programs from prekindergarten through grade 12 should enable all students
Representation is central to the study of mathematics. Students can develop and deepen their understanding of mathematical concepts and relationships as they create, compare, and use various representations. Representationssuch as physical objects, drawings, charts, graphs, and symbolsalso help students communicate their thinking.
Representations are ubiquitous in the middle-grades mathematics curriculum proposed
here. The study of proportionality and linear relationships is intertwined
both with students' learning to use variables flexibly in order to represent
unknowns and with their learning to employ tables, graphs, and equations
as tools for representation and analysis. Middle-grades students who are
taught with this Standard in mind will learn to recognize, compare, and
use an array of representational forms for fractions, decimals, percents,
and integers. They also will learn to use representational forms such
as exponential and scientific notation when working with large and small
numbers and to use a variety of graphical tools to represent and analyze
Students in the middle grades solve many problems in which they create and use representations to organize and record their thinking about mathematical ideas. For example, they use representations to develop or apply their understanding of proportionality when they make or interpret scale drawings of figures or scale models of objects, when they connect the geometric notion of similarity with numerical ratios, and when they draw relative-frequency histograms for data sets. While solving challenging problems, students might use standard representations, but they can also develop and use nonstandard representations that work well for a particular problem.
When solving problems involving proportionality, students can create representations
that blend visual and numerical information to depict relationships among
quantities. Consider the following problem:
The Copy Cat printing shop has a printer that uses only black, red, and blue cartridges. All the cartridges print the same number of pages. The black cartridges are replaced 4 times as often as the red ones. And during the time in which 3 red cartridges need to be replaced, 5 blue cartridges will also need to be replaced. »
- What fraction of Copy Cat's printing is in black?
- What percent of the printing is in blue?
- In a month, 60 black cartridges are used. What is the total number of red and blue cartridges used in that month?
Students can use a variety of approaches to represent and solve this problem, including both standard and nonstandard forms of representation. Some students will find it natural to develop and use a discrete model like the one shown in figure 6.40. With such a representation, students can merge the information from the two ratios in the problem statement. They can see that in every set of 20 cartridges used at Copy Cat, 12 are black, 3 are red, and 5 are blue. They can conclude that 12/20 (or 6/10, 3/5, or 0.6) of the printing is in black, which answers the first question. To answer the second question, a student might imagine replicating this set of 20 five times to see that Copy Cat uses 25 blue cartridges in every 100 used. Thus, 25 percent of the printing is in blue. To answer the third question, students could note that a set of 60 black cartridges comprises 5 sets of 12 cartridges and that a total of 8 red and blue cartridges are used in the time that 12 black cartridges are used. Thus, it follows that 40 red and blue cartridges15 red and 25 blueare used in the time that it takes to use 60 black cartridges.
A group of students has $60 to spend on dinner. They know that the total cost, after adding tax and tip, will be 25 percent more than the food prices shown on the menu. How much can they spend on the food so that the total cost will be $60?
Of the various ways this problem might be represented, many students would find the representation in figure 6.41 useful. In this figure, a rectangular bar represents the total of $60. This total must include the price of the food plus 25 percent more for tax and tip. To show this relationship, the bar is segmented into five equal parts, of which four represent the price of the food and one the tax and tip. Because there are » five equal parts and the total is $60, each part must be $12. Therefore, the total price allowed for food is $48. This type of visual representation for numerical quantities is quite adaptable and can be used to solve many problems involving fractions, percents, ratios, and proportions (see, e.g., some problems in Curriculum Development Institute of Singapore 1997, or Bennett, Maier, and Nelson 198899). For example, the representation in figure 6.41 could also help students see and understand that when one quantity is 125 percent of a second quantity, then the second is 80 percent of the first.
Students also need to examine relationships among representations for linear
functions. The use of graphing calculators or appropriate computer software
can greatly facilitate such an examination and can allow students to see
such important relationships as the one between the value of k in
the equation y = kx and the slope of the corresponding
Students will be better able to solve a range of algebra problems if they can move easily from one type of representation to another. In the middle grades, students often begin with tables of numerical data to examine a pattern underlying a linear function, but they should also learn to represent those data in the form of a graph or equation when they wish to characterize the generalized linear relationship. Students should also become flexible in recognizing equivalent forms of linear equations and expressions. This flexibility can emerge as students gain experience with multiple ways of representing a contextualized problem. For example, consider the following problem, which is adapted from Ferrini-Mundy, Lappan, and Phillips (1997):
A rectangular pool is to be surrounded by a ceramic-tile border. The border will be one tile wide all around. Explain in words, with numbers or tables, visually, and with symbols the number of tiles that will be needed for pools of various lengths and widths.
Some students would solve this problem by using a table to record the values for various lengths and widths of rectangular pools and for the corresponding number of tiles on the border. From the table, they could » discern a generalization and then express it as an equation, as is suggested in the response and the accompanying work shown in figure 6.42.
I drew several pictures and saw this pattern. You need L + 2 tiles across the top and the same number across the bottom. And you also need W tiles on the left and W tiles on the right. So all together, the number of tiles needed is T = 2(L + 2) + 2W.
I pictured it in my head. First, place one tile at each of the corners of the pool. Then you just need L tiles across the top and the bottom, and W tiles along each of the sides. So all together, the number of tiles needed is 4 + 2L + 2W.
You can find the number of tiles needed by finding the area of the whole rectangle (pool plus tile border) and then subtracting off the area of the pool. The area of the pool and deck together is (L + 2)(W + 2). The area of the pool alone is LW. So all together, the number of tiles needed is (L + 2)(W + 2) LW.
These three responses differ in the way in which particular geometric
(visual) features are considered. For example, in the first two solutions,
the tile border is related to the perimeter of the large rectangle comprising
the tiles and the pool but with different decompositions of the perimeter.
In contrast, the third response considers the area of the large rectangle
and derives the number of tiles as the area of the border, which is legitimate
because the tiles are unit squares.
By working on problems like the "tiled pool" problem, students gain experience in relating symbolic representations of situations and relationships to other representations, such as tables and graphs. They also see that several apparently different symbolic expressions often can be used to represent the same relationship between quantities or variables in a situation. The latter observation sets the stage for students to understand equivalent » symbolic expressions as different symbolic forms that represent the same relationship. In the "tiled pool" problem, for example, a class could discuss why the four expressions obtained for the total number of tiles should be equivalent. They could then examine ways to demonstrate the equivalence symbolically. For example, they might observe from their sketches that adding two lengths to two widths (2L + 2W) is actually the same as adding the length and width and then doubling: 2(L + W). They should recognize this pictorial representation for the distributive property of multiplication over additiona useful tool in rewriting variable expressions and solving equations. In this way, teachers may be able to develop approaches to algebraic symbol manipulation that are meaningful to students.
Finally, it is important that middle-grades students have opportunities to use
their repertoire of mathematical representations to solve relatively large-scale,
motivating, and significant problems that involve modeling physical, social,
or mathematical phenomena. The goal of this sort of mathematical modeling
is for students to gain experience in using the mathematics they know
and an appreciation of its utility for understanding and solving applied
problems. For example, students might decide to investigate problems associated
with trash disposal and recycling by collecting data on the volume of
paper discarded in their classroom or home over a period of weeks or months.
After organizing their data using graphs, tables, or charts, the students
could think about which representations are most useful for illuminating
regularities in the data. From their observations, the students might
be able to offer thoughtfully justified estimates of the volume and types
of paper discarded in their entire school, school district, or city in
a week, month, or year. Drawing on what they have learned in science and
social studies, they might then make recommendations for reducing the
flow of paper into landfills or incinerators.
Mathematics teachers help students learn to use representations flexibly and appropriately by encouraging them as they create and use representations to support their thinking and communication. Teachers help students develop facility with representations by listening, questioning, and making a sincere effort to understand what they are trying to communicate with their drawings or writings, especially when idiosyncratic, unconventional representations are involved. Teachers need to use sound professional judgment when deciding when and how to help students move toward conventional representations. Although using conventional representational forms has many advantages, introducing representations before students are able to use them meaningfully can be counterproductive.
Teachers play a significant role in helping students develop meaning for important
forms of representation. For example, middle-grades students need many
experiences to develop a robust understanding of the very complex notion
of variable. Teachers can help students move from a limited understanding
of variable as a placeholder for a single number to the idea of variable
as a representation for a range of possible values by providing experiences
that use variable expressions to describe numerical data (Demana and Leitzel
Teachers need to give students experiences in using a wide range of visual representations and introduce them to new forms of representations » that are useful for solving certain types of problems. Vertex-edge graphs, for instance, can be used to represent abstract relationships among people or objects in many different kinds of situations. Take a situation in which several students might be working in different groups (for math review, history research, and a science project) or involved in different activities (basketball team and band). Each group wants to arrange a different meeting time to accommodate the students who are involved in more than one group. To help solve this scheduling problem, a teacher might suggest that students make a graph in which the vertices represent the groups and an edge between two groups indicates that there is some student who is a member of both groups. Figure 6.43 shows a possible vertex-edge graph involving the five groups, where an edge represents a relationshipcommon membership. This graph illustrates that no student participates in both the math-review group and history-research group (e.g., there is no edge joining those vertices) and that at least one student participates in both the math-review group and band practice (e.g., there is an edge between those vertices). The information in this graph can identify the potential scheduling conflicts so they can be avoided by scheduling all connected activities at different times.
Another new type of representation that teachers might wish to introduce their students to is a NOW-NEXT equation, which can be used to define relationships among variables iteratively. The equation NEXT = NOW + 10 would mean that each term in a pattern is found by adding 10 to the previous term. This notational form can be used as an alternative to the equation form of the general term when a recursive relationship is being highlighted. The data in figure 6.44 can be represented in a summarized form both as y = 10x (where x must be a whole number) and as NEXT = NOW + 10.
NEXT = NOW + 10
Fig. 6.44. Values
for the terms in a pattern represented by a
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