What is Pre-Algebra?

Pre-Algebra is the study of everything after long division, up until you get just before polynomials in algebra. So it actually should include some algebra basics.

According to some math advisors whom we have consulted with - teachers and professors who have become frustrated with the "new math" or the Common Core math that is being pushed on school kids - there is an orderly, logical progression in learning mathematical skills. No matter how "relative" the new math publishers try to make things with funny videos or word problems using faddish lingo, you can't force the skills to be mastered. Some children might be able to "get through" a lesson or exam, they may even be able to score well on an assignment or exam, but retention is very difficult if concepts are taught out of order. The human brain processes information in specific ways, and again, math concepts need to be in the logical order that the brain will process the information - which will then produce the most retention. Retention is critical for advancing to higher math without frustration.

Go for mastery, versus a passing grade! It is critical when you are teaching math that you do not move forward until your child has mastered the concept. If your child gets a "B" in math, it doesn't mean they understand what they are doing. And if that B is because they missed a critical step, then they will likely struggle in their math career. Getting a "B" essentially is a signal that your child does not understand about 15 to 20 percent of the concepts.

Also remember that if your child does not do well in 6th grade math, it does not mean they are behind. Go back to these lists, and start from the beginning, and think of the progression as mastery, versus grade levels. Once they have mastered all three sections (A, B, and C) of the Pre-Algebra concepts below, they are ready for Algebra, which is 9th grade. So don't let grade levels intimidate you or tell you what your child cannot do.

BEFORE PRE-ALGEBRA

Here is the logical progression which must be mastered before pre-algebra:

- Counting and Sorting
- Addition
- Subtraction
- Multiplication
- Division

From our director:

"When I taught remedial math to homeschoolers (kids who had been moved to Algebra who were struggling)...nearly 100% of them had to back all the way up to long division or right after long division when they were introduced to fractions. That is where they were rushed through..."

Keep in mind, that word problems or discussion questions involving math are absolutely essential to have that retention. From one of our members: *"When I was in school, my instructors never made me do the word problems because I did so well on the drills (you know, 50 or 60 number problems in a chapter??) This caused a hindrance for me when I got into high school and was preparing to take the pre-college exams. I simply went blank on the word problems and I did not do well. I hated math. Hated it. Because I could not reason with it."*

Drills, drills, and more drills will only teach your children to expect more drills. It doesn't mean they are liking what they are learning, which in our opinion interprets to - it doesn't mean they will retain. Discussion and application will strengthen retention. Now, the ACT and SAT exam (and most state exams) are primarily word problems. As of this date, those exams are still very relevant for kids wanting to get into college and to do well. So never skip the word problems! As a matter of fact, our experts suggest that you have the child do the word problems first, then the drills if they are needed. They should be doing way more word problems (sometimes called story problems) than worksheet drills.

As students work through division, be sure they fully understand long division BEFORE you introduce fractions. Fractions are essentially division problems, they just use a different order and different symbols than regular division or long division. There are some people who say you can teach a toddler fractions. Yes, you can to a degree. However, they don't understand fractions as a division problem because they haven't learned multiplication yet. Therefore later on, our experts say that they could have difficulty with more complicated fractions. So again, master long division before fractions.

Once your child has mastered long division, you are ready for pre-algebra (not just a passing grade, but they have mastered it), the logical order of pre-algebra is as follows, and can be divided into three sections. Some kids can do all of this in less than a year, for some it takes 2-3 years.

Master everything in this list - in order - before you go to list B.

NEW! List A and List B now have videos and worksheets as supplements for PLUS GOLD and Plus Legacy members to teach the concepts on this page - it has been extremely helpful for mastering pre-algebra as well as reviewing important concepts for students who are further along in algebra but are stuck. See below for the link to access the Pre-Algebra Mastery videos and worksheets!

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- Fractions - what are they?
- Adding and Subtracting Fractions
- Least Common Multiple
- Least Common Denominator
- Equivalent Fractions
- Greatest Common Factor
- Reducing Fractions
- Improper Fractions

- Decimals - What are they?
- Adding and Subtracting Decimals
- Multiplying and Dividing Decimals
- Converting Fractions to Decimals
- Ordering Fractions
- Adding Mixed Numbers and Improper Fractions
- Subtracting Mixed Numbers and Improper Fractions

- Converting Decimals to Fractions
- Converting Fractions to Percents
- Multiplying Fractions
- Dividing Fractions
- Multiplying/Dividing Mixed Number Fractions

After Part A, you will need to introduce probability and statistics, which again are essentially fractions. So it simply isn't logical to introduce probability and statistics before learning about fractions. This is also part of pre-algebra and is in order as follows:

- Identify Certain and Impossible Events
- Making Decisions with Probability
- Showing Probability with Fractions
- Showing Probability with Decimal/Percent
- Comparing Probabilities
- Comparing Compound Events
- Tree Diagrams and Lists
- Making Predictions with Probability
- Making Predictions with Data
- Bar and Line Graphs
- Scatter Plots
- Types of Numbers (Natural, Whole, Integers, Rational, Irrational, Real and all the relationships)
- Operations of Numbers and introduction to variables
- Like and Unlike Terms
- Definition of a coefficient
- Addition and subtraction of numbers with variables
- Multiplying with variables
- Dividing with variables
- Zero as an operation on the different number systems

- Order of Operations basics
- The number line
- Adding/subtracting integers
- Multiply/divide integers
- Absolute Value and additive inverse property of addition
- Signed integers and symbols in Order of Operations
- Properties of Numbers
- Identity properties
- Cummutative properties
- Associative properties
- Inverse and reciprocal operations
- Multiplicative Inverse operations
- Distributive property
- Distributive property with variables
- Exponents
- Identifying base and exponent
- Using base ten with exponents
- Scientific Notation explanation
- Converting Scientific Notation to standard form
- Negative exponents

- Negative exponents and scientific notation
- Converting standard form back to scientific notation
- Advanced negative exponents
- Exponential expressions
- Multiplication rule for exponents and exponential expressions
- Dividing expressions with same base
- Dividing exponential expressions
- Raising a power to a power
- Add/Subtract exponential expressions
- Distributive property and exponents
- Square roots introduction
- Perfect squares
- Add/subtract square roots
- Simplifying roots and radicals
- Simplifying radicands that are not perfect squares
- Multiply/Divide radical expressions
- Cube roots and other indices
- Roots and variables

A final area of what we consider pre-algebra is the ability to see patterns and to turn the concepts into graphs. Do it in this order as follows:

Note: This final section can be skipped if you are going to have your child do high school Algebra, because it's included in those texts to slow kids down in high school. The following list is considered "Algebra" in high school. However, if you are going to go from Pre-Algebra to college level Algebra, then you must master Section C. We teach more about dual credit college courses in our Homeschooling High School and Beyond Course for Plus members.

- Generalizing patterns
- Patterns in number lists
- Input/Output tables
- Algebraic expressions and equation introduction
- Solving simple linear equations
- Problem solving strategies
- Using formulas in word problems
- Using linear equations to solve problems
- Graphing linear equations

- Slope and intercept
- Distance formula
- Midpoint Formula
- Slope and intercept in graphing
- Functions and variations
- Systems of equations and methods
- Solving inequalities
- Graphing inequalities
- Disjunctions and Conjunctions

- Operations with monomials
- Operations with polynomials
- Rationalizing denominators
- Binomial factoring
- Solving quadratic equations
- Rational expressions
- Complex fractions
- Rational equations

List A and List B now have videos and worksheets as supplements for PLUS GOLD and Plus Legacy members to teach some of the concepts on this page - it is not a complete curriculum, but has been extremely helpful for mastering pre-algebra as well as reviewing important concepts for students who are further along in algebra but are stuck. Videos and worksheets for list C are not available at this time. This content is for Plus Gold and Plus Legacy members only.

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Once the child has mastered all of the above, then and only then are they ready to proceed to algebra. According to our experts, if your child has not mastered all of the above, then they will struggle in algebra and higher math. These concepts will also make geometry so much easier!

The following videos and articles are offered to PLUS members to help you teach math. They are a collection from guest speakers, our online conferences, and special articles in our archives - Enjoy!