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Category Archives: Lesson Reflections

Project-Based Learning via Javascript

I’m joining the maker movement!! I’ve been feeling tugged for a long time to teach mathematics using computer science – learning programming as a means to accomplish math goals.  Some of the factors at work are:

– The current math curriculum has been around since long before the computer, and the nature of mathematics in society has changed to become more computer-based.
– Millions of computer science jobs may go unfilled due to a lack of skilled programmers
– In our school district, in a tech hub of northern Colorado, a child can actually choose, in the year 2013, to never take a programming course beyond their sixth-grade tech class.  Ever.  They can actually opt out of learning to program.
– Yet students cannot opt out of learning to solve systems of inequalities with a pencil.  This seems backwards to me.
– My own understanding of mathematics deepened when I learned to program computers.

I made a promise to myself to incorporate programming into my common-core-based seventh-grade math class this year.  This is a heavy promise.  I am not teaching programming as an end in itself (even though I believe that is worthwhile).  I need to integrate the common core standards and teach programming with a purpose.  It is not easy.  Yet I was really proud of the first unit we did and the first unit project my students pulled off.

Our first unit was on congruence transformations.  I gave students the task of creating a computer program that created a design with symmetry.  It had to include two transformations, and they could choose from translations, rotations, and reflections.  They had to describe their transformations in a write-up and explain how they created them.

The platform I chose was Khan Academy’s Javascript. Any programming environment would work.  I considered Scratch, but eventually chose Javascript because of Khan Academy’s easy integration with Google accounts and the class management tools given to teachers.

Here are a few examples of the incredible artwork my students produced with computer programs.  It took several days but was WELL WORTH IT.  I was so impressed with the quality of their writing and their ease with using the difficult math vocabulary.  They really had to think hard about transformations to make their projects work, and the results were nothing short of amazing.

Student work from our programming project on congruence transformations!

Student work from our programming project on congruence transformations!

I didn’t expect the results to be as good as they were, frankly, but it turned out to be a great project not just as a way of introducing programming, but also to really understand congruence transformations deeply.

Here is the rubric I used for the project.

Khan Academy Programming Project

I would really encourage you to try it.  I’ve been emboldened to do more with computer science in math – the key being to hold on to the mathematical purpose behind it.

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Flipped lesson plan – intro to equations

This is about week 3 of flipping the classroom.  We are just starting our last unit of the year, which is on equations, equivalent expressions, and inequalities.  I decided to start this unit with a lesson on writing expressions and equations.  The students had previously learned the distributive property, so I thought I could extend that into distributive property with negatives, and combining like terms.

Common Core State Standards:

7.EE:

Use properties of operations to generate equivalent expressions.

1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

In a flipped lesson, I have the students watch a video for homework that gives them background information or reviews old topics.  I made this video.  It’s not an amazing video, but I wanted them to review their understanding of profit, income, expenses, and how variables are used in equations.

http://www.youtube.com/watch?v=TDcII5U-7bw

In class, I checked to see if they had taken notes on the video.  About half the class did.  NOT GOOD!  This is a classic struggle of flipped teaching, isn’t it?  I had planned that if this happened, I would go ahead and do the opening problem anyway.  This one was designed to be *less* accessible without the video, but not completely inaccessible.

legoproblem

The students worked the problem while they also logged onto their Netbooks.  After a few minutes, I asked for answers.  They called out…. “350.”  “2100.”  “negative 50.”  “I also got negative 50.”  “1500.”  “Wait. I changed my answer.  Negative 50.”  “Negative 50.”  “Negative 50.”  I said we seemed to be reaching a consensus.  I asked for a few students to share how they got negative $50 for the profit.

One student said: “First, I had $1000, and then I multiplied $100 by 5 and got 500 and added that on.  Then, I took the $1200, and I multiplied $70 by 5 and added that to the $1200 and got $1550.  Then I subtracted $1500 minus $1550 and got the answer.”  I wrote: (1000 + 100×5) – (1200 + 70×5) and she said it captured her method.  I said she must have watched the video!  We discussed whether parentheses were necessary and the class decided they were.

I asked if anyone got the answer a different way.  Another student volunteered: “I got the $1500 to start, but then I just subtracted the $1200 and got $300 left to spend on T-shirts and legos.  But the T-shirts and legos for 5 students costs $350 and so that’s more than you have.”  I wrote: 1000 + 100(5) – 1200 – 70(5) and asked if that basically explained it.

A third student offered that we basically start $200 “in the hole” and so we would need to make up the $200 with the student fees.  But if you take the -200 and add 500 for the fees and then take away 350 for the other expenses, you still end up $50 behind where you started.  This expression was: -200 + 100(5) – 70(5).  I told them I thought it was a clever strategy to combine the fixed income and expenses into one number.

We took note of the different strategies and it was time for one more quick mini-lesson before diving into work time.  The students went into Google Docs and opened a spreadsheet.  I taught them directly how to create a table for the function y=30x+12.  We created a column of x values, increasing by 5 each time.  I showed them how to enter a formula for the y-values and drag down the table to complete it.  Most students had not done this for a long time and needed the review.

spreadsheet1

For fun and spiraling, 🙂 I asked volunteers for a real-world situation that might match the equation y = 30x + 12.  One student offered (I loved this answer):  “You have a countertop.”  I said, “Describe the countertop.”  He suggested, “It, like, has a part sticking off the side you could set your baby on, and it’s the 12.  Then the big part of the countertop is 30 wide and the height is x.”  He drew:

countertop

Another student offered: “You have 12 dollars now.  And you’re selling something, and the thing you’re selling is 30 dollars each.”  I said, “So what’s x?” And she said, “However many of the thing you sell.”

It was time for the cookie problem.  I love this problem because it doesn’t have an easy mental-math solution – the spreadsheet is a very handy tool for reasoning out how many cookies you need to sell.  I presented the problem of owning a business making cheap, processed cookies and told students they should write down important information.  Students asked if they could take pictures of the problem with their phones, and I said they could.

cookie

I showed the students just how to get started with “Cookies” and “Income” columns in their spreadsheet, and I gave them the instructions to use the spreadsheet to figure out how many cookies they should sell to break even.  As students worked on their spreadsheets, one young lady waved me over and asked “Don’t you basically make 7 cents per cookie, but since you have $2000 and you pay $6000, you just need to see how many times 7 cents goes into $4000?”  She was combining like terms in her head! I said that she should give it a try and see if her answer comes out reasonable.  I also asked if she thought she could calculate profit for any number of cookies using what she said.  She nodded and started in on her spreadsheet.

Many students had a spreadsheet that looked like this after a while, so we did a catch-and-release to see what everyone thought and where we were going next.

spreadsheet2

Their income was not getting close to the expenses very quickly.  The students could tell they were not making a profit yet, because the expenses were so high.  Some couldn’t tell if the cookies would ever make a profit.  Others had used the formula “0.5 * A2 + 6000” for expenses, making their expenses grow faster than their income, which was really confusing.  I asked the student who combined like terms to share how many cookies she thought we needed to sell.  She explained that 7 cents goes into $4000 over 50,000 times.  Students wondered if they had to add that many rows!  I suggested they just increment x by a different number.  Perhaps 1,000 or 2,000?  Who could figure out the exact number of cookies needed to break even?  Some kids did make a third column for “profit”, subtracting the expenses from the income. This was helpful to them as they looked for the break-even point.

I released them and they worked a bit longer.  Eventually almost everyone determined the number of cookies would be between 50,000 and 60,000, and some students even figured out the answer precisely, although by different means.  One student figured out if he made x increment by intervals of 584, he could get within $6.00 of the break-even point.  Another student played with two rows of the spreadsheet until the profit got as close to zero as she could get.  A few more students used calculators and divided $4,000 by 0.07 to find the exact number of cookies needed.

spreadsheet3

This student figured out if he incremented by 583.2 cookies, he could get very close to the break even point.

The lesson was overall a very good one and the students were really engaged.  I can tell when a lesson is a good one by the number of kids that approach me for bathroom breaks.  NONE!

I would like them to play with the ideas of combining like terms, distributing a negative, and solving equations more tomorrow – and then they can start learning how symbolic manipulation can help them solve equations more efficiently… and that is when it makes sense to use symbolic manipulation!  I’ll have to look around for videos! I know there are a lot available already.

 
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Posted by on April 26, 2013 in Lesson Reflections

 

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?
 

 
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Posted by on February 7, 2013 in Lesson Reflections

 

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!

 

 

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.

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!

 
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Posted by on January 19, 2013 in Lesson Reflections

 

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.

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.

base_ten_fraction2

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.

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.

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!

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!

 
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Posted by on January 12, 2013 in Lesson Reflections

 

Animation and Integer Operations

For anyone who missed it, I’m experimenting with adding some computer-based math in my seventh-grade pre-algebra curriculum.  I see value in having students write computer programs to apply math concepts they learn in class, and to use computer programming as a creative and artistic medium using mathematics!

Previously, students created geometric designs in four quadrants using JavaScript.  This week, we are learning how to do operations on rational numbers, starting with integer addition and subtraction.  This is contained in some of the bullets under the Common Core standards 7.NS.1, or Colorado’s 1.2a.  To facilitate understanding of integer addition and subtraction, students will learn how to created animated drawings in JavaScript.  Animated drawings move about in four-quadrant space, so students will understand what addition and subtraction of signed numbers do in terms of moving on a number line.  It doesn’t hit on other models of addition and subtraction, and that is one of its weaknesses.  However, I like that it will help them understand what addition and subtraction actually do.  Regardless of your first operand, if you add a positive number, you move in the positive direction on the number line.  If you add a negative number, you move in the negative direction on the number line.  Subtraction does exactly the opposite, and so we need to get to the point where students realize that all addition operations can be rewritten as subtraction and vice-versa.  It is the movement that is important, and reinforcing that internal number line in students’ heads.

I have a sample program here which has the basics of animation, but also has some flaws.  🙂  Part of the joy of programming comes from pointing out those flaws and imagining ways to fix them.
http://www.khanacademy.org/cs/animate_circle/1031737006

This sample animated circle moves down and to the right, and leaves a trail behind it. They can see how changing variables and operations affects its movement.

The students’ task will be to create an animated drawing that has at least one object moving strictly horizontally or vertically, one object moving toward the lower right, and one object moving toward the upper right.  I’ve added bonus challenges for making objects bounce and switch direction.

This one is more complicated, with two shapes bouncing around the screen. Perhaps this is my differentiation strategy. Advanced students can learn about the “if” statement and make their objects bounce – although I really hope everyone gets there.

 

The kids LOVE programming, and several of them already have told me that they log on to Khan Academy on their own time and work on their programs.  How often do students go home and do more math than is required?  This is a side benefit I was really hoping for, and I’m pleased that it’s coming to fruition.  I want the kids to be really engaged in learning math and creating wonderful things.  I certainly hope that all students get to this point, and not just some.

I still do not have an assessment plan or differentiation strategy in place, and I know these are areas I need to work on. Coming soon!!  Perhaps tonight I should write rubrics based on how I *think* I would assess students, and give it a non-graded test run.

How should I assess them?