Solutions

• Find a pattern (formula) by Guess and Check.
• Use a spreadsheet.
• Add all the numbers with a calculator or pencil and paper.
• Use a geometric solution (the area of a triangle).
• Use the first and second differences to help find a pattern.
• Use a polynomial regression, along with info from the first and second differences, to find a formula on a spreadsheet or graphing calculator.
• Use matrices or a system of three equations and three unknowns to find the formula.
• Regroup the numbers and find a general pattern.
•  

Introduction / Overview
 

This is a summary of some of the problem solving strategies available to students when attacking traditional problem, such as finding the possible combinations of two, or patterns which reduce to counting consecutive integers. Problems such as these are being used at the middle school level in Oregon's schools in preparation for state-wide assessments each year. While this type of problem is certainly not new, I would consider it quite challenging for the majority of 8^th grade students, especially under testing situations. The type of problem where students may be required to find a pattern by guessing and checking, does not lend itself well to the goal of making all students successful. Problems which can be attacked by finding patterns, guessing and checking, and making tables naturally fit within the realm of computer spreadsheets. We should consider technology as a viable problem solving tool at all times. Not all teachers may agree that technology or a spreadsheet by itself represents an actual problem solving strategy, for example a spreadsheet may only be seen as an extension of making a table or guessing and checking. This is certainly true and as we know, a computer's actions are restricted to directions from the user. However we should also recognize how a computer can change our approach to using different strategies. How many of us would have considered using a lengthy session of guessing and checking or making a long table as viable strategies before familiarity with spreadsheets. We were more enamored and restricted by problems with nice concise generalizations which could be easily simplified, provided you were a teacher and had already seen the solutions, and not a typical student. The proliferation of technology now compels us to revisit and apply strategies which many students have always naturally employed when solving problems, despite our best efforts to teach them "better" methods, and make them more sophisticated students. We can begin to see the power of computer spreadsheets begin to empower more students. This is especially true for middle school students who lack the mathematical experience necessary to arrive at quick accurate solutions, and when generalizations are called for as part of a solution.

Some of the open-ended problems which I've seen circulating in Oregon, intended to be used as practice for state assessment, are rich examples of mathematics and provide great opportunities to teach a diverse range of problem solving strategies and develop students' conceptual understanding. The Third International Mathematics and Science Study (1996) noted that in higher achieving countries such as Japan and Germany, eighth grade teachers will cover less subjects but go into much greater depth than most United Sates classrooms-less breadth but more depth. The state requirement that students must be able to effectively attack open-ended problems, may assist in driving our curriculum in the direction of greater depth. Much energy in our state has been devoted to generating and assessing problems, and we may have taken for granted that teachers have the time, knowledge, or inclination to develop a variety of strategies to solve these problems. Unfortunately, the most misassigned teachers in schools are secondary mathematics teachers (The Condition of Education, 1996). Teachers do not need more problems from cryptic mathematics texts, nor do they have the luxury of working out detailed strategies, plans and lessons when attacking new problems.

This article will attempt to take two problems, show that they are essentially the same problem, and outline some strategies which can be used to solve them. I will more than likely not cover all possibilities, but at least this can help serve as a starting point for teachers to build up a repertoire of solutions, to accompany their library of problems. As we expand our students' choices of problem solving strategies, we will begin to open the door to understanding for more students.

A PRACTICE PROBLEM- HOW MANY BRICKS ARE NEEDED?

Problem I involves a stack of bricks (See Figure 1). The problem is to figure out how many total bricks there would be if the pyramid was 50 bricks high with 50 bricks on the bottom row. We can see by counting that there are 10 total bricks if the stack has four bricks on the bottom.

One obvious method of attacking this problem is to create a table (See Figure 2).

With a table it may be possible establish a pattern. Upon examination we can see that the problem involves adding consecutive integers.

The problem can be reworded as The Total # of bricks =1+2+3+4+5+6+...+50. So we have a problem exactly the same as adding consecutive integers from I to 50.

Solution I

Use a pencil and paper, or calculator, and add the integers.
 

Solution 2

Use a spreadsheet to add the consecutive integers (See Figures 3 and 4).
 

Solution 3

Guess and check to find a pattern.
 

Solution 4

Regroup the numbers and simplify. This is a good strategy for middle school students without formal knowledge of combinations.

1+2+3+4+...+47+48+49+50 (50+1)+(49+2)+(48+3)+(47+4)+...+(27+24)+(26)

=51+51+51+...+51, (51 will be added 25 times) =25*51=1275.

If this pattern is repeated for students with a different number of bricks, it should lead to the generalization of n/2*(n+ I) =sum of the first n integers.
 
 

Solution 5

Take a geometric approach. Assume that each brick is I unit by I unit. Then finding the area of the bricks is equivalent to finding the number of bricks (See Figure 5). The tricky part here is ignoring the original picture and our conception of the typical rectangular shape of a brick. There was no stipulation that the bricks could not be another shape, which is what I find particularly creative about this solution. It should also be noted that the sequence of 1,3,6,10,15,21.... are the triangular numbers, which could lead into the development of this geometric solution.

The total area of the bricks=area inside the triangle + area outside the triangle.

The area inside the triangle =1/2* base*height=.5*50*50. =25*50=1250.

The area outside the triangle= 1/4* 1 00= 100/4=50/2=25.

So total area = 1250+25= 1275, which means there are 1275 bricks.

This can also be easily generalized to 1/2*n*n+n/2=(n²+n)/2=n(n+ I)/2=sum of the first n integers.
 
 

Solution 6

We use the first and second differences to give us a clue about the formula (See Figure 6). The constant second difference suggests a second degree polynomial. We can thus set up a system of three equations and three unknowns.

A general quadratic is given by ax²+bx+c=y. You may choose different notation, since we are using integers, such as an ² +bn+c=f(n)=a_n.

We now choose any three points, for this example we use (1, 1), (2,3), (3,6).

(1, 1) a( 1)²+b(l)+c= 1 la+lb+lc=l

(2,3) a(2)²+b(2)+c=3 4a+2b+lc=3

(3,6) a(3)²+ b(3)+c=6 9a+3b+lc=6

This system is equivalent to the matrix equation of:

=

This can be solved by your favorite method, mine is using my TI-85. Or you may choose to row

reduce the matrix:

The solution of this method will yield the correct solution of f(n)=.5n²+ .5n.
 
 
 
 

Solution 7

Use a calculator or statistics program and use a polynomial regression (See Figure 7). This example uses Microsoft Excel (1995) and the built-in Trendline feature. The constant term of 4E-12 is close enough to zero to be ignored and we are left with the correct solution of y=.5X²+ .5x. Students with graphing calculators can easily generate the same solution. To check the solution, a column can be set up using the computer's suggestion for the function and the same domain values. Using a graphing calculator, the user could generate a sequence using built-in list features and check the output against the original sequence.
 

THE SAME PROBLEM BUT WITH DIFFERENT WORDS!

An ice cream store has 31 flavors of ice cream. How many different two scoop cones are possible? If we do not allow two scoops of the same flavor, we have a standard combination problem of 31 things taken 2 at a time -C . If we do allow two scoops of the same flavor, as is more realistic, than it will soon become clear that the problem is the same as the sum of the first 31 consecutive integers.
 
 
 

Solution 8

For students without formal knowledge of how to solve combinatorial problems, a good approach could be to solve simpler analogous problems and try to develop a pattern. For instance, a matrix of flavors could be used, C=chocolate, V=vanilla, S=strawberry, M=mint, (See Figure 8).

The sequence generated from this is 1,3,6,10.... which is the same pattern of triangular numbers in the sum of the consecutive integer problem. We can therefore continue using solutions to that problem. Using a matrix is certainly not the only way for a teacher to model combinations, but it does work well for combinations of two.
 
 
 
 

Solution 9
The number of different combinations of 31 flavors, two at a time is 31!/(2!29!), if two of the same flavor are not allowed. This reduces to (31*30)/2=465, and the similarity between this and the sum of the first n consecutive integers is becoming clear. We will have to add another 3 1 possible cones if we allow for a double scoop of each of the flavors, which gives us a total of 496 possible different scoops.
 
 
 
 
 
 

SUMMARY

We have looked at two problems which at first may seem dissimilar, but actually amount to solving the same type of problem. Strategies which were reviewed included guess and check to find a pattern, regrouping, geometry, linear algebra, a spreadsheet, regression, and combinatorial theory. As I worked the ice cream problem mentally, my knowledge of combinations was a little rusty and I was never 100% certain if I needed to add the additional 3 1 flavors. I had to get a pencil and generate some patterns to be sure. What I actually did to reassure myself was to use a spreadsheet approach, where I was able to find an answer within a few minutes. Since the most obvious solution is to count the integers from one to fifty, knowledge of using a spreadsheet can provide a quick, accurate solution long before most people will be able to find an answer by other means. Spreadsheets have the potential to make more students successful in attacking this type of problem. Middle school students with enough practice in using spreadsheets and creating formulas will easily solve this problem and be able to verify their work. The limitation to achieving deeper conceptual understanding is that the spreadsheet will not generalize the pattern, unless the user understands enough algebra to select a second degree polynomial regression. Although a spreadsheet can generate more examples of specific instances of the pattern and with more accuracy than pencil and paper techniques. I especially like the geometric solution, which was suggested by David Walker, a sixth grade teacher at our school. If students can remember how to find the area of a triangle it should not be hard to for them to derive a general formula.

References

The Condition of Education [W-WWI. (1997). Washington, DC: National Center for Educational

Statistics. Available : Hostname: www.ed.gov Directory: www.ed.gov/nces/pubs

Microsoft Excel, Version 5.0, (1995). [Computer software], Redmond, WA: Microsoft Corporation.

Third International Mathematics and Science Study[W'WW]. (1996). Washington, DC: National

Hostname: www.ed.gov Directory:

Center for Educational Statistics. Available

www.ed.gov/nces/timss
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

OBJECTIVES