Category math

Mar 18 2021

How to Find Sine, Cosine, and Tangent of 15° and 75° (without using a Trig Identity)

SINE, COSINE, AND TANGENT OF 15° AND 75°

It is possible to find the sine, cosine, and tangent of 15° and also 75° without using a trig identity (and without using a calculator).

The trick is to begin with a 75°-75°-30° isosceles triangle, as shown above. Let AB = AC = 2. These are the sides opposite to the 75° angles. The remaining side, BC, is unknown at this point.

With side AB serving as the base, draw an altitude down from point C, as shown below. This divides the isosceles triangle into two right triangles. Triangle ACD is a 30°-60°-90° triangle and triangle BCD is a 75°-15°-90° triangle.

It is well-known that the sides of a 30°-60-°90° triangle come in the ratio 1:sqrt(3):2, with 1 opposite to 30°, sqrt(3) opposite to 60°, and 2 as the hypotenuse. Since AC = 2 is the hypotenuse, this means that CD = 1 and AD = sqrt(3).

Now we know that the altitude of triangle ABC is CD = 1, which is one side of triangle BCD. We can find side BD by subtracting AD from AB. Since AB = 2 and AD = sqrt(3), we get BD = 2 – sqrt(3).

The Pythagorean theorem can then be used to determine BC, which is the hypotenuse of triangle BCD.

Recall that BD = 2 – sqrt(3) and that CD = 1. After applying the Pythagorean theorem, apply the “foil” method from algebra: (x – y)² = x² – 2xy + y². Recall that (sqrt(3))² = sqrt(3) × sqrt(3) = 3.

To solve for BC, take the square root of both sides of the equation. Factor out the 4. Recall that sqrt(ab) = sqrt(a) × sqrt(b). Note that sqrt(4) = 2.

The above answer has one square root inside of another. There is a clever way to rewrite this without using a nested square root. Rewrite 8 as 6 + 2. The reason behind this is that (sqrt(6))² = 6 and (sqrt(2))² = 2. If you “foil” out (sqrt(6) – sqrt(2))², you get 6 – 2sqrt(12) + 2 = 8 – 4sqrt(3) because sqrt(12) = sqrt(4) × sqrt(3) = 2 sqrt(3) such that 2sqrt(12) = 2(2)sqrt(3) = 4 sqrt(3). If you’re still not convinced, note that sqrt(8 – 4sqrt(3)) is approximately 1.03527618 on a calculator, and that sqrt(6) – sqrt(2) is also approximately 1.03527618 on a calculator.

Since BC = 2 sqrt(2 – sqrt(3)) and BC = sqrt(6) – sqrt(2), it follows from the transitive property that 2 sqrt(2 – sqrt(3)) = sqrt(6) – sqrt(2). Divide both sides by 2 to get the following:

The following forms for BC are equivalent, but the right expression is considered to be “standard form.”

Now that we know BC, BD, and CD, we can easily find the sine, cosine, and tangent of 15°. In triangle BCD, note that BD is opposite to 15°, CD is adjacent to 15°, and BC is the hypotenuse. For the sine of 15°, note that (2 – sqrt(3)) / sqrt(2 – sqrt(3)) = sqrt(2 – sqrt(3)) for the same reason that x/sqrt(x) = sqrt(x). For the cosine of 15°, the answer 1/(sqrt(6) – sqrt(2)) is equivalent to the answer (sqrt(6) + sqrt(2))/4. Both answers approximately equal 0.965925826 on a calculator, but the rightmost expression is considered to be “standard form” because it has a rational denominator. We multiplied the numerator and denominator each by (sqrt(6) + sqrt(2)) in order to rationalize the denominator. This is called “multiplying by the conjugate.” The conjugate of sqrt(6) – sqrt(2) is sqrt(6) + sqrt(2) because the product of these two conjugate expressions is rational. When applying the “foil” method, the irrational terms cancel out.

Note that the sine of 15° is equivalent to the cosine of 75°, and that the cosine of 15° is equivalent to the sine of 75°. What is “opposite” for 15° is “adjacent” for 75°, and vice-versa. For the tangent of 75°, we again multiplied by the conjugate in order to rationalize the denominator.

Below is a summary of our final answers.

CHRIS MCMULLEN, PH.D.

Copyright © 2021 Chris McMullen, author of the Improve Your Math Fluency series of math workbooks

Newest releases:

  • Plane Geometry Practice Workbook with Answers (Volumes 1 and 2)
  • 101 Involved Algebra Problems (includes full solutions)
  • Trig Identities Practice Workbook with Answers
  • Logarithms and Exponentials Essential Skills Practice Workbook (with Answers)
  • Master Essential Algebra Skills Practice Workbook (with Answers)
  • Word Problems with Answers
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Jul 28 2018

The Answers Are Correct (Why Some Students Don’t Realize It)

CHECKING YOUR ANSWERS:

Many math books post the answers to selected questions in the back of the book. Most of my workbooks include the answers to every question in the back of the book.

A few students get a different form of the same answer, and mistakenly believe that the answer in the back of the book is wrong, when it’s really correct.

One common example occurs when the student gets a squareroot in the denominator, like:

The above answer isn’t in standard form. Most math teachers and authors would rationalize the denominator in order to express the answer in standard form. To rationalize the denominator, multiply both the numerator and denominator by the squareroot of 2, like the example below. Note that the squareroot of 2 times itself equals 2.

If you didn’t realize this, there is a simple way around it. Use a calculator. If you enter 1/sqrt(2) on your calculator, you will get 0.707106781. If you enter sqrt(2)/2 on your calculator correctly, you will get 0.707106781. Now you can see that the two seemingly “different” answers are exactly the same.

Unfortunately, many of the popular algebra solvers that you can find online don’t express their answers in standard form. So if you solve an algebra problem and get 1/sqrt(2) as your answer, an online algebra solver might give you the same answer, 1/sqrt(2), and then you might feel “confident” that the book’s answer, sqrt(2)/2, must be wrong because it’s different. But the book’s answer is actually the same answer. In fact, the book’s answer is better because it is expressed in standard form. It’s “better” in the sense that many teachers prefer for their students to express their answers in standard form.

Here is another common example. Suppose that a student obtains the following answer:

This answer isn’t in standard form. Most math teachers and authors would factor out the perfect square. Note that 8 equals 4 times 2. The number 4 is a perfect square, since the squareroot of 4 is 2 (put another way, 2 squared equals 4). You can factor out the perfect square as follows:


Again, you can check this with a calculator. Whether you enter sqrt(8) or 2*sqrt(2) on your calculator, either way you will get 2.828427125.

As a third example, suppose a student arrives at an answer of 9/6, but the book’s answer is 3/2. Once again, the student sees something different and assumes that one of the two answers must be incorrect. However, 9/6 equals 3/2. The difference is that the fraction 3/2 is in reduced form, whereas the answer 9/6 can be reduced. To see this, divide both the numerator and denominator of 9/6 by 3. You will get 9/6 = (9/3) / (6/3) = 3/2. If you use a calculator, you can check that 9/6 and 3/2 both equal 1.5.

This has been a problem for me with my workbook, Algebra Essentials Practice Workbook with Answers. An occasional review suggests that some of the answers are wrong. However, in every case that a student has mentioned a specific problem, and in every case that a student has contacted me by email to inquire about an answer, it has turned out that the student’s answer was, in fact, equivalent to the answer in the back of the book. The problem has usually been that the student didn’t rationalize a denominator, factor out a perfect square, or reduce a fraction.

As an author, it can be frustrating to know that the answers are correct, yet see a book review on Amazon suggesting otherwise. When I made the updated 2014 edition, an international math guru and I independently checked every single answer in this book with painstaking care. The answers are correct. The challenge is getting students to realize this.

On the product page, I worked out the full solutions, step by step, for some common questions that students have asked about, in order to help show that the answers are, in fact, correct. If students have questions, they are encouraged to contact me and give me an opportunity to teach them something that they might not have realized. Unfortunately, reviews that suggest that the answer key has mistakes discourage some students from using the book. Some customers would believe a random stranger’s review over a teacher with 20+ years classroom experience with a Ph.D. (If you think about it, if the student bought an algebra practice workbook, that student is probably not yet an expert on the subject of algebra.)

If you use any of my math workbooks and come to trust the answer key, please post a review to help other potential customers. If you suspect any of the answers may be wrong, please contact me (there is a contact me button my blog, for example) to ask about the answer. (It would also help if customers who know that the answers are reliable would vote on reviews as being helpful or not helpful.)

For my newest workbooks, like 50 Challenging Algebra Problems (Fully Solved), I’m now working out the full solution to every problem in the book so that students can see how I arrived at each answer. In many cases, I also offer alternative forms of the same answer (but it’s impossible to list every possible answer that a student might obtain). I hope that the full solutions will be helpful to students, and help them realize that the book’s answers are correct.

I have two new workbooks on the subject of calculus coming soon.

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
  • calculus (coming August, 2018)
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
Oct 25 2015

Mathematical Pattern Puzzle: Fill in the Missing Numbers

Puzzle Square

NUMBER PATTERN PUZZLE

Here is an exercise in pattern recognition.

It’s not a linear pattern.

This is an array, so there is a slight geometric element to the pattern.

See if you can figure out the missing numbers in the above puzzle.

Study the four arrays.

See if you can recognize the pattern.

Once you identify the pattern, apply it to the fifth array.

Spoiler alert.

If you scroll down too far…

You will run into the answer.

So stop scrolling down…

If you would like more time to solve the puzzle.

Ready or not.

Here it comes.

First the answer:

Puzzle Square Answer

Now the solution.

Begin with the top left number.

Double the top left number. That makes the top right number. 5 doubled = 10.

Now multiply the top two numbers. That makes the bottom left number. 5 times 10 = 50.

Now add the bottom left number to the top right number. That makes the bottom right number. 50 plus 10 equals 60.

CHRIS MCMULLEN, PH.D.

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

Newest releases:

  • 300+ Mathematical Pattern Puzzles
  • Basic Linear Graphing Skills Practice Workbook
  • Systems of Equations: Simultaneous, Substitution, Cramer’s Rule

Improve Your Math Fluency. Build fluency in:

  • arithmetic
  • long division
  • fractions
  • algebra
  • trigonometry
  • graphing
Sep 13 2015

Five Math Puzzles (pattern recognition): Can You Solve Them?

MATH PUZZLES

Here is a math puzzle challenge.

Hint: Each of the 5 patterns below has something in common.

Directions: See if you can figure out which numbers go in the blanks.

  • 1, 2, 4, 6, 10, 12, 16, 18, 22, _, _
  • 4, 6, 10, 14, 22, 26, 34, 38, _, _
  • 3, 7, 13, 19, 29, 37, _, _
  • 4, 9, 25, 49, 121, 169, 289, _, _
  • 5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, _, _

If you need help, you can find hints below.

But don’t scroll too far or you’ll run into the answers and explanations.

PUZZLE HINT

Each pattern above has something in common.

They all involve prime numbers.

A prime number is only evenly divisible by two integers: 1 and itself.

For example, 7 is a prime number because the only integers that can multiply together to make 7 are 1 and 7.

In contrast, 6 isn’t a prime number because 2 x 3 = 6 (in addition to 1 x 6).

Here are the first several prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37

Each of the puzzles above relates to these prime numbers.

When you’re ready, you can find answers and explanations below.

PUZZLE ANSWERS

Here are the answers and explanations to the math puzzles:

  • 28, 30. Explanation: Subtract 1 from each prime number: 2 – 1 = 1, 3 – 1 = 2, 5 – 1 = 4, 7 – 1 = 6, 11 – 1 = 10, etc.
  • 46, 58. Explanation: Double each prime number: 2 x 2 = 4, 3 x 2 = 6, 5 x 2 = 10, 7 x 2 = 14, 11 x 2 = 22, etc.
  • 43, 53. Explanation: Every other prime number: 3 (skip 5) 7 (skip 11) 13 (skip 17) 19 (skip 23) 29 etc.
  • 361, 529. Explanation: Square each prime number: 2² = 4, 3² = 9, 5² = 25, 7² = 49, 11² = 121, etc.
  • 78, 84. Explanation: Add consecutive prime numbers together: 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21, 13 + 21 = 34, etc.

WANT MORE MATH PUZZLES?

One way is to follow my blog. I will post occasional math puzzles in the future.

Another way is to check out my newest book, 300+ Mathematical Pattern Puzzles.

It starts out easy and the level of challenge grows progressively so that puzzlers of all abilities can find many puzzles to enjoy.

A wide variety of topics are covered, including:

  • visual patterns
  • arithmetic
  • repeating patterns
  • Roman numerals
  • Fibonacci sequence
  • prime numbers
  • arrays
  • analogies
  • and much more

The cover was designed by Melissa Stevens at www.theillustratedauthor.net.

CHRIS MCMULLEN, PH.D.

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

Newest releases:

  • 300+ Mathematical Pattern Puzzles
  • Basic Linear Graphing Skills Practice Workbook
  • Systems of Equations: Simultaneous, Substitution, Cramer’s Rule

Improve Your Math Fluency. Build fluency in:

  • arithmetic
  • long division
  • fractions
  • algebra
  • trigonometry
  • graphing
Aug 23 2015

Tips for Finding the Slope of a Straight Line

Graphing

HOW TO FIND THE SLOPE OF A STRAIGHT LINE

Slope is a measure that indicates how steep or shallow a straight line is:

  • A line with greater slope is steeper.
  • A line with less slope is shallower.
  • A horizontal line has zero slope.

Slope Sign

Slope can be positive or negative:

  • A line with positive slope slants upward.
  • A line with negative slope slants downward.
  • A line with zero slope is horizontal.

Slope Sign B

The slope of a straight line equals rise over run.

  • The rise between two points is vertical. It’s the change in y.
  • The run between two points is horizontal. It’s the change in x.

Slope Eqn

TIPS FOR FINDING SLOPE

From the graph of a straight line, determine the slope as follows:

  • Mark two points on the line.
  • Read the x- and y-coordinates of the two points, (x1, y1) and (x2, y2).
  • Subtract y2 – y1 to get the rise.
  • Subtract x2 – x1 to get the run.
  • Divide the rise by the run.

Slope Triangle

Here are a few tips:

  • When choosing the two points, try to find points where both x and y are easy to read without interpolating. This isn’t always possible: In that case, at least one coordinate should be easy to read without interpolating.
  • Choose two points far apart. This reduces the relative error in interpolating.
  • Make sure that both points lie on the straight line. Don’t choose a point that’s close to the line, but doesn’t lie on it.

EXAMPLE OF HOW TO DETERMINE SLOPE

Example: Find the slope of the straight line in the graph below.

Slope Example

Solution: First, choose two points on the line. Ideally, these points should be far apart and easy to read.

In this case, it’s easy to read both the x- and y-coordinates for the leftmost and rightmost points shown in the graph. So let’s choose those.

Slope Example 2

  • The leftmost point has coordinates (0, 3).
  • The rightmost point has coordinates (10, 8).

Subtract the y-values to determine the rise:

y2 – y1 = 8 – 3 = 5

Subtract the x-values to determine the run:

x2 – x1 = 10 – 0 = 10

(In coordinate graphing, recall that x is horizontal and y is vertical.)

Divide the rise by the run to find the slope:

Slope Example 3

The slope of the line is 0.5.

Check: You can check your answer as follows.

Look at the graph. Starting from (0, 3), the next point that’s easy to read is (2, 1).

From (0, 3) to (2, 1), the line goes one unit up (vertically) and 2 units over (horizontally).

The ratio of the rise to the run is 1 to 2. Divide the rise (of 1) by the run (of 2). The slope is 0.5. ♦

CHRIS MCMULLEN, PH.D.

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

Newest release:

  • Basic Linear Graphing Skills Practice Workbook

Related books:

  • Trigonometry Essentials Practice Workbook with Answers
  • Learn or Review Trigonometry Essential Skills
  • Algebra Essentials Practice Workbook with Answers
  • Systems of Equations: Simultaneous, Substitution, Cramer’s Rule
Aug 12 2015

Memory Tip for Sine, Cosine, and Tangent of Special Angles (Trigonometry)

Trig Table

TRIGONOMETRY MEMORY TIP

There is a simple way to remember the sine, cosine, and tangent of special trigonometry angles.

The special trig angles are 0º, 30º, 45º, 60º, and 90º. What makes these angles special? The 30º-60º-90º triangle is one-half of an equilateral triangle, while the 45º-45º-90º triangle is one-half of a square. In both cases, the trig functions (sine, cosine, and tangent) can be expressed as simple ratios.

Here is the trick for quickly working out the sine, cosine, and tangent of 0º, 30º, 45º, 60º, and 90º.

STEP 1: Special angles.

Write the special angles in order.

Trig Table top

STEP 2: Integers.

Write the integers 0 thru 4 in order.

Trig Table numbers

STEP 3: Squareroots.

Squareroot each number.

Trig Table roots

STEP 4: Find the sine of theta.

Divide each number by 2.

Trig Table sine

These are the sine of 0º, 30º, 45º, 60º, and 90º.

It’s that simple. Here’s a recap:

  • Write the numbers 0 thru 4.
  • Squareroot each number.
  • Divide each number by 2.

STEP 5: Find the cosine of theta.

Just write the previous numbers in reverse order.

Trig Table cos

Why does this work? Because the sine of theta equals the cosine of the complement of theta: sin(θ)=cos(90º–θ). What’s opposite to theta is adjacent to its complement.

STEP 6: Find the tangent of theta.

Divide sine theta by cosine theta.

Trig Table tan

TRIG CHART

This chart shows all of the steps together.

Trig Table Tips

  • Write the special angles.
  • Write the integers 0 thru 4.
  • Squareroot each number.
  • Divide each number by 2. This gives you sine of theta.
  • Write the numbers in reverse order. This gives you cosine of theta.
  • Divide the previous two rows (sine over cosine). This gives you tangent theta.

NOTE

There are two different, yet equivalent ways, to write the above chart.

That’s because of the following properties of irrational numbers:

Roots

So, for example, there are alternative ways to express the following trig values:

Trig Values

The chart on this blog uses standard form. Most math courses use standard form, which means that there no irrational numbers (like root 2) in the denominator.

Sometimes you find the trig table in another nonstandard form. In that form, you see 1 over root 2 in place of root 2 over 2, and you see 1 over root 3 in place of root 3 over 3.

It’s important to realize that both forms are correct. The standard form, however, is expected in most math courses.

CHRIS MCMULLEN, PH.D.

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

  • Trigonometry Essentials Practice Workbook with Answers
  • Learn or Review Trigonometry Essential Skills
  • Trigonometry Flash Cards (for Kindle)
  • Algebra Essentials Practice Workbook with Answers
  • Systems of Equations: Simultaneous, Substitution, Cramer’s Rule
  • Other volumes cover fractions, long division, arithmetic, and more
  • Also look for books on the fourth dimension, astronomy, conceptual chemistry, and more
Jun 7 2015

Cursive Handwriting for Math Lovers

Cover designed by Melissa Stevens at http://www.theillustratedauthor.net.

 

CURSIVE HANDWRITING FOR MATH LOVERS

Julie Harper is the author of a popular series of handwriting workbooks.

I recently had the opportunity to coauthor a handwriting workbook with her, geared toward math lovers.

This unique book combines math terms and concepts with handwriting practice.

Find this book here:

Find Julie Harper’s handwriting books:

Find Chris McMullen’s math workbooks:

CHRIS MCMULLEN, PH.D.

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

Jan 10 2015

Practice Arithmetic with Geometric Dice

Dice

MATH DICE

My daughter recently received a huge pack of cool geometric dice in several different colors:

  • An icosahedron (20-sided polyhedron) with the numbers 1 thru 20.
  • A dodecahedron (12-sided polyhedron) with the numbers 1 thru 12.
  • A decahedron (10-sided polyhedron) with the numbers 0 thru 9, and another with the numbers 0 thru 90.
  • An octahedron (8-sided polyhedron) with the numbers 1 thru 8.
  • A cube (6-sided polyhedron) with the numbers 1 thru 6.

These dice turned out to be really handy for learning addition and multiplication facts.

ARITHMETIC DICE GAMES

You can easily practice addition facts and multiplication facts with these dice.

Here are some examples:

  • ADDITION/MULTIPLICATION. Roll two decahedra, marked 0 thru 9. Add or multiply the two numbers to practice addition or multiplication facts 0 thru 9.
  • SMALLER NUMBERS. Roll two cubes, marked 1 thru 6. Add or multiply the two numbers to practice addition or multiplication facts 1 thru 6. The cubes let students focus on the smaller numbers first, before working with 7, 8, and 9. (If you want more basic practice, find tetrahedra—4-sided polyhedra—marked 1 thru 4.)
  • FOCUSED FACTS. Roll one decahedron, marked 0 thru 9. For example, suppose you want to practice your multiplication table of 4’s. Simply multiply the die by 4. This lets you concentrate on a single number’s addition or multiplication facts at a time.
  • 11 THRU 20. Advance to dodecahedra or icosahedra to practice the facts 1 thru 12 or 1 thru 20.
  • SUBTRACTION. Practice subtraction facts using an icosahedron and a decahedron. Be careful to subtract the smaller number from the larger number; sometimes, the number on the 10-sided die will be larger. (Advanced students who are learning about negative numbers can use these to sometimes subtract the larger number from the smaller number.)
  • TENS. Multiply powers of 10 using one decahedron with 0 thru 90 and another with 0 thru 9. Or roll one decahedron and multiply that by 10 for more basic tens practice.
  • POWERS. Roll a tetrahedron and a decahedron together to learn about powers. Let the tetrahedron serve as the exponent.
  • FRACTIONS. Roll four decahedra to learn about fractions. These will give you the numerators and denominators of two fractions. Then you can add them, multiply them, divide them, compare them (figure out which is bigger), or subtract them (but first find out which is larger).
  • DICE WAR. If you have several dice to divide equally, you can play dice war with a friend. Each player rolls two dice. Either add or multiply the numbers (choose one before the game begins). The higher sum or product collects both dice.

HANDS-ON GEOMETRY

Another cool thing about using a variety of geometric dice to play math games is that kids get to hold various geometric solids in their hands, see how they look, get a feel for them, and after much use remember how many sides each shape has.

Better than just being told or shown what a dodecahedron is… hold one in your hands, roll it, play with it for months. Then you’ll ‘know’ that solid when you hear its name. (It helps when someone learns and uses the correct names while using the dice.)

Many of these dice packages are sold with role-playing games in mind, but there is no reason that you can’t use them for math practice instead.

TERMINOLOGY

  • Polyhedron: a three-dimensional solid.
  • Polygon: a two-dimensional object, not a solid; it’s flat.
  • Polyhedra is plural, polyhedron is singular.
  • Dice is plural, die is singular.

CHRIS MCMULLEN, PH.D.

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