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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.
Annnnnddd we’re back! What an exciting first couple of days back to school!! Today, we constructed the water cup origami for the first part of the class. The link to this is www.wipapercouncil.org/origami.htm I give them the instructions and then come around and pour water in their cup to see if they folded it correctly. It’s not hard origami, so it doesn’t take very long. After this exercise, we discussed Venn Diagrams. Venn Diagrams are a part of PASS for sixth grade math. The students were already in groups of two based on the seating for the day. I showed them an example using two of the kids and used funny items to compare and contrast them. I gave the students each a colored piece of paper had them each come up with five items (for a total of 10) and then place it on the Venn Diagram in the appropriate place. Most of the students knew Venn Diagrams already, so they were able to do it with ease. A couple of groups had a tougher time, but with further explanation I was able to get them on track. The best part was when there was an odd number of students, I had them make a triple Venn Diagram. After everyone was finished with the project, I asked different groups to go on the board and reproduce their diagrams. It was a great way for students to learn about each other as well as learn their names. I know this was probably one of the best “back to school” projects I’ve done in order to remember their names. Overall, a very successful day! So glad to be back, and I’m excited to get on with next week…….three dimensional shapes!!!
Have you ever had one of those teaching days when you’re like, “Wow! This is really working!”? Well, I did today. For years, I have always wanted to set up stations in my classroom to allow students to move around the room, work in groups, and do different things throughout the hour. I was lucky enough to find a book on eNasco.com that is built around “stations” for math. Because of my class sizes, I set up two sets of four stations, a set of “A” stations and a set of “B” stations. The stations were four groups of desks, a manilla file folder with the station number, and whatever materials were needed for that station. The following is a list of the stations and their required materials:
Station 1: Worksheet, one dice, a set of 5 index cards with the following expressions written on them:
3n + 4, 2m^2, 15 – x, 60 / p, 2x – 2
Station 2: Worksheet
Station 3: Worksheet
Station 4: Worksheet, set of 5 index cards with the following expressions written on them:
2x + 2, 5 + x, 36 / x, 16 – x, 2x^2
set of 5 index cards with the following values of x written on them:
x = 2, x = 3, x = 4, x = 5, x = 7
Give the students ten minutes at each station. The worksheets explain how to conduct each station, but I still had to help a few of the groups. The following pictures are of the worksheets for each station.