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Cramer, K., (2001) Using Models to Build Middle-Grade Students' Understanding of Functions. Mathematics Teaching in the Middle School. 6 (5), .

 

 

From the time that students enter kindergarten and throughout their early elementary school years, they should have multiple experiences exploring patterns. The study of patterns for middle school students should shift to the study of functions (NCTM 1989). The question that this article addresses is bow to plan and organize instruction for middle-grades students to help them develop an understanding of function.

Middle school students' experiences with function should be based on problems that incorporate manipulative materials and involve students in collecting data. The problems should lead students to build connections between the concrete model and the numerical patterns or functional relationships observed in the data. Students should work through multiple problems that build on one another and enable students to abstract significant mathematical ideas from the work.

The teacher's role in this type of instruction is first to identify good problems and organize them in a sequence that builds on previous problems. Teachers need to lead students in discussion, asking questions to help clarify the mathematics and draw connections among different problems. The problems should be rewarding to do, but students must move beyond the fun with the hands-on aspects of the activities and see the mathematics. The role for the middle school teacher is essential: to help make the mathematics explicit.

The three problems presented here, taken together, foster an understanding of the characteristics of linear, quadratic, and exponential functions. These problems ask students to examine patterns in tables, find function rules, look at graphs, and find similarities and differences within and across the three types of functions. The problems presented in this article were adapted from materials used in two teacher-enhancement projects to help primary and middle school teachers develop a deeper understanding of functions. The problems were adapted by teachers for use with middle school students. Although this article addresses activities for middle school students, functions can be made accessible to students on a number of levels. Regardless of the grade level of the students, the mathematics should be embedded in problems that involve concrete models, and students should use informal language to describe patterns and functional relationships before using symbolic notation. Students should reflect on the similarities and differences among these problems.

Mrs. Lin's classes in a Minneapolis middle school recently worked on the problems described in this article. The examples from students' work are included.

Linear Function Problem

The problem in Figure 1 is an example of a manipulative-based exercise that models a linear function. Figure 1 also shows the format for all problems.

 
 

Number Patterns from Cutting String

Fold a piece of string in half. While it is folded, make 1 cut. How many pieces of string do you have? Continue with another piece of string folded in half, making 2, 3, 4, and 5 cuts. Complete the table below.

# of cuts
0
1
2
3
4
5
# of pieces
 
  
 
 
 
 

Questions

1. Describe patterns that you observe in the table.

 

 

2. Without cutting the string, use the pattern from the table to determine the number of pieces for 6 cuts, 7 cuts, and then 8 cuts. Describe how you use patterns in the table to do this. [Find more than one way to extend the table.]

 

 

3. It is possible to predict the number of pieces given the number of cuts? Describe in words how to determine the number of pieces for 20 cuts.

 

 

4. What if you had 21 pieces, how many cuts did you make? Describe how you solved this problem.

 

 

 

5. Extension: Fold the string like this:

Predict how many pieces you would have if you made 1 cut, 2 cuts, 3 cuts. Verify your predictions by actually cutting string. Build a table of data, and record the patterns that you see in the table. Find the pattern that will predict the number of pieces, if you know the number of cuts.

Adapted from Sobel and Maletsky (1975)

Fig. 1 Problem modeling a linear function
 

Students are presented with a task, and they collect data using manipulative materials. Guiding questions ask students to describe multiple patterns using informal language, then to consider the functional relationships represented by the data. Figure 2 shows a sample of William's work on this problem.

Most of the middle school students in this class described similar patterns. Notice that William describes patterns going across the table ("bottom numbers are all odd and go up by 2"), as well as the function rule ("if you do the top x 2 + 1 it equals the bottom number"). Other examples of students' language to describe patterns for this string activity follow. Note that the quotations show exact student wording and spelling.

• "Theirs always 2 pieces more every time we cut a piece of string."
• "Each time you cut one more it goes up by two. Times the number of cuts by 2 plus one. The number of cuts plus the next number of cuts equals pieces for the original."
• "It skips by 2's @ the bottom, and not @ the top."
• "The number of pieces =the number of cuts x 2 + 1 more."
• "#of pieces = # of cuts + (# of cuts + 1)."
• "The difference between # of cuts and pieces keeps going up by 1."

Fig. 2 Student work for the Cutting String Problem
 

After students complete the problem in cooperative groups, they should have opportunities to share their work with the class. The problem is open ended to allow all students to participate at some level. One method of sharing is to record on a large chart the patterns that students find. During this whole-group discussion, students should see similarities and differences among the patterns that each group describes. The teacher can ask students to use their patterns to predict what the data set would look like if it continued to follow the patterns. A number of ways to make this prediction will come out of the discussion. The teacher can ask students to explain the connection between the patterns they described and the concrete model. The classes that worked on this problem made the following connections during whole-group sharing times:

• "The number of pieces is always odd because of the extra piece at the end."

• "The number of pieces increases by two each time - 1 cut makes 2 pieces and you are cutting it one more time."

• "The number of pieces is # of cuts x 2 + I since you fold it the piece where it's folded is one piece. The string is folded when you cut it so it's two times."

The teacher should focus students' attention on two important patterns. First, the teacher should highlight the pattern going across the table, that is, as the number of cuts increases by one, the number of pieces increases by two. When students examine other linear relationships, they should see that an increase by some constant amount is a characteristic of all linear functions. The other important pattern to highlight is the symbolic function rule. This rule generalizes the number of cuts to all possible cases. Teachers can build on the students' language and help them translate their words into algebraic symbols. Any of the students' descriptions of this function rule that are noted above can easily be translated to P = 2C + 1, where P = number of pieces and C - number of cuts.

Once the symbolic function rule has been established, teachers can ask students to identify which of the many patterns that they discovered can be used to extend the table to 100 cuts. Students will conclude that the function rule, P = 2C + 1, is the most efficient pattern. Asking how many cuts would be needed to yield 309 pieces focuses students' attention on the related function rule for the data set, that is, C - (P - 1) /2. Students usually divide first, but when they think about the problem concretely, they can see that to work backward one must subtract the extra piece first, then divide by 2.

     

Number Patterns from Trains of Equilateral Triangles

Using green pattern blocks, form the first three triangles shown below. Notice that the triangles formed from the equilateral green triangles are also equilateral.

Questions

1. Construct the fourth triangle in the series. If the unit area is 1 green triangle, what is the area of each of the four triangles built? Record the resulting areas in the table below:

Triangle number
1
2
3
4
5
Area in green triangle
 
 
 
 
 

2. Describe patterns that you observe in the table.

 

 

3. Predict the area of triangle 5. Verify by building it.

 

 

4. It is possible to predict the area of the triangle, given the triangle number. Describe in words how you could determine the area of the 20th triangle in the series.

 

 

5. Translate the rule into an algebraic equation for the nth triangle in the series.

 

 

6. The area of an equilateral triangle is 441 green triangles. To what triangle number would this area be matched? Explain how you determined the answer.

 

 

Adapted from Roper (1988)

Fig. 3 Problem modeling a quadratic function
 
 

Quadratic Function Problem

The problem in Figure 3 generates a Quadratic relationship, A= T2, where A = area and T = triangle number. Figure 4 shows Anya's response to the problem. Other students gave similar responses when asked to describe patterns going across the table. Examples include the following:

• "The numbers in between the area in the green triangles keep adding 2."
• "the numbers go odd, even, odd."
• "Goes in odd numbers - l, 3, 5, 7, 9 [in terms of added triangles]. The amount of increase is an odd number."
• "Growing faster than other tables."
• "Triangles grew: 1   1 + 3    1 + 3 + 5."
• "Odd numbers and the next odd number 1 +3 + 5 + 7 + 9 etc."

Students noted important patterns. They explained that the areas of the triangles grew at a faster rate than the numbers of pieces did in the previous string problem. Growth was connected with the concrete model; each new layer for the next larger triangle was the next odd number. For example, because triangle 2 has a bottom row of three green triangles, triangle 3 will have a bottom row of five green triangles, which is two more than triangle 2. Triangle 4 will have a bottom row of seven, which is two more than five. This pattern continues as each larger triangle is built. The area did not grow at a constant rate, as the number of pieces did in the string problem, but students noted that the area grew by consecutive odd numbers.

Fig. 4 Sample of student work on the Triangle Problem
 

In a whole-group discussion, the teacher can build on students' patterns to clarify the differences between patterns in tables for linear functions compared with those for quadratic functions. For linear functions, the dependent variable grows by a constant amount. Quadratic functions grow at a faster rate. In this example, the dependent variable grew by the pattern 1, 3, 5, 7, 9; a constant difference is observed in this growth. This second order constant difference is a characteristic of quadratic functions.

Students should also observe that the area values are square numbers. Students described this function rule in a variety of ways:

•  "Area - number2"
•  "The area is the triangle number square."
•  "The top number multiply by itself to get area in green triangles."
•  "1x1=1; 2x 2=4; 3x3=9; 4x4=16; 5x5=25   the top # times itself = the bottom #"

 

This quadratic rule is easy for students to discover, and the connection with recording the rule symbolically comes quickly. Students should be asked to compare the rules from the string problem, noting that the rule for the triangle problem involves an exponent.

At this point, students should graph the functions from both problems. Spreadsheets and graphing calculators can facilitate the graphing component of the lessons. In their mathematics journals, students can describe what the function rule for each problem looked like when graphed on a coordinate graph. Students can then summarize the differences among the graphs, patterns found in the tables, and rules for each problem.

Exponential Function Problem

The paper-folding problem in Figure 5 is an example of an exponential function. As students solve this problem, they notice that the patterns in this table differ from patterns examined in each of the previous problems. They observe how quickly the number of regions grows. Students may also recognize a doubling pattern in the table and wonder why no constant differences can be found as they look down the data in the column labeled "Number of regions."

 

Paper-Folding Patterns

Take a piece of paper, and fold it in half as many times as you can. After 1 fold, there will be 2 regions.
How many regions will occur after 3 folds? Four folds? How many folds are possible? Complete the table below. Imagine that there is no limit to the number of folds possible.

# of folds
# of regions
0
 
1
    
2
  
3
 
4
 
5
 

 

Questions

1. Describe patterns found in the table.

 

 

2. Describe how the number of regions is related to the number of folds. Translate this relationship into an algebraic rule. Use this rule to determine the number of regions given 18 folds.

 

 

3. Add another heading to the table: Area of the smallest region. Complete the table under this heading.

 

 

4. Describe patterns observed in the table. How is the area related to the number of folds? Describe the relationship algebraically.

 

 

5. Graph both relationships. How are they alike? Different?

 

 

Adapted from Phillips (1991)

Fig. 5 Problem modeling an exponential function
 

Students will likely comment that the number of folds is multiplied by 2 each time. Students from Mrs. Lin's class described this pattern in a variety of ways:

• "# of regions doubles every time."
• "Each region double it up by 2 each time."
• "Except for 0, # of folds x 2 = # of regions."
• "It doubles every time the number is folded in half therefore the regions double."

 

Some students related this idea to actual paper folding, as shown in the last two students' comments. To translate their understanding of this doubling pattern into a symbolic function, the teacher may need to connect the idea to the paper folding, as follows:

If I start with no folds, then I have one region and, with one fold, I have two regions (1 x 2). If I fold again, each previous region is doubled; this idea is the same as (I x 2) x 2. If I fold again, each previous region will be doubled; this idea is the same as (1 x 2 x 2) x 2. What happens if I fold a fourth time? A fifth time? What numerical relationship seems to hold true between the number of folds and the number of total regions? Describe this relationship in your journal.

The students in these classes were able to describe the function in words: "# of 2 multiplied is same as the number of folds," "2 to the exponent of the # of folds = # of regions," "2 (power of # of folds)." The students found that translating their words into symbols, y = 2x, was difficult, however, and had to be led to this new type of symbolic notation by the teacher. This kind of instruction was usually not needed for the function rules in the previous examples.

 

Again, students should graph the function; record in their journals the characteristics of the patterns, symbolic function, and graph for the exponential function; and describe differences among all three examples. Students will need help to understand how to evaluate 20 and to graph values for x < 0. A graphing calculator would work well for this purpose.

Once students have had experiences with the three types of functions, they should look at other examples. Figure 6 shows four other problems. Patterns with Squares: Area and Perimeter provides examples of linear and quadratic functions. Developing Pick's Theorem is an example of a linear function, the Tower Puzzle is an example of an exponential function, and the Peg Game is an example of a quadratic function.

 
Fig. 6 Other sample problems that develop function concepts with concrete models
 

Students' investigation of each problem should be followed by leading questions that ask students to describe multiple patterns and function rules in their own words and to translate the function rules into algebraic notation. Teachers should guide students to see connections among concrete models, the symbolic function rules, and other patterns observed in the data tables.

 

To help students bring all their new knowledge together, they can complete the summary data form shown in figure 7. The summary data form includes eight sections, one for each of the problems presented in this article. In the appropriate section, students consolidate each problem's data table, rule, and graph. Then students can cut out the eight sections, sort them into three piles according to their similarities, and write about the similarities in each set and the differences among the sets. The observations about similarities and differences should be based on an examination of graphs, rules, and patterns.

 

Fig. 7 Summary Sheet
 

Summary

Middle-grades students can learn about functions by exploring multiple concrete examples that ask students to talk and write about ideas informally. Symbolic notation is more meaningful when it is connected with physical representations and informal language. By exploring multiple examples and by making comparisons within and between problem types, students can develop a deep understanding of function.

The examples presented here are not new. In the 1960s, Robert Davis developed activities for elementary school children to explore function with his Madison Project materials (1966). The Tower Puzzle, Developing Pick's theorem, and the Peg Game were adapted from those materials. The string activity was adapted from a wonderful resource, Teaching Mathematics: A Sourcebook of Aids, Activities and Strategies, by Max Sobel and Evan Maletsky (1975). The equilateral triangle and area and perimeter problems were adapted from Moving On with Pattern Blocks (Roper 1988). The paper-folding example was adapted from one of the NCTM's Addenda series booklets (Phillips 1991). Good problems such as these are available for teachers to use and, surprisingly, have been around for the last thirty years.


The work reported in this article was supported in part by the National Science Foundation under grant number ESI-9254455. All opinions expressed are solely those of tire author.

 

References

Davis, Robert, developer. Madison Project Independent Exploration Materials. Danbury Conn.: Math Media, 1966.

National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989.

Phillips, Elizabeth. Patterns and Functions. Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5-8. Reston, Va.: National Council of Teachers of Mathematics, 1991.

Roper, Ann. Moving On with Pattern Blocks: Intermediate Problem-Solving Activities. Sunnyvale, Calif.: Creative Publications, 1988.

Sobel, Max A., and Evan M. Maletsky, Teaching Mathematics: A Sourcebook of Aids, Activities, and Strategies. Englewood Cliffs, N.J.: Prentice-Hall, 1975.


KATHLEEN CRAMER, kathleen.a.cramer@uwrf.edu, teaches mathematics education courses at the University of Wisconsin-River Falls, River Falls, WI 54022. She is especially interested in developing mathematics content courses for elementary school teachers.