Category mathematics

Apr 23 2018

Challenging Algebra Problems

CONTENTS

This article includes the following sections:

  • Ways to Challenge Advanced Students
  • How to Make Challenging Algebra Problems
  • A Couple of Sample Problems (from my latest book)
  • 50 Challenging Algebra Problems – Fully Solved (my latest workbook)

WAYS TO CHALLENGE ADVANCED STUDENTS

Some students catch on quickly, while others struggle to understand.

So how can you engage and challenge the better problem-solvers without blowing the minds of the other students?

  • If you’re a teacher, you can include a challenging bonus question on an assignment, quiz, or exam.
  • If you’re a parent, you can search for supplemental material with challenging problems.
  • If you’re a student, you can try solving harder unassigned problems, reading ahead in your textbook, or looking for supplemental material.

HOW TO MAKE CHALLENGING ALGEBRA PROBLEMS

For teachers (or very knowledgeable parents) who would like to make challenging problems, here are some ideas for how to go about it:

  • Disguise the quadratic equation. One way is to make more than the usual three terms and put some on both sides of the equation, so that students need to combine like terms. Another way is to substitute y = xc, where c is a power of your choice. For example, if you let c = 2, your equation will look like a quartic, but if students make the substitution y = x2, the quartic will turn into a quadratic. Another example is to let c = 1/2 and replace y = x1/2 with a squareroot. When students see that squareroot, it won’t seem like a quadratic. Tip: Think of what you want the answers to be, like x = 2 and x = 4/3, and then f.o.i.l. this into the quadratic, like (x – 2)(x – 4/3) = 0. (In this example, I would multiply both sides by 3 to get rid of the fraction.)
  • Make a system of equations look different from normal. Students are used to seeing things like 3x + 2y = 21 and 4x – 5y = 5. Instead, you might write the second equation like xy = 15, or y = 15/x. Another variation is to replace x with 1/x and y with 1/y. In my example, you would get 3/x + 2/y = 21 and 4/x – 5/y = 5. If students define t = 1/x and u = 1/y, they could then solve for t and u like usual, and then solve for x and y from t and u. Tip: When making a system, first decide on the answers, like x = 9 and y = 12, next make the coefficients, and lastly determine the constants.
  • Put squareroots in problems they should know how to solve, but which usually don’t have squareroots in them. For example, if you have squareroot(3) and 1/squareroot(3) in the same equation, the terms can be combined by rationalizing the denominator to rewrite 1/squareroot(3) as squareroot(3) / 3. (Multiply the numerator and denominator by the squareroot of 3, and use the rule that the squareroot of 3 times itself equals 3.) Then factor out squareroot(3) to combine the terms.
  • Put variables inside of squareroots. For example, you could write something like squareroot(x + 8) = x + 2. When you square both sides, you f.o.i.l. out the (x + 2) squared. There are many other ways to write a solvable equation with a variable in a squareroot. You could have things like squareroot(x) or 1/squareroot(x).
  • Include an extra variable that cancels out. If students count equations and unknowns, it will seem like the problem can’t be solved. They’re partly right: The variable that cancels out will be indeterminate. But since one variable cancels, in this case it will be possible to solve for the other unknown(s).
  • Use fractional exponents like 2/3 or 3/4. For example, if you isolate x to the power of 2/3, you can solve for x by raising both sides of the equation to the power of 3/2. You can make the numbers work out so that a calculator isn’t needed for students who understand the fractional powers. For example, if you raise 8 to the power of 2/3, you can do this in your head, since the cube root of 8 equals 2 (the power of 1/3 means to take the cube root, and you can think of 2/3 as involving a square and a cube root in any order). Of course, you can easily check your answer with a calculator, just to be sure.
  • Put the unknown in a denominator. It’s amazing how many students freeze up over this, yet it’s not uncommon in science for variables to appear in the denominators of formulas. The progression of this occurs when students need to make a common algebraic denominator to solve for the unknown.
  • Give problems that involve inequalities, especially when there are minus signs. For example, make a system of inequalities that involves substitution.
  • Look for applications of algebra in higher-level math, physics, engineering, chemistry, economics, astronomy, and other subjects. (Sometimes, the problem isn’t really that hard, but it just seems hard because it’s out of context. But there are also challenging applied algebra problems in many subjects.)
  • Make word problems that apply algebra. There is great variety in what can be done in the way of algebraic word problems.
  • Prepare an elaborate solution that has an intentional mistake and ask students to figure out what is wrong. Ideally, the mistake would seem subtle, and would be a common mistake that many students tend to make on their own (like not distributing a minus sign correctly).

There are two types of problems that especially appeal to me:

  • One kind where the problems look harder than they really are. I love it when students feel convinced that the problem is impossible based on what they know, but when they eventually figure it out on their own. This happens sometimes when a problem appears out of context, involves an application of algebra, or when a simple substitution makes something seem a lot more complicated than it really is. Word problems often fall in this category, too.
  • The other kind is where the problem looks much easier than it really is. The structure might be so simple that at first glance you expect it to be a piece of cake, but there is some subtle aspect to it that it turns out to be much harder than it seemed. The catch is that the problem still needs to be reasonably solvable, so that the top students can still figure it out given enough time and sufficient background. It’s surprising how many mathematical problems look easy, but turn out not to be nearly as easy as they look.

A COUPLE OF SAMPLE PROBLEMS

Here are a couple of sample problems from my latest math workbook.

50 CHALLENGING ALGEBRA PROBLEMS (FULLY SOLVED)

My latest math workbook includes 50 challenging algebra problems, space to try and solve them (in the paperback edition), and full solutions on the page following each problem (you won’t be able to see the full solutions until you turn the page).

There is a healthy variety of problems, involving a range of techniques. There is also a good range in the level of difficulty from problem to problem.

50 Challenging Algebra Problems (Fully Solved)

Paperback ISBN: 19416912344

Kindle ASIN: B07C7WHV3R

CHRIS MCMULLEN, PH.D.

Copyright © 2018 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Newest releases (in math):

  • 50 Challenging Algebra Problems (Fully Solved)
  • Fractions Essentials Workbook with Answers
  • 300+ Mathematical Pattern Puzzles

Improve Your Math Fluency. Build fluency in:

  • arithmetic
  • long division
  • fractions
  • algebra
  • trigonometry
  • graphing
  • physics
Feb 13 2018

What Should Students Learn about Fractions?

ESSENTIAL FRACTION SKILLS

Not everyone loves fractions, but fraction skills are important for a number of reasons.

  • It helps to be fluent with fraction skills when you take algebra, trigonometry, and calculus courses. When you solve for an unknown in algebra, the answer doesn’t always turn out to be a whole number. One reason that cross multiplication is difficult for many students is that the problems inherently involve fractions. In calculus, simple polynomial anti-derivatives naturally involve fractions (like the integral of x squared, which is one-third of x cubed plus a constant).
  • We also use fractions in science courses like chemistry and physics. So if a student is struggling with fractions in math courses, these struggles also impact science courses. On the other hand, if a student takes the time to master fractions in math, this becomes an asset when fractions appear in other courses. Decimals and percentages (which are basically just other forms of fractions where the denominator is a power of ten) are very common forms of fractions in science and engineering.
  • Fractions are actually pretty common. Almost every day, we see something happen a few times, and start to wonder how often that happens. Maybe your cell phone is doing something funny, or maybe you notice a quirk in somebody’s personality. If you give attention to this, you’ll realize that it’s common to wonder, “How often does that happen?” Many people just guess at it, but their guesses aren’t always realistic (especially when something obviously isn’t extremely common, but they say something like 90% of the time). If you have a small sample, like it happened 3 out of the last 8 times, you can use the fraction (in this case, 3/8) as a projection. You might convert 3/8 to a percentage and see that it equates to 37.5%, for example.
  • If you have a ruler marked in inches, you will see fractions in many common measurements. The fraction will be something like 5/12, 7/8, 3/4, or 1/2. If you have a good feel for fractions, it helps to interpret these numbers. For example, if you measure the diameters of two different balls, and one ball is 3/8″ while the other is 1/3″, can you tell which is bigger just from the numbers?

With the importance of fractions in mind, following is my list of essential fraction skills, based on my experience helping students learn how to apply their math skills in science classes and laboratories.

  • Visual association. Students should be able to draw pie slices to represent fractions, or should be able to write down a fraction to represent a pie slice.
  • Terminology. You can’t discuss fractions with anybody or understand a lecture about fractions if you don’t understand what the different words and phrases mean, like numerator, denominator, reciprocal, common denominator, reduced fraction, greatest common factor, decimal, percentage, proper fraction, improper fraction, mixed number, ratio, and proportion.
  • Reducing fractions. Students should be fluent in reducing fractions down to their simplest form. For example, 8/12 reduces to 2/3 if you divide the numerator (8) and denominator (12) both by 4.
  • Common denominators. Given two different fractions, like 4/7 and 3/5, students should be able to find a common denominator.
  • Mixed numbers. Students should be able to convert between improper fractions and mixed numbers.
  • Addition and subtraction. Students should be able to find a common denominator in order to add or subtract fractions.
  • Multiplication. Students should be able to multiply fractions. (It’s easier than addition or subtraction.)
  • Reciprocals. Students should know how to find the reciprocal of any fraction or whole number.
  • Division. Students should know that dividing two fractions is equivalent to multiplying by the reciprocal of the second fraction.
  • Decimals. Students should be able to convert fractions into decimals or decimals into fractions.
  • Percentages. Students should be able to convert decimals into percentages or percentages into decimals.
  • Repeating decimals. Students should be familiar with repeating decimals and how they relate to fractions.
  • Word problems. You know you understand the concepts well and can apply them when you can solve a variety of word problems that involve fractions.

CHRIS MCMULLEN, PH.D.

Copyright © 2018 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Newest releases (in math):

  • Fractions Essentials Workbook with Answers
  • 300+ Mathematical Pattern Puzzles

Improve Your Math Fluency. Build fluency in:

  • arithmetic
  • long division
  • fractions
  • algebra
  • trigonometry
  • graphing
  • physics
Jul 16 2015

Multiplication Facts 0-10 Memory Tips

Multiplication Fives

MULTIPLICATION FACTS 0-12

When students learn multiplication, it starts out easy.

Too easy, as 0 times anything equals 0 (e.g. 0 x 8 = 0, while anything times 1 equals itself (e.g. 4 x 1 = 4).

But it doesn’t stay easy. Most students struggle to remember multiplication facts where one of the numbers is 6, 7, 8, or 9.

However, there are some tips to aid in memorization.

For one, mirror images don’t matter. The word for this is commutative.

This means 2 x 8 = 8 x 2, for example. Both equal 16.

So if you know 6 x 8 = 48, you also know that 8 x 6 = 48. You don’t need to memorize both.

One of the tricky multiplication facts is 7 x 8. But it’s easy if you remember the trick.

Remember 5678. These are 5 thru 8 in order. 56 = 7 x 8. Piece of cake, huh?

The 9’s are easy. The answer is one decade less than multiplying by 10, then make the two digits add up to 9.

For example, 9 x 7 is in the 60’s (because 10 x 7 = 70, one decade less is 60). It’s 63 since 6 + 3 = 9.

Another example is 5 x 9. It’s in the 40’s (one decade less than 5 x 10 = 50). It’s 45 since 4 + 5 = 9.

To do the trick, after you figure out the decade (by multiplying by 10 and then subtracting 10), subtract the tens digit from 9 to get the units digit.

For example, consider 9 x 6. Multiply 10 x 6 to get 60, and subtract 10 to make 50 (one decade less). Now subtract 5 (the tens digit of 50) from 9 to get 4. Therefore, 9 x 6 = 54.

Once you get the hang of it, this makes remembering the 9’s easy. Try out all the 9’s to get some practice.

If you’re good at doubling numbers quickly, try writing 6 as 2 x 3.

Then 6 x 7 = 2 x 3 x 7. If you know 3 x 7 = 21, double 21 to get 42.

Similarly, for 6 x 4 = 2 x 3 x 4, start with 3 x 4 = 12 and double 12 to make 24.

You can use the doubling trick for the 8’s, too. Just double the number 3 times.

For example, consider 5 x 8. Double 5 three times: 10, 20, 40. So 5 x 8 = 40.

Try 8 x 6. Double 6 three times: 12, 24, 48. Therefore, 8 x 6 = 48.

It works with the 4’s, also. Just double twice.

With 4 x 9, double 9 twice: 18, 36. So 4 x 9 = 36.

That leaves 7 x 7 = 49. You should know 7 x 9 from the 9’s trick.

You can make 7 x 6 and 7 x 8 from the doubling tricks. (The latter you can also know from the 5678 trick.)

5 and under are easier. So to complete the 7’s, you really just need to memorize 7 x 7 = 49.

This covers the 6 thru 9’s, which tend to be the trickier multiplication facts.

The 10’s and 11’s are easy. For the 10’s, just add a zero, as in 8 x 10 = 80 or 6 x 10 = 60.

For the 11’s, with 1 thru 9 just double the digits, like 3 x 11 =33 or 11 x 8 = 88. Get 11 x 10 = 110 from the 10’s trick (add a zero). Then you just need to memorize that 11 x 11 = 121 to complete the 11’s.

You can get the 12’s by doubling the 6’s. For example, knowing that 6 x 5 = 30, you can find that 12 x 5 = 60 by doubling 6 x 5.

Do you know any other tips for remembering multiplication facts 0-12? If so, please share them in the comments.

CHRIS MCMULLEN, PH.D.

Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

  • arithmetic facts
  • multi-digit arithmetic
  • long division with remainders
  • fractions, decimals, and percents
  • algebra and trigonometry
Jul 4 2015

Fibonacci Sequence & a Cool Pattern

Image from ShutterStock.

Image from ShutterStock.

FIBONACCI SEQUENCE

The Fibonacci sequence adds consecutive terms:

  • 1
  • 1
  • 1 + 1 = 2
  • 2 + 1 = 3
  • 3 + 2 = 5
  • 5 + 3 = 8
  • 8 + 5 = 13
  • 13 + 8 = 21
  • 21 + 13 = 34
  • 34 + 21 = 55
  • 55 + 34 = 89

Since the last two terms were 55 and 89, we would add these together to get 89 + 55 = 144.

Then you would add 144 and 89 to make 233, and so on.

I saw a cool pattern involving the Fibonacci sequence recently at the Mathemagical Site:

Fibonacci Triples via Mathemagical Site

This involves Fibonacci triples.

A Fibonacci triple consists of three consecutive numbers from the Fibonacci sequence, such as:

  • 1, 1, 2
  • 1, 2, 3
  • 2, 3, 5
  • 3, 5, 8
  • 5, 8, 13
  • 8, 13, 21
  • 13, 21, 34

As shown on the Mathemagical Site, the square of the middle number is always one less or one more than the product of the first and third numbers:

FT

Here are a few examples:

  • (2, 3, 5): 3 x 3 = 2 x 5 – 1
  • (3, 5, 8): 5 x 5 = 3 x 8 + 1
  • (5, 8, 13): 8 x 8 = 5 x 13 – 1

Curious about this, I’ve been working through the algebra, and finally came up with an algebraic proof, which follows.

My proof is algebraic and not necessarily obvious, but since I worked it out, I thought I would share it here. 🙂

We begin with two facts about the Fibonacci sequence:

FT3

These are two different ways of saying that if you add two consecutive numbers from the Fibonacci sequence, you get the next number in the sequence.

Now solve for xn in each sequence:

FT4

Multiply these together:

FT2

Foil this out:

FT5

Recall that

FT6

Plug this into the first term on the right-hand side of the previous equation:

FT7

Distribute:

FT8

Two of these terms cancel (the remaining terms are rearranged):

FT9

Believe it or not, this basically concludes the proof. The remainder is basically interpreting this result.

This is a recursion relation that relates the square of the n-th term to the square of the previous term (xn-1 times itself).

Following is the Fibonacci sequence, labeling values of n:

  • n = 1 is 1.
  • n = 2 is 1.
  • n = 3 is 2.
  • n = 4 is 3.
  • n = 5 is 5.
  • n = 6 is 8.
  • n = 7 is 13.
  • n = 8 is 21.

Let’s plug in n = 3 and see what happens:

FT10

If instead you plug in n = 4, you get:

Ft11

Now just plug in these two expressions (x3x3 – x4x2 and x4x4 – x5x3) into the previous recursion relation and you can prove that all of the Fibonacci triples satisfy one of the following relations:

FT

That is, if x3x3 – x4x2 = 1 and x4x4 – x5x3 = -1, the previous recursion relation gives similar expressions for x5x5 – x6x4, x6x6 – x7x5, and so on.

CHRIS MCMULLEN, PH.D.

Copyright © 2015 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

  • Algebra Essentials Practice Workbook with Answers
  • Trigonometry Essentials Practice Workbook with Answers
  • Learn or Review Trigonometry: Essential Skills
Sep 14 2014

Breaking It Down Doesn’t Mean Dumbing It Down

Shattered

TEACHING & COMMUNICATION

I’ve taught from both sides of the spectrum:

  • high-ability, highly motivated math/science students in advanced courses
  • students struggling to survive the most basic physical science classes

Yet, in both cases, I’ve seen two common factors:

  • even the brightest students benefit when the material is broken down
  • it really doesn’t help anyone to dumb it down

WHAT’S THE DIFFERENCE?

Breaking it down is different from dumbing it down:

  • Breaking it down means showing how a more advanced concept is built from more basic concepts.
  • Dumbing it down means accepting a simpler, but not as correct or complete, concept as a substitute for a more advanced concept.
  • (Dumbing it down sometimes also refers to the way that this is done, making the person feel inferior, i.e. unable to understand the more advanced concept.)

Breaking it down can be a very helpful skill for communicating effectively:

  • It’s crucial to master the fundamentals in order to understand more advanced concepts.
  • First, teach the fundamentals very clearly.
  • Then build up more advanced concepts from these fundamentals.

Even the most advanced students can better understand advanced concepts by understanding how they are built up from more fundamental concepts.

Very often, when an advanced student makes a mistake applying an advanced concept, it’s because the student doesn’t fully understand the underlying fundamental concepts or doesn’t fully understand the relationship between those fundamental concepts and the more advanced concept from which it is built.

Advanced concepts are more accessible to struggling students, too, when they are broken down into more fundamental concepts.

(I would go so far as to say that when you can’t explain an advanced idea in simpler terms, then you really don’t understand that advanced concept as well as you should. However, there are occasions when it would take an enormous amount of time to fill in the gaps between the simpler terms and the more advanced terms. For example, try breaking down string theory so that anyone can understand it without sacrificing the material in any way. It’s a formidable task, and a few authors have done an excellent job with this, but there sure isn’t a suitable five-minute explanation to be found.)

BREAK IT DOWN, DON’T DUMB IT DOWN

The keys to breaking it down without also dumbing it down are:

  • Don’t sacrifice the content. Let the fundamentals build up to the final product.
  • Don’t oversimplify the advanced concept, losing important aspects of the complete, correct formulation.
  • Don’t settle for a lower level of understanding in the final stage.
  • Don’t make the student feel inferior in the process.

My goal as a teacher is to make it possible for every student to understand the material well in its most complete, correct formulation.

Not every student may reach this level of understanding along the same path, however. The first step is to master the fundamentals, and then build more advanced concepts up from this foundation.

(Of course, learning also involves motivation and diligence on the part of the student. My goal is to make learning possible, not to offer students a path of no effort. Unfortunately, not every student will choose to follow the path to success as you suggest it.)

EXAMPLE

I will illustrate the distinction between breaking it down and dumbing it down with an example.

I will use Newton’s first law of motion to illustrate this.

Objective: Students understand Newton’s first law when it is phrased, “Every object has a natural tendency to maintain constant momentum.”

Unfortunately, it is common for physical science textbooks and courses to simply dumb this down. Here are a couple of examples of how it can be dumbed down:

  • Every object has a natural tendency to maintain constant speed.
  • An object in motion tends to stay in motion, while an object at rest tends to stay at rest.

I call this dumbing it down because these statements aren’t equivalent to the original. The original statement is more complete and more precise. In practice, the distinction is important.

The original statement can be understood well, without sacrificing the content in any way, simply by breaking it down.

The way to do this is to first teach the fundamentals and then show how to build the main idea up from the fundamentals:

  • Velocity is a combination of speed and direction.
  • Momentum is mass times velocity.
  • All objects have a natural tendency to maintain constant momentum.
  • For an object with constant mass (that’s most objects), constant momentum means constant velocity.
  • Constant velocity means traveling in a straight line with constant speed.
  • So, for most objects, maintaining constant momentum means traveling in a straight line with constant speed.
  • For a rocket, mass isn’t constant because it’s ejecting steam. Rockets have a natural tendency to maintain constant momentum, but not a natural tendency to maintain constant velocity. That’s why it’s more precise to say momentum than velocity. For most objects, however, this distinction isn’t important.

There isn’t any point in this breakdown that a basic student can’t grasp, which means there is no reason that every student shouldn’t be able to grasp the full meaning of Newton’s first law of motion.

If you’re wondering what’s wrong with, “An object in motion tends to stay in motion, while an object at rest tends to stay at rest,” it’s because “motion” is quite vague, and, in fact, includes situations that are inconsistent with Newton’s first law. Acceleration, for example, is a kind of motion, but objects in motion do not tend to have acceleration. It’s more precise to say constant momentum or constant velocity than to say “stay in motion.”

(Of course, illustrating the concept by showing how to apply it to a variety of examples also aids greatly in helping students understand it. Also, some students will need additional help fully understanding specific points on the list above. However, with effort and assistance, everyone can master Newton’s first law of motion.)

DUMBED DOWN

Imagine a teacher has two classes. One is a physical science class; the other is a physics class.

The instructor teaches the physics class the proper definition of Newton’s first law, but dumbs this down for the physical science class.

Now, imagine that one physical science student and one physics student get together and chat.

They happen to see another student pass by on a skateboard. The skateboard strikes a curb and the student falls forward.

The two students excitedly begin describing how this relates to Newton’s first law.

The physical science student describes how the skateboarder stayed in motion.

The physics students corrects the physical science student, describing how the skateboarder maintained constant momentum.

“What’s momentum?” wonders the physical science student, suddenly feeling dumb. But why? There is no reason that the physical science student can’t grasp this same definition in the same terms.

It’s true that they are enrolled in different classes. Physics will inherently involve much more math. That doesn’t make the physical science student stupid, of course. (In fact, I’ve found some very capable math students in my physical science classes, and sometimes there are physics students with very rusty algebra skills.) It’s just the difference in the nature of the courses, including their goals. Physical science is focused more on the concepts (and also includes some chemistry, and perhaps geology or astronomy, in exchange for less physics coverage), and may contain more range and less depth. Physics is strongly oriented toward how to apply mathematics to solve problems (which requires understanding the fundamental concepts very well). But pick any concept that’s common to both courses, and there isn’t anything that a physical science student can’t learn just as well (in principle).

Let me take this a step further and flip the table.

Instructor X teaches physical science.

Instructor Y teaches physics.

Instructor X teaches the proper definition of Newton’s first law. Instructor Y, feeling the need to put more time on the math, dumbs down Newton’s first law in the interest of time.

Now imagine the previous example with the skateboarder. The physics student won’t be happy feeling that the physical science student has come to understand this law on higher terms!

CHRIS MCMULLEN

Copyright © 2014 Chris McMullen, author of the Improve Your Math Fluency Series