routine and non routine problem solving involving factoring

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Solving & Factoring Polynomials: Examples

Solving Factoring Examples

These exercises can be very long, so I've only shown three examples so far. Here are a few more, for practice:

Find the real-number solutions to x 6 + 9 x 5 + 11 x 4 − 22 x 3 − 9 x 2 − 11 x + 21 = 0 .

They've given me an equation, and have asked for the solutions to that equation. So I'll be finding x -values, rather than factors.

First, I'll try the trick with 1 and −1 . Trying x  = 1 , I get:

1 + 9 + 11 − 22 − 9 − 11 + 21 = 0

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Excellent! So x  = 1 is one of the zeroes. Trying x  = −1 , I get:

1 − 9 + 11 + 22 − 9 + 11 + 21 = 48

Okay; so that one isn't a zero. But, to reduce my polynomial by the one factor corresponding to this zero, I'll do my first synthetic division:

So my reduced polynomial is equation is:

x 5 + 10 x 4 + 21 x 3 − x 2 − 10 x − 21 = 0

This is so nasty... I'm gonna try the trick with x  = 1 again, just in case it's a root twice:

1 + 10 + 21 − 1 − 10 − 21 = 0

Nice! Okay; here's my second synthetic division:

Alright. My new polynomial equation now is:

x 4 + 11 x 3 + 32 x 2 + 31 x + 21 = 0

All the coefficients are positive, so +1 cannot be a zero again. Now it's time for the Rational Roots Test:

x = ±1, 3, 7, 21

Hm... I've already scratched off ±1 directly. Because all the coefficients are positive, then I know that +3, +7, and +21 are out, too. I'd rather stay small, if I can, so I'll try −3 next:

So now my polynonial equation is:

x 3 + 8 x 2 + 8 x + 7 = 0

From this reduced polynomial, I can see that I can cross −21 and −3 off the list of possible roots; they clearly won't work in this reduced polynomial. So I guess I'm left with −7 :

And now I'm down to a quadratic, which I can easily solve:

x 2 + x + 1 = 0

x = (−1 ± sqrt[−3])/2

Because there's a minus inside the radical, I know that the solutions of this quadratic are complex numbers; it has no real zeroes. Since they asked for just the real-valued roots of the original polynomial, then I can ignore these last two zeroes. Then my answer is:

x = 1, −3, −7

I didn't check the graph when I did the above, but it does confirm my answer:

The intercepts at x  = −7 and at x  = −3 are clear. The intercept at x  = 1 is clearly repeated, because of how the curve bounces off the x -axis at this point, and goes back the way it came.

Note: This polynomial's graph is so steep in places that it sometimes disappeared in my graphing software. I had to fiddle with the axis values and window size to get the whole curve to show up. When using your calculator, don't stick only with the default screen for graphs; play with the axis values until you get a picture that's useful.

Factor completely: 2 x 5 − 3 x 4 − 9 x 3 + 3 x 2 − 11 x + 6

They've given me an expression rather than an equation, and have told me to factor. So I'll be finding factors rather than x -values, and I'll need to keep track of everything I pull out, from beginning to end.

There is no factor common to all terms, so there is nothing to pull out yet. I'll check for zeroes of the associated polynomial equation (setting the original expression equal to zero), and see what I can find. Then I'll convert the zeroes to factors, and pull them out.

First, I'll try the usual shortcut with ±1 ; the positive first:

2 − 3 − 9 + 3 − 11 + 6 = −12

No joy. I'll try the negative now:

−2 − 3 + 9 + 3 + 11 + 6 = 90

That's even worse. Okay, now I'll use the Rational Roots Test to create a list of maybe-solution values:

±(1, 2, 3, 6)/(1, 2)

= ±1/2, 1, 3/2, 2, 3, 6

I already know that I can ignore ±1 . Descartes' Rule of Signs tells me that there will be four, two, or zero positive roots; and one (definite) negative root. So I'll start with the negative integers:

Okay; I've found that x  = −2 is a root, which means that x  + 2 is a factor. Also, I've reduced the expression still needing to be factored to:

2 x 4 − 7 x 3 + 5 x 2 − 7 x + 3

The constant term is 3 , so I know that ±2 cannot be a solution to what's left, nor can ±6 . Also, I've already found the one negative zero. So this leaves me with:

1/2, 3/2, 3

I'm trying to avoid fractions, so I'll try the last integer possibility:

The last row above is a four-term polynomial that looks like it can be factored in pairs:

(2 x 3 − x 2 ) + (2 x − 1)

x 2 (2 x − 1) + 1(2 x − 1)

(2 x − 1)( x 2 + 1)

The quadratic factor is the sum of squares, so it isn't factorable. This means I'm done, and my complete factorization is:

( x  + 2)​( x  − 3)​(2 x  − 1)​( x 2  + 1)

The method for answering the two exercises above is the method that I learned, back in the olden times when dinosaurs ruled the world and calculators were made with bear skins and stone knives. It's probably at least similar to the method that you've seen in your book, and your instructor likely expects you do show work along the lines of what I did above.

However, if you have a graphing calculator (and nearly everybody does, nowadays), you can avoid wasting quite so much time on maybe-solution values that turn out not to work.

Find all zeroes of y = 8 x 5 − 58 x 4 + 137 x 3 − 118 x 2 + 33 x + 18

Before doing anything else, I'll do a quick graph:

Looking at the graph, I know to check x  = 3 twice:

The original polynomial was degree-five. I've found one zero of multiplicity two, which leaves at most three more zeroes. Looking at the polynomial represented by the last row above, the Rational Roots Test says that any remaining "nice" zeroes will be among these:

±(1, 2)/(1, 2, 4, 8)

= ±1/8, 1/4, 1/2, 1, 2

Now I can see that there's a common factor of four that can be divided out and discarded, leaving me with:

2 x 2 − 3 x + 2 = 0

x = (−(−3) ± sqrt[(−3) 2 − 4(2)(2)])/(2(2))

= (3 ± sqrt[9 − 16])/4

= (3 ± sqrt[−7])/4

They asked me for all of the zeroes, not just all of the real-valued ones, so I have to include these two roots that don't show up on the graph. However, by checking the graph first, I was able to save a lot of time in arriving at my answer:

x = −(1/4), 3, (3 ± sqrt[−7])/4

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There is a variant of these exercises, where they provide one or more factors (of an expression) or zeroes (of an equation or function), and they want you to find the rest of them. I've got examples of how this works in the last page of the lesson on synthetic division. Varient exercises are often a bit messier and, to answer them, you're expected to have a deeper understanding of how the Quadratic Formula generates solutions in pairs, because of the " ± ". Otherwise, they work in pretty much the same way.

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Mathematical Mysteries

Revealing the mysteries of mathematics

How to Solve Math Problems: Non-Routine Problems

Nonroutine Problem Solving , stresses the use of heuristics [3] and often requires little to no use of algorithms. Unlike algorithms, heuristics are procedures or strategies that do not guarantee a solution to a problem but provide a more highly probable method for discovering a solution. Building a model and drawing a picture of a problem are two basic problem-solving heuristics. Studying end-of-game situations provides students with experiences in using the heuristics of reducing the problem to a similar but simpler problem and working a problem backwards, i.e. from its resolution to its initial state. Other heuristics include describing the problem situation, classifying information, and finding irrelevant information. [1]

There Are Two Categories of Nonroutine Problem Solving: Static and Active

Static-Nonroutine  problems have a fixed, known goal and fixed, known elements that are used to resolve the problem. Solving a jigsaw puzzle is an example of a Static- Nonroutine problem. Given all pieces to a puzzle and a picture of the goal, learners are challenged to arrange the pieces to complete the picture. Various heuristics such as classifying the pieces by color, connecting the pieces that form the border, or connecting the pieces that form a salient feature to the puzzle, such as a flag pole, are typical ways in which people attempt to resolve such problems. [1]

Active-Nonroutine  problems may have a fixed goal with changing elements, a changing goal or alternative goals with fixed elements, or changing or alternative goals with changing elements. The heuristics used in this form of problem-solving are known as strategies. People who study such problems must learn to change or adapt their strategies as the problem unfolds. [1]

What is non-routine problem-solving in math?

A non-routine problem is any complex problem that requires some degree of creativity or originality to solve. Non-routine problems typically do not have an immediately apparent strategy for solving them. Often times, these problems can be solved in multiple ways.

Incorporating non-routine problem solving into your math program is one of the most impactful steps you can take as an educator. By consistently allowing your students to grapple with these challenging problems, you are helping them acquire essential problem-solving skills and the confidence needed to successfully execute them. [2]

Step 1: Understand

This is a time to just think! Allow yourself some time to get to know the problem. Read and reread. No pencil or paper necessary for this step. Remember, you cannot solve a problem until you know what the problem is!

• Does the problem give me enough information (or too much information)?
• What question is being asked of me?
• What do I know and what do I need to find out?
• What should my solution look like?
• What type of mathematics might be required?
• Can I restate the problem in my own words?
• Are there any terms or words that I am unfamiliar with?

Step 2: Plan

Now it’s time to decide on a plan of action! Choose a reasonable problem-solving strategy. Several are listed below. You may only need to use one strategy or a combination of strategies.

• Draw a picture or diagram
• Make an organized list
• Make a table
• Solve a simpler related problem
• Find a pattern
• Guess and check
• Act out a problem
• Work backward
• Write an equation
• Use manipulatives
• Break it into parts
• Use logical reasoning

Step 3: Execute

Alright! You understand the problem. You have a plan to solve the problem. Now it’s time to dig in and get to work! As you work, you may need to revise your plan. That’s okay! Your plan is not set in stone and can change anytime you see fit.

• Am I checking each step of my plan as I work?
• Am I keeping an accurate record of my work?
• Am I keeping my work organized so that I could explain my thinking to others?
• Am I going in the right direction? Is my plan working?
• Do I need to go back to Step 2 and find a new plan?
• Do I think I have the correct solution? If so, it’s time to move on to the next step!

Step 4: Review

You’ve come so far, but you’re not finished just yet! A mathematician must always go back and check his/her work. Reviewing your work is just as important as the first 3 steps! Before asking yourself the questions below, reread the problem and review all your work.

• Is my answer reasonable?
• Can I use estimation to check if my answer is reasonable?
• Is there another way to solve this problem?
• Can this problem be extended? Can I make a change to this problem to create a new one?
• I didn’t get the correct answer. What went wrong? Where did I make a mistake?

[1] “Pentathlon Institute Active Problem-Solving”. 2023.  mathpentath.org . https://www.mathpentath.org/active-problem-solving/ .

[2] Tallman, Melissa. 2015. “Problem Solving In Math • Teacher Thrive”. Teacher Thrive. https://teacherthrive.com/non-routine-problem-solving/ .

[3] A heuristic is  a mental shortcut commonly used to simplify problems and avoid cognitive overload .

Additional Reading

“101 Great Higher-Order Thinking Questions for Math”. 2023.  elementaryassessments.com . https://elementaryassessments.com/higher-order-thinking-questions-for-math/ .

⭐ “Developing Mathematics Thinking with HOTS (Higher Order Thinking Skills) Questions”. 2023. saydel.k12.ia.us . https://www.saydel.k12.ia.us/cms_files/resources/Developing%20Mathematics%20Thinking%20with%20HOTS%20Questions%20(from%20classroom%20observations)PDF.pdf .

“Higher Order Thinking Skills in Maths”. 2017.  education.gov.Scot . https://education.gov.scot/resources/higher-order-thinking-skills-in-maths/ .

“How to Increase Higher Order Thinking”. 2023.  Reading Rockets . https://www.readingrockets.org/topics/comprehension/articles/how-increase-higher-order-thinking .

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

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Improving Students’ Problem-Solving Flexibility in Non-routine Mathematics

• Conference paper
• First Online: 30 June 2020
• Cite this conference paper

• Huy A. Nguyen   ORCID: orcid.org/0000-0002-1227-6173 13 ,
• Yuqing Guo 13 ,
• John Stamper 13 &
• Bruce M. McLaren 13  

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12164))

Included in the following conference series:

• International Conference on Artificial Intelligence in Education

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1 Citations

A key issue in mathematics education is supporting students in developing general problem-solving skills that can be applied to novel, non-routine situations. However, typical mathematics instruction in the U.S. too often is dominated by rote learning, without exposing students to the underlying reasoning or alternate ways to solve problems. As a first step in addressing this problem, we present a cognitive task analysis study that investigates how students without a mathematics-related background solve novel non-routine problems. We found that most students were able to identify the underlying pattern that yields the final solution in each problem. Furthermore, they tended to use various forms of visualization in their draft work, but occasionally made computational mistakes. Based on these results, we propose our plan for developing an instructional platform that leverages learning science principles to train students in problem-solving abilities.

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Opportunity to learn problem solving in dutch primary school mathematics textbooks.

Reflections on Problem-Solving

Designing opportunities to learn mathematics theory-building practices

• Problem-solving flexibility
• Non-routine mathematics

1 Introduction

The ability to tackle non-routine problems – those that cannot be solved with a known method or formula and require analysis and synthesis as well as creativity [ 9 ] – is becoming increasingly important in the 21st century [ 5 ]. However, when faced with a non-routine problem, U.S. students tend to apply memorized procedures incorrectly rather than modify them or develop new solutions [ 8 ]. One possible source for this difficulty is the typical instructional focus in U.S. schools on memorization and application of routine procedures [ 2 , 6 , 7 ]. Such an approach makes students proficient at executing rote procedures, but it does little to help them understand the conceptual basis for the procedures or to think creatively about novel problems - both of which are essential for developing problem-solving flexibility.

An important first step in addressing this issue is to assess how students currently approach non-routine problem solving, so that we can design the appropriate learning interventions. In this work, we present an empirical cognitive task analysis where participants were asked to think aloud while solving a series of non-routine problems from discrete mathematics. We chose this domain because discrete math problems can often be tackled from multiple perspectives while not requiring any advanced background beyond the high school curriculum [ 3 ]. Based on the findings from this study, we propose our plan for developing a tutoring system for non-routine problem-solving ability. Then, we discuss the system’s broader implications and the challenges we need to address in deploying this system at scale.

2 Assessing Students’ Problem-Solving Skills

We conducted interview sessions with three students at a private university in a midwest US city. None of the students had a mathematics-related background. The participants were asked to solve three non-routine mathematics problems on paper in one hour. They were also encouraged to think aloud and write down their draft work. The three problems in our study, taken from [ 3 ], and a brief summary of their sample solutions, are as follows.

In an air show there are twenty rows. The first row contains one seat, the second three seats, the third five seats, the fourth seventh seats, and so on. How many seats are there in total ?

Sample solution: In the first row there is 1 seat. In the first two rows there are 1 + 3 =  4 seats. In the first three rows there are 1 + 3 + 5 =  9 seats. In the first four rows there are 1 + 3 + 5 + 7 =  16 seats. In the first five rows there are 1 + 3 + 5 + 7 + 9 =  25 seats. Based on this pattern, in the first k rows there are k 2 seats. In our case, there are 20 rows and therefore 400 seats in total.

Find all integers between 1 and 99 (inclusive) with all distinct digits.

Sample solution: there are 99 integers between 1 and 99 in total, and 9 of them have non-distinct digits, namely 11, 22, 33, …, 88, 99. Hence, the remaining 90 integers have distinct digits.

What is the digit in the ones place of 2 57 ?

Sample solution: Looking at the sequence of powers of 2–2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, … – we see that the corresponding sequence of digits in the ones places is 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, … In other words, this sequence is a cycle of length 4. Therefore the last digit of 2 57 is that of 2 53 , which is that of 2 49 , …, which is that of 2 1 , which is 2.

We then analyzed recordings of the participants’ think-aloud and their draftwork, from which we derived the following insights:

Pattern Identification.

Participants were aware that they had to find a pattern or formula to solve the problems, because it was not feasible to directly compute the final answer. All participants were able to identify the expected pattern for each problem as outlined above, except for one student who failed to do so for Problem 1 . While this participant realized that the number of seats on row k is the k-th positive odd number, this pattern alone was insufficient to solve the problem.

Visualization.

Participants tended to visualize the problem by drawing examples and making lists or tables (Fig.  1 ). They expressed that these visualizations were crucial in helping them identify the correct pattern and solve the problem.

Participants’ attempts at visualizing the problem in their draftworks.

Computation.

Participants occasionally made computational mistakes while calculating the initial sequence values, especially in Problem 3 . As a consequence, they could not identify any pattern based on the wrong values, and took some time to realize the mistake. All students who corrected their mistakes were able to subsequently solve the problem.

In summary, we found that participants were aware of the idea behind identifying patterns, and they all did so via some kind of visualization. On the other hand, computational mistakes, while not directly related to our learning objectives, can be detrimental to the overall problem-solving process. From these insights, we propose the following next steps.

3 Developing a Tutoring System for Flexible Problem-Solving

Moving forward, our plan is to iteratively conduct more cognitive task analysis interviews and develop a prototype of the system. Our initial conceptualization of how the system will work is as follows. A single round of exercise in the system incorporates four learning stages, all of which are built on established learning principles: 1) Reviewing a worked example of a non-routine mathematics problem, 2) Explaining the worked example to a partner, 3) Solving a new problem which is isomorphic to the worked example problem, and 4) Explaining the isomorphic solution to a partner. Between rounds, the student can review previous solutions, look at materials related to the problem space, or practice basic math skills. This design is intended to (1) formally introduce students to a complete solution through worked examples, (2) reinforce their understanding of the worked example through self-explanation, and (3) assess students’ learning through an isomorphic problem. Our hypothesis is that through the learning system, students will get a better sense of how to approach a novel non-routine problem, so that in case they have not yet found the solution – for example, like the participant in our study who did not identify the true pattern in Problem 1 – they can still adopt a different viewpoint and explore other strategies.

We have already begun mapping the problem space by developing a non-routine problem-solving flowchart and identifying sets of potential non-routine problem solutions. Once we have tested our solution space, we will develop and pilot a low fidelity paper prototype version of the system with college students to further refine the mathematical content and identify areas for revision to the design. We are also looking at which technological features could be useful for students learning in this domain. As a first step, our system will include a canvas for students to perform their draftwork on, as well as a simple calculator interface with basic arithmetic operations to help students avoid computational mistakes. An important follow-up question is whether students’ draftwork can be analyzed to infer their thinking process, which could in turn guide the design of appropriate feedback mechanics. While this task has previously been performed manually by domain experts [ 1 ], employing a machine learning technique to automate it to some extent would greatly enhance the system’s adaptive support functionality and scalability.

4 Conclusion

This research will provide concrete, generalizable evidence about the utility and implementation of worked examples, multiple solutions, and self-explanation to promote skills in non-routine problem solving. Results will inform future tutoring system design by identifying how and when the instructional features are most beneficial for developing problem-solving skills. We also intend to have a practical impact by distributing a tutoring system that is accessible to a wide range of students, including lower-performing students who would typically not be exposed to these types of problems and strategies [ 1 , 4 ]. In addition, we will provide a teacher’s guide to support educators in using the system adaptively to support their instructional goals.

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Huy A. Nguyen, Yuqing Guo, John Stamper & Bruce M. McLaren

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Nguyen, H.A., Guo, Y., Stamper, J., McLaren, B.M. (2020). Improving Students’ Problem-Solving Flexibility in Non-routine Mathematics. In: Bittencourt, I., Cukurova, M., Muldner, K., Luckin, R., Millán, E. (eds) Artificial Intelligence in Education. AIED 2020. Lecture Notes in Computer Science(), vol 12164. Springer, Cham. https://doi.org/10.1007/978-3-030-52240-7_74

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Non-routine Algebra Problems

(a new problem of the week).

Last week I mentioned “non-routine problems” in connection with the idea of “guessing” at a method. Let’s look at a recent discussion in which the same issues came up. How do you approach a problem when you have no idea where to start? We’ll consider some interesting implications for problem solving in general, with an emphasis on George Pólya’s outline.

First problem: mindless manipulation?

This came to us in March, from a student who identified him/herself as “J”:

Hi, Recently I had to solve a problem If (a + md) / (a + nd) = (a + nd) / (a + rd) and (1 / n) – (1 / m) = (1 / r) – (1/n) , then (d / a) = -(2 / n) i.e. Given the two expressions above I need to prove the last equality. I don’t understand problems like these. Basic Algebra books talk about problems like equation solving or word problems, but those are easy because there’s always some method you can use . For example regarding equation solving you move x’s to the left, numbers to the right; word problems can be solved using equalities like distance = rate * time. But a problem like the one above it seems has no method; it seems like you’re supposed to just manipulate the symbols until you get the answer . For example I tried to solve it like this: Regarding the first expression, after multiplying numerator of the first fraction by the denominator of the second I get (d / a) = ((m + r – 2n) / (n^2 – mr)) and 2mr = nm – nr then substitute for mr in the first expression. I reached the solution by luck; I just manipulated the symbols and it took me a lot of time . So is there a more efficient way to solve problems like these? How to think about these problems? Am I supposed to just mindlessly manipulate the symbols until I get lucky? Finally are there any books that deal with problems like these ? Because like I mentioned it seems like most precalculus books talk about equation solving etc., problems which have a clear method. Thanks.

The solution

Before we deal with the question, let’s look more closely at his solution.

We are given two equations:

$$\displaystyle\frac{a + md}{a + nd} = \frac{a + nd}{a + rd}$$

$$\displaystyle\frac{1}{n} – \frac{1}{m} = \frac{1}{r} – \frac{1}{n}$$

We need to conclude that

$$\displaystyle\frac{d}{a} = -\frac{2}{n}.$$

J gave only a brief outline of what he did; can we fill in the gaps?

My version is to first “cross-multiply” in each equation to eliminate fractions, and do a little simplification:

The first becomes $$(a + md)(a + rd) = (a + nd) (a + nd),$$ which expands to $$a^2 + rda + mda + mrd^2 = a^2 + 2nda + n^2d^2,$$ then $$rda + mda – 2nda = n^2d^2 – mrd^2,$$ which factors to yield $$(r + m – 2n)da = (n^2 – mr)d^2.$$ Dividing, we get $$\displaystyle\frac{d}{a} = \frac{r + m – 2n}{n^2 – mr}.$$

(You may notice here that in dividing both sides by d , we obscured the fact that the line before is true whenever d = 0. I’ll be mentioning this below.)

The second equation, multiplied by \(mnr\), becomes $$mr – nr = nm – mr,$$ which easily becomes $$2mr = nm + nr.$$ (J had a sign error here.)

Now, replacing \(mr\) with \(\displaystyle\frac{nm + nr}{2}\), we get $$\displaystyle\frac{d}{a} = \frac{r + m – 2n}{n^2 – \frac{nm + nr}{2}} = \frac{2(r + m – 2n)}{2n^2 – nm – nr} = \frac{2(r + m – 2n)}{-n(r + m – 2n)} = -\frac{2}{n}.$$

How to solve it

Taking the question myself, I replied:

I tried the problem without looking at your work, and ended up doing almost exactly the same things. That took me just a few minutes. So probably it is not your method itself, but your way of finding it , that needs improvement. In my case, I did the “obvious” things (clearing fractions, expanding, factoring) to both given equations, keeping my eyes open for points at which they might be linked together , and found one. It may be mostly experience that allowed me to find it quickly. That is, I didn’t “mind lessly manipulate”, but “mind fully  manipulated”. And the more ideas there are in your mind, the more easily that can happen. So maybe just doing a lot of (different) problems is the main key.

I added a few more thoughts about strategies:

There may be a better method for solving this, but finding it would take me a longer time than what I did. So perseverance at trying things is necessary , regardless. Solutions to hard problems don’t just jump out at you (unless they are already in your mind from past experience); you have to explore . The ideas I describe for working out a proof apply here as well: Building a Geometric Proof I like to think of a proof as a bridge, or maybe a path through a forest: you have to start with some facts you are given, and find a way to your destination. You have to start out by looking over the territory, getting a feel for where you are and where you have to go – what direction you have to head, what landmarks you might find on the way, how you’ll know when you’re getting close. (By the way, in my work I also found that d/a = 0 gives a solution, so that if d=0 (and a ≠ 0), the conclusion is not necessarily true. Did you omit a condition that all variables are nonzero?) You are probably right that too many textbooks and courses focus on routine methods, and don’t give enough training in non-routine problem solving . They may include some “challenge problems” or “critical thinking exercises”, but don’t really teach that. One source of this sort of training is in books or websites (such as artofproblemsolving.com ) that are aimed at preparation for contests. Books like Pólya’s   How to Solve It  (and newer books with similar titles) are also helpful. Here are a few pages I found in our archives that have at least some relevance: Defining “Problem Solving” Giving Myself a Challenge Preparing for a Math Olympiad Learning Proofs What Is Mathematical Thinking? Others of us may have ideas to add.

Some these have been mentioned in previous posts such as How to Write a Proof: The Big Picture and  Studying Math: Want a Challenge? .

Another problem: following Pólya

The next day, J wrote in with another problem, having already followed up on my suggestions:

Hi. I posted here recently asking about problem solving and algebra and I was recommended a book called “How to solve it” by Pólya . I bought that book and now I am trying to solve some algebra exercises using it. Today I came across this problem If bz + cy = cx + az = ay + bx and (x + y +z)^2 = 0 , then a +/- b +/- c. (The sign +/- was a bit confusing to me since it’s not brought up anywhere in the book besides this problem, but Wikipedia says that a +/- b = 0 is a + b =0 or a – b = 0.) In the book “How to solve it” Pólya says that first it’s important to understand the problem and restate it . So my interpretation of a problem is this: If numbers x, y, z are such that (x + y + z)^2 = 0 and bz + cy = cx + az and bz + cy = ay + bx, then the numbers a, b, c are such that a + b + c = 0 or a – b – c =0 Next Pólya says to devise a plan . To do that he says you need to look at a hypothesis and conclusion and think of a similar problem or a theorem. The best I could think of is an elimination problem, i.e. when you’re given a certain set of equations and you can find a relationship between constants. Can you think of any other similar problems which could help me solve this problem?

I first responded to the last question:

Hi again, J. I would say that the last question you asked was “similar” to this, so the same general approach will help. That’s essentially what you said in your last paragraph, I think. I know that isn’t very helpful, but it’s all I can think of myself. You’d like to have seen a problem that is more specifically like this one, such as having (x + y + z) 2  = 0 in it, perhaps, so you could get more specific ideas. I only know that I have seen a lot of problems like this involving symmetrical equations (where each variable is used in the same ways), and I suspect those problems can be solved by similar methods. But I don’t know one method that would work for this one.

I’ll get back to that question. But let’s focus first on Pólya.

Here is what Pólya says (p. 5) when he introduces his famous four steps of problem solving:

In order to group conveniently the questions and suggestions of our list, we shall distinguish four phases of the work. First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan . third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it.

This process is then explained in more detail, and used as an organizing principle in the rest of the book. It can be amazing to see how many students jump into a problem before they understand what it is asking, or do calculations without having made any plans . On the other hand, it would be wrong to think of these four steps as a routine to be followed exactly; often you don’t fully understand a problem until you have started doing something , perhaps carrying out a half-formed plan and then realizing that you had a wrong impression of some part.

Understanding the problem

And J has here a good example of a misunderstanding. This problem uses the plus-or-minus symbol (±) in a rare way, which in this case requires asking (not explicitly one of Pólya’s recommendations, but valuable!).

The problem says this:

$$\text{If } bz + cy = cx + az = ay + bx \text{ and } (x + y + z)^2 = 0 \text{, then } a \pm b \pm c.$$

(No, that doesn’t quite make sense! We’ll be fixing that shortly.)

What does it mean when there are two of the same symbol? The Wikipedia page J found says, “In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the + or − symbols, allowing the formula to represent two values or two equations.” They give an example (the quadratic formula), where either sign yields a valid answer; then an example with two of the same sign (the addition/subtraction identity for sines) in which both must be replaced with the same sign ; and third example (a Taylor series) where the reader has to determine which sign is appropriate for a given term. Later they introduce the minus-or-plus sign (\(\mp\)), which explicitly indicates the opposite sign from an already-used ±.

But here, we have two ±’s with no clear reason why they should be the same, or should be different. Is this a special case? J has assumed they are the same, so that it means “\(a + b + c = 0\) or \(a – b – c = 0\)“. This is the first issue I had to deal with:

First, though, did you mean to say that the conclusion is a ± b ± c  = 0 ? That wouldn’t quite mean what you said about it, because the two signs need not be the same. Rather, it means that  either  a + b + c = 0, or a + b – c = 0, or a – b + c = 0, or a – b – c = 0: any possible combination  of the signs.

Now, how did I know that, when it goes against what Wikipedia seems to be saying? I’m not sure! There is actually some ambiguity; really, we just shouldn’t rule out this possibility . But I saw from the start that if the two signs are the same, then the problem has an odd asymmetry , requiring b and c to have the same sign in this equation, but not a . That simply seems unlikely, considering the symmetry elsewhere.

Sometimes we discover, as we proceed through the solving process, that we have to interpret the statement one way or another in order for it to be true – an example of my comment that understanding can come after doing some work. (That was actually the case here. But the problem really should have been written to make this clear!)

Hints toward a solution

What this means is that we don’t know the signs of the numbers. One thing that suggests is that we might be able to show some fact about a 2 ,  b 2 , and c 2 , so that we would have to take  square roots , requiring us to use ± before each of a, b, and c. It’s also interesting that they said that (x + y + z) 2  = 0, which means nothing more than x + y + z = 0. That also makes me curious, and at the least puts squares into my mind for a second reason.

Here I am just letting my mind wander around the problem, pondering what the givens suggest. This is part of both the understanding phase, and the “looking for connections” Pólya talked about.

Not even being sure of the conclusion, I just tried manipulating the equations any way I could, just to make their meanings more visible; and then I solved x + y + z = 0 for z and put that into my derived equations, eliminating z. That took me eventually to a very simple equation that involved a, b, x 2 , and y 2 . And that gave a route to the ± I’d had in mind.

We could say that my initial plan is, as I suggested at the top, to explore ! We can refine the plan as we see more connections. (As I said, Pólya has to be followed flexibly.)

There’s a lot of detail I’ve omitted, in part because much of my work was undirected, so you may well find a better way. But the key was to have some thoughts in mind before I did a lot of work, in hope of recognizing a useful form when I ran across it . The other key was perseverance , because things got very complicated before they became simple again! (I suspect that as I go through this again, I’ll see some better choices to make, knowing better where I’m headed.) I don’t think you told us where these problems came from; they seem like contest-type problems, which you can expect to be highly non-routine. As I said last time, until you’ve done a lot of these, you just need to keep your eyes open so that you are learning things that will be useful in future problems! I am not a contest expert, as a couple of us are, so I hope they will add some input.

Since we never got back to the details of this problem, let’s finish it now. Frankly, I had to look in my stack of scrap paper to find what I did in March, because I wasn’t making any progress when I tried it again just now. Clearly I could have given a better hint! I was hoping that just the encouragement that it could be done would lead to J finding a nicer approach than mine.

But here’s what I find in my incomplete notes from then. First, I rewrote the equality of three expressions as two equations, and eliminated c; I’ll use a different pair of equations than I did then, with that goal in mind: $$cx + az = ay + bx\; \rightarrow\; c = \frac{ay-az+bx}{x}$$ $$ay + bx = bz + cy\; \rightarrow\; c = \frac{ay-bz+bx}{y}$$

Setting these equal to eliminate c, $$\frac{ay-az+bx}{x} = \frac{ay-bz+bx}{y}$$

Cross-multiplying, $$ay^2-ayz+bxy = axy-bxz+bx^2$$

Solving \(x + y + z = 0\) for z and substituting, $$ay^2-ay(-x-y)+bxy = axy-bx(-x-y)+bx^2$$

Expanding, $$ay^2 + axy + ay^2 + bxy = axy + bx^2 + bxy + bx^2$$

Canceling like terms on both sides, $$2ay^2 = 2bx^2$$

Therefore, $$\frac{x^2}{a} = \frac{y^2}{b}$$

We could do the same thing with different variables and find that this is also equal to \(\frac{z^2}{c}\). So we have $$\frac{x^2}{a} = \frac{y^2}{b} = \frac{z^2}{c} = k$$

Now we’re at the place I foresaw, where we can take square roots: $$x = \pm\sqrt{ak}$$ $$y = \pm\sqrt{bk}$$ $$z = \pm\sqrt{ck}$$

Therefore, since \(x+y+z=0\), we know that $$\pm\sqrt{ak}+\pm\sqrt{bk}+\pm\sqrt{ck}=0$$

and, dividing by \(\sqrt{k}\), we have $$\pm\sqrt{a}\pm\sqrt{b}\pm\sqrt{c}=0$$

In March, it turns out, I stopped short of the answer, thinking I saw it coming. But in fact, I didn’t attain the goal! I hoped that a , b , and c would be squared before we have to take the roots. We seem, however, to have proved that they must all be positive , which makes the conclusion impossible!

I’m wondering if the problem, which was never quite actually stated, might have been different from what I assumed. In fact, armed with this suspicion, I tried to find an example or a counterexample, and found that if $$\begin{pmatrix}a & b & c\\ x & y & z\end{pmatrix}= \begin{pmatrix}1 & 4 & 1\\ 1 & -2 & 1\end{pmatrix}$$ satisfies the conditions, with $$bz + cy = cx + az = ay + bx = 2,$$ but no combination of signed a , b , and c add up to 0. So the real problem must have been something else …

Remembering how to solve a problem

At this point J abandoned that path, and closed with a side issue:

Hi Doctor. I have one more question about problem solving. I spent some more time on the problem we discussed then I skipped it and decided to focus on other problems instead. I managed to solve a few of them but then I took a long break when I came back I couldn’t remember the solutions without looking at my work . I don’t know if you read How to solve it by Pólya. I ask since at the beginning of that book Pólya gives an example of a mathematical problem. The problem in question is this: Find the diagonal of a rectangular parallelepiped if the length, width, and height are known. He asks the reader to consider the auxiliary problem of finding the diagonal of the right triangle using Pythagoras theorem. I am telling you this because the solution to this problem is very clear; I can recall it even long after I finished reading. I do not feel the same about algebra problems. I solve them, do the obvious things, and then I almost immediately forget. Does that happen to you? If not how do you remember the solution? I just want to know if you find these algebra problems as unintuitive as I do.

My memory is as bad as anyone’s! I replied,

I wouldn’t say that I remember every solution I’ve done, or every solution I’ve read. The example you give is a classic that stands out, particularly the overall strategy. Others are more ad-hoc and don’t feel universal (in the sense of being applicable to a large class of problems), so they don’t stick in the memory. I don’t have my copy of Pólya with me (I’ve been meaning to look for it), but I recall that one of his principles is to take time after solving a problem to focus on what you did and think about how it might be of use for other problems. This is something like looking around before I leave my car in a parking lot to be sure I will recognize where I left it when I come back from another direction. I want to fix the good idea in my mind and be able to recognize future times when it will fit. But even though I do have that habit, there are some problem types that I recognize over and over, but keep forgetting what the trick is. (Maybe sometimes it’s because I’ve seen two different tricks, and they get mixed up in my mind.) So you’re not alone. For me, though, it’s not such much being unintuitive , as just not being memorable , or being too complex for me to have focused on them enough to remember.

So Pólya recognized the likelihood of forgetting (failing to learn from what you have done), and the need to make a deliberate effort there!

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Routine and non routine problems

This document discusses and provides examples of routine and non-routine problem solving. Routine problems involve using arithmetic operations to solve practical problems and have clear, straightforward procedures. Non-routine problems require more creativity, originality, and higher-order thinking skills as they may have multiple solutions or approaches that are not immediately obvious. Ten examples are given that illustrate the differences between routine and non-routine problems of increasing complexity. Read less

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• 1. Routine and Non-routine Problem
• 2. Routine and non-routine problem solving We can categorize problem solving into two basic types: routine and non-routine. The purposes and the strategies used for solving problems are different for each type.
• 3. Routine problem solving From the curriculum point of view, routine problem solving involves using at least one of the four arithmetic operations and/or ratio to solve problems that are practical in nature. ion.uwinnipeg.ca Routine and non-routine problem solving
• 4. Example: Sofia had _____ dimes. She game some to her friend. Now she has ______ dimes. How many did she give to her friend? whatihavelearnedteaching.com Routine problem solving
• 5. Non-routine problem solving A non-routine problem is any complex problem that requires some degree of creativity or originality to solve. Non-routine problems typically do not have an immediately apparent strategy for solving them. Often times, these problems can be solved in multiple ways. aRif [email protected]
• 6. Example: There are 45 questions in an exam. For every correct answer 5 marks awarded and for every wrong answer 3 marks are deducted. Melissa scored 185 marks. How many correct answers did she give? youtube.com/inspired Learning Non-routine problem solving
• 7. Differences between routine and non- routine problems ROUTINE QUESTIONS  Do not require students to use HOTS.  Use clear procedures. NON-ROUTINE QUESTIONS  Require HOTS. Increase the reasoning ability. Use answers and procedures that are not immediately clear. Encourage more than one solution and strategy. Expect more than one answer. Challenge thinking skills. Produce creative and innovative students.
• 8. Differences between routine and non- routine problems ROUTINE QUESTIONS  Do not require students to use HOTS.  Use clear procedures. NON-ROUTINE QUESTIONS  Require solutions that are more than simply making decisions and choosing mathematical operations. Require a suitable amount of time to solve. Encourage group discussion in finding the right solution. researchgate.com
• 9. Illustrative examples: 1. Sofia had 42 dimes. She game some to her friend. Now she has 17 dimes. How many did she give to her friend? 2. Ali eat 2 piece of cakes. 5 minutes later, he eat 1 more piece of cakes. How many piece of cakes that Ali eat? 3. There are 45 questions in an exam. For every correct answer 5 marks awarded and for every wrong answer 3 marks are deducted. Melissa scored 185 marks. How many correct answers did she give? 4. A watch cost $25 more than a calculator. 3 such watches cost as much as 8 such calculators. What is the cost of each calculator?
• 10. 5. At a stadium, there were 243 women and 4 times as many men as women. There were 302 more children than adults. How many children were at the stadium? 6. Peter had $1800. After he gave $400 to John, he had twice as much money as John. How much money did John have at first? 7. Abbey and Ben had some money each. The amount of money that Abbey had was a whole number. Abbey wanted to buy a watch using all her money but she was short of $90.50. Ben wanted to buy the same watch using all his money but he was short of $1.80. The total amount of money that both of them had was still not enough to buy the watch. How much was the watch? Illustrative examples: youtube.com/Mr. Matthew John ; youtube.com/Danny Lim
• 11. 8. In a farm there are twice as many chickens as cows. If there are 980 legs altogether. Find the number of chickens in the farm. Illustrative examples: youtube.com/inspired Learning
• 12. 9. Let f be a function, with the domain the set of real numbers except 0 and 1, that satisfies the equation 2𝑥𝑓 𝑥 − 𝑓 𝑥 − 1 𝑥 = 20𝑥. Find 𝑓 5 4 . Illustrative examples: PEAC-InSet 2018
• 13. 10. If (a, b, c, 2c) are real numbers such that (a)(b)(c)≠ 0 and given condition 𝑏𝑥+ 1−𝑥 𝑐 𝑎 = 𝑐𝑥+ 1−𝑥 𝑎 𝑏 = 𝑎𝑥+ 1−𝑥 𝑏 𝑐 then, prove that a = b = c. Illustrative examples: youtube.com/Raveena Chimnani
• 14. THANK YOU www.slideshare.net/reycastro1 @reylkastro2 reylkastro

Grade 4 Mathematics Module: Solving Routine and Non-routine Problems Involving Perimeter

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

Word problem involving perimeter is just an application of basic math skills but there are also some problems involving perimeter that are complicated. We have to follow a step by step solution, make an illustration, choose a formula and make use of different strategies to come up with the right answer.

After going through this module, you are expected to: solve routine and non-routine problems in real-life situations involving perimeter of squares, rectangles, triangles, parallelograms, and trapezoids.

Grade 4 Mathematics Quarter 3 Self-Learning Module: Solving Routine and Non-routine Problems Involving Perimeter

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