Category Archives: Pre-Algebra

Writing Linear Equations from Word Problems – Foldable

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6th and 7th grade students do not like to write equations for word problems. They just want to scratch out some calculations and give me their answer. Often, their answer is correct, but I would love for them to develop how to write an equation from a word problem at this level.

I made this four step foldable to visually slow them down, and illustrate the steps they need to take when writing an equation from a word problem. The students caught on very easily when I did this in class with them. For the first time, they seemed to realize that they needed to complete several steps in order to solve a word problem (by writing an equation). I emphasized the importance of the equation.

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This is the example that I wrote on the board to illustrate how to write an equation and solve it.

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PDF and Word File of the Foldable.

Sorry for the upside down picture!  I was trying to post using my iPad.  That app needs work!

Foldable for Functions and Graphing (Graphic Organizer)

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Our Functions chapter is full of great information that I would love my students to remember – even more than usual!  I created this Graphic Organizer in the form of a foldable.  It is stuffed full of information including…

  1. NAGS – the four ways to represent a linear equation.  Numerical/analytical, Algebraic equation, Graph, Sentence.  (Thanks @pamjwilson for this idea!)
  2. Functions – Function notation, Domain, Range, Vertical line test.
  3. GRAPHING – Three ways to graph, Table, y=mx+b, using intercepts
  4. Slope – words, formulas, parallel, and perpendicular slopes
  5. Inequalities – Graphing inequalities in two variable.  Dotted line vs. shaded line, test point.

On the back I put a place for their name and extra notes.  The notes section was a new addition to this type of foldable for me and one of my favorite parts!

I go over the graphic organizer in class, giving them examples to work for most of the sections.  After we were finished, I had them turn the graphic organizer over and write notes on the back.  I suggested that they could take notes of things they wanted to remember, or about the topics they felt they needed to focus on when studying.  It was good to do this after the review because what they were unclear about was fresh in their heads.  I then had a few students share what they wrote.  This was helpful to other students.  Some of them wrote extra items as other students read their notes out in class.

I had much higher than usual scores on this test (and more students scoring higher as well).  Considering that this is one of our more difficult topics of the year, I feel that this foldable really helped the students!

Note:  I did not include how to write linear equations because my students are solid in this area.  We added examples when we went over this in class.

To use:  This is a double sided foldable.  You will need to print both side, being sure to line the correct word up with the inside when you print it out.  Students will need to only fold the outside two flaps and then cut on the dotted lines.

File to print out:  7th Chapter 8 Review Foldable – Graphing PDF  or one you can edit from Box.

My Students were Published in Mathematics Teaching in the Middle School!

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We are so excited!  My students’ answers to a “Solve It!” were published in the November edition of Mathematics Teaching in the Middle School.  This is a national publication produced by the National Council of Teachers of Mathematics.  The students collected data about the number of jeans owned by every student in their class.  Working in groups of three or four, they used Google Spreadsheets to compare their class average to the national average and to do further analysis of the data.  They created graphs to visually present their findings.  They also worked collaboratively on a Google word document to write their analysis of the data.  After they came up with conclusions, they created wiki pages to present their findings.  Their wiki pages were creative and beautifully formatted.  They contained uploaded graphs of their data, detailed analyses, and even some pictures they took with their cameras or computers.

We had already covered the unit on averages when the “Solve It!” question came about.  Therefore, I gave few instructions to the students for this project.  I showed them the question in the article and let them work independently in their groups.  I encouraged them to be creative.  They loved this project and they reported that much of their enthusiasm was due to the freedom they were given.

Check out the wiki pages they created  and read the Solve It! Article.

Jeans Solve It Published

Graphing Stories

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I absolutely loved Dan Meyers Graphing Stories! I rounded them all up into a single Powerpoint presentation, and watched them with my 7th grade class last fall. All I forgot was popcorn (…next year)!

After the viewing, I told them that they were all going to get to make their own Graphing Stories.  I put them into groups of four, and then assigned each group a graph characteristic, either vertical, horizontal, increasing, or decreasing.  (About fifty percent of their graphing story needed to be their “assigned” characteristic.)

After the groups brainstormed and planned, we hit the outdoors to film!  We filmed their 15 second stories around our beautiful campus with my iPhone. I uploaded their videos to YouTube and then they went to work!  It was the beginning of the year, and they were all new students to me thus I had just introduced them to Google Spreadsheets.

Graphing Story Requirements:

  • The presentation must be done on Google Presentations (so that the four group members could easily share and work at the same time).
  • Fifty percent of the graph needed to be the assigned graph (vertical, horizontal, increasing, or decreasing).
  • Use Google Spreadsheets to create the “answer” graph.
  • Create a “blank” graph with label axis so that students could label their graphs.
  • Create an activity sheet with a blank graph and summary questions.
They worked so hard – and this was a very difficult project for them because they were new at Google Spreadsheets and had never used Google Presentation before.  It took them longer than I had anticipated.  The end results were beautiful and creative.  Some of the graphs were not as “exact” as they could have been, some bumps were left out.  But, I was so proud of them that I did not even take off that many points for the errors.  I have included links to their movies below.
Now, seven months later Dan has created a Graphing Stories website and very generously offered to create beautiful Graphing Stories videos from stories we all send in!  I am excited about this extension of their graphing stories for my students.  This is perfect timing for them as we have just begun reviewing material in preparation for their final.  Going over their graphs in order to submit their stories will be a great way to review.  And I am so excited that their Graphing Stories are going to the next level.
7th Grade Graphing Stories
I can’t wait to watch all of the new graphing stories that Dan pumps out next year with my rising 7th grade.  I think it will be fun for kids to watch them as homework, and then present their favorites the next day.  I know it will be fun for them to watch their class mates as well!

Math Wiki Project to Develop Interactive “Concept Help” Resources

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One of my primary goals this year has been to develop easily accessible resources for students to get additional help outside of the classroom.  Students, parents, and tutors ask me for extra material for the students to work on.  Additionally, sometimes students are at home and stuck on a problem, need help with an entire concept or lesson, or miss a few days of school and need to catch up.  I wanted to be able to point students (and parents and tutors) to an interactive “re-teaching” resource that they could benefit from.  As a bonus, it would be nice if these “re-teaching” resources varied from my original instruction, were interesting, or even fun!

I started this project solo.  But, there are many, many concepts in the two classes that I teach.  Also, I did not feel that I was finding enough of a variety of resources.  So, I decided to enlist student help.

I created a project for my 7th grade pre-algebra students called “Concept Help Pages”.  I assigned each student three

concepts.  I picked the three concepts for each student from their lowest personal concept scores.  For each concept, students had to create a wiki page on our class wiki and fill it up with four different types of resources.

  • Online video explaining their concept
  • Online worked out practice problems that illustrated each step and had the answers.
  • Online “interactive” problems for students to work and get immediate feedback
  • Online game
Part of the requirements were that the videos and games had to be interesting and fun.  They needed to view the videos and play the games.  I wanted the pages to be visually appealing as well so I offered a small amount of bonus points for including a picture or illustration that directly related to their concept.  I also had the students do “Peer Reviews” of each other’s pages to check the links, play the games, and give suggestions on how to make each others pages better.
This is a work in progress.  I plan on having other classes add to the help pages in the future.  As a project for 6th grade, I am going to assign them each one of the wiki help pages for them to explore.  I will then see if they feel that they can find additional (and maybe even better) resources to include on each page.  In this way, I hope to make the pages very rich in helpful resources.
Going forward, I would like to use these pages during after school help sessions, especially when I have multiple students that need help on several different concepts.  When this happens after school I am usually scrambling to help everyone.  I make up several problems for several students and then try to work with them all individually (at the same time).  The students are rarely at the same level.  Usually, everyone ends up waiting on me while I work with one student.  If all students could pull up the online interactive problems on the concept that they need to work on, they wouldn’t have to wait for me to make up a problem for them OR check to see if their answer is correct.  I would have more time to walk them through the problems and help them when they got “stuck”.  This would enable me spend after school help time circulating between students, helping each student with exactly what they needed help on, instead of spending time making up problems and giving out answers.

Factoring with X-Puzzles and Boxes

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Factoring is an important concept in Algebra I, and often a difficult one. My pre-algebra class flew through polynomial multiplication, so I decided to give them some exposure to factoring.  I was going for exposure more than mastery just so they could have some factoring experience going into Algebra.  As my students are young and just beginning to develop abstract thinking, I decided to try more concrete/ visual methods for factoring.

Naturally I though of my fabulous Algebra teacher friend who gave me the Math Hunt idea. She swears by factoring with “the box”, especially with students who tend to struggle more.  Even though I had never factored using the box, multiplying polynomials with the box was a big hit so I decided to look into it and give it a go!

The GCF
We started with the GFC, how to find it and un-distributing (or de-clawing) it from Polynomials.  I made a graphic organizer relating what they knew with what they were learning. I gave them a slew of homework and they SLAYED it.  Looking back, I should have also put GCF problems using the box too. It would have helped a few of my strugglers in the coming days.

X Puzzles
Riding on the good GCF vibes I found a great x puzzle worksheet (more commonly known as diamond puzzles) that I gave to my students.  I really, really love the Internet!   Some of these puzzles were HARD.  But since they were puzzles they LOVED them. Go figure.

Putting It All Together
I then showed them what came to be know in our classroom as “x-box” factoring.  Since they were very familiar with polynomial multiplication with the boxes, showing them factoring was almost effortless. Puzzle + boxes = factoring.  I had them do a few in their notes so that they would have examples to go by for homework, and then I pulled out a dry-erase template that I made.  I printed out the templates and they put them in page protectors.  They love anything that they get to do with dry-erase markers (and I will do anything to make factoring less painful) so it was win-win!
http://www.box.net/embed/ped744bvi3nvv5h.swf

Just because it went very smoothly for them during practice does not mean that they would remember it.  So, after several practice problems I had them write down their own steps to x-box factoring.

After somewhat of a rocky start, writing their own steps went REALLY well.  This showed me that I definitely need to have them do more of their own step writing in the future.  They came up with some great (and very detailed) steps!  We combined their steps as a class to come up with a condensed list.  Here is a pic I took of their first few steps.

The best thing about the x-box factoring is that we stared with the “hard” problems first.  Only a few days later did I show them the simple method for factoring trinomials with a leading coefficient of one.  Some kids took to it, but most stuck with the x-box.  I did not do as many games as I should have this unit.  I was too focused on making sure that each student could get the factoring down.  I will definitely because (too late) I noticed waning attention on the last day we practiced.  A game would have definitely livened things up!  I haven’t begun to look for games for this unit next year but @merryfwilliams shared one that she tried called Algebra Connect.

More diamond puzzles, worksheets, and activities for factoring can be found here.  Happy Factoring!

Amazing Algebra Tiles

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Not like I’m opinionated or anything, but if you are teaching students younger students (or struggling students) algebra and you are not using Algebra Tiles, then you are missing the boat!  Not that I can be too judgmental, I had them on my shelf and missed fabulous opportunities to use them all year.

I bought them solely for multiplying polynomials with my 7th graders.  I only bought a few sets because I didn’t want to waste too much money if we all hated them.  I have taught older students Algebra (Pre-Algebra, Algebra IA, Algebra IB, Alg II for seniors) many times in the past and never used them.  With those classes the kids had seen Algebra before, sometimes many times, and I didn’t think it would benefit them very much.  The tiles seem very elementary, not to mention a lot of trouble, and I was afraid it would be, “too little, too late”.  So yeah, I was WRONG again.

I pretty much bought them for one purpose.  I wanted to show 7th graders who had never multiplied x by x to SEE the difference between

x + x = 2x     and    x times x = x^2.

Plus, they are young (12-13) and thus more concrete thinkers so I felt it was a good tool to use for this age group.  I wanted to explore taking them from the concrete to the abstract with Algebra Tiles.  I was skeptical but hopeful.  I’ll try anything once!

However, once I pulled these things out though there was no stopping us!  For the first time ever, my students did not mix up 2x and x^2.  W-O-W.  If you’ve taught Algebra before, you know that is big.  My students knew not only what it meant, but what it looked like.  They knew why the 2x and x^2 were different.  They did not ask me a gazillion times if it was 2x or x^2.  And if they did, all I had to do was pull out three tiles and viola – instant understanding.  I mean, HELLO – Look at them!  They look nothing alike!  You cannot ask for a better explanation than that!

One thing that I discovered that really helped my students is how you can represent negatives with the tiles.   This was wonderfully helpful with multiplying a negative through a polynomial.  My kids always stumble over that or even “forget” it.  However, since we have been doing the tiles they can see what they are doing and are no longer forgetting it.  Bonus!

I bought them to teach polynomials to 7th grade, but once I got them out, I have not put them away!  The tiles put ACTION with the MATH.  More great uses for the tiles:

  1. Illustrating the distributive property.  As fun as “The Claw” is, seeing is really believing for my students.
    3(x + 2) is just THREE sets of x and three 2’s.   (Ahhh,… so THAT’s what we’ve been doing all year.)
  2. Negative Operations – The opposite of a positive number is it’s negative.  For the tiles, to get the negative (or opposite), we flipped the tile over to the other side.  (one side is yellow – positive, the other is red – negative).  This can even illustrate basic concepts, like – (-4).  What is the opposite of  -4?  Flip it over and see!
  3. Multiplying a negative + distributive property.  – (x – 3)  or  -2(x – 4).  When you see   – (x – 3) you are taking the opposite of (flipping) everything inside of the parenthesis.
  4. Equation solving – I draw a line down the center of the paper and use the algebra tiles to solve equations.  When kids tried to subtract 4 from 6x, I would take out 6 x-tiles and 4 constant tiles and say, “Now, how can I combine these?”  The answer, “Oh you can’t, they aren’t like terms.”
  5. Equation solving – divide by a number.    3x = 6.  Three x’s = 6, so one x must be 2.  Then, I would split the x’s and line up the constants on the other side, dividing the x’s.
  6. Equation solving – FRACTIONS.    x/2 = 3 .  If HALF of an x = 3, then what does an entire x equal?
  7. Multi-step equation solving – This was especially helpful when I was teaching multi-step equations.  With tons of tiles spread out all over the paper, the students could easily SEE that of course I needed to combine like terms before trying to get x alone on one side!

(I used Hands On Equations at the beginning of the year, but I think that the Algebra Tiles beat them “hands” down.  They are just more intuitive, especially when it comes to the negatives.)

Of course I still do my crazy songs, dances and rhymes so that math will be stuck in their brains – FOREVER.  (And yeah, I just LOVE to sing and dance!)  But, I think that it is important for every student to know what they are doing, and not to just be  following random steps!   So, one day in the future, when they can’t get our annoying equation song out of their head, maybe they will also recall the Algebra Tiles, and remember WHY x times x = x^2 and NOT 2x.

Links:

Algebra Tiles Video

Interactive Online Algebra Tiles

The Data of Germs and Hand Washing

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Flu season is upon us – and we are working on percents!  What better time to estimate how many germs we have on our hands and investigate how long it really takes to get rid of them?  The CDC (Center for Disease Control and Prevention) defines proper hygiene as vigorous hand rubbing with soap for 20 seconds.  Most of my students were familiar with the 20 second rule, but not convinced of it’s effectiveness.  So, we decided to collect the data and find out for ourselves!

The Set-Up:
I gave them each a squirt of glowing germ simulating lotion which they rubbed onto their hands. We went into the bathroom, turned out the lights, and I shined the black light onto each students hands. This was FUN!  They were quite disgusted with the amount of “germs” on their hands. They were even more surprised to see so many “germs” on their faces, which they had obviously been touching way too much in the five short minutes since they had applied the lotion. It was easy to estimate at this point that the germ count on their hands was at 100%. Let the hand washing begin!

Hand Washing:
The students washed their hands with soap for 5 seconds and then rinsed. I turned the lights off again and we inspected them again with the black light. They were shocked (and disgusted) to see that their hands were still VERY dirty!  Each student then estimated the percent of germs still on their hands.

Rinse and Repeat:
Our goal was to analyze germ data at 5 second intervals for up to 20 seconds.  So we washed, analyzed, estimated, recorded and repeated three more times.

Results:
It took a solid 20 seconds to eliminate the majority of the “germs” from the students hands. The most difficult places to clean were the fingernails and the creases in the palms. The most often missed spot was right around the wrist.  Most of the girls were able to get the estimated percentage of germs down to about 2% while the boys seemed to get stuck at about 10%. The boys found this quite amusing.

Analysis:
After we finally got back to the classroom we loaded all our data into Google Docs and created line graphs.  We then uploaded them to the Handwashing page on our class Wiki.


After we graphed our data in Google Docs we discussed our findings. The kids were surprised that it took so long to wash all of the germs off of their hands even though most of them had either heard about the 20 second rule or knew of a “hand washing” song.  We then picked one of the songs, “Twinkle, Twinkle, Little Star” and sang it as a class while I timed us.  We all enjoyed singing and it took us 23 seconds to sing the song.

In Summary:
This was an extremely engaging and interesting activity!  It did take an entire class period as we only had four sinks and one black light.  But it really brought some fun to a cold winter’s day math lesson.

Looking Forward:
Next year I would like make this like a “Myth Busters” activity. I have since heard Myth Busters did a hand washing episode but could not find it. I would also like the students to do more data analysis and possibly even some comparative analysis.  For this, I am thinking of percent of change between the boys and girls or for each hand washing interval.

Uh-oh! I’m Forgetting What They Mess Up!

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I love that I am a HS teacher teaching 6th and 7th grade. I am their first stop in Algebra. And, I know what they really need to remember. However, I am afraid that I may forget what they mess up on a regular basis! Some things are obvious – like canceling out individual terms in expressions. But, some are more forgettable.

Tonight on Twitter, druinok posted
how to fix kids from doing 3|x+4| = 3x+12??

Ok, totally forgot about that one!

Yes, I did teach them the distributive property last week using, “THE CLAW!!!” and then two days later taught them absolute value and told them that they could not distribute into the | |. Give it one day, take it from them the next. The question is, are they EVER going to remember that random bit of miscellany until they do it for a few years? Doubtful. So, what do we do as teachers? Make it memorable! How am I going to do that? Well, since we aren’t graphing yet I can’t make them believe that. However, I can tell them that the BARS keep out the CLAW. I am going to tell them this ALOT and we are going to do ALOT of problems. Hopefully it will stick!

So, all of you Algebra teachers out there, if you want to comment to me what drives you CRAZY as you come to it so I won’t forget, I would really appreciate it! Hopefully, I can find a way to permanently imprint the correct way into their heads! : )

Wiki’s, Google Docs, and Population Density in Math!

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** All links I refer to are listed at the end. **

On Friday I did an integration unit with social studies.  They are studying Japan and population density.  We got wifi in our building late Wednesday night, so I decided it was time to “break out” the laptops on Friday!

I used a fabulous lesson that I found on NCTM Illuminations called “Five’s A Crowd”.  It was a great activity so I only had to modify it slightly.  It covered population density, ratios, and even estimation!  And, it had a game element so it was very fun! I used their grid worksheet but modified their game sheet.  Of course it was my first time doing anything like this so we had some bumps in the road, but next year it will be so much better!

1)  First the students distributed rice onto grids to visually compare population density of two communities, and to create a definition of population density.  The only thing I had at home that was small enough for the grids was arborio rice.  So, I used colored paper so the rice would show up better.  The kids loved the rice (apparently it was tasty and of course I let them eat it after the activity was over).  But, it was very hard to move around because it was so small.  Next year I am going to enlarge the grids and use a small dried bean – like black beans.

2)  Then we talked about the density of the United States.  We wrote down the exact population and area of the US and then I had them estimate the population density without using a calculator using the definition that we found in number 1.

3)  Next, I gave them the “game sheets” and they had 5 minutes to try to estimate the 5 densest countries in the world using lists I had compiled and put on the wiki.
**Big Bumps Here:

  • I used Box to put the files on the wiki and some kids could not download the files (it was freezing).  NEXT year I will just put the PDF on the wiki.
  • My kids needed longer than 5 minutes.  I had them write down the exact population and area of the places we picked and THEN estimate (not calculate).  So, it took them longer than I thought.  10 minutes next year!

4)  When I finally made them stop (and they didn’t want to), we were seriously running out of time.  So, even though I had planned to have them enter all five countries into a Google Document spreadsheet that I had created, I only asked them to enter their MOST dense country.  They were to enter their country, their exact data, and their estimate.  Then, I had a column that calculated the exact density so we could see how close their estimate was.  Then, I was going to sort their list and pull in the top 20 most densely populated countries to see how good their estimates were.

This is when my lesson went HAYWIRE.

First of all, I didn’t even know if the Google Docs were going to work.  I didn’t want to spend time having students sign up for accounts or sign-in so I made my document editable by everyone.  If they couldn’t enter the info, we would have skipped this part.  But, voila!  It worked!!

Since I didn’t know if it was going to work, and this was our first time using it (including MY first time), I did NOT prepare them for what would happen.  And to be honest, I didn’t even anticipate the snafu’s that we would encounter.  The kids were simply AMAZED that what they entered showed up on the projector and everyone else’s computer.  They had never seen or used this sort of interactivity.  They were very new at this and very excited, and this caused several problems:

  1. They entered the data in crazy places (ie – they deleted my title column “Countries” with their county’s name).
  2. Since they were doing it all at the same time, everyone clicked on ROW 1 to enter their data -at the same time.  Thus, one kid would start typing and then another kid would accidentally erase it.
  3. I saw this quickly and told kids to go on different lines
  4. Some students thought the deleting thing was too hilarious to resist and started deleting each others entries, it really became a “deleting war” at the end.  They thought it was funny – I (and some students) did not appreciate that.

With all of the craziness that ensued I barely got to throw up all of the amazing population density graphs I had found AND play with the SORTABLE wikipedia population density by country data that I found.  I would have liked to spend so much more time on this stuff!

Overall, though I still think the lesson was a smashing success because it was interesting, fun, and even exciting!  It was very enriching to add real life statistics and integrate their current social studies lessons into pre-algebra!  I NOW know what to expect when using Google Docs with a whole class of students at once and can prepare them, in advance!  ALL of these snafu’s were completely my fault and could have been easily prevented with proper instruction on HOW to use Google Docs to the students.  Too bad for me I just didn’t know those instructions in advance.  But, I think that we all learn best by doing it ourselves and I learned so much from this lesson!

I can’t wait to use Google Docs again in the future!

LINKS: