Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
|Introduction | Defining Representation | Connecting Representations | The Teacher's Role | Your Journal|
Students benefit from a balanced instructional program in which they grapple with their own interpretation of the use of a particular representation, participate in discussions, and share ideas with their teacher and peers. This can lead to learning about common conventions of representations, and experimenting with uses of representations as suggested by others. In addition, as students compare and discuss connections between representations, they deepen their understanding of mathematical ideas.
As a follow-up to the Representing Data activity from Part A, the teacher asked a student, Cody, to gather data on the weight of pennies. She was hoping to bring up connections to previous science work with scales and to the use of decimals to report measures. Cody used the classroom scale and created a table as he weighed different numbers of pennies. He chose to use a coordinate graph to show his data. Because Cody had a large range in his data, the teacher guided him in his choice of scale and the labels for his graph. She also helped him understand that some variation in his numbers was to be expected when using real pennies and a real scale.
Here is Cody's data:
Later, while discussing Cody's table and the graph with the whole class, the teacher was able to extend the lesson to include algebraic representation. Several students were able to look at the line on the coordinate graph and, with the help of the teacher's questions, notice that the number of grams steadily increased. The class stated this observation in a verbal rule: "Every time you have 10 more pennies, you get about 25 more grams."
The teacher continued the discussion and asked about the general relationship between the data in the Number of Pennies row and the data for the corresponding number of grams. Several students said, "It's times two and a half!", and the class went on to record another rule: "The number of grams is (about) 2.5 times as much as the number of pennies." They then shortened their rule to "g = 2.5p" by using the letter g to stand for the number of grams and the letter p to stand for the number of pennies. This equation is an example of an algebraic symbolic expression, a linear equation showing constant growth.
The discussion also included acknowledgement that this formula doesn't perfectly predict the weight of a number of pennies. For example, 100 pennies only weighed 248 grams, not 250. Their teacher showed them the "is about equal to" symbol and wrote "g ≈ 2.5p" before returning to the central purpose of the lesson: developing connections between various forms of representation for relationships in sets of data.
Notice that the development of the equation was built on a series of translations among various representations. The situation was used to generate a table of values, related points were plotted on a coordinate grid, and a verbal rule was stated. The rule was then translated into a symbolic form.
Through such lessons, upper elementary students gain understanding that will support their success with more formal algebraic ideas and skillful use of representations during the middle grades.
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