## Mathelogical

Home Non-Routine Mathematics

## Introduction to Non-Routine Mathematics / Non-Routine Problem Solving

- It’s an engaging and interesting way to introduce problem solving to kids and grown-ups.
- Its helps boost the brain power.
- It encourages us to think beyond obvious and analyse a situation with more clarity.
- Encourages us to be more flexible and creative in our approach and to think and analyse from an extremely basic level, rather than just learning Mathematical formulae and trying to fit them in all situations.
- Brings out originality, independent thought process and analytical skills as one must investigate a problem, reach a solution, and explain it too.

How to Analyse a Non-Routine Problem :

- Read the problem well and make note of the data given to you.
- Figure out clearly what is asked or what is expected from you.
- Take note of all the conditions and restrictions . This will help you get more clarity.
- Break up the problem into smaller parts , try to solve these smaller problems first.
- Make a note of data and properties or any similar situations ( faced earlier)
- Look for a pattern or think about a logical way of reaching a solution. Make a model or devise a strategy .
- Use this strategy and your knowledge to reach a solution.

Not really! A small child and a Non-Math student can solve it too.

Logic: Each piece of furniture has at least 3 legs (stools-3 legs, chairs -4 legs).

Chairs have 4 legs i.e., each extra leg belongs to a chair.

(we have already taken 3 legs of each chair and stool into account)

Number of extra wooden legs in the restaurant

= total number – minimum possible number of legs for 50 pcs of furniture

Each extra leg (4 th leg) belongs to a chair.

Therefore, the number of chairs in the restaurant = 30

So, number of stools = 50-30=20

For more on this topic: Solving without Simultaneous Equations

A cube is painted from all sides. It is then cut into 27 equal small cubes. How many cubes

There are 6 faces in any cube.

Therefore, 6 cubes have only one side painted.

b)The small cubes at the corners of the big cube have 3 sides painted.

There are 8 small corner cubes in the big cube.

Therefore, 8 cubes have 3 sides painted.

- Worksheet 1 - Rabbit and Chicken
- Introduction - Regions made by intersecting lines
- Intro Number of trees planted along a road

## More Related Topics

- Using Models to Solve Problems
- Number of Trees Planted Along a Road
- How to Solve Without Algebraic Equations
- Man-Hours Word Problems
- Regions Made by Intersecting Circles
- Regions Made by Intersecting Lines

## Routine and Non-Routine Problems in Mathematics

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## Non-routine Algebra Problems

## 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.

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

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

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

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

## 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 Polya’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.

## 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 Polya.

Here is what Polya 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.

## Understanding the problem

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

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.

## 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.

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.

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.

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

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 $$\pm\sqrt{a}\pm\sqrt{b}\pm\sqrt{c}=0$$

## 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 Polya 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.

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## What is non routine and routine in math?

- 1 What is non routine and routine in math?
- 2 What are number routines?
- 3 What are examples of routine problems?
- 4 What are some instructional routines?
- 5 What is number sense examples?
- 6 Which is an example of a math language routine?
- 7 How does the ways to make a number routine work?

What is an instructional math routine?

## What are number routines?

What are routine problems in mathematics?

## What are examples of routine problems?

## What are some instructional routines?

## What is number sense examples?

Do you have any math routines you use in your classroom?

## Which is an example of a math language routine?

What is the purpose of mathematical routine 2?

## How does the ways to make a number routine work?

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## Routine and non routine problems

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Routine and Non-routine problem solving

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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

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## NON-ROUTINE PROBLEMS IN PRIMARY MATHEMATICS WORKBOOKS FROM ROMANIA

Please select one or more items.

Keywords: mathematical problem solving; non-routine problems, textbook.

- Adding and substracting natural numbers;

- Multiplying natural numbers between 0 and 10 (multiplication table);

- Dividing natural numbers (divisions which can be calculated using the multiplication table).

We classified the problems using two different classifications.

- routine exercises, which require application of a known algorithm.

In the second classification we grouped the problems in three categories:

- based on an operation chain;

- based on a graphical representation;

Dividing tasks in routine, gray-area and puzzle-like categories

Dividing tasks by how the operations needed to carry out are given

The number of tasks where pupils have to formulate word problems is very limited.

[13] Mayer, R. E. (1992). Thinking Problem Solving, Cognition. New York: Freemann.

[22] Pólya, G. (1945). How to Solve It. Princeton University Press

[23] Pólya, G. (1957). How to Solve It. Doubleday

[25] Schoenfeld, A. H. (1985). Mathematical problem solving. San Diego: Academic Press, Inc.

Iuliana Marchis, Babes-Bolyai University, Cluj-Napoca (Romania).

This research was founded by Domus Hungarica research grant, contract number DSZ/34/2012.

Copyright Babes Bolyai University, Didactics of Exact Sciences Chair 2012

## Suggested sources

## Professor of Mathematics, University of North Georgia

## 3. Non-routine Problem Solving

The best non-routine problems are:

- Open-ended,
- Allow multiple, correct solution paths, and
- Are completely new to the person trying to solve it.

International Conference on Artificial Intelligence in Education

AIED 2020: Artificial Intelligence in Education pp 409–413 Cite as

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

- Huy A. Nguyen ORCID: orcid.org/0000-0002-1227-6173 13 ,
- Yuqing Guo 13 ,
- John Stamper 13 &
- Bruce M. McLaren 13
- Conference paper
- First Online: 30 June 2020

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

## 1 Introduction

## 2 Assessing Students’ Problem-Solving Skills

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

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

## Pattern Identification.

## Visualization.

Participants’ attempts at visualizing the problem in their draftworks.

## Computation.

## 3 Developing a Tutoring System for Flexible Problem-Solving

## 4 Conclusion

## Author information

Carnegie Mellon University, Pittsburgh, PA, 15213, USA

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

You can also search for this author in PubMed Google Scholar

## Corresponding author

Correspondence to Huy A. Nguyen .

## Editor information

Federal University of Alagoas, Maceió, Brazil

Prof. Dr. Ig Ibert Bittencourt

University College London, London, UK

Carleton University, Ottawa, ON, Canada

University of Malaga, Málaga, Spain

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## NON-ROUTINE MATHEMATICAL PROBLEMS: PHENOMENOLOGICAL ANALYSIS OF POSITIVE AND NEGATIVE LEARNING IMPACT

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Creative problem solving or Non-Routine Mathematics involves finding solutions for unseen problems or situations that are different from structured Maths problems. There are no set formulae or strategies to solve them , and it takes creativity, flexibility and originality to do so.

Non-routine problem fRoutine problem defined as a problem in mathematic lesson that involves easy and simple problem solving. It present a question to be answered with out need certain strategies. It means, the routine problem can be solved by direct application of previously learned algorithms. fExample for routine problem:

• 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. Oftentimes, these problems can be solved in multiple ways. -Melissa from teacherthrive.com

(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?

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. What is an instructional math routine?

Solving routine and non routine problems involving multiplication without or ... EMELITAFERNANDO1 4.4k views • 24 slides Math 7 Curriculum Guide rev.2016 Chuckry Maunes 18.9k views • 26 slides K to 12 math curriculum Melvin Rey Alim 28.2k views • 50 slides Grade 7 Learning Module in MATH Geneses Abarcar 585.3k views • 261 slides

This video shows the difference between routine and Non-routine Problems in Mathematics with some examples for each type of problem. Changing Improper Fraction into Mixed Number and Vice Versa...

Articulate the importance of Mathematics in one; Harnessing Inclusive Opportunities of Globalization around the world in making things happen; ... 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 ...

My Research and Language Selection Sign into My Research Create My Research Account English; Help and support. Support Center Find answers to questions about products, access, use, setup, and administration.; Contact Us Have a question, idea, or some feedback? We want to hear from you.

a non-routine problem. Leading math educators argue that non-routine problems are indispensable for the development of students' problem-solving and reasoning skills (e.g. Polya, 1957; Schoenfeld, 1992). Non-routine problem-solving strategies can be defined as procedures used to explore, analyze and examine aspects of non-routine problems to ...

Non-Routine Problems 1. Jean thinks of a number. If he multiplies it by 30, then divide by 5 then subtract by 250, he will have an answer of 62. What was Jean's number? 2. Mr. Thomas raise chickens and cows in a farm for a total of 32 heads in all. How many of each does he have if there are 98 feet?Lesson 2

the mathematics teacher candidates in non-routine problems at different difficulty levels. The descriptive survey model was preferred because it can be used to summarize the characteristics (capabilities, preferences, behaviors, etc.) of the study group, as it is intended to describe a situation that exists in the past or still

This study examines several activity tasks incorporated with non-routine problems through the use of an emerging mathematics framework, at two junior colleges in Brunei Darussalam.

Non-routine problems are those for which most of us have no idea what the solution is before we start. Scientists, mathematicians, doctors, lawyers, teachers - no one has a head start on these problems. The best non-routine problems are: Open-ended, Allow multiple, correct solution paths, and Are completely new to the person trying to solve it.

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 [] - is becoming increasingly important in the 21st century [].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 [].

The findings of this study underscore the importance of implementing writing as an integral part of the mathematics curriculum and emphasize the need for additional research on writing in mathematics. The present study is related to seventh and eighth grade students' solutions to non-routine problems.

Specifically, non-routine problems were given emphasis in this study to understand education students' dispositions, appreciations, and experiences in applying mathematics in practical life situations and, consequently, will employ a more personalized teaching technique which suits their mathematical learning styles especially in doing problem …

The Inside Problem Solving problems are non-routine math problems designed to promote problem-solving in your classroom. Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

In an effort to inculcate higher order thinking skills (HOTS) among students, the form of mathematical problems given to them should be changed from routine to nonroutine problems. To achieve...