Activities
Activity One - Meeting Polyominoes
In preparation for the activity, I will pair my students with a partner (or two if I have an odd number of students) for collaborative work. I will be making a concerted effort to form partnerships based on mathematical skill level. I aim to pair high performing students with middle range students and struggling students with middle range students as well. I would like to avoid pairing struggling students with high performing students, as this combination may not be conducive to equitable roles during math talks. Once teams are made, I will reintroduce my students to the square tiles. They will be familiar with these tools from the beginning of the school year when they were available to them for free explorations. Each team will be given only 4 square tiles and some graph paper. I will then introduce the term tetromino (a shape made of 4 connected tiles) and explain how to make them according to conditions i) and ii) (see above). During this explanation, I will be modeling these conditions on the board. I will then pose this challenge: “connect your 4 tiles to find as many different shapes as you are able.” I will then require them to record each new shape that they find, by coloring boxes on the graph paper.
At this point, the students will begin collaborating on the task. I will be circulating throughout the room, observing the work. During this time, I plan on asking questions, guiding explorations, or encouraging/praising effort dependent upon the progress of my teams. I anticipate that these teams will be able to find some if not all of the tetromino shapes. I also expect my students to include multiple copies of similar shapes on their paper. I imagine that many will not recognize rotated or reflected shapes as “being the same” at this point.
At the close of the task, the class will gather on the carpet and prepare to share out and engage in math talk. I will begin by asking teams to share shapes that they found. As I record these shapes on the board, I will invite other teams to add to their papers if they are missing one. When a shape is mentioned that is congruent to one already listed, I will record it on the board but then open a notice and wonder session about the two congruent shapes. This will be a great time to introduce the terms rotation, translation, and reflection, as well as the term congruent. On the board I will show how different looking shapes can actually be similar. I’ll then invite the students to search their papers for similar shapes that have undergone a rotation or reflection. Students will continue to share out their ideas until we no longer have new shapes to add. If all unique tetrominoes are accounted for, there will be a total of 5. If class ends and we have not found all 5, I would revisit the activity the next day with the challenge of finding the missing shapes.
Activity Two - Counting
I will begin the activity by introducing the term “attribute”. The students will have likely heard the term before, as I use it in science class frequently. I will talk about how the number of straight sides of a shape is an attribute of that shape. I will then model the counting of sides of an assortment of shapes (I’ll draw some irregular polygons.) Students will also be given the opportunity to count the sides of shapes. When I’m confident that the kids understand the procedure, I will introduce the terms “vertex” and “vertices”. I will then model counting all the vertices of an irregular polygon. I will make certain that I include some polygons with interior vertices. Typically my kids can easily count the exterior “corners” but they infrequently encounter the interior vertices in our work with regular polygons. The students will have a chance to count vertices at this time also.
Each student will receive their graph paper containing shapes (from activity 1), 4 square tiles, and a new polyomino attribute-recording sheet. They will return to tables to collaborate with their partner as they count and record the numbers of straight sides and vertices for each tetromino. As students work, I will circulate from team to team asking them to count aloud. This will allow me to assess their understanding of the activity, the term vertices, and their proficiency with basic counting. A quick glance at their papers will also afford me the opportunity to assess their ability to write numerals. I would expect a fair amount of numeral reversals in their writing. Reversals are very typical of kindergarten students.
As the teams complete their job, they return to the carpet prepared to share their data. As students report their findings, I will fill in the data on the board. Should any disagreements arise we will of course discuss our ideas and come to an agreement as to the correct response. Teams that may have mistakes or incomplete portions of their paper will be encouraged to correct or complete any answers that require it.
“Notice and wonder” will provide the class with an opportunity to engage in math talk. The focusing question of this strategy is: “what do we notice about the number of straight sides and the number of vertices.” Does anyone notice anything or wonder anything about what our data shows? After examining all 5 or our tetrominoes, we should note that the number of sides and vertices are always the same. If the class is unable to make this connection on their own, I will guide them there. This fact is a take away I intend my students to leave the activity understanding. I will conclude the activity by collecting papers for safe keeping until activity 3.
Activity Three - Measuring
The activity begins with a review of the term “attribute”. I will give the students the opportunity to share attributes with which they are familiar. I anticipate them answering with attributes like color or size. I would also expect them to respond with number of straight sides and the number of vertices. If the class does not list these two as attributes I will be certain to get them included. I will then introduce the term “perimeter” as a new type of attribute. As a means of introduction, I will share a picture of a fenced-in yard. This image will help me define perimeter for the students. At this point I will explain the difference between counting sides and measuring perimeter. I will demonstrate the difference by finding both the perimeter and the number of sides for a polyomino made of 3 tiles (in which the amount of these attributes is different) on the board. After modeling the finding of perimeter, I will ask the class if they notice anything different between counting and measuring. I do not expect the students to find this difference without help. At their level, I would like the kids to understand that measuring is also a form of counting, but that it is the counting of units of the same size. I can point this out on the board. In the counting of “sides” activity, the lengths of the sides are not the same. Some sides may be long and other could be short. This is different than when measuring. After this distinction is made, I will introduce the term area and demonstrate how we measure it. In this unit, the measuring of area is simply counting the (squares) units used to create our polyominoes. When the students are working with tetrominoes, the area will always be 4, as tetrominoes are always made of 4 square tiles. I will explain that when we report the area of a polyomino, that we will state the area as a number of square units, since we are counting units that are square.
I will then return 4 square tiles and the graph paper of tetrominoes to the teams and have them begin measuring and recording the perimeters and areas of each shape. As the students work through this stage of the activity, I will circulate about asking students to find the perimeters for me. By observing this I can again check on understanding of perimeter as well as assess their skill at counting. I will follow up by asking them what the areas are of the different tetrominoes. If they reply with 4, I will remind them that they need to include the units, and that the proper answer is 4 square units.
As the class finishes the task, they will return to the carpet to report our measurements. I will record the class’ measurements on the board and encourage all teams to correct or complete any mistakes or incomplete sections of their paper.
Activity Four- Critical Thinking
A review of the tetrominoes and their attributes will start this activity. I will then relate the story of my three dogs (found above in Content.) Following this story, I will provide the teams with a recording sheet and the 4 square tiles (should they wish to use them) for the challenge. The challenge consists of the teams matching tetromino shaped yards to the needs of the specific dog.
The teams will compare the needs of the dogs to the attributes of the tetrominoes. Students will discuss the matching of tetrominoes to dogs and be prepared to justify these pairings by using data about the attributes of each shape. While these choices are being made, I will circulate about and listen in to the conversations. If teams are not considering the attributes of our shapes during their decisions, I will guide them in the direction to do so.
As a final step to the activity and an assessment of my students’ learning, I will have them state their choices and explain their thinking. There are many ways to have the students do this. For example, they could report to the teacher one team at a time. Another option would be to have the class record their thoughts with an I-pad. I will have my students use an App called Seesaw. With the Seesaw app, the students can record a video of themselves sharing their ideas about the final challenge. They will then upload this video to my computer and I will be able to view their responses at a time of my choosing.
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