6.2: Other Rules - Mathematics

6.2: Other Rules - Mathematics

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Let’s play the dots and boxes game, but change the rule.

The 1←3 Rule

Whenever there are three dots in single box, they “explode,” disappear, and become one dot in the box to the left.

Example (PageIndex{1}): Fifteen dots in the 1←3 system

Here’s what happens with fifteen dots:


The 1←3 code for fifteen dots is: 120.

Problem 2

  1. Show that the 1←3 code for twenty dots is 202.
  2. What is the 1←3 code for thirteen dots?
  3. What is the 1←3 code for twenty-five dots?
  4. What number of dots has 1←3 code 1022?
  5. Is it possible for a collection of dots to have 1←3 code 2031? Explain your answer.

Problem 3

  1. Describe how the 1←4 rule would work.
  2. What is the 1←4 code for thirteen dots?

Problem 4

  1. What is the 1←5 code for the thirteen dots?
  2. What is the 1←5 code for five dots?

Problem 5

  1. What is the 1←9 code for thirteen dots?
  2. What is the 1←9 code for thirty dots?

Problem 6

  1. What is the 1←10 code for thirteen dots?
  2. What is the 1←10 code for thirty-seven dots?
  3. What is the 1←10 code for two hundred thirty-eight dots?
  4. What is the 1←10 code for five thousand eight hundred and thirty-three dots?

Think / Pair / Share

After you have worked on the problems on your own, compare your ideas with a partner. Can you describe what’s going on in Problem 6 and why?

The current tax rate for social security is 6.2% for the employer and 6.2% for the employee, or 12.4% total. The current rate for Medicare is 1.45% for the employer and 1.45% for the employee, or 2.9% total. Refer to Publication 15, (Circular E), Employer's Tax Guide for more information or Publication 51, (Circular A), Agricultural Employer’s Tax Guide for agricultural employers. Refer to Notice 2020-65 PDF and Notice 2021-11 PDF for information allowing employers to defer withholding and payment of the employee's share of Social Security taxes of certain employees.

Additional Medicare Tax applies to an individual's Medicare wages that exceed a threshold amount based on the taxpayer's filing status. Employers are responsible for withholding the 0.9% Additional Medicare Tax on an individual's wages paid in excess of $200,000 in a calendar year, without regard to filing status. An employer is required to begin withholding Additional Medicare Tax in the pay period in which it pays wages in excess of $200,000 to an employee and continue to withhold it each pay period until the end of the calendar year. There's no employer match for Additional Medicare Tax. For more information, see the Instructions for Form 8959 and Questions and Answers for the Additional Medicare Tax.

Example for 6 rules of the Law of Indices

Example for Rule 1:

Example for Rule 2:

Example for Rule 3:


Example for Rule 4:

Example for Rule 5:

Example for Rule 6:

Examples of How to Round Numbers

  • 1,000 when rounding to the nearest 1,000
  • 800 when rounding to the nearest 100
  • 770 when rounding to the nearest 10
  • 765 when rounding to the nearest one (1)
  • 765.4 when rounding to the nearest 10th
  • 765.37 when rounding to the nearest 100th
  • 765.368 when rounding to the nearest (1,000th)

Rounding comes in handy when you are about to leave a tip at a restaurant. Let's say your bill is $48.95. One rule of thumb is to round to $50 and leave a 15 percent tip. To quickly figure out the tip, say that $5 is 10 percent, and to reach 15 percent you need to add half of that, which is $2.50, bringing the tip to $7.50. If you want to round up again, leave $8—if the service was good, that is.

We use the negative exponent rule to change an expression with a negative exponent to an equivalent expression with a positive exponent. The rule states that any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power. In other words, an expression raised to a negative exponent is equal to 1 divided by the expression with the sign of the exponent changed.


This unit introduces algebra by examining similar models. You should be able to read a problem and create a table to find an equation that relates two variables. If you are given information about one of the variables, you should be able to use algebra to find the other variable.

Signed Numbers:

Adding or subtracting like signs: Add the two numbers and use the common sign.

Adding or subtracting unlike signs: Subtract the two numbers and use the sign of the larger, (more precisely, the sign of the number whose absolute value is largest.)

Multiplying or dividing like signs: The product or quotient of two numbers with like signs is always positive.

Multiplying or dividing unlike signs: The product or quotient of two numbers with unlike signs is always negative.

Order of operations: Please Excuse My Dear Aunt Sally
1. Inside Parentheses, ().
2. Exponents.
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

Study Tip: All of these informal rules should be written on note cards.

Introduction to Variables:

Generate a table to find an equation that relates two variables.

Example 6. A car company charges $14.95 plus 35 cents per mile.

Simplifying Algebraic Equations:

Distributive property:

Solving Equations:

1. Simplify both sides of the equation.
2. Write the equation as a variable term equal to a constant.
3. Divide both sides by the coefficient or multiply by the reciprocal.
4. Three possible outcomes to solving an equation.
a. One solution ( a conditional equation )
b. No solution ( a contradiction )
c. Every number is a solution (an identity )

Applications of Linear Equations:

This section summarizes the major skills taught in this chapter.

Example 9. A cell phone company charges $12.50 plus 15 cents per minute after the first six minutes.

a. Create a table to find the equation that relates cost and minutes.

c. If the call costs $23.50, how long were you on the phone?

If the call costs $23.50, then you were on the phone for approximately 79 minutes.

Literal Equations:

A literal equation involves solving an equation for one of two variables.


Write percentages as decimals.

Example 11. An English teacher computes his grades as follows:

Sue has an 87 on the short essays and a 72 on the research paper. If she wants an 80 for the course, what grade does Sue have to get on the final?

Sue has to get a 78.36 in the final exam to get an 80 for the course.

Study Tips:

1. Make sure you have done all of the homework exercises.
2. Practice the review test on the following pages by placing yourself under realistic exam conditions.
3. Find a quiet place and use a timer to simulate the test period.
4. Write your answers in your homework notebook. Make copies of the exam so you may then re-take it for extra practice.
5. Check your answers.
6. There is an additional exam available on the Beginning Algebra web page.
7. DO NOT wait until the night before the exam to study.

Famous Math People

There are countless famous math people that have helped shape the mathematics that we use today. In fact, many of the discoveries of these famous mathematicians have roots in the science, medicine, and technologies that are now common place.

Grade A has a short list of some of the most well known mathematicians and some of their contributions to the field of mathematics.

You may also want to learn about some of the world's most famous black mathematicians or famous women mathematicians.

Please view the pictures below. A short description is also listed, but please click the links or pictures for more details about their fascinating contributions

For more information on any famous math people, please click on their pictures, or the name below. A short description of each mathematician is given.

Ren é Descartes : Most known for contribution to the coordinate plane. In fact, it is sometimes referred to as the Cartesian coordinate because of Descartes.

Albert Einstein: Perhaps the most famous known scientist of all time. His theories on relativity were groundbreaking and are still used today.

Leonhard Euler: Well known for his contributions to mathematical vocabulary and notation. In particular, he is consider the founder of function notation.

Fibonacci: His given name is Leonardo of Pisa. He is most known the number sequence 1, 1, 2, 3, 5, 8, 13, 21. which was eventually named after him as the Fibonacci Numbers.

Carl Friedrich Gauss: Considered a child prodigy that eventually realized his true potential. He made monumental contributions in the areas of set theory, statistics, and many others.

Sir Isaac Newton: Shares in the credit as the developer of Calculus!

Blaise Pascal: Contributed in several areas of mathematics, but his nameis most recognized with its connection to Pascal's Triangle.

Mathematicians share a great diversity - who knows - maybe you can be next great mathematician?

Explore the Grade A homepage to learn about some of the topics that these famous people contributed to!

If $( P ightarrow Q ) land (R ightarrow S)$ and $P lor R$ are two premises, we can use constructive dilemma to derive $Q lor S$.

$egin ( P ightarrow Q ) land (R ightarrow S) P lor R hline herefore Q lor S end$


“If it rains, I will take a leave”, $( P ightarrow Q )$

“If it is hot outside, I will go for a shower”, $(R ightarrow S)$

“Either it will rain or it is hot outside”, $P lor R$

Therefore &minus "I will take a leave or I will go for a shower"

Developing the Concept: Order of Operations

Materials: Whiteboard or way to write for the class publicly

Prerequisite Skills and Concepts: Students should be familiar with order of operations and feel prepared to practice it.

As you continue teaching your students about parentheses, be sure to demonstrate that parentheses do not always change the value of an expression, though they often do.

  • Ask: What operation do I perform first in the expression (3 + 5 imes 8) and why?
    Write the expression publicly. Make sure students understand clearly that the order of operations requires them to perform multiplication before addition.
  • Ask: What happens if I want to add 3 and 5 before I multiply by 8?
    Allow students to discuss ideas of how to override the order of operations. Do not tell students that they are right and wrong. Instead, encourage mathematical discourse and compare differing opinions in order to correct misconceptions. Note that there are many possible answers! For example, the problem could explicitly say "add 3 and 5 first," or historically, there have been other ways of grouping, such as using horizontal bars over the expression. If they don't mention parentheses, remind them of what you did in the first lesson.
  • Say: By putting parentheses around (3 + 5) we are saying that we must add 3 and 5 first, then multiply by 8. Today we're going to practice finding the value of expressions with and without parentheses and see what difference the parentheses make.
  • Write the following three expressions publicly for all students to see.
    • (3 + 6 imes 2)
    • ((3 + 6) imes 2)
    • (3 + (6 imes 2))
    • ((8 div 4) - 2)
    • (8 div (4 - 2))
    • ((3 + 4) imes 1)
    • (3 + (4 imes 1))

    Wrap-Up and Assessment Hints
    It is important that students can remember the rules for order of operations both with and without parentheses. Avoid giving worksheets of rote practice. Instead, look for math problems that naturally result in expressions that need to be evaluated, for example substituting values into a formula, and have students practice order of operations in the context of other problems.

    Looking to grow student confidence in mathematics, beyond practicing the math rules of order of operations? Explore HMH Into Math, our K–8 core mathematics solution.