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!









Monday

Ratios and Proportions Problem Solving - Problem of the Week 11

Happy Monday!
It's a rainy Monday here this morning....and a bit darker than usual with the time change, so I really wanted to stay in bed today!

We are working with ratios and proportions in 6th grade math this week, SO this week's problem of the week is a pizza-themed ratio problem solving activity.

I hope you can use it! You can click below either image to download:-)
Have a great day!

Problem solving to help students practice setting up and solving proportions.
Click to download


Problem solving to help students practice setting up and solving proportions.
Click to download.








Friday

Using the Ladder Method - GCF, LCM, Factoring

I'm just writing to express (again) how much I love the ladder method! :-)

Using the ladder method to find GCF, LCM and to factor.Last spring, I wrote a guest post about the ladder method on Rachel Lynette's blog, so if you're interested in reading all the details, check it out.

Otherwise, the latest ladder method item I've created is a new poster/anchor chart. I had a few different ones last year, so I decided to consolidate!

The link to my ladder method fold-it-up is below, if you'd like to download it.

What I really like about this method is that the process is the same for each use, but the outside numbers are used differently.
I like the fact that the continued use of the ladder method (for various reasons) leads to the students making greater connections between numbers.....finding factors seems to come more easily.

I haven't shown the students how to reduce fractions using the ladder method, but they'll see it on the poster next week. Then when we discuss fractions, it will already be there!

Using the ladder method to find GCF, LCM and to factor.
Using the ladder method to find GCF, LCM and to factor.
Click to download

Using the ladder method to find GCF, LCM and to factor.

















Monday

Math Fun - With Dates!


Writing dates as different mathematical expressions.Last week I started throwing a little extra math into my classes, homeroom and last period (homeroom students again) - by using the date! I went to a conference years ago and this was one of the ideas in the book we were given.

I haven't used the idea in a while, so I brought it back in two ways:

1) Use all the digits in the date to create an equation. The digits should stay in the same order they are in the date, and any operation signs can be added in between any digits. The equal sign can also be placed between any digits.
Digits can be used as exponents, as in the example shown, and you could add square roots signs if you can find a way to use them.

Writing dates as different mathematical expressions.2) The other way I used the dates was to write the date so that students have to solve an expression for each number in the date.

It's been fun to see some students writing these in the corner of their notebooks during class! Others have asked to write their equations on the board during the last period of the day.

Writing dates as different mathematical expressions.
I'm looking at the date-writing as another way to introduce notation my students haven't seen before, like the cube root.... I loved today's date!
Writing dates as different mathematical expressions.
 How would you write some of these dates?











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