Background
I think of math as a bunch of blocks laying around in a big messy pile. Each block represents a topic that helps mathematical understanding begin to take shape, as students become proficient with a topic, a block gets added to the structure. If the base of this structure is solid, the blocks that are added on top are stable. If gaps or holes emerge early in a student’s learning, the entire structure becomes wobbly and unpredictable.
Figure A
If I focus my instruction solely on procedures and formulas, my students’ understanding can get wobbly, quickly. This is why I allow students to investigate, manipulate, and contextualize math. I believe that by providing students with rich and varied representations of mathematical concepts, I am helping students make sense of the math problems. I need to connect the mathematical concept to my students and give them a way to understand the process for determining perimeter and area. My students often confuse the concepts of perimeter and area because the two topics are almost always presented together as a set of procedures and they misinterpret the measurement ideas. In this unit, my students will develop an understanding of perimeter as they construct pens for pets, create frames for their art work, and determine how much fencing we need to enclose our recess yard to prevent the ball from rolling into the street. I will highlight the concept of length unit to measure the length of a path that fits around a shape. Additionally, students will develop an understanding of area as they tile rectangles, create posters to cover a variety of classroom surfaces, and move to measuring the base and height of rectilinear shapes.
Foundations
Students begin in kindergarten to count to tell the number of objects. They are developing one-to –one correspondence, as they point to an object and they state a whole number, the stated number thus represents the number of objects present. They move on to master the skill of counting forward beginning from a number within a known sequence instead of having to always begin at 1. As first graders, they move to express the length of an object as a whole number of length units by laying multiple copies of a length unit end to end with no gaps. This foundational work occurring in kindergarten and first grade helps to set the stage for measuring the side length of an object by using non-standard units and then standard units, using tools such as rulers, as second graders. When these students arrive in my third grade classroom, they should have experience with using tools such as rulers to measure the length of an object. They can compare the lengths of two objects using both non-standard units such as paperclips, or standard units such as inches. One object that has more paperclips or inches is longer than the other object. I will build on these understandings or provide activities and re-teaching lessons to sure up these essential base building blocks. If my students have gaps in their learning with these topics, it will lead to larger gaps, misconceptions, and frustration.
Perimeter
Perimeter is made up of the Greek prefix “peri” which means “around”, “about’, ‘enclosing, “surrounding’ and the Geek root “meter” with the meaning “measure”, thus making up the definition, to measure around. Therefore, this leads us to the understanding that perimeter is a measure of the length of a side or path that surrounds or outlines a two dimensional shape. To find the perimeter of a figure, one measures the lengths of all the sides, and adds them up. My students are going to explore this topic using a variety of manipulatives and different polygons. As they participate in the investigation, they will be using problem solving steps to practice Mathematical Practice 1; make sense of the problem and persevere while solving it. As students work through these steps, they are connecting to the problem, making and testing conjectures, and determining what works or doesn’t work and why. This process is what is helping them make connections to their previous learning and experiences and is helping to solidify understanding.
Our initial investigations will involve different manipulatives that will enable students to measure side lengths of a variety of items found around our classroom. Students will choose an item and a manipulative and work to measure the item by placing these non-standard units end to end and then counting the manipulatives used. This exploration time will help me to assess students’ ability to understand the process of measuring end to end without any gaps, it will give students time to freely investigate and begin to form theories about their findings. After a set amount of time, about 10 minutes, students will participate in a class discussion that is student led. Prior to this unit, students in my classroom have participated in many math discussions, which I will describe more thoroughly later in this unit. Another activity that will help to introduce the concept of perimeter to my students uses dot grid paper. Students will be asked to draw a horizontal line segment connecting two dots. This line segment represents 1 unit of length. The students will draw a horizontal line segment 4 units long. Next they will draw a vertical line segment 3 units long. Students will participate in math discussions to explain how many line segments they have drawn. Students will be provided with an addition chart and will utilize this tool to add the line segments. Then students will be instructed to draw to more line segments to form a rectangle. Again a math discussion will take place as students determine the total length of line segments used to draw the rectangle. Some students will add each length as they make their way around the rectangle, 4+3+4+3=14 units. Others may group the horizontal line segments together then the vertical line segments, 4+4+3+3=14 units. Some students may add the horizontal line segment to the vertical line segment and then multiply by two, 4+3=7 7x2=14. Or, students may double the horizontal line segments and then the vertical line segments and add the two products together, 4x2+3x2=14. All of these possibilities will help to solidify that students are calculating the sum of the line segments.
I will also have my class work with polyominoes. A polyomino is a collection of unit tiles, put together using some simple rules:
- Any two tiles that touch either touch only at a corner, or along a whole edge of each; and
- The figure is connected, in the sense that you can go from any tile to any other tile by passing through edges (not corners).
A domino has 2 tiles in the collection, a triomino has 3 tiles in the collection, a tetromino has 4 tiles in the collection, and a pentomino has 5 tiles in the collection (that is as far as we will go in this unit but the possible number of tiles is limitless). The purpose of using these “ominoes” allows us to create figures of a given area, but possibly different perimeter. Students can create polyominoes with different shapes. I will model the activity using four tiles to create tetrominoes, with students following along at their work stations. This will give me an opportunity to discuss some geometric terms like symmetry and translations, yet this will not be a focus of the lesson. Shapes that can be made to coincide using reflections, translations and rotations are considered the same. In Figure B, you will see the five different possible tetrominoes.
Figure B: Tetrominoes
Students will investigate and calculate the perimeter and the area of each of these figures and discuss similarities and differences they notice. Students then will continue the investigation with 5 tiles, pentominoes. After the investigations with the ominoes, students will move on to measuring items with rulers to determine the side lengths and then perimeter. As we make this switch, it is important to assess the students’ abilities with these tools and provide re-teaching activities to students who need it. One common error is to measure starting at the 1 on the ruler and not at the 0.
Area
Area is the amount of space inside the boundary of a flat, two dimensional object, such as a rectangle or a polyomino. For most of the figures in this unit, this can be calculated by tiling a shape with unit squares; the area is the number of unit squares enclosed by the figure. The tiles should not overlap, and should completely fill up the figure, leaving no uncovered parts. I will present my students with larger polyominoes of the same area but different shapes and different perimeters, so that they can see that very different looking shapes can have the same area. I will also have them find perimeters and see that they are different. This will help establish that area is distinct from shape, and also from perimeter.
After some work with irregular shapes, I will have my class study rectangles with small whole number side lengths. We will see that they can be paved with regular arrays of squares. I will have students find the side lengths of the rectangles, and we will establish that the area is the product of the side lengths. One of my major intentions is to connect much of the work of third grade to multiplication and develop their conceptual understanding and fluency. Area of rectangles and squares can be helpful tool toward this goal.
Jo Boaler encourages teachers to use an inquiry version of common area tasks, “Instead of asking students to find the area of a 12 by 4 rectangle, ask them how many rectangles they can find with an area of 24.”2 Doing this makes students think about spatial dimensions and relationships and then begin to think about what happens when one dimension changes. Students are making connections, conjectures, and are engaged and excited about their thoughts. I try to use this approach as much as I can.
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