OBJECTIVE: (3)This activity is designed for third or fourth graders as a tool to try to get them thinking about the concept of "canceling". The beginning of the visualization of how something can "cancel out" something else will develop as they design various routes between two points on a grid. By realizing that opposite moves (left and right or up and down) "cancel each other out", the students will be able to more easily relate to negative numbers when it is time to learn about them. This activity could also be used with fifth graders as an opening exercise and discussion at the beginning of introducing negative numbers. The negative number maze should be appropriate for all three grade levels and can be made either easier or more challenging based on the direction of the discussion that follows the actual activity.
(2) The prerequisites that need to be mastered before experiencing this
activity are the ability to recognize patterns and distinguish between them.
The students must also have the cognitive ability to think abstractly and
see the "canceling out" in the activity. Otherwise, since this
is used as introduction activity to a new concept, there are not too many
prerequisites.
MATERIALS NEEDED: (9)This activity calls for the use of tape to make the grids on the floor, red construction paper dots to make points A and B, graphing paper to record the routes taken by the students, colored pencils, a chalkboard and chalk to record various routes during class discussion, and white paper for students to jot down any patterns or ideas they have during the activity.
DESCRIPTION OF ACTIVITY: (6 and 11)
The teacher will prepare a large grid on the floor with masking tape (If a life size grid can not be used, a representation can be made onto graph paper and will work just as well). An example of the grid with two points(points A and points B) labeled can be seen here:
The children will be divided into many groups of two. (1) The teacher will show the students the grid and will ask them to work together in their pairs to find three different ways of getting from point A to point B, staying on the lines by using up, down, left, and right moves only. The students will also be told that there is no "right answer". There are an unlimited number of routes that can be taken between the two points, so any three that they choose are fine. Taking turns, one student should physically complete the route moves while the other student in the pair transcribes the sequence of moves. The moves should be recorded in colored pencil tracing the routes taken onto the graph paper and also should be written as a pattern using letters and numbers. (15) The teacher should be sure to use a pattern that does not really work and to tell the students that it does not work so that students will not be tempted to feed off of it. Here is an example of what a student's drawing and route pattern may look like:
1D 1R 1R 1U 1R 1R 1D 1D 1D 1R 1R 1U 1U 1R 1R 1D
1D 1D 1D 1D 1L
(10) Next, the teacher will explain to the students that there is a mathematical
purpose to this activity, but that they will have to discover for themselves
what they think that it is. They have several tools, including the movement
of their bodies, their graph paper and the blank white paper to help them
"think" about what they are doing. The teacher will ask the students
to be very observant during this activity and to take notice of any patterns
they see when they are coming up with their various routes. Also, the teacher
may want to include in the directions for the students to try to make one
of their three routes to be done in the least amount of moves possible (this
may help them to see the difference between the shortest route and a longer
one, and they will more easily notice that opposite moves will cancel each
other out).
After the instructions have been clearly explained by the teacher, the
students should have plenty of time to freely explore with their grids.
The teacher should walk around the room and monitor as the students come
up with their various routes from point A to point B. The teacher should
encourage all students to record their rates and to jot down patterns and
conclusions resulting from their experience during the activity.
Once sufficient time has been given for free exploration with the maze,
the students should be called back together by the teacher to discuss their
findings. Hopefully some of the students will notice that opposite moves
"cancel each other out", but to help move the discussion in this
direction, in the case that they do not notice this on their own, there
are ways that the teacher can guide this discussion.
CLASS DISCUSSION: (8) Following is an example of a possible class discussion model for the teacher to go by:
First, the teacher should ask to see some examples of the different routes taken, but not the shortest ones yet. Next, he/she should ask the students if they can see any similarities between all of the example routes that have been displayed by their classrooms. Are there certain patterns of moves that can be found in all the examples? The teacher should then have the students show their shortest route examples. If the students come up with different shortest routes, he/she should discuss with the class to determine what the true shortest route is (there may actually be several different directions that the shortest routes can take but they will have to have a certain number of moves). Here is an example of a shortest route:
1D 1R 1D 1R 1D 1R 1D 1R 1D 1R 1R 1R
After this, the teacher should have the students look at the shortest route compared to all of their longer examples. Do the students notice anything about the comparison? If not, the teacher should ask them if they can find the shortest route embedded in one of the longer examples. The kids will check it with all of their own long examples. They should be able to find the short route inside all of them. Once this is established, the teacher will ask them what they notice about what is left over in their route patterns once the short route moves have been crossed out. They can also be shown that for every left move left over there is also a right move and that for every up move left over there is also a down move. Finally, the teacher should try to encourage the students to put some thought into what they think that this means. (7)At this point the kids should begin to see that these moves "cancel each other out". Here is an example of what it may look like to find the shortest route in a long one and to notice that what is left over "cancels each other out".
CONTINUING THE ACTIVITY: (13)This activity can end
with an introductory discussion of negative numbers, or it can be continued
with another canceling generative activity. One such activity is the sticks
activity where toothpicks are used to represent positive and negative numbers
and addition and subtraction can be done with understanding. (12) The negative
number maze is a flexible activity that can be molded to fit whatever time
schedule that the teacher would like to allow. There should not be a problem
using the entire class period if enough time is allotted for exploration
and discussion. The maze discussion and extension could also last several
days if it is extended, stretched and thoroughly explored.
THE REAL TEST: I tried this activity out with a third grader in my student teaching classroom. Clay is a student who exhibits above grade level work in mathematics. Since I was not sure about the grade level that I recommended for this activity , and since I only have access to third graders right now, I decided that I would be best off experimenting with the highest level child that I could find.
Clay had a really hard time with the activity. I actually ended up having to hold his hand and walk him through it. To start out the activity, I had him come up with his paths. He had no trouble creating two original route paths and one shortest route path. Then I showed him how to record his paths as patterns. This he also did with ease. However, when I asked him if he noticed anything about the patterns that was unusual, or if he could see similarities in them, he did not see anything. I went ahead and had him find his shortest route path embedded in his longer paths and had him put dots by the corresponding moves. After he had done this, I had him circle what was left over in each of the long routes. I, then, asked him if he noticed anything about what was left over. He thought maybe that he would find that they both contained the same moves. but then ruled this out when he realized that if that were true, they would both be the same path. He then looked and thought very hard about it before he said, "I don't see anything." So, I pointed out to him that for every left move left over there is also a right move and for every up move left over there is also a down move. He could see this and find it in his other example once I pointed it out, but he saw no significance in the fact and therefore was not very inspired by this activity.
Because of my experience with Clay, I would say that I probably greatly
underestimated the grade level for this activity. It would most likely be
better for fifth or sixth grade students. Also, since the relevance of this
activity was very hard for a child who has no concept of negative numbers
to see, I decided that perhaps this activity would be more effective if
it were used after the introduction of negative numbers so that students
are more likely to see the connections and think that they are valuable.
Otherwise, this activity is great and very fun to do with students!
Clay's responses are below:
