Measuring with Manipulatives and Addressing Misconceptions
Using manipulatives is a critical math strategy in the primary grades. There are many types of manipulatives that can be used for different mathematical concepts.
“Manipulative materials, such as geoboards, pattern blocks, chip trading boards, counters, algebra tiles, attribute pieces, fraction bars, and Cuisenaire rods, have been employed to teach children and adolescents a variety of math concepts. These manipulative materials have been used to teach students counting, place value, word problem solving, basic computation, numeration, and equation solving skills.”18.
Different manipulatives serve different purposes. This unit will focus on using square tiles, rulers, and eventually pre-made plastic polyominoes. We would start off by using a non-standard unit such as a square tile (with sides that are one inch long). My students would eventually move on to using a ruler. Towards the end of the unit when they are familiar with the different pentomino shapes, they would get to explore using the plastic pre-made versions of them. They will use these instead of constructing them on their own out of square tiles.
Square Tiles
My students will begin this curricular unit measuring with square tiles. They will be measuring area while exploring pentominoes. And, they will be measuring length with the one inch edges of the squares when determining perimeter. These concepts will be new for my students and I will need to give explicit directions when explaining the different activities. I would not want my students to get confused over area and perimeter because I was not clear in my directions. There are some important factors to consider when choosing which type of square tiles to use. I decided to use square tiles that have an area of one square inch. This means that each side is one inch long. While working in groups, my students will be assigned a specific color to work with. Eventually, when they move from measuring to constructing pentominoes they will need tiles of two different colors. In order for my students to see the polyominoes and encompassing rectangles they will need to construct the polyominoes in one color and fill in the remaining area with a different color. This will help them to visualize the pentomino shape and the rectangle that encloses it.
Misconceptions while using Square Tiles
When working with square tiles (or any other manipulative) for measurement I will have to address what gaps and edges are. When students are measuring both length using square tiles, they must make sure there are no gaps between the tiles. The edges are the sides of the square, one side must directly line up with and touch the side of the next square. It is important for the tiles to be lined up edge to edge is so that the students are not including any extra spaces. If there are gaps between the tiles, they will be measuring the space between the tiles as well as the length of the object. Students must ensure there are no gaps so that they can be sure they are only measuring the actual length of the object.
By placing the tiles directly next to each other, edge to edge, they are showing that they understand that measuring length is really measuring distance. They are measuring the distance from the beginning of an object to the end of an object. When they count the tiles to see how long the object is they are showing that they understand that counting numbers are related to measurement as they are counting how far away the end point is from the start point.
Rulers
I know that if I want my students to be successful in this unit, they will need to learn how to use a ruler properly. Unfortunately, when many students first hold a ruler, they automatically want to use it as a sword. Using a standardized unit of measure is critically important when measuring but is not expected at the first grade level. Some specific criteria my students will need to learn in order to use a ruler is: that they lie flat, they must be flush with the edge of the object being measured, that they count on by ones to twelve, and that they must make sure that the end of the object is lined up with the zero on the ruler, since the numbers on the ruler are telling the distance from zero. Since my students would be unfamiliar with rulers, I feel sure that they would be excited to use them. Due to their excitement I will have to be very careful and clear when explaining how to use a ruler as a tool for counting and measurement. I will start by having them line up the one inch squares against the ruler, starting at the zero on the ruler. My students and I will observe that the number at the end of the line just tells the number of squares that have been placed.
Misconceptions while using a Ruler
This unit will be my students first exposure to using a ruler. They will need to get explicit instructions on how to use a ruler properly. There are certain misconceptions that will need to be addressed. Students will need to learn that rulers must lie flat, with the numbers facing the person who is doing the measuring. They must also be made aware that it must be lined up with the object being measured. These concepts must be taught in order for students to use rulers appropriately.
Sometimes the zero appears at the edge of the ruler, and other times the zero is embedded a little in from the edge. Students must be able to identify where the zero is on their rulers. They will need to learn that regardless of where the zero is, this is where they must start measuring from. Some students may wish to start at one instead of zero. It will be important for them to realize that since they are measuring length (which is distance) that they are determining how far away from zero the object extends. It is critical for them to understand that a number on a ruler is telling them how many inches it is away from zero. It is describing distance in multiples of one inch. If they begin at one instead they are measuring the distance from one and therefore will not calculate the proper number of units. Ensuring that students know where to start measuring from is critical to their success in this unit and as they continue to use measurement in their lives. For this reason I will tell my students that when we measure with a ruler, we start at the zero on the ruler. The activity where we line up the squares against the ruler will help them with this.
Finally, while measurement is clearly correlated to counting, students may become familiar with measuring with a ruler and not applying counting strategies at all. This would demonstrate that they have truly mastered how to effectively use a ruler as a measurement tool. At first students may wish to count all the inches between zero and the far end of the object since they are familiar with one-to-one correspondence when counting. As students become more familiar with using a ruler, they will not need to count all the inch marks. However, as they become more accustomed to using a ruler they will be able to identify the length as the number that it is lined up with. When they are doing this, they are determining the distance from zero to the edge of the object. Therefore, they are finding the length in terms of how far away one edge is from the other in inches.
Plastic Pentominoes
My curriculum unit will require my students to use pentominoes when planning out walkways and a plot for our garden. I realized that when they are planning these different elements it might be challenging to continuously recreate them using square tiles. Pre-made plastic pentominoes already exist as a manipulative. I was able to order all twelve pentominoes in six different colors for a total of 72 pentominoes. The different colors make it easy to see where one pentomino ends and the next begins. I knew that this manipulative would help my students visualize aspects of our garden.
Since the pentominoes are pre-made, students would not need to worry about accidentally touching one tile and changing the whole design, an issue they might experience when using seperate square tiles. By using these manipulatives that are already in the pentomino shapes students will be able to move, flip, and rotate the pentominoes to see the different ways that they fit together. These manipulatives will be a great asset for them to see how the pentominoes can be used to create rectangles with different areas, but are still composed of squares, further emphasizing the array structure.
By using this tool and exploring the concepts of area and perimeter, my students will be able to relate counting to geometry and arithmetic. Even though the pentominoes are already in their twelve different designs, you can clearly see that they are still composed of squares. I will check that my students can also see this. Due to this, students will be able to count all the edges to determine the perimeter and count all the squares inside the shape to determine area. These will be a great resource to help further the understanding of these above grade level concepts. As these manipulatives have not been used before they will breathe life into the curriculum, and engage my students in a new way.
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