A New Resource Book I’ve Found

Discovering Decimals by Laura Candler

If you’ve followed my blog at all, you know that I am addicted to finding new resource books. Last year, after already teaching decimals, I found a great book called “Discovering Decimals”. It is a Kagan book, for those of you who have heard of Kagan. I personally had never heard of Kagan until I came to Midwest City. A few of the math teachers had attended a Kagan workshop the year before, but they didn’t really elaborate on the details. After getting this book and a couple of other Kagan books, I am anxiously awaiting an opportunity to attend a workshop. Kagan is extremely hands-on, movement oriented, and stresses cooperative learning. It is everything I strive for my classroom to be like.

“Discovering Decimals” is authored by Laura Candler. The other day, I decided to Google her name to see if she had any other resources. I was happy to see that she has her own blog. Here is a link to her blog for those that would like to check it out  http://www.lauracandler.com 

The book is designed for grades 3-8, so it covers all aspects of decimals. The beginning of the book discusses cooperative learning structures, creating a cooperative classroom, investigating decimal concepts, teaching decimals with manipulatives, motivating reluctant problem-solvers, enriching decimal instruction with centers, and goes into detail about the different cooperative learning structures. The rest of the book has specific decimal activities. There were so many activities for the specific PASS we were learning, that I had to decide which one would the students benefit most from. They were all amazing activities with lots of hands-on manipulatives, games, and cooperative groups. There are black-line masters for the worksheets you need and there are several activity cards that you can copy on card stock and cut out. I will be discussing some of these activities in later posts. If you teach decimals at all in your classroom, I highly recommend it. As I said before, it is for grades 3-8. I ordered the book from www.enasco.com for $23.50, and it’s worth every penny and more.

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Comparing and Ordering Decimals

Base 10 Blocks

As crazy as it sounds and as bad as I hate to admit it, I had never taught using Base 10 Blocks. No teacher had ever taught me using them, and when I taught 9th grade algebra the books never talked about them. It wasn’t until I started teaching 6th grade math that I was introduced. Needless to say, I’ll never teach decimals without base 10 blocks again. Last year I made the base 10 blocks out of paper. For this year, I ordered plastic blocks and what a great investment!

For teaching “comparing and ordering decimals”, I used a couple of different resources. The first resource I used was the “Hands-On Math!” book. There is a review of this book in my “Resource” tab.

Day 1:

Activity 1: “Match Up”:

 I placed the students in pairs. I gave each pair a set of base 10 blocks that I had prepared ahead of time. 4 flats, 18 rods, 18 small cubes. I explained base 10 blocks and that 1 flat was equal to a whole, that it took 10 rods to make up a whole so the rod is .1 of a whole, and that it takes 100 small cubes to make up a flat so a small cube is .01. I then wrote the numbers 2.34 and 2.61 on the board and told the groups to build those numbers with their blocks. The first time took a little help for some of the groups, but they quickly caught on. I explained that in order to compare them that they had to start with the largest blocks first. Both numbers had two flats. They then move on to the rods and compare them. Six is more than three, therefore 2.34 < 2.61. I put up another few problems as in 1.8 and 1.75, 0.95 and 1.3.

Activity 2: Count to Match:

This activity moves from the concrete materials to drawing pictures. This process helps them transition to the more abstract action of comparing decimals without the help of manipulatives. Still in partners, I have the students get out a sheet of paper and I write the decimals 4.35 and 4.38 on the board. I ask the students to each draw the models of those numbers using the following symbols: 

Starting with the largest place value, they are to begin marking out the symbols the two numbers have in common until they have a set of blocks that are not equal and circle those that are left over. Do this a couple of more times with numbers such as 3.8 and 3.75, 0.98 and 1.2, and 5.04 and 5.

Activity 3: Who Has More?

This is the “game” activity that the students will play. The activity in the book has them continue to use the base 10 blocks, but I didn’t feel my students still needed that help. I kept the groups in pairs for this game as well. The book suggests to put them in teams of four and have them pair up. I suppose if I taught younger students, I would maybe do this but my students worked just as easily by themselves. The students were to each have a piece of paper and draw a line down the middle and put their names at the top of each side in order to keep score. I gave each student a ten sided dice numbered 0-9.

10-Sided Dice

They were to each roll their dice three times to form a decimal number. I had them use 0 as their whole number each time. The first roll was the tenths place, second roll was the hundredths place, and the final roll was the thousandths place. They were to write their numbers down in the respective column and compare them and circle the larger number. They played this until I called time.

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Divisibility Rules

The folded divisibility rule foldable.

Divisibility rules are not a PASS for Oklahoma sixth grade, but it is very useful especially for prime factors which is a PASS. Because it’s just a helpful tool, we only spend one day on it. Here is how I taught it this year…………

To begin, I gave each student a colored sheet of paper. I told them to fold it “Hot Dog” style. They then folded it in half, three times. Open it all up and cut down the creases to make eight “tabs”. The following pictures show exactly how to fold and cut it…………

Start with a piece of paper.
Fold the paper like a "Hot Dog"Fold the paper in half.Fold it in half againand fold it one more timeYou should have eight sectionCut each crease on one side of the foldable

The following are the different divisibility rules that I go over and have them place in their tab book.

Tab 1: Divisibility Rules

Tab 2: Divisible by 2 if:   the number is even

Tab 3: Divisible by 3 if:   the sum of the digits is divisible by 3

Tab 4: Divisible by 4 if:    the number formed by the last two digits is divisible by 4

Tab 5: Divisible by 5 if:    the number ends in a 0 or 5

Tab 6: Divisible by 6 if:     the number is divisible by 2 and 3

Tab 7: Divisible by 9 if:     the sum of the digits is divisible by 9

Tab 8: Divisible by 10 if:    the number ends in a 0

The folded divisibility rule foldable.The inside of the foldable

After writing each tab section on the board, I show a couple of examples that I have them write in their tab book also.

The students were grouped in pairs. I gave each group a “dice in dice” dice. I drew a table on the board and had them copy it on a clean sheet of paper. They were to roll the dice. The outside dice formed the tens digit and the inside dice formed the ones digit. They were to write the number down and then put a check mark in the numbers that it was divisible by. You can actually come up with many ways to get the numbers. In one of my classes, I gave each student a 30-sided dice and they were to add the two dice together to form their number. You could use regular dice as well. Just any creative way to form a number. This process is much more interesting than giving them a worksheet with a bunch of numbers on it.

The table that the students copy on their sheet of paper.

Update: I am participating in a linky-party with Laura Candler on her amazing blog called Corkboard Connections. Here is a link to the linky party where you can get amazing foldable freebies!!

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Area of a Circle

 

 

In order to teach finding the area of a circle, I used and modified a couple of activities out of the “Hands on Math” book. Look under the “Resources” tab for a picture and review of this book. It is an excellent resource. Before I teach finding the area of a circle, I first teach my lesson on finding the circumference of a circle. Doing this, they are already familiar with pi and know that it is “three plus a little bit more.” The first activity we do is called “Circle Cover-Up.”  For this activity, the students are groups of two but they each do the activity themselves. The reason for pairing them is so that they can watch the other person to make sure they are doing the activity correctly. In the “Hands on Math” book, there are black-line masters of the pages I use. You print the pages out in two different bright colors so that they stand out. Below is a picture of the two pages we use. One is a centimeter grid and the other is two different sized circles with a radius drawn and tick marks to show the measurement of the radius.

I tell the students to cut out squares that have the same side length as the length of the radius. For example, if the radius is five units, then you would cut out a square that is five units long and five units wide. I usually go ahead and tell them to cut out four of those five by five squares out. They are actually suppose to cut each of the larger squares into the smaller squares and place each individual square on the circle. I found it much easier and less time consuming to have them cut the larger squares into strips or as big of pieces as they can. They are to glue these on the circle and make note of how many larger squares it takes to fit on the circle. This is kind of difficult and the students won’t all get the same answer. Most usually only fit three on the circle. This is fine because I look through the class and find the best example to show the students that they should have been able to get three larger squares plus a little bit of the fourth square.

The students are to do this with the second circle on the page. That circle has a radius of four, so they are to cut out squares that are four by four. Again the students should use three squares plus a little bit of the fourth one. I ask them if they remember that from learning circumference. By now the students have already figured it out that pi has something to do with it. I draw the following diagram on the board using the measurements from the first circle.

I explain to them that the meaning of the area of a circle is how many unit squares will actually fit onto that surface. I then draw an example of the squares we just did on that same circle. I know, I know, I am no artist, but the kids get the point of my drawings.

I ask them how many small squares are in the one larger square on that circle. They say twenty-five and I make sure they understand they easily find this by taking the side times the side. I question them and make sure they understand that the side length is also the same as the radius. I then draw that same square on the other three sections of the circle. I ask them if we actually was able to fit all four of those large squares on the circle. No, then how many? Three plus a little bit more. What do we call that three plus a little bit more? Pi. Very good. So if there are twenty-five in one square then there would be twenty-five in each of the other squares too right? How many big squares were we able to fit on? Pi. How did we calculate how many small squares were in the big square? Side length times the side length. How did we determine the side length? The radius. So you could also say the radius times the radius. So if there are twenty-five small squares in this large square, how many are there in another larg square? Twenty-five. How many in the other large square? Twenty-five. So how many altogether in those three large squares? Three times twent-five which is seventy-five. Did we use all four squares? No. Three plus a little bit more. Sooooooo…… I could find out how many little squares there are on the circle if I took the radius times the radius times three plus a little bit more (pi). We also call radius time the radius, radius squared (sorry, I can’t make the little squared number so I’m just going to have to write it out.). So we could make our formula…… A = r^2 * pi. Wow, that was quite a dialogue I just had with myself. I hope it didn’t confuse you. Maybe reading a few times will help you. It’s much easier when I’m teaching it to actually human beings who respond back to me. I hope you got the main point of it and are able to use this information in your classes.  

We work several examples using different measurements for the radius. For the first several, I draw a circle and the squares in them until the students catch on and are able to understand it without the pictures.

The activity we play to review finding the area of a circle is the same game I used when finding the circumference of a circle only making them find the area instead of the circumference. This game consists of the following modified paper with two spinners on it, a large paper clip, and a pencil.

They spin the paper clip on the top spinner to get what dimension they have. They spin the paper clip on the bottom spinner to get the measurement and find the area using that information. I have found that doing this activity and the activity with circumference, the students actually understand the meaning and reasoning behind what those two measurements really are. When you combine area and circumference on the same worksheet is when most students become confused because they forget which formula goes with which measurement. I am able to remind them that with circumference, we had the string and wrapped it around the circle and then saw how many times the string could go all the way across the circle which is the diameter. With area, we cut up the squares to put on the circle and we needed the radius to find out how many squares were actually on the larger square. Hopefully this gives you some insight on a good way of teaching area. This activity is the best I’ve found so far.

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The Different Ways We Play Integer War

One group of students playing Integer War with the "Dice in Dice". Usually they are in groups of two, but it was an odd numbered class.

Integer War is probably the most common game to review integers. The concept is exactly like the card game of War. I have the students grouped in pairs. They each have a piece of paper that they draw a line down the center and place their name on one side and their partner’s name on the other side of the line. They get their two numbers in the fashion that I explain below. They either add, subtract, or multiply (depending on the operation we are working on) those two numbers. Each player works both problems each “round” and they check their answers with each other. The player with the larger value wins that round and they put a star next to the number. This gives students several problems of solving the integer problem as well as practicing “comparing integers” as well.  In order for the students to play the game several times without actually getting bored with it, I find different ways to come up with the numbers. Here are some of what we’ve done this year………

One way that we get numbers is by using a deck of cards. I have the students remove the jokers. The cards are all dealt out. Each student flips over two cards. The student adds, subtracts, or multiplies (whichever operation you are reviewing) those numbers. The person with the highest value wins that round. Here are a list of the values for the cards……

red = negative

black = positive

Jack = 10

Queen = 11

King = 12

Ace = 1

Another way that we used to get the numbers for the problems was to use 30-sided dice. I give each student a 30-sided dice. This year is the first time I used them, and I gave them each a different color and told them to choose one color as the negative and one color as the positive. This limited the types of problems they did. Next time I do this activity, I will give them a two color counter that they must flip as well as the roll the dice. The yellow side of the counter is positive and the red side of the counter is negative. With the 30-sided dice, this gives a larger variety of numbers the students are able to work with. I love, love, love the 30-sided dice. I will be buying other types of dice in the near future (next payday).

Another way I used to come up with numbers was with “dice in dice”. Yes, that is exactly what they are…..a dice with a dice inside. I also recently noticed that there are polyhedral dice in dice as well. I will be purchasing those. With these dice, I gave each student a dice and a two color counter. I told them that the inside dice was always going to be negative and they were to flip the two color counter in order to determine the sign on the outside dice. I did this to save time on the whole flipping thing. You could have them flip the counter twice, once for the inside dice and once for the outside dice. Whichever you prefer.

The last way that we play Integer War is with cards that I found out of my “The Middle School Mathematician” book. If you look in my “Resources” section, I give a review on this book.  I always use these cards for reviewing “division of integers” for sure. I do this because in sixth grade, we are just introducing operations with integers and I want nice even numbers. I suppose you could do any of the above mentioned methods and give them a calculator, but I find these cards work much easier. It is played exactly like “War” with the deck of cards, only they give you two numbers on each card so they only flip one card over at a time.

This is not “Integer War”, but it’s a great way to review adding or subtracting integers as well as ordering integers. It’s called “What a Hand”. I got it out of my “The Middle School Mathematician” book. The actual hands out of the book have no numbers on them, I wrote those on each finger myself so these hands could be used for many different activities. You give each student a “hand”. They are to add or subtract (which ever you prefer) all of the numbers on the hand and then they are to all order themselves fron least to greatest in the front of the room. It’s a fairly quick activitiy that the kids always enjoy.

These were just a few ways that you could play “Integer War”. Use your imagination for other ways. The kids LOVE playing these games, and you can get them to do a ton of problems without a single complaint. It is truly amazing to listen to the kids argue over what the answers are.  Have fun!!

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