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Tag Archives: hands on math
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…………
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
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.
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.
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!!
Integers are probably my most favorite thing to teach! There are soooooo many great tools, activities, and games to use when teaching integers. The following is the best way I’ve found to teach adding integers.
To begin, the students are grouped in pairs so that they can easily view what others are doing to make sure they are correct. I give each pair a handful of two-color counters.
I explain that the red side is negative and the yellow side is positive. I write a problem on the board, example: -6 + 4. I explain that we are going to make a model of this problem. I ask them how we would model -6 and then have them do it. I then say that we are going to add four positives and how would we do that. I explain what zero pairs are and show them that the cancel each other out and that we remove them from our model. I then ask them how many they have left over. We do several examples of this in all of the different ways you can add integers. Then I give them a colored sheet of paper (I always like to use colored paper when I want them to write down important notes, because I feel like the think it is more important if they get special paper for it). I take up the counters and tell them that now we are going to do the same thing only drawing the model instead of using the actual counters. A circle with a negative sign in it represents a negative and a circle with a positive sign in it is a positive. I give them examples and have them write down the problem and then draw a model of the problem and crossing out the zero pairs.
Of course by this time, several students will have seen the pattern and will be able to do it in their head. I always tell them not to tell the others our secret if they’ve figured it out. Then I give them a big number like -450 + 25. They start griping about having to write all of those circles. I ask them if they saw any pattern in the other examples that would help them do the problem without drawing all of those circles. I ask them to imaging drawing 450 negative circles and then twenty-five positive circles. I ask them how many they would be able to cross out. They usually get it by now and can easily answer. I then ask them what they would be left with, and they would answer 425 negatives. I give them a few problems that I ask them not to draw circles for. After each problem, we talk about the steps we take. For instance, -2 + -5, I would say “What would I have drawn for the -2?” and then “What would I have drawn for -5? Would I have any zero pairs to cancel? If I don’t have any zero pairs to cancel, what do you have?” If they were different signs, I would ask, “Which did you have more of, the positives or the negatives?” After making sure they understand the why and how comes of adding integers, I introduce the song. I always make a big deal of the Integer Song. I pick song leaders and let them make their own beats and what have you. If you look on the side bar, under links, you can find a couple of my classes that performed the song. The words to the song go like this:
Same signs add and keep.
Different signs subtract.
Take the sign of the higher number, then it’ll be exact.
One game we play for “Adding Integers.” :
1. Integer War: This game is very common for integers, and you can play it with all operations except for division. I put them in groups of two. Pass out a deck of cards for each group. Here are the values of the cards:
Red cards= negative numbers
Black cards = positive numbers
Ace = 1, Jack = 10, Queen = 11, King = 12
Each student needs a piece of paper. They are to draw a line down the center of the paper and place their name on one half, and their partner’s name on the other half. One person deals out all of the cards. Both students turn over two cards out of their pile. On the first line of their paper, they need to write down their problem and their partner’s problem in the appropriate places. Each person works both problems and they check their answers when finished. They each put a star next to the answer with the highest value. This also strenghthens their ability to compare integers.
I will discuss some different games in later posts.
This is the second year that I have taught finding the circumference of a circle using the lessons out of the “Hands On Math!” book. Refer to my “Resources” section to find this book. Here is a picture of the book though……
The “Hands On Math!” book is set up by objectives. Each objective contains three different activities. The first activity is very concrete, the second lesson is pictorial (they are usually drawing or coloring something), and the third activity is a cooperative learning game. The lesson I used for teaching circumference of a circle starts on page 355. Before I started these activities, I gave them a colored sheet of paper and we drew a circle and labeled the diameter, radius, center, and we wrote along the margin that the circumference is the distance around the circle.
For these activities, I had the students grouped in pairs. The first activity is called “All Wrapped Up.” The actual activity calls for assorted plastic lids, but I didn’t think about saving up lids (maybe I’ll start saving now for next year’s group. I’ll make a note). Instead of actual lids, I drew three different sized circles on a piece of paper and made copies for each student. While they are in pairs, I still wanted each student to actually do this exercise themselves but still look at their partner for “security” making sure they are doing the activity correctly. This helps because as there are twenty-something students in the class, there is only one teacher.
I also gave them cotton twine (not stretchy) long enough to at least go around the largest circle.
The students were asked to, as accurately as they could, put the string around the medium sized circle and then mark with their fingers where the end of the string meets the rest of the string after it wraps around once. Basically they are measuring the circumference of the circle with the string.
Then ask them to see how many times that marked off string will go across the center of the circle (the diameter).
Go around the room asking the students how many diameters they were able to get out of the marked off string. Hopefully they will get “three plus a little more.” After several of the students saying three plus a little more, then you can explain that this “three plus a little more” actually has a name in math. That name is pi. I draw the symbol on the board and tell them that the actual number is 3.14………
Second Activity: Around and Across
With this activity, I give each pair a copy of the worksheet in the book, adding machine tape, centimeter rulers, and a calculator.
The students are to wrap the adding machine tape around the circle (a little easier since it already wraps).
They should mark the adding machine tape with a pencil at the place where the end meets the rest of the tape. They then need to measure the marked off piece of the tape to see the measurement of the circumference of the circle to the nearest cm. You may have to explain how to measure with a ruler. They then place that measurement in the appropriate place in the table on the back of the worksheet. Then they need to measure the diameter with the ruler and record that in the table. Using the calculator, they need to type in the circumference divided by the diameter. They need to do that will all of the circles. After everyone has completed, go around the room asking for what there circum/diam was. Hopefully most of them will say three point something. I always emphasize the “three plus a little bit more”. I then ask them if that sounds familiar, and they always yell out pi! This is where I go into the discussion and I question them until they start realizing that the distance around the circle (the circumference) is the same as three plus a little bit more diameters. Drawing pictures on the white board is always beneficial in my classes. I then tell them that the actual formula for the circumference of a circle is C=pi * d (sorry, I don’t know how to type the pi symbol on here). We also talk about how it takes two radius to make a diameter, so we also may need C= 2 * pi * r.
Activity Three: Circlespin
This is a pretty cool “game”. Still in pairs, I give each group a copy of the spinners, a large paper clip, and they need a pencil.
This is not the original spinner that came out of the book. I used white out and changed it to fit our sixth grade PASS. First of all, we don’t use decimals with circumference and area, and they won’t have to find the diameter or radius given the circumference. Because of this, I changed the “circumference” on the spinner to “both” and changed the numbers to all be whole, even numbers. The students then flick the paper clip once for each spinner. Both students must find the circumference based on the information they are given by the spinner. For instance, if the paper clip landed on “radius” on the top spinner and “6″ on the bottom spinner, both students would find the circumference of a circle with a radius of six. They are to then check each others answers to see if they are the same. In sixth grade, PASS only asks them to find the circumference to pi and not multiply it out. Because of that, this game should not take very long at all. I usually ask them to do ten problems all together. Each pair’s paper should look identical when they turn them in.
I have different worksheets that I give them if I feel they need a little practice. I usually give them at least one homework assignment for finding the circumference.
Today we started learning about coordinate graphs. I told my first hour that graphing on a coordinate graph was so much fun, and a couple of students said, “Mrs. Kerr you say that about everything.” My reply was, “Have I ever lied to you?” The majority of the room replied, “NO!”
I gave them all a yellow piece of paper (bright yellow to be exact) and we went over all of the parts of a coordinate plane…..the x-axis, y-axis, origin, each of the four quadrants, discussed ordered pairs and how to graph them.
They were given two worksheets that I copied out of a book called “The Basic Not Boring Middle School Grade Math Book.” The worksheets were called “Creature Coordinates” and “A Startling Meeting”. These were excellent worksheets and I highly recommend that book for anyone who teaches middle school math. It is actually a series of books that covers many topics. Today was an excellent day in which I believe the vast majority of students enjoyed. If they didn’t enjoy, they are pretty good actors. Tomorrow we play “Coordinate Grid Hangman” and Friday we play “Battleship”. It’s going to be a fun week!!