The Case For Doing Hard Things
We ask hard questions because so many of the problems worth solving in life are hard. If they were easy, someone else would have solved them before you got to them. This is why college classes at top-tier universities have tests on which nearly no one clears 70%, much less gets a perfect score. Theyâre training future researchers, and the whole point of research is to find and answer questions that have never been solved. You canât learn how to do that without fighting with problems you canât solve. If you are consistently getting every problem in a class correct, you shouldnât be too happy â it means you arenât learning efficiently enough. You need to find a harder class.
The problem with not being challenged sufficiently goes well beyond not learning math as quickly as you can. I think a lot of what we do at AoPS is preparing students for challenges well outside mathematics. The same sort of strategies that go into solving very difficult math problems can be used to tackle a great many problems. I believe weâre teaching students how to think, how to approach difficult problems, and that math happens to be the best way to do so for many people.
A Closer Look At Sat Math Sections
When working on SAT, the third and fourth parts are dedicated to mathematics. The math part has 54 mathematics questions: 10 student-produced response questions and 44 multiple-choice questions.
The ten multiple-choice questions require students to workout math problems and select answers from the choices provided. But the student-produced response questions are some of the most hard math questions because students have to calculate the right answers: there are no answer choices.
Notably, the SAT math questions are drawn from four main mathematics areas: algebra and functions number and operations data analysis, probability, and statistics and geometry and measurements.
Therefore, you better be well prepared in all of the areas if you want to pass and join college.
C What Do I Already Have
You know the question. You know what you need in order to solve it. Now you can simply fill in the equation with what you have already been given.
Dont get lost in unimportant details. Math word problems are notorious for giving you too many details. Thats why this step is the last of the three questions.
Some students try to figure out what all they have first. They read the problem, write out all the details they have been given, and then expect to solve it from there. Instead, they often experience detail-overload.
But you will save yourself an enormous amount of time if you know you are looking to answer first. Knowing the question is more important than knowing what details you have. Only when you know the question you are answering and what you need in order to answer it can you then find the right details to answer it correctly.
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How To Easily Make Your Own Math Word Problems & Word Problems Worksheets
Armed with 120 examples to spark ideas, making your own math word problems can engage your students and ensure alignment with lessons. Do:
- Link to Student Interests: By framing your word problems with student interests, youll likely grab attention. For example, if most of your class loves American football, a measurement problem could involve the throwing distance of a famous quarterback.
- Make Questions Topical: Writing a word problem that reflects current events or issues can engage students by giving them a clear, tangible way to apply their knowledge.
- Include Student Names: Naming a questions characters after your students is an easy way make subject matter relatable, helping them work through the problem.
- Be Explicit: Repeating keywords distills the question, helping students focus on the core problem.
- Test Reading Comprehension: Flowery word choice and long sentences can hide a questions key elements. Instead, use concise phrasing and grade-level vocabulary.
- Focus on Similar Interests: Framing too many questions with related interests — such as football and basketball — can alienate or disengage some students.
- Feature Red Herrings: Including unnecessary information introduces another problem-solving element, overwhelming many elementary students.
A key to differentiated instruction, word problems that students can relate to and contextualize will capture interest more than generic and abstract ones.
Solving Equations Using The Division Property
Consider the equation
3x = 12
The solution to this equation is 4. Also, note that if we divide each member of the equation by 3, we obtain the equations
whose solution is also 4. In general, we have the following property, which is sometimes called the division property.
If both members of an equation are divided by the same quantity, the resulting equation is equivalent to the original equation.
Example 1 Write an equation equivalent to
-4x = 12
Solution Dividing both members by -4 yields
In solving equations, we use the above property to produce equivalent equations in which the variable has a coefficient of 1.
Example 2 Solve 3y + 2y = 20.
We first combine like terms to get
5y = 20
Then, dividing each member by 5, we obtain
In the next example, we use the addition-subtraction property and the division property to solve an equation.
Example 3 Solve 4x + 7 = x – 2.
Solution First, we add -x and -7 to each member to get
4x + 7 – x – 7 = x – 2 – x – 1
Next, combining like terms yields
3x = -9
Last, we divide each member by 3 to obtain
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B What Do I Need In Order To Find The Answer
After you know what youre being asked, you can then think about what it will take to get that answered. You should have some idea at this point of the equation that will be needed to find a solution.
Specifically, were talking about equations here and the most important variables.
If you know what you are looking for and you can then name the pieces you need to find, even the most difficult problems become extremely manageable.
For a simple example, lets say youve been given a question and you realize you are being asked how tall a ladder youll need to paint a wall .
After discovering whats being asked of you the length of the ladder you realize that the Pythagorean Theorem is what youll need to solve it. This means our final question needed to solve this math word problem will be super easy.
How To Solve Squares
You can re-write a square equation into numbers that are easier to deal with using this formula
n^2 = + d^2
where n is the number to be squared, and d is the difference
Heres an example
57^2 = + 3^2# we add 3 to 57, as 60 is easier to multiply than 57, and subtract 3 from the second 57-> 60 x 54 + 9 = 3000 + 240 + 9 = 3249
The ultimate example is when you are squaring a number ending in 5, then round one number up to the nearest 10, the other number down to the nearest 10, and add 25.
65^2 = + 5^2 = 4200 + 25 = 4225
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Best Ways On How To Learn Math
Practice, Practice & More Practice
Practice makes a man able to solve the problems with perfection. Most of the students think that they can solve the math problem by just paying attention to the class. What do you think? Is it possible to solve math problems by just reading and listening? Of Course not, because math is not similar to other subjects. You find problems in math, and you have to solve it using some math formulas.
To solve any problems in math, you need to do a lot of practice. The more you practice to solve the math problems, the more you get better. There are lots of ways to solve math problems in different ways. Each math problem has its characteristics. Therefore these problems have different ways to solve it. Thus the more you explore the best ways to solve the problems, the better you perform in math exams. There is no shortcut to solve math problems without doing the proper practices to solve them.
Using 1000 Subtraction Rule:
Its not just addition, when it comes to maths, its like everything that is related to mathematical is troubling.
For example, while subtracting a large number from a thousand, keep note that a person need to subtract the first number from nine and second numbers from nine but except last number.
Instead of subtracting the last number from nine, make use of ten while subtracting.
Suppose the number is 665, now while subtracting it from 1000 make sure that a person use 9 each time, except for the last time and that is 6-9=3Therefore, the answer is 1000-665=335
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Strategies For Difficult Math Problems And Beyond
Here are a few strategies for dealing with hard problems, and the frustration that comes with them:
Do something. Yeah, the problem is hard. Yeah, you have no idea what to do to solve it. At some point you have to stop staring and start trying stuff. Most of it wonât work. Accept that a lot of your effort will appear to have been wasted. But thereâs a chance that one of your stabs will hit something, and even if it doesnât, the effort may prepare your mind for the winning idea when the time comes.
We started developing an elementary school curriculum months and months before we had the idea that became Beast Academy. Our lead curriculum developer wrote 100â200 pages of content, dreaming up lots of different styles and approaches we might use. Not a one of those pages will be in the final work, but they spurred a great many ideas for content we will use. Perhaps more importantly, it prepared us so that when we finally hit upon the Beast Academy idea, we were confident enough to pursue it.
Simplify the problem. Try smaller numbers and special cases. Remove restrictions. Or add restrictions. Set your sights a little lower, then raise them once you tackle the simpler problem.
Focus on what you havenât used yet. Many problems have a lot of moving parts. Look back at the problem, and the discoveries you have made so far and ask yourself: âWhat havenât I used yet in any constructive way?â The answer to that question is often the key to your next step.
Probability And Data Relationships Word Problems
Best for: 4th grade, 5th grade, 6th grade, 7th grade
89. Understanding the Premise of Probability: John wants to know his classs favourite TV show, so he surveys all of the boys. Will the sample be representative or biased?
90. Understanding Tangible Probability: The faces on a fair number die are labelled 1, 2, 3, 4, 5 and 6. You roll the die 12 times. How many times should you expect to roll a 1?
91. Exploring Complementary Events: The numbers 1 to 50 are in a hat. If the probability of drawing an even number is 25/50, what is the probability of NOT drawing an even number? Express this probability as a fraction.
92. Exploring Experimental Probability: A pizza shop has recently sold 15 pizzas. 5 of those pizzas were pepperoni. Answering with a fraction, what is the experimental probability that he next pizza will be pepperoni?
93. Introducing Data Relationships: Maurita and Felice each take 4 tests. Here are the results of Mauritas 4 tests: 4, 4, 4, 4. Here are the results for 3 of Felices 4 tests: 3, 3, 3. If Mauritas mean for the 4 tests is 1 point higher than Felices, whats the score of Felices 4th test?
94. Introducing Proportional Relationships: Store A is selling 7 pounds of bananas for $7.00. Store B is selling 3 pounds of bananas for $6.00. Which store has the better deal?
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A For What Am I Looking
This is the biggest question. It shapes your entire time answering the question. Take the following situation for example:
A plane leaves Toronto, Ontario , heads to Newark, New Jersey, and then heads to Seattle, Washington. Along the way to Seattle, a storm forces the plane to head north in order to get around it. The plane ends up crossing the Canadian border, but then has some engine trouble. When the plane is at 30,000 feet, an engine fails, and the plane has to attempt an emergency landing. Unfortunately, the plane crashes and it does so directly on the Canada-United States border. Where will they bury the survivors?
Well let you think about that question for a few minutes. You can see the answer at the bottom of this post if youre curious. But I recommend rechecking the most important detail before you guess: For what I am looking?
Dont get lost in details. Get the question right before anything else. You must know what your math word problem is asking.
If you dont know what you are looking for, youll end up missing it every time.
Develop The Plan To Solve It
There are four simple steps which one needs to go through in order to develop a plan to solve it. The steps are as mentioned below:
- Firstly one needs to figure out the formula you will need to solve the problem. Here you need to spend some time reviewing the concepts in your textbooks which will help you solve the problem
- You need to write down your need in order to get the answer to your problem. For this, you need to make a step-by-step list of the things which you need to solve the problem and also help you to stay organised
- In case there is an easier problem which is available then you could probably work on that first to solve it. Sometimes, the formulas are repetitive for solving both the problems. This will give you some more time to solve the difficult problem
- You can make an educated guess about the answer so that you can try and get the estimate the answer before you start to solve it. Here you can identify the number and other factors as well that will contribute to the same. Lastly, review the estimate and then check if you haven’t left out on anything
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Solve A Series Of Simple Math Problems
Keep breaking it down. You work the math. Dont let the math work you. Victor writes:
This seems like a lot of steps to go through, but the point is that you can quickly simplify the problem so you can solve it easily in your head. Often is is much easier to solve a long series of simple math problems than a short series of complicated ones. You may not need to break down this problem into as many simple parts as I did, but the process of doing so is important when it comes to quantitative assessment tests.
How To Solve Trigonometric Equations
Trigonometric functions can be used for various purposes, like for planning urban issues. To solve the basic trigonometric problems, you need to follow the below-mentioned steps:
- First of all, understand the trigonometric identities and reference angles.
- Grasp all necessary ratios that lie between the 0 degrees to 360 degrees of a graph.
- Proceed by using the substitution formula.
- Solve the math problems to get the answer in degrees or radiations.
Solve sin + 2 = 3 in interval 0° x < 360°.
sin + 2 = 3
sin = 3 2 = 1
We know that sine 1 lies in the first quadrant that is between 0 degrees to 90 degrees. Moreover, we are well versed that sin = 1. Therefore, the answer will be:
x = 90 degrees.
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Multiplying Even And Odd Numbers By Five:
Whenever there is a multiplication problem by five, there is one simple rule or trick that can get one the answer without wasting much time.
And that would be while multiplying any number from five, remember to follow the trick.
For example, while multiplying five with an even number, take the even number and make half of it and add zero at the end of the number to get the answer.
And on other hand, with an odd number, deduct one from the odd number and make half of it, then just add five at the side of the number to get the answer
With an even number, suppose the number is 5*6Take even number 6, half of it is 3Add 0 aside, therefore, the answer is 5*6=30Trick: With an odd number, suppose the number is 5*7Take odd number, deduct 1 from the number 7-1=6Make half of 6, half is 3Add 5 aside, therefore the answer is 5*7=35.
Round Off Numbers While Adding:
Most of the time adding two or more than two numbers seems difficult and to make addition simple a person can consider rounding off those two numbers and try to get the answer.
For example, if there are numbers which is 535 and 346. While looking at the number it is not that hard to solve, but there is a simple way to solve the problem and that is first, round off both the numbers like this
540 and 350, now apply addition and the answer is 540+350=890. Then, 540-535=5 , 350-346=4 i.e, 5+4=9Now deduct 9 from 890= 890-9=881 and the answer is 881.
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From Group To Independent Problem
After rereading the above-grade-level problem as a class, each student receives a word problem printout differentiated based on their ability. Students work through a five-step problem-solving procedure based on the reading protocol they use as a class. They work independently, or in small groups if they need more support. A checklist of the steps guides them through the problem.