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Different Variable Usages

14 Jan

I remember there being a time when I had identified at least 3 different ways to use a variable, but for now, I can only think of two.

  1. Using variables to represent an unknown.
  2. Using variables as a pattern summary.

Well, w/ Common Core right around the corner, here are a couple of activities I loved, but rarely found the time to do well in the past. However, with my double Algebra this year, I have twice as much time w/ them to go over the same material so I got to work on these much more thoroughly… and I LOVE ’em even more!! I truly believe the lack of understanding variables in the different ways is what makes Algebra hard and not something that should make solving problems easier.

1. Variables to represent an unknown.

Usually used in solving simple equations. x + 5 = 9. “Some number plus 5 will give me 9”. I did a LOT more mental math to teach this part and I really do believe it went significantly better than ever before. Even up to equations like -2x + 6 = 0.

Another big way of using variables as an unknown that we neglect often, is in word problems. So this year, I have used a couple of worksheets that my former professor created which contains just word problems. After going over them pretty thoroughly for the first time this year, I fell in love with it even more because my students came up w/ different ways of writing the equations! Here we come, Common Core!

One example:
Lunch Money. Daisy’s mom gives her lunch money every weekday. On Monday and Tuesday she gave her an extra dollar. The rest of the week, Daisy received the regular amount. For one week, she received a total of $42 for lunch. How much money does Daisy get for lunch each day normally?
Here are the 3 different answers my students came up with:
Once I got them to start drawing pictures to represent the problems prior to this, they did this one on their own. =)
Execution Tips: Only pass out the first two pgs of Amazing Stories first (no answers). We did the problems, one by one, up to “Pocket Change.” Starting “Lunch Money” (the posted example), they were supposed to do it by themselves.
Once they started working by themselves, after a few minutes, I projected all the possible answers so that they had to find their equation (or the equivalent) up on the board. They got a copy of the answers in order to finish the worksheet at home for homework.
The next day, I gave them the follow-up worksheet of “Life of a High School Student” for them to get a few more practice problems on their own, without having a bank of answers.
I saved a few problems from “Life of a HS Student” to use as an exit ticket and on future quizzes.
                 2. Variables as a pattern summary
This is the type of variable usage where all those darn rules come from. For example, if 2^3 * 2^5 = 2^8
4^9 * 4^7 = 4^16
what does x^m * x^n = ?
Anyways, here is the activity on practicing this type. Again, I love it all the more due to the multiple solutions that are possible. I copied down my entry below.
Materials: Cubes. Preferably not too small. And the worksheets.
Students are then asked to build certain objects that build on top of each other and asked how many cubes it would take to build a structure of 10 layers… then 100 layers.
For example:
First question after this would include: “How many cubes will it take for 10 cows? Explain.
Next they would be asked, “How many cubes will it take for 100 cows? Explain.
You can then summarize the patterns by writing an equation, with the input (x) representing how many cows, in this case, and the output (y), representing the number of cubes.
TimingWe started Pattern 1 and 2 in the last half hour of one class, continued the next day for about an hour, and then they were doing Pattern 5 by themselves by the next day’s 50 minute class. Amazing how many students who do not understand Algebra actually are really good at figuring these puzzles out.
Anyways, to get a better feel, you have to take a look at the multiple solutions that these problems have to offer. I have written out what my students had come up with this year. Yep, these were my intensive students coming up with these different solutions. We also took time to go over how to write a good explanation also. I went around giving them feedback and having them re-write a lot of what they had written…
Again, here are the links to the two types of variable practice.

Bad Math!

17 Dec

Bad Math

Bad math! Bad math! Bad math! 

I really hate this workbook! I tutor middle school kids once a week and this is what they came in with. They pace themselves so this is what was supposed to be teaching them? This shouldn’t even get onto paper!!

Bad math! Bad math! Bad math!

And you wonder why this kid was confused!

For all you non-mathy folks, let me tell you what to watch out for when your kid ends up in middle school…

  1. First and foremost, and what trips up the students when they get to Algebra: wrong usage of that dangnabit EQUAL SIGN! (<– Oh, Xanga days! How I miss thee). 0.05 x 100 does NOT equal to 0.05. This is why they have SUCH  a hard time remembering that you NEVER add 6 to one side twice…
    Shoot me now.
  2. A rule w/out any kind of explanation will not stick.  “To multiply by 100, move the decimal two places to the right” is no explanation at all.
    Bah. Humbug.

Publishing companies are the devil.  To math. I’m sure of it. Draw a happy pencil and math does not magically get easier.

Totally Underrated!

11 Oct

It’s INTENSE alright. Not the advanced kind, ladies and gentlemen, the other kind. With all the love and frustration I have for this class though, it’s right up my alley. =)

I basically have the privilege of teaching the 9th graders weakest in Algebra for a block period, 5 days a week!! I’m seriously very happy to be teaching this class, though I was thrown in with almost no materials or prep.

Now, I’ve taught Algebra a ridiculous amount of times considering I came from a 4 by 4 school back in the days. However, I stopped teaching it for the last several years due to intense overload and hatred of life from teaching Algebra 1A up to 4 times a year. Now that I’m back, things seem more clear and I feel like the fogginess of how to order teaching/introducing Algebra is starting to get clearer.

We’ll see though…

Anyways, first insight of the year (and feel free to tell me what you think), but I think the COMMUTATIVE PROPERTY and the ORDER OF OPERATIONS are waaayyy underrated!!

On the commutative property

  • I think when students don’t recognize you’re distributing -3 in 4-3(x+9), it’s a lack of knowing the commutative property
  • I think when students can’t seem to decipher which term is negative or positive in simplifying expressions like 4 – 6x + 9 – 5 + 3x, I think it’s a lack of knowing the commutative property
  • I think when students can’t seem to take 4/7 of a number on their calculator, it’s a lack of knowing the commutative property
  • I think when students don’t recognize that they can simplify a product of fractions before multiplying across, it’s a lack of knowing the commutative property

On the order of operations:

  • Very related to most of the previous issues, I think the order of operations in these damn textbooks don’t emphasize enough how multiplication and division are essentially on the same level and addition and subtraction are on the same level. Therefore the commutative property should NOT be only the commutative property of addition and multiplication by this point, but apply to all 4 basic operations!
  • The focus in Algebra should not be on the FIRST order of operations but now seeing the LAST order of operations. (The light bulb was turned on for me by this author and book.) Students should be able to chunk long expressions by the last order of operations.

Lastly, more things that I highly underrated: writing expressions from patterns and connecting mental math to the algebra.

Lessons I’ve got so far on focusing on these to be continued…

Movies in Math. Math in Movies.

18 Nov

Here is a website with a compilation of short movie clips on math and physics that you can freely download.

I use the Shrek clip as the introduction to inverses and contrapositives in Geometry.(This one was also on YouTube.) The PowerPoint I used is in the Box widget.

Drill, drill, drill!

17 Nov

Finished! I have 27 pages of drills on 1) multiplying, 2) number sense-ing (working in groups of 10), and 3) adding/subtracting integers.

They have been combined into 2 PDFs in the Box widget, now moved to the left of the page.

These drills are to be used for the Timed Folders and all of them were created to be finished under 2 minutes. Time yourself to see how if you might want to vary the pass times.

Most of these (after the regular times tables) are new this year so PLEASE let me know what you think! =)

Really kid? You need fingers for 8+2?

15 Nov

Several years ago at my very first math conference, I accidentally walked in to a session for elementary teachers (before learning that they were categorized by grade levels for you). It was on  Singaporean Math, where I finally understood what “number sense” meant.

Out of all the epiphanies and eye-openers that I was absolutely amazed to learn, I have some strategies that have been working with my high school students that I’d like to share here.


 But first, if you don’t know what the difference between having number sense and counting is, read this following section. If you already do, you can skip on past the next line.

When we add 3+2 in our heads or add 7+3 in our heads, even 6+7, do you actually count out the numbers? Of course not. Most people have developed what is called number sense, which to me, is a type of visualization of the numbers.

What does good number sense look like in:

  • young children (kinder-1st)?: If I flash 3 fingers and pull them away immediately, can they identify that it was 3 fingers without having been given time to actually count them? If so, they are developing good number sense.
  • early elementary?: Can they add 7+8 in their heads without counting? Can they immediately say what 26-6 is?
  • Late elementary: Do they conceptually understand fractions? Fractions inherently require number sense. For example, can they draw and understand 4/3?

So, back to the high school setting. These activities are geared more towards the classes packed with students who lack number sense completely. Classes like pre-Algebra, Algebra support, or if you’re tutoring one-on-one.

#1. Finger Flashes (particularly for those students who count 8+2 on their fingers)
This works best when tutoring a student one-on-one (or for early elem kids).

Flash 1, 2, 3, 4, or 5 fingers on one hand and pull them away immediately and ask them how many fingers you had just held up. They should not have needed any time to count them. They should have just seen them.

After they master seeing up to 5 fingers immediately (usually less than 10 tries), move on to randomly flashing 6, 7, 8, 9, and 10 fingers. It is important that one hand always has the full 5 fingers up (no 4 fingers up on one hand and 3 fingers up on the other for 7). Remember, the whole goal is to get them to see the groupings of 5.

Once they master this, this is when you start asking them, at random, what 5+1, 5+2, 5+3, 5+4, and 5+5 is. Teach them explicitly to visualize the 5 on one hand and the second number on the other.

Then move on to asking them to do 10+1, 10+2, 10+3, etc. Then maybe add in 13-3, 15-5, etc. You get the idea? Always work in groups of 10.

#2. Up/Down (for working in groups of 10 again)

This is a verbal game that you can play with them for about 5 minutes a day. When I call out a number and “UP” or “DOWN”, like “14 UP”, the kids should respond with how many steps it takes UP to the next group of 10. So in this case with “14 UP” the answer would be “six” because it takes 6 steps UP to the next group of 10, which is 20.

To make sure everyone gets a fighting chance, make sure to train your class to only respond after you say “GO”. Threaten them with the fact that you will call them out individually if they do not respond out loud.


I say:      “37 UP… GO”
They say:      “3”

I say:      “28 DOWN… GO”
They say:      “8”

I say:      “94 UP… GO”
They say:      “6”

I say:      “42 DOWN… GO”
They say:      “2”

Based off of this, with my high schoolers, I’ve actually added a series of saying numbers in the hundreds and still asking how many steps to the next group of TEN. I then graduated them to saying negative numbers with UP/DOWN.


I say:      “-37 UP… GO”
They say:      “7”

I say:      “-28 DOWN… GO”
They say:      “2”

I say:      “-94 UP… GO”
They say:      “4”

I say:      “-42 DOWN… GO”
They say:      “8”

After playing this in the beginning of the school year while they are working on their times table in their folders, I then add worksheets based off these 2 activities for after they pass all their times tables…

These worksheets are in development and a post on them shall be made one day. In the meantime, I’ve just dropped what I’ve got into the Box widget to the right.





website to generate own worksheets

Better than the “Mad Minute” Drills

15 Nov

It doesn’t surprise me at this point, but I have many 9th graders who

  1.  can’t multiply… by 2, let alone by 8.
  2. count 12-2 or 8+2 on their fingers. Hell, I’ve even seen 3+2 need fingers.
  3. forget how to do 4-3 when taught the existence of a 3-4.

So what do we do as high school math teachers? Ignore this and hand them a calculator? (Btw, the answer is NO. Speed does matter by this point and it is not enough that “at least they can do it.”)

Well, if you have an entire class of these students (like my current Algebra support class or my previous Twilight classes), try using the following FOLDER METHOD (for lack of a cooler name). It’s similar to the 1-Minute Madness (a much better name), but so much better! It’s cleaner logistically and it forces them to work faster and faster.

THE FOLDERS:Everyday that students walk in, I quickly pass out their folders with their names on them (which are already in order because they always pass them up in order.)

In each of their folders is one page each of their times tables*.

I put 5 minutes on the clock* and when I say GO and start the clock, everyone opens their folders and works on finishing just the first page as fast as they can. When they finish, they raise their hands and my TA or I will walk around writing the time at which they finished it in red. They then must close their folders and wait for everyone to finish.

After about 5 minutes, if anyone is not done, simply end the time and have them pass up the folders in order, placing their folder on top of the stack being passed to them. (The first row then passes the folders down all the way to the left/right in order the same way, placing their stack on top of the stack being passed to them. The last person w/ the stack of all the folders then hands it to my TA.)

GRADING:My TA then quickly corrects them (with transparencies that have the answers on them) and logs the time in which they finished directly onto their folders (see picture to right).

Each worksheet is created to be finished UNDER 2 MINUTES and to save paper, the same times table is copied on to the front and back of each page (they only work on one side).

From here, I go through the folders and do one of two things:
  • If the student PASSED: remove the page he/she just did. If the back side is still blank, I put it back into my pile of extra copies to be re-used for someone else. If the back is already complete, toss it.
  • If the student DID NOT PASS: just turn the paper over if the back was blank, or replace the page with a blank one from the pile of extra copies.
LOGISTICS: After having gone through it for the second year this year, I recommend a few quick things.
  • The order for the times table should be the twos (2), fives (5), tens (10), then perfect squares. Then move on to the 3, 4, 6, 7, 8, and 9s.
  • Having the students write their own times on their papers only works the first couple of times before they realize they can LIE. Try to have a TA or maybe another teacher w/ a conference period next door to just help you out the first 5 minutes of class. It’s timed so it never takes longer than 5 minutes. I have about 3 students who are done w/ all that I have and so they help write the times when my TA is absent.
  • Make sure they close their folders after they get their times written. Often, they want to go back and check and fix all their mistakes, defeating the purpose of timing them. It’s also a very good indicator to see when the entire class is done.
  • For the 2, 5, 10s, pass them only if they finish under 1:45 instead of the complete 2 minutes. If they take 1:59, they’re probably just professional finger counters.
  • See how I logged their times in the picture? I recommend just writing “twos” once and using the ditto sign (“) underneath for all the subsequent ones. I think this will be visually much easier for the kids (as well as the teacher) to see how many times they are needing to repeat before moving on.
  • I’m in the process of purchasing one of these to help me organize all my extra copies.
What do they do AFTER they finish the times tables?
So this is something that I have been working on and is still an ongoing process (feedback more than welcome!).

Well, first off, I think I should add an extra mixed times table page, and a second mixed times table page with negatives and positives. THEN give them the Algebra Intro* and maybe add another Integer Algebra Intro.

THEN I want to move them on to helping them with number sense, and then of course with adding and subtracting integers.
Anyways, all that I have is in the Box widget to the right. I will post more on number sense and the development and meaning of those sheets next.

*These are all provided in the Box widget to the left.

11/17 UPDATE here

Baffling Decimals

19 Apr

As I am catching up on reading Steven Strogatz‘ brilliant articles on math, this one in particular is really making me want to teach elementary school math again! I dream of being Deborah Ball and creating sassy lil kids who debate why odd numbers are odd. I also dream of being Jaime Escalante and randomly yelling “GANAS!” to kids while they’re taking integrals…

Anyways, have you heard the Verizon billing debate? I want to share this with everyone I know, so that they can share in some of the pains of teaching math.

For example, this week, we started talking about the Quadratic Formula, therefore, we had to discuss square roots. I listed the square roots in order going down.

When we established that these were in ascending order and that the square root of 1 was 1 and the square root of 4 was 2, we got to approximating what the square root of 2 was.

After testing 1.4 and 1.5 and figuring out that it was a number in between, I asked the class to choose a number between 1.4 and 1.5.


“O… k… What if I add this?,” I say as I write ‘$’ in front of each number, “and this?,” as I add a zero at the ends.

“OH!” “1.45!” “1.47!” “1.42!”…

So then we get to seeing that square root of two is between 1.41 and 1.42, which then bodes the next question, “What number is between 1.41 and 1.42?”

**silence** x2

And then someone yells, “There no such thing!”

Ah yes, my future Verizon workers…

Fraction Fail

13 Sep

SIX days of 95 minute block periods spent on HOW TO USE A RULER and all but ONE failed the 5 question test. Not only can they not find the fractional part of an inch, they are still reading 3 inches as 4 if I don’t start the ruler from 0!!


Ikea Rulers

9 Sep


Ikea rulersConfession: While all this debate is brewing about Ikea changing their font, I snuck in and took a stack of their paper rulers. Actually, I was just the driver who kept the engine running while a couple of my kids who graduated last year went in to do the job. =P Hehe.

Well, this year, I finally had(have) the chance of teaching a couple of concepts I had shockingly found that many of my students did know. I had found this out in the last couple of years while teaching juniors and seniors!!

Shocking discovery #1: I knew for awhile now that many students did not know how to use a protractor to measure an angle, but I did NOT know how many did not know how to use a ruler! I had asked my, albeit remedial, geometry class to draw me a 4 inch line only to find that it looked a bit short. Three out of the 10-12 that showed up that day drew me a 3 inch line instead. They had started from the ‘1’ instead of from the ‘0’.

Shocking discovery #2: In starting to explicitly teach using rulers now, I started to see that most of my kids also did not understand why using centimeters was easier than using inches.

To try to teach these concepts, we went through 2 block days of coming up with our own unit of measure and then moving on to use the official foot and inch.

So far, the best thing about this entire assignment were these Ikea rulers!! (Thanks yet again Ji!)

Basically, before class I went through all of them and tore off a random piece of the beginning and end. When we finally got to needing a ruler, I told the class that all this poor teacher could find were these janky Ikea rulers. Once they all received theirs, I started to hear the joyous commotion begin about how they couldn’t use them- that they were “broken.” I couldn’t help but to smile. I simply told them they weren’t “broken” and to deal with it. That’s when I started hearing some great discussions in their groups.

The point in which I had to intervene was not actually in finding the fraction anymore, it was in counting. I went around the room folding a random length less than one foot and asked them to find me what fraction of a foot that was. (Another beauty about these rulers, you can fold them!!). They found me the fraction correctly, assuming they counted correctly. However! 4 out of the 5 groups counted wrong! Even in that one group that counted right, I discovered that only one in that group really understood!

As you might’ve guessed, they were counting the NUMBERS instead of the actual INCH-SPACES in between.

More results from this class to follow.