth if they’re ready.  Since the topic of study this week was unit rates, we gave the kids a pre-assessment and then grouped them into three groups.

Support:  These kids struggled with the idea of a unit rate, and correctly labeling which unit rate they calculated (such as misunderstanding ounces per dollar, versus dollars per ounce).  We gave them this activity.  unit_rate_flex_group

Target:  These kids seemed to understand some concepts of unit rates, but struggled some with how to find the correct unit rate to make a valid comparison.  We wanted them to work with ideas around comparing and scaling rates, using this activity.  Making Comparisons of Two or More Rates

Enrichment:  These students understood the how and why of calculating and comparing rates.  With these kids, we dug deeper and used technology so they could understand table/graph/equation relationships.  Comparing Water Usage Investigation

My team felt this was a pretty successful day of flexible grouping, where all groups of kids made some growth toward their targets.  Another big success was that we went paperless!  With class sets of netbooks in every classroom, we posted our activities on our websites and had students access them online, edit, save, and submit.  We hope to do this more and more!

 

 

Gender Equity… Dance vs. STEM

In my spare time, okay, not that I have spare time, but in the couple of hours a week that I carve out for being responsible for my own health, I take clogging lessons.  I got interested in it while helping my daughters practice, and when their studio offered an adult class, I jumped on it and haven’t looked back.  I like making exercise and rhythm part of my life.  I’m well aware of how beneficial the arts are to a well-rounded education, and I feel it’s made me a more creative, thoughtful, energetic, and resourceful person.

The gender gap in dance is one that is obvious, and glaring, and hasn’t budged for decades.  My adult dance class is one of the few with a male dancer in it.  Most classes have no boys at all.  The hip-hop dance classes are the least unbalanced, but that means there will be two or three boys in the class with seven or so girls.

For me, it leaves a question in the air.  Why don’t we care?

The gender gap in STEM education and tech jobs spurs a call to action.  We create girls’ computer camps.  We create science field trips just for girls.  We analyze our data and fret over the unfulfilled potential of our girls.  Why is there no similar call to action to get more boys involved in dance?

I did just a tiny bit of research today online to find out about the gender gap in dance.  The few tidbits I found out are that the gender gap in dance is wide and hasn’t budged in decades, and that interestingly, and these pieces of info were fascinating, the wage gap in the performing arts favors men, and boys tend to get more attention and are called on more often in dance class.

This contrasts to the STEM gender gap, in which the boys outnumber the girls, especially in computer science and physics, but are still paid more and get more attention and are called on more often.  Or at least, that’s the way it used to be.  Is it still true?

We don’t apply the same level of urgency to getting boys into the arts as we do to getting girls into STEM.  Presumably, it’s because the job market in the arts isn’t perceived as growing as quickly or having as much earning potential.  Does that mean we’re right to apply no energy into getting boys to enjoy dance?

As a point of reflection, I’ve done a project a couple of times when I felt my classes (and I) were in need of a movement break.  Instead of doing math warmups, we planned a flash mob.  We’d find a funny dance on youtube and practice it for ten minutes a day, and then plan a secret day and time to play the music and have the math classes coalesce and just start dancing.  The students LOVED it – boys and girls alike.  It helped create community, a sense of purpose, got us some much-needed physical activity, and made us laugh.  I have no data to inform whether this is a good educational practice or not – so is it?  How important is a dance break for a student’s educational well-being?

I welcome your thoughts!

 

Computer Science and Math

Our school has a 40-minute intervention/enrichment period at the end of every day, and students can take a different class each quarter.  This quarter, we made a decision for me to teach a computer science enrichment.  We invited students to join it, with the intent of targeting certain needs.  We wanted to target students who had potential to grow in math if they learned it in a little different way.  The invitation list consisted of some partially-proficient math students who might make a big leap if they learned math in a non-traditional class, and some gifted students who might enjoy an extension to what they knew.  We created the class to be balanced between girls and boys.

I’m enjoying writing a curriculum that gives kids an intro to programming but also weaves in math topics. I am making things up a little as I go, based on what kids want to learn next.  Here is what we have done so far, and I’ll include the links to the demo programs I made.

1)  Introduction to drawing.  Students learned to create shapes on the screen, and how to plot objects, thinking of the screen as a coordinate plane. We also explored RGB color – how colors on a screen are a combination of Red, Green, and Blue lights, and by turning on the lights in different proportions, you can make any color of the rainbow.
http://www.khanacademy.org/cs/flower/1285404246

2)  Variables and Expressions.  Students learned how to “anchor” a small drawing at certain x,y coordinates, and how to create the rest of the shape around those anchor points using variables.  We discussed the relationship between these expressions and algebra expressions, and how the computer uses substitution to replace a variable with a value.
http://www.khanacademy.org/cs/here-kitty/1298014903

3) Binary Numbers.  I really appreciated place value when I learned and understood the world of binary, so we spent a day and a half working on building that understanding.  I mainly used worksheets from Computer Science Unplugged.  The worksheet on this webpage is great and very kid-friendly.
http://csunplugged.org/binary-numbers
I demonstrated the conversions with a Javascript program, although Javascript doesn’t have simple ways of representing binary numbers.  I haven’t used this in class yet, because I wanted to get into data types and functions before getting here.
http://www.khanacademy.org/cs/printbinary/1289019650

4)  Functions.  In elementary school, students get used to a representation of a function as an “in/out” machine.  In middle school, we extend that to equations that include variables, and they are related to these old “in/out” rules of the past.  In computer science, a function is also an in/out machine.  It takes parameters as inputs, does something to them, and outputs something new.  I showed the students how they could make their little character from Lesson 2 appear on the screen multiple times by wrapping it in a function.  The computer again uses substitution to use the input parameters.  For advanced students, they learned how to make their drawing appear randomly by substituting a random number instead of a fixed one.
http://www.khanacademy.org/cs/multiple-kitties/1298760197

Next, we will work on creating algebra expressions with our variables to make our drawings animate.  We’ll also learn about “if” statements and how they can be used to make decisions.

Loads of fun!  I’m loving it.

 

 

Levels of Comprehension, and Math!

I had a lightbulb during class this week that I wanted to share.  In context, we’re in the middle of a unit on rates and ratios, but our math team is simultaneously working with the kids on creating data displays and writeups for a big science project.  I decided to extend our rates unit into graph/table/equation relationships, because it would dovetail nicely into the science discussion.

The students solved a rate problem involving calculator prices, and I tasked them with creating a line graph.  Many students produced something like this.

Pretty straightforward line graph.

Some students finished early while others were still working, and the work extended over a second day.  Due to procrastination on my part, I ended up at my desk five minutes before students were to come in, asking myself, “okay, what do I have the fast workers do while the other kids are finishing their graphs??”  So I decided to have everyone write their own questions that could be answered from their graphs.  Remembering a little from staff meetings past, I asked kids to write some Level 1, Level 2, and Level 3 questions.  They shared their questions in whole-class discussion.

Level 1:  These questions could be answered right from the graph. The answers were literally right there.  They wrote questions such as:
– How much do 20 graphing calculators cost?
– How much is 1 fraction calculator?
– How many scientific calculators could you buy for $80?
– What is on the x axis?

Level 2:  These questions could be answered with a little problem-solving if you used the graph to start with.  They definitely had a right answer, but they weren’t right in front of you as the Level 1 questions were.  They had to be inferred.  Students wrote:
– How much would 100 graphing calculators cost?
– What’s the difference in price between 20 graphing calculators and 20 fraction calculators?
– How much would 10 graphing calculators cost in pesos?
– How much would you spend on 10 fraction and 5 scientific calculators?

Level 3:  These questions didn’t necessarily have a right answer.  They are opinion or judgment questions that make real-world connections with the graph, or they extend the thinking beyond the graph.  Students had a lot of fun with these and came up with some clever and off-the-wall questions.
– Who is my audience for this graph?
– What kind of features do graphing calculators have that make them so expensive?
– Why isn’t there a calculator that does all three functions?
– Who would buy 20 graphing calculators?
– What happens if you buy half a calculator?
– Do infinity graphing calculators cost more than infinity fraction calculators?
– Why don’t you go to the dollar store?
– Why not just use the app on your phone?
– Can I have a discount if I buy more than 20 calculators?
– If we save money by getting scientific calculators instead of graphing calculators, can we use the saved money for donuts?

I considered how this would blend into their science report task ahead of them, and I like where it’s headed.

In a science fair report, you generally have your data table / graph, a “results” writeup, and a “conclusions” writeup.  I know now how I am going to structure the kids’ writing for “results” and “conclusions”.

“Results”:  This section should be Level 1 and some Level 2 observations about their science data.  They should start with specific, low-level observations, and then move into general patterns they notice by problem-solving with their graph and data table.

“Conclusions”:  In this section, you start with Level 2 inferences about how well your data supports your hypothesis, and you move into Level 3 references about what your data means in a real-life context, what your sources of error could be, and what you should study in the future.

That five-minute “oh shoot what do I next” moment ended up giving me the gift of some pretty good structure for the kids’ science lab reports.  After they create their graphs, they can craft some statements in each level, and then weave them into a narrative.  Hooray!

 

Long Division and Decimals

Fraction-decimal-percent conversions are a fundamental building block for seventh-grade math, and after I browsed the Common Core standards and noticed this in 7.NS.2.d, I decided to do things a little differently this year.

d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

Don’t judge, but I didn’t fully understand why long division worked until well into my adult years.  I suspected my students didn’t either.  So, I created this lesson with these targets in mind – students will be able to:
– Explain why long division results in a decimal equivalent of a fraction.
– Predict when a fraction will result in a repeating decimal or terminating decimal.

The lesson is a workshop model, with a mini-lesson, cooperative investigation, catch-and-release, and summary.

I have a small set of base-ten blocks, but for this lesson, I found some terrific virtual base-ten blocks here. I liked them because with a click on the corner, the bigger blocks can be broken into ten smaller blocks.
http://www.glencoe.com/sites/common_assets/mathematics/ebook_assets/vmf/VMF-Interface.html

I asked the students if they liked to understand why things work, or if they just like to know how to do something and don’t care how it works.  The poll results were interestingly split about fifty-fifty (why??) – but I said “for those of you that like to know why things work, this lesson is for you.”

I put a base-ten block on my head and told them “You need to know something about me.  I have a super power.  My secret identity is Decimal Woman.  My super power is that I can break things into ten equal pieces.  LOOK! It’s our special guest today, PEYTON MANNING!”  And I switched the bigger block out for ten flats. “Oh, Sorry you missed it. See how that works?  Ten equal pieces, each a tenth of the original.  Unfortunately, I’m a one-trick pony.  I can break anything, but I can ONLY break things into ten equal pieces.  It’s all I do.  Cakes, pizzas, enemies… ten equal pieces only.”

I put a place-value chart on the board and reviewed decimal place values.  Thousands, hundreds, tens, ones.  Each was explained in words and exponential form.  I moved to the tenths (10 to the negative first power), hundredths (10 to the negative second), thousandths (10 to the negative third), and so on down.  I told them my super power is useful in a base-ten world, where every place value is ten times as small as the one next to it.  Every time I break something into ten pieces, I can think of it one place value down.  I did some quick random questioning:  what’s a tenth of one cake?  A tenth of a tenth of a brownie?  A tenth of a hundredth of my enemy?

I explained that long division is a manifestation of my power, and I used the virtual manipulatives to put a cake on the board.  I picked four students.. let’s call them, Emma, Jackson, Caleb, and Grace… and offered to split the cake evenly between them.  How much cake does each person get?

One cake – would love to cut it into fourths! However, I can only cut it into tenths.

The students were excited. “One fourth!  One fourth!”  I stopped them. “I would love to give them one fourth of the cake, but I can’t!  I can only break things into ten equal pieces.  So I’m going to cut the cake, but cut it into tenths.  How many tenths does each person get?” The students thought about it and told me each person gets two tenths, but there would be two left over.  I divided up the tenths, put the results in the division problem above, and had the two tenths left over.

Everyone has two tenths, for a total of 8 that have been given away. But there are two tenths left over!

The students were excited again.  “Cut them in half!”
I stopped them again.  “I can’t cut them in half.  Remember my super power?   I can only cut them into…”
“Ten pieces!  Then give everyone five!”  They started yelling out the answer.  So I broke the tenths into twenty hundredths and gave each student five of them, adding the results to the long division problem.

The cake is successfully divided into fourths, simply by breaking it into tenths and then hundredths and dividing them up.

I pointed out that even though I can only break the cake into ten pieces each time, I could successfully divide up the cake evenly with my method.  Everyone gets two tenths, and I broke the leftovers into hundredths, which everyone got five of.  This is the heart of how long division works.  You divide up the pieces and then break the leftovers into smaller pieces.. ten times smaller each time.

We did the same experiment with a cake and only three students – poor Grace would not get any cake this time.  I had to use my super power to break a cake into thirds among Emma, Jackson, and Caleb, using the same method.  I could only break it into ten equal pieces every time.  We started with the tenths of a cake.

One cake.. attempting to divide into thirds using Decimal Woman’s power of breaking pieces into tenths.

We went through the same procedure as before. Everyone got three tenths, and there was one tenth left over.  I broke the tenth into tenths, which the students identified as hundredths.  Everyone got three of those, and there was one left over.  Even though everyone could see where the pattern was going and some students were yelling it out, I divided the leftover hundredth into thousandths and divided them up… each student got three of them, with one left over.  At this point, the base-ten blocks can’t be broken anymore, but the students saw where the pattern was going.

Three tenths and three hundredths and three thousandths… one left over every time!

This left a question hanging in the air, so Decimal Woman asked it.  Can you predict when a decimal will repeat and when it will terminate?  What kinds of numbers produce each kind of decimal?  I introduced the word “Conjecture” as something between an educated guess and proven statement.  It’s a well-reasoned statement that has patterns and evidence to back it up.  Their group’s product would be a conjecture on what makes a repeating decimal vs. a terminating decimal.

I passed out the group work, which I’ll attach.  The students were asked to use long division to find decimal equivalents of a lot of fractions, from halves all the way through twelfths.  I told them to divide up the work and look for interesting patterns, which would help them create their conjecture.

Worksheet:patterns_of_decimals

We had been practicing long division for a day or two, but many students were rusty on converting remainders into decimal equivalents.  Most students picked this up fairly quickly and the teaching was minimal.  The classroom management was not as big a challenge as I expected, because most students found the task really engaging, to my surprise!  They got excited when they found an interesting pattern, and when they discovered that a decimal repeated or ended.  A few students did not pull their fair share of work, but it was a better cooperative activity than usual, all things considered.  After I let them churn through decimals for 15-20 minutes, I did a “catch” and had them report out some of the things they found.  They reported:

– Eighths are interesting because you always end up with forty of something left over, which divides evenly into five – so all the eighths terminate with a 5.
– Tenths and fifths simply have one terminating decimal.
– Sixths all repeat except for three-sixths.
– Most twelfths start with a sequence and then repeat a single digit.
– Elevenths all have a two-digit repeating sequence, and that sequence is a multiple of 9.
– Ninths are cool because the numerator is the repeating decimal.  This led to a great discussion on whether nine-ninths was 0.999999… or simply 1.
– Sevenths have a six-digit repeating sequence, and the sequence starts from a different number each time.
– If you can reduce a fraction to one of the terminating decimals, then it terminates.

Students determined that halves, fourths, fifths, eighths, and tenths all terminated.  Thirds, sixths, sevenths, ninths, elevenths, and twelfths mostly repeated.  The students were very confused about the general trend that would make a decimal repeat vs. terminate.  It didn’t seem to be related to odds vs. evens, or primes.  I asked them for factors of ten, and the told me the factors were 1, 2, 5, and 10.  I pointed out to them that it was interesting that if I broke something into tenths, I could only divide it up evenly if I could then split it up into groups of 10, 2, or 5 – nothing else – and I prodded them to use that idea in coming up with their conjecture.

I gave them a few minutes more to finalize their conjectures and then we finished with a summary and a round-robin of what they learned.

I know not all students achieved the second objective, and could not immediately tell me without calculating if they had a terminating or repeating decimal.  However, I do think they are well on their way to being able to compute decimal equivalents easily and explaining why the long division method works – and they were engaged and excited with what they did.  I was really pleased with the lesson!