Middle School Math in Tucson
These courses (6th Grade Math, 7th Grade Math, and 8th Grade Math, i.e., Pre-Algebra), represent what could be called the final ‘bricks’ in the foundation of the skills needed to work with the functions that constitute high-school-level algebra. As outlined by the Arizona Math Standards, students extend the number system into rational & irrational numbers, work with proportional relationships, develop understanding of systems of equations, and begin to work with functions.
Contents
1. The Challenge of Middle School Math
The difficulty with these courses could be said to be two-fold. First is increasing levels of assumption of automation-level fluency of basic arithmetic; by 6th Grade, the final few arithmetic skills are being covered, (e.g., with addition & subtraction fluency being reached in 4th Grade, and multiplication fluency in 5th Grade). And, second, is dealing with the high-level cognitive demands of topics including: negative numbers, multi-modal representations of relationships (e.g., with graphing), roots, simulations, and the definition of a function.
2. Prerequisites & Course Topics
In my view, the primary goal of Middle School Math is to further develop and coalesce skills in rational number arithmetic, basic algebra, and graphing into the ability to work with linear functions. The main prerequisites for this – covered in Elementary-level math – are: basic arithmetic (the four operations), fractions, and decimals; (a slightly more exhaustive list is detailed below).
Prerequisites to Middle School Math
K-5 Math
- Number Basics
- (Kindergarten) Connect counting to cardinality (i.e., connect numbers said when counting to a number of objects).
- (2nd Grade) Decompose 3-digit numbers into hundreds, tens, and ones.
- (2nd Grade) Count within 1000 (being able to skip count by 5/10/100).
- (4th Grade) Read and write any multi-digit number using base-ten numerals, number names, and expanded form.
- Arithmetic
- (3rd Grade) Memorize all multiplication facts through 10×10.
- (4th Grade) Fluently add and subtract multi-digit numbers.
- (5th Grade) Fluently multiply multi-digit numbers.
- Fractions
- (3rd Grade) Represent fractions on a number line.
- (4th Grade) Generate equivalent fractions.
- (5th Grade) Add and subtraction fractions with unlike denominators.
- (5th Grade) Multiply with fractions.
- Decimals
- (5th Grade) Read and write decimals to the thousandths using base-ten numerals, number names, and expanded form.
- (5th Grade) Round decimals to any place.
- (6th Grade) Fluently perform arithmetic with decimals.
- Expressions
- (1st Grade) Apply the commutative & associative properties of addition.
- (3rd Grade) Apply the commutative, associative, and distributive properties of multiplication.
- (5th Grade) Evaluate numerical expressions involving parentheses.
- (5th Grade) Write simple expressions that record calculations with numbers.
- Equations
- (1st Grade) Understand the meaning of the equal sign & be able to verify addition/subtraction equations for “true-ness”.
- (3rd Grade) Use a letter to represent an unknown quantity.
- Graphing
- (5th Grade) Understand the basic idea of a coordinate plane, know the vocabulary words “axes”, “origin”, “coordinates”, & work with points in the first quadrant.
- Negatives
- (2nd Grade) Represent whole numbers on a number line. (*Negative numbers are not formally introduced until 6th Grade, but this can be considered a foundational step towards understanding and working with them).
Middle School Math Topics
Below are dropdowns of summarized course topics. For a fully exhaustive list, see the Arizona Standards Database on this website or the Arizona Department of Education standards documents.
6th Grade Math
Arithmetic
- GCF & LCM: Greatest common factor and least common multiple – putting multiplication fact fluency to the test.
- Exponents: Writing and evaluating expressions with whole-number exponents.
Rational Numbers
- Fractions: Quotients of fractions (the final piece of the fraction puzzle).
- Decimals: Extending from working with decimals in the hundredths and thousands to all multi-digit decimals.
- Negative Numbers: The first introduction to negative numbers including real world interpretations, number lines, absolute value, and negative coordinates.
Algebra
- Expressions: Identifying parts of expressions, translating verbal statements into mathematical expressions, applying properties, evaluating, and factoring out a GCF.
- Understanding Equations: Grasping the dual interpretations of a variable and practicing verifying the “truthfulness” of equations.
- Solving Equations: Solving one-step linear equations, interpreting what a “solution” is, and verifying solutions using substitution.
Miscellaneous
- Other: Solving percent problems, finding area by composition/decomposition, plotting inequality solutions, manipulating units, finding surface area, and using rate language.
7th Grade Math
Rational Numbers
- Negative Numbers: Performing arithmetic involving negative numbers, including situations with “double-signs”, e.g., 5+(-4).
- Arithmetic: Adding and subtracting rational numbers.
Algebra
- Expressions: Adding, subtracting, factoring, and expanding linear expressions.
- Solving Equations: Solving linear equations in the form px+r=q and p(x+r)=q.
- Graphing: Identifying the constant of proportionality (slope).
Miscellaneous
- Other: Solving percent increase/decrease problems, finding area & circumference of a circle, and solving multi-step problems involving complementary/supplementary/adjacent/vertical angles.
8th Grade Math
Arithmetic & Rational Numbers
- Exponent Properties: Applying properties of integer exponents to generate equivalent expressions.
- Roots: Evaluating square roots up to 225 and cube roots up to 1000.
- Scientific Notation: Using numbers in scientific notation to approximate very large and very small quantities.
Algebra
- Linear Equations: Fluently solving and determining number of solutions.
- Systems of Equations: Solving pairs of linear equations algebraically and graphically.
- Graphing: Dealing with non-unit slopes and graphing lines.
- Functions: Definition of a function, comparing functions, modeling, rate of change, initial value, qualitative descriptions, and sketching.
Miscellaneous
- Other: Pythagorean theorem, volume of 3D shapes, scatter plots, and compound probability.
3. Support Strategy
With the further delving into algebra characteristic of Middle School Math, as alluded to, arithmetic fluency becomes, largely unavoidable. Attempting to recall multiplication facts or algorithmic procedure for multi-digit subtraction, while already engaging with new and high-level concepts, can overload the working memory and lead to cognitive fatigue. As such, emphasis is placed – however poorly it may currently fare in collective estimation – on rote practice, automation, of basic arithmetic.
Additionally, to meet the increasing complexity of topics, emphasis is placed on: (1) finding the right balance between, and ordering of, procedure-based explanations and conceptual explanations (e.g., a 60-minute lecture on what a function is likely best broken up and interspersed with practicing a relevant algorithm for the sake of engagement and an experience of “success” with the topic), and (2) explicitly or “implicitly”, for the purpose of cognitive “schema” formation, articulating the ‘big picture’ of the progression of topics that effectively culminate in 8th Grade Math (namely, whole-number arithmetic → arithmetic with rational numbers → equations + graphing → functions).
4. Additional Supports for Alternative Tracks (Homeschool, Learning Pods, & Micro-schools)
Show details.
The following additional services are offered for those in ESA learning contexts. (ESA-Approved Vendor – $0 out-of-pocket with ClassWallet).
Option A: Course Consulting + Validation Report ($1260 per semester)
This support option is best for those wanting to self-teach or utilize established online platforms, but want to add an extra layer of individualization, structure, and quality control (i.e., for ensuring well-rounded, solidified understanding and alignment to state standards e.g., for ACT/SAT and college admissions).
- Initial Diagnostics: 60-minute personalized assessment to identify math knowledge and skill deficits relevant to current course progression.
- Progress Evaluations: Five, hour-long blocks spaced throughout the semester to review completed course materials, analyze any written work, and assess for proficiency (e.g., for early identification of difficulties that could compound).
- Midterm & Final Exams: In-person proctoring of Arizona standards-based midterm and final exams.
- Signed Validation Report: Issuance of a course completion report, upon meeting appropriate criteria, displaying state-certified Educator Identification Number (EIN) to lend an extra layer of legitimacy for state audits and college admissions.
Option B: Full-Course Instruction ($2400 per semester)
This support option is best for those wanting to delegate math course instruction fully. This option entails individually tailored 6-8th Grade Math instruction from a former accelerated-track middle school math (and high school math) classroom teacher. (Learn more about Matt here).
- Full-Course Instruction Includes:
- Live Instruction: 2x per week (60 minutes each block) (in-person or virtual, see in-person service area in table below);
- Assignments & Videos: Daily assignments with companion how-to videos;
- Assessment: Weekly exams & progress reports for quantified growth;
- Course-Completion Report: Proctored midterm and final exams & signed course completion report.
- Additional Information
- Self-Contained: No outside materials or subscriptions needed – all necessary materials provided;
- AZ Standards Based: Course is built upon the Arizona math standards, and as such, prepares students for the math portions of the ACT & SAT, and is in alignment with the Arizona Board of Regent’s (ABOR) college admissions policies;
- Self-Paced: In the case of meeting course requirements prior to instructional year-end, funds can be applied to subsequent math course without penalty.
Math Tutoring Information
| Pricing | $65 per 60-minute session |
| Format | In-person and virtual tutoring are available |
| Scheduling | Contacting via text, call, or email is the best way to check current availability |
| Service Area (In-Person) | Catalina Foothills, Midtown Tucson, Northwest, Northeast (& Tanque Verde) |
| Contact | Email: matt@azmathematics.com Phone & Text: (520) 433-2811 Location: Tucson, AZ (Note: Inquiries received Monday-Friday are typically answered within 24 hours) |