Paying Attention to Attributes

Grade Level: Sixth or Seventh Grade

Areas of math engaged: Classification, Multiplication

Central ideas: Ability to pay attention to specific attributes regarding shape, size, and shade, and the ability to use this information in an effort to understand the process of multiplication.

Length: Entire class period of at least 45 minutes or more (12).

Materials needed: a) A class set of the following 24 attribute pieces: Circles, Triangles, and Squares (9). (Note: These pieces can be made with construction paper and lamented in order to keep them in relatively good condition.)

b) The instructor will need an overhead set of the 24 attribute pieces in order to serve as a model for the class.

Prior Knowledge: Prior to this activity, students must be able to recognize the difference between shapes and shades. They must have also had practice justifying answers through logical reasoning and explanation (2). In an effort to recall some of this knowledge, the following could be used as a possible lead in to the activity:

First, the instructor should hand out the attribute pieces to the students, allowing them to tell what it is they see in front of them. The answers to be looking for should include "shape," "size," and "shade." At this point, I would also suggest that the students get together in groups of approximately three to four students in an effort to allow students to brainstorm ideas. I would then hand them the following activity, carefully reading through the directions with the class as a whole (6).

Directions: A set of four attribute pieces is given. In each set, one of the pieces is different from the others. For each problem, decide which piece does not belong in the given set. Justify your answer.

* A possible idea to mention is to tell the students that there could be more than one solution (1). However, it may be something that the teacher wishes for the students to discover on their own. Because this activity is a "warm-up" to the activity that will follow later, it is important to note that the space is not too large, yet it will allow certain central ideas to emerge (5).

As the students begin working on their activity, the teacher should be circulating the room from group to group in order to check for the student's understanding. For each of these questions, the students should be placing the shapes in front of themselves in order to provide them with a concrete example of what is being presented (10). Questions regarding the activity should be asked in a manner that will require the students to justify their answers through reasoning. For example, in number one of the activity, the square does not belong to the set because all of the others are circles. Its shape is different from the set because it has four sides. Also, the last circle is not a part of the set because it is not shaded whereas all of the other shapes are shaded (7). Note, too, that the various solutions will challenge the students to think about exactly what they are looking for, and why they came to the conclusions that they have (3).

In number two, the square does not belong in the set because all of the other shapes are circles. The square has four sides, whereas the circle is round. Second, the small circle does not belong because its size is different from the other shapes. All of the shapes are big except for the small circle. Third, the large circle with stripes does not belong in the set because all of the other shapes are lacking a shade and this particular attribute piece is shaded.

For problem number three, some possible responses might be 1)the big square with no shade does not belong because all of the other shapes have a shade, and this one does not, or 2)the small circle does not belong in the set because its shape is small while all of the other shapes are big.

At this point, the teacher should regain the attention of the groups and discuss as a whole what discoveries were made. It is important to listen for the groups understanding of why they choose the answers that they do, and to be able to explain these answers to the rest of the class. Check for students' understanding of the meaning of the words size, shade, and shape. (This will usually be covered in their explanations). After discussing the various answers, then go onto the next activity.

Generative Activity: Students should still be in their respective groups. Next, hand out the activity and read it to the class as a whole before they begin discussing it within their groups (6).

Directions; The picture below is an example of an attribute train. The first car of the train, the striped circle, was "given" to the person who built the train. There is one attribute difference from the first car to the second car making it a "one-difference" train.

Note: You can only use a given piece in a train once. For example, once you put the large, striped circle in the train, you cannot pull it out and use it in the same train a second time.

Begin a "one-difference" attribute train with any piece you want. Keeping that first car the same for all trains, how many different three car, "one-difference" attribute trains can you build? Find all possibilities and justify your conjecture by convincing someone that no train has been missed, nor has any train been duplicated (11).

In order to answer this problem, the student must first choose one attribute piece to act as the first shape. After making this discovery, the student must then discover what piece will be chosen as the second attribute piece. There are only six attribute pieces that will work in the sequence that will maintain the "one-difference" train (4). If the students are having difficulty, the teacher could provide this information in an effort to help the students along (15). After this discovery is made, the students must then figure out all the available choices for the third attribute piece without duplicating any pieces (4). The following pattern is what the students should have discovered.

Note that in the first column, there is one possible piece that can fit in that place of the set. In the second column there are six possible pieces that are available for that place in the set, and there are five possibilities per "one-attribute" train in the third column (4). After going over the possibilities, the students should come together to discuss the results as a whole. The big theme to focus on as a class whole is the patterning sequence in which the attribute pieces were placed. It should be noted that the sequence starts out with one piece, then six, and then the choice of five attribute pieces. The total number of possibilities in the third column are equal to thirty. When set up in the form of a multiplication problem, one can see that after finding the first, second, and third number of possibilities for "one-attribute" difference trains, the final number of attribute pieces in the third column can be obtained by multiplying column one, by column two, and the possibilities of choices in column three. The concrete pieces, along with this explanation, will give the student with an answer of thirty, and a greater understanding of how the answer is obtained (8). After the discussion is over, the students may continue on with other activities along the same lines in an effort to build on the same concepts (13).

*Is it possible to build a one-difference attribute train which uses ALL the attribute pieces in the set? (You do not need to start with the train pictured in the directions above). If so, draw the train. If not, explain why it cannot be done, and justify your answer.

*Is it possible to build a three-difference attribute train which uses ALL the attribute pieces in the set? If so, draw the train. If not, explain why and justify your answer.