Wednesday

Integers Fold it Up Freebie!

Click to download!
Wow, the summer has just flown by (as always)!

Among other things this summer, I've been working hard on creating new units for 6th grade math. I've made an effort to include different "colorable" sections on many of the pages, and all of the units include a couple different Fold it Ups that I've created or tweaked over the years. I just love the Fold It Ups, as I'm sure many of you do.

This one is from my Rational Numbers unit - (it's also in my Fold It Up book, but the labels on that one are black rather than "colorable").
I especially like this one because it gets glued onto a second page, which has a number line. This allows for some flexibility when teaching about negatives, positives, and opposites. The number line can be used during the introduction of the concepts, to label vocabulary on the number line....make it a visual. Or, it can be used during review, when preparing for a test or quiz....students can label when self-quizzing (or quizzing each other)!

Or click here!
The download also includes notes for the inside:-)

Hope you can use it!






Font in the graphic by KG Fonts.

Sunday

Fun with Math Dates, 2016-17

It's almost August! I'm always amazed at how quickly the time goes!
This is just a quick post to share what I'm planning with "math dates" this year. Last year, I got such a positive response when I posted ways to write the dates as mathematical expressions. So I decided to get a little more organized with it this year, and plan the week of dates on Sundays, instead of thinking of them one day at a time.

Every Sunday, I'm going to post the week's dates on Instagram and Facebook, so if you'd like to use them, you might want to follow me one of those platforms (if you aren't already). I'll post the dates here sometimes as well, but not as regularly.

I loved using expressions as the dates last year - it kept students fresh with their exponents/exponent rules, exposed them to roots (we don't really study these yet in 6th grade), and encouraged them to figure out their own expressions.


Do you already use something like this in your classroom?






**Fonts in images by KG Fonts

WHY the Butterfly Method When Adding and Subtracting Fractions?

The "butterfly method" for comparing, adding, and subtracting fractions -WHY teach this? I don't
understand why this method is being used.

I understand that it works and that it can make finding an answer easier for students (for simple fractions), but I am discovering definite drawbacks to this method. I don't want to seem negative, but I don't like to see students adding and subtracting fractions this way! Let me explain-

I used to have students come to me, having learned the butterfly method for comparing fractions.....they often had no idea why it worked....they knew the trick better than they knew how to find a common denominator...they didn't seem to understand that the products from the cross-multiplying were the actual numerators they would get if they made certain equivalent fractions (figure 1); they just knew it worked.This bothered me, because I believe they should understand why things work. So I always made sure to explain why the method worked.

But this year, I had students tell me they were taught to use the butterfly method to add and subtract fractions (cross multiply and add those products, then multiply the denominators together -figure 2). But again, they didn't really have a conceptual understanding of WHY it works. It seems to me that many students are being taught "hacks" like this, to make learning fraction operations "easier and fun." In reality, they aren't learning what it means to add or subtract fractions. (And, really, why wouldn't we want them to see that 6 is the LCD in the problem in figure 2? Why would we want to them to use a larger denominator and then have to do more reducing...?)

I recently gave students problem solving that required them to use all fraction operations. Since adding and subtracting is in the 5th grade curriculum (and I teach 6th), I did just a brief review of adding and subtracting fractions before students worked on these problems - to see what they remembered. This is when I found out that many of them had been taught the butterfly method, among others.

In the problem solving, students had to add 5/6, 2/3, 7/12, and 7/10. And here's where the butterfly method totally fails the students who have learned to rely on it, not only because they don't understand why it works, but because it becomes so cumbersome! They couldn't use the butterfly method to add 3 or 4 fractions at a time, so they added two fractions at a time. Instead of finding a common denominator for all 4 fractions, they found a new common denominator each time they added on the next fraction. For example, they added 5/6 and 2/3, getting a denominator of 18, as in figure 2. Then they added 27/18 and 7/12, as in figure 3,  getting a denominator of 216. From there, they added 450/216 and 7/10, shown in figure 4. They ended up with a HUGE denominator that they then had to work really hard to reduce! It might seem crazy that they continued this process to get such huge numbers, but some of them did, because this was the method they learned. I was so shocked to see this....and this is the first year I saw this method used in this way.

When we as teachers (or parents) find certain tricks that work for simple math problems, we need to look ahead to what our students will experience in future years. We need to try these methods with more complicated problems, to see if they will still be effective. We need to think about whether the "hacks" teach them math concepts, or number sense, or number connections....or do they just teach short-cuts? I have no problem with teaching short-cuts once the conceptual understanding is there. But short-cuts before understanding is detrimental to our students. They are capable of understanding the concepts and we need to have faith that they can "get it" without the tricky methods.

In the video linked below, Phil Daro stresses the value of teaching mathematics in greater depth and avoiding "clutter" in the curriculum - one of his examples includes the butterfly method.
Don't Leave Out the Math 

Thanks for reading!






Tuesday

Table Tennis and Math

I love playing table tennis (but I really call it ping pong all the time)! I played it a lot as a kid and I play occasionally as an adult....we have a table in the basement:-) I would never claim to be a SERIOUS player, but I'm not bad!

I was playing with my daughter the other day, and it occurred to me that playing ping pong is a great way for younger children to practice their addition facts and some multiples of 5 (good for older kids too, if they don't know these facts very well). Now, this idea is based on the "serving rules" that we used when I was growing up. It appears (after I searched for info) that these are not the official rules any more, but since I'm not a professional, I'm ok with playing by the unofficial rules!  The way we played is that the server switches every 5 points, and we played to 21 points.

So, here's where the math comes in....when you're playing, you need to know when to switch who's serving, so you need to know what adds up to the multiples of 5. When the score is 5-0, 4-1, or 3-2, serving switches. To switch servers at 10 points, players  need to know that the score would be 10-0, 9-1, 8-2, 7-3, 6-4, or 5-5. When serving switches at a total of 15 points, the score possibilities are 15-0, 14-1, 13-2, 12-3, 11-4, 10-5, 9-6, 8-7. At 20 points, the score would be 20-0, 19-1, 18-2, 17-3, 16-4, 15-5, 14-6, 13-7, 12-8, 11-9, 10-10. The repetition of these facts throughout many games can really help kids learn them.

Over the years, I have noticed that students (in general) seem less aware of, and less automatic with the digits that will add to 10. Playing ping pong is a great way for kids to practice these facts without thinking that they're practicing math (math in real-life!).

This is great for parents to do with their kids, but also - a mini ping pong table in the classroom sounds like fun!!
 





**Fonts in graphic by KG Fonts

Saturday

Fraction Division - Another Way?

How often have you taught fraction division to your students only to find them "flipping" the wrong number? You may have taught them to "skip, flip, flip," "invert and multiply," or "multiply by the reciprocal." You may have listed out the steps, or taught them a nifty song, but somehow they still flip the wrong one or they forget to flip at all.
OR they change a mixed number into an improper fraction and seem to subconsciously think that since they did something to that mixed number, the flipping had already occurred...and then they don't flip anything.

Why does this happen? I'm going to say that it happens because they don't see the sense in it - it doesn't mean anything. Yes, you CAN PROVE to them why multiplying by the reciprocal works, but at the age they learn this process, the proof still doesn't really seem to mean much.

So, I have another way to teach fraction division - perhaps you've heard of it, or you use it. I never learned it this way as a child, but I like it and it makes more sense to some students. I learned this method when I had a student teacher a few years back. She was teaching the fraction unit, and when her supervisor came in to observe and discuss, she asked if I had ever taught fraction division using common denominators. Having only learned (and then taught) to multiply by the reciprocal, of course, I said no.
The next time she visited, she brought me a page from a textbook that explained dividing fractions using common denominators. These are the steps:

Step 1: Find common denominators, just as when adding and subtracting and then make equivalent fractions (students are already used to doing this - hopefully).
Step 2: The answer is the first numerator over the second numerator.
Done (unless you need to reduce)!
I was shocked - it seemed SO simple!


Check out this example - it's a simple one, for starters:
5/6 divided by 2/3.
1) Find the common denominator of 6. This gives you 5/6 divided by 4/6.
2) The first numerator (5) becomes the numerator in the answer. The second numerator (4) becomes the denominator.  Then reduce.


Let's look at another one, with mixed numbers:
1 and 4/7 divided by 1 and 3/4.
1) Convert the mixed numbers to improper fractions, which gives you 11/7 divided by 7/4.
2) Find the common denominator of 28 and make equivalent fractions. This gives you 44/28 divided by 49/28.
2) The first numerator (44) becomes the numerator in the answer. The second numerator (49) becomes the denominator.  No reducing, in this case.

I've shown both methods to my sixth-graders. Some really like it. Others stick to the flipping method - but I don't know if this is because they like it better or because it was the first way they learned it.....most of them had been taught something about fraction division in 5th grade.

What do you think? Do you see any advantages or disadvantages to teaching fraction division using common denominators?




**Fonts in graphic by KG Fonts

Focus & Fun with the Array Game, Using Polyhedral Dice

Have I mentioned that I love Jo Boaler’s books and site, Youcubed.org? Well, I do! She shares so much fantastic research and so many wonderful ideas.

So, I was reading her book Mathematical Mindsets this week, and read about the “array game” (called How Close to 100), which I’ve seen all over Pinterest and thought was very cool. I tried it with my classes last year during a little bit of down time, and they liked it. I hadn't really thought of using it this year, but last week I noticed the baggie of polyhedral dice that I've had for a looooong time and thought it would be cool to use the dodecahedron dice for the array game. With these dice, the students could use numbers up to 12, rather than 6.

To set up their game, students each outlined a 20 by 20 area on their own graph paper. They took turns rolling their dice and creating arrays to represent the multiplication problem they had rolled. It was very interesting to observe the way students arranged their arrays. Some started in the corner and worked their way out, while others started on one side and worked their way across. Some made the arrays touch, if possible, while others left a row between each one. Some just drew their first few arrays anywhere and then discovered that they didn't have a lot of room to fit additional ones. The "winner" was the student with the fewest number of boxes left (some did get to zero left). The students really had fun with this!

Of course, some finished their games earlier than others. In these cases, I asked students to create arrays that used different numbers than the numbers they rolled, but represented the same area. For example, if they rolled 12 and 5, their arrays could be 10 by 6, 15 by 4, or 20 by 3 (not 30 by 2, we discussed, because the grid is only 20 by 20). If they rolled a number that couldn't be represented by a whole-number array, they could then use an irregular shape, or a triangle - anything they could find the area of. It was interesting to see how some students got stumped when they tried to draw an irregular shape to represent a number like 81.
Most students enjoyed this twist (we continued it the next day so they all got to play this version), but a few complained that it made their heads hurt! That's ok...I know they were really thinking and growing mathematically!


The next extension for early finishers (only a few) was to use the icosahedron (20-sided) dice, and have students create area models to cover their grids and find the answer to the multiplication problems. This required a larger grid, so I had them tape 2 pieces of graph paper together and create 20 by 40 grids. Using the icosahedron dice gave a mix of 1-digit by 1-digit, 1 by 2-digit, and 2 by 2-digit problems to model and solve. Most students didn't get very far with this before we ran out of time, but I think this is a great way to them to visualize what multiplying by a two-digit number means. I'd like to revisit this one!


I'm so glad I thought about using those polyhedral dice! I can't remember where I bought my polyhedral dice; as I mentioned, it's been so long since I bought them. I did find some on Amazon, though, if you need to buy some yourself!
Click to see on Amazon







Have you used polyhedral dice in your math classroom? If so, please share how!




Friday

Playing "Decimal Dice" - Converting Fractions to Decimals

Do your middle school math students like to play math games? Mine do, but over the past few years I've noticed that many of them aren't familiar with some of the games I played when I was a kid, like Yahtzee, for example.

So, as we started working on converting fractions and decimals, I decided to create a game to make practicing the conversions more fun AND give them some more game experience! I based it on the idea of Yahtzee:-)

Here's how it works:
Students roll four dice, and pair the dice up to create "target numbers" that are decimals or whole numbers.

For example, a student rolls 1, 2, 4, and 6. From these dice, the student may create any two of the following decimal (or whole) numbers:
Check out this decimal dice game to help students practice converting fractions to decimals!
  ½ = 0.5                   4/1 = 4

  ¼ = 0.25                 4/2 = 2

  1/6 = 0.1666...        4/6 = 0.666...

  2/1 = 2                    6/1 = 6

  2/4 = 0.5                 6/2 = 3

  2/6 = 0.333...          6/4 = 1.5


Once a player has chosen two target numbers, he or she finds the score by adding the dice that were used for each decimal. If the player chose to use 1 and 4 to get 0.25, he or she adds 1 + 4 for a sum of 5 to place in the score column. The second decimal choice uses 2 and 6, to equal either 0.333...or 3. The sum of 8 would go in the appropriate column as the score.

Check out this decimal dice game to help students practice converting fractions to decimals!On the next roll, this student rolls 1, 1, 3, and 5. This student can pair 1 and 1,
to get 1, and pair 3 and 5 to get either 0.6 or 1.666...  The score for 1 is 2 (1 + 1) and the score for 0.6 is 8 (3 + 5).


In many cases, students' scores will be the same, but some of the decimals can be found with different combinations of numbers (1 and 3 = 0.333..., and so do 2 and 6, so students could score either 4 or 8). Some students will notice this sum difference and go for the combination that will give them the higher score.

The students have really enjoyed playing this game. They do need a few examples at the start, to understand exactly how the game works, so if you decide to try the game, be prepared to go through a few turns together.

Check out this decimal dice game to help students practice converting fractions to decimals!
Click to see on TPT
Detailed instructions are included, and a complete answer key of highest and lowest possible scores for each target number are included as well. This is handy to quickly check student score cards as you check in on their games.

If you give it a try, please let me know how it goes!









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