Syllabus Prealgebra 1
Prealgebra 1 includes a thorough exploration of the fundamentals of arithmetic, including fractions,
exponents, and decimals. We introduce beginning topics in number theory and algebra, including
common divisors and multiples, primes and prime factorizations, basic equations and inequalities, and
ratios.
This course is specifically designed for high-performing students and draws material from many
programs for top middle school students in the country. Our philosophy is that students develop more
by learning to solve problems they haven’t seen before, as opposed to offering repeated drills that
students can memorize their way through. In this way, our classes are structured much more like
courses at top-tier colleges.
Book: Prealgebra, Chapters 1-7, by Richard Rusczyk, David Patrick, and Ravi Boppana
Time Commitment: This 50-lesson self-paced course includes 60-80 hours of total work, corresponding
to a half-year course.
Grading: Grading is based on student performance on Alcumus homework, Short-Answer Challenge
Problems, Writing Challenge Problems (proofs), and student progress in our online lessons. Alcumus is
the AoPS online learning system that adjusts to student performance to deliver appropriate problems
and lessons.
Content:
Chapter Topic
1 Properties of Arithmetic
Rigorous definition of arithmetic basics
Clever applications of arithmetic rules to make seemingly-complicated calculations simple
2 Exponents
Exponentiation and powers
Exponent laws
Zero as an exponent
Negative exponents
3 Number Theory
Multiples and divisibility
Primes, composites, and prime factorization
Fundamental Theorem of Arithmetic
Least common multiple (LCM) and greatest common divisor (GCD)
4 Fractions
Rigorous definition of fractions
Arithmetic with fractions and mixed numbers
Fraction comparison and simplification
Challenging word problems with fractions
5 Equations and Inequalities
Expressions and equations
Linear equations
Applications of linear equations to word problems
© 2019 AoPS Incorporated 1
Syllabus Prealgebra 1
Principles of inequalities
Linear inequalities
6 Decimals
Definition of decimal notation
Rigorous exploration of arithmetic with decimals
Decimal comparison and approximation
Conversion between fractions and decimals
Rational numbers and their decimal representation
7 Ratios, Conversions, and Rates
Definition of ratio and proportion
Proportional thinking
Problem-solving with part-to-part and part-to-whole ratios
Variables in ratios and proportions
Unit conversion with conversion factors
Relationship between speed, distance, and time
Joint work, relative speed, and average speed
Common Core State Standards (Middle School):
Grade Domain Standards
Ratio & Proportional Relationships 1, 2, 3bd
6 The Number System 1, 4, 5, 6a, 7a
Expressions & Equations 1, 2abc, 3, 4, 6, 7, 8, 9
Ratio & Proportional Relationships 1, 2c
7 The Number System 1cd, 2abcd, 3
Expressions & Equations 1, 3, 4ab
8 Expressions & Equations 1, 7b
Common Core State Standards (High School):
Domain Subdomain Standards
Number and Quantity Quantity 1, 2
Seeing Structure in Expressions 1ab, 2, 3
Algebra Creating Equations 1, 4
Reasoning with Equations and Inequalities 1, 3
© 2019 AoPS Incorporated 2
Syllabus Prealgebra 1
Sample Problems:
I Jack finds the product of three different prime numbers. Is it possible for the sum of the digits of
Jack’s product to be 18? Why or why not?
I Kayla adds the same number to both the numerator and denominator of the fraction 10
1
. Her resulting
fraction equals 3 . What number did she add to both the numerator and denominator of her original
2
fraction?
I What is the smallest positive integer k such that k
660 can be expressed as a terminating decimal?
I Five workers together can build a road in 20 days. Suppose every worker works at the same rate.
If three workers work on the road for 10 days before eleven more workers join them, then how long
total will it take to build the road?
© 2019 AoPS Incorporated 3
Art of Problem Solving is an ACS WASC Accredited School.
