How to Use This Book

Grade 6 Mathematics · 2023 Virginia Standards of Learning

This book covers every Grade 6 mathematics standard Virginia expects students to learn, in a sequence built for teaching rather than for filing. It is free to read, print, and share.

For students

Each chapter opens by telling you exactly what you will be able to do when you finish it. Work through a lesson in this order:

1. Read the narrative. New words appear in bold the first time they are used. If a word stops making sense later, it is in the glossary.
2. Follow the worked examples. Every step is shown. Cover the answer, try the step yourself, then check.
3. Do the guided practice. These are scaffolded — the earlier ones give you more help than the later ones.
4. Do the independent practice. Each set includes at least one real-world application and one reasoning problem where you explain your thinking. The explaining is not extra credit; it is the part that makes the skill stick.
5. Take the exit ticket. Four questions. If you miss one, go back to the matching part of the lesson before moving on.

Answers to everything are in the answer key. Use them to check your work after you try a problem, not before.

About the calculator

Virginia assesses some Grade 6 standards without a calculator. Chapters covering those standards are marked, and their practice is designed for mental math and estimation. In those chapters, reaching for a calculator will cost you on the test even when it gets the right answer today. The standards assessed without a calculator in Grade 6 are:

• 6.NS.1 — equivalency, comparing, and ordering fractions, decimals, and percents
• 6.CE.1a–c — multiplying and dividing fractions and mixed numbers
• 6.CE.2a–b — adding, subtracting, multiplying, and dividing integers

For teachers

Sequence. VDOE presents the standards grouped by content strand in numeric order and states explicitly that local curricula and pacing guides determine instructional sequence. This book sequences arithmetic foundations first, then proportional reasoning, then the coordinate plane and algebraic representation, then geometry, then data. Chapters are self-contained enough to reorder, with two dependencies worth respecting: the integer chapters (1–3) come before the coordinate plane (10), and ratios (8) comes before proportional relationships (9).

Chapter-to-standard mapping. Every chapter closes with a coverage table showing which Knowledge and Skill letter is taught in which lesson and practiced in which exercise set. The volume's full SOL coverage index lists all fifteen standards and where each is addressed.

Where standards are split. Two standards are taught across two chapters each, because their Knowledge and Skills bundle procedural fluency with contextual problem solving:

• 6.CE.1 — fluency in Chapter 5 (parts a–c), contextual problem solving in Chapter 6 (parts d–e)
• 6.CE.2 — addition and subtraction in Chapter 2, multiplication, division, and absolute value expressions in Chapter 3

Modeling is required, not optional. The Grade 6 standards name specific representations: number lines, algebra tiles, colored chips, balance scales, grid paper, and circle graphs. Where the standard names a model, this book teaches with that model rather than jumping to the procedure. Lessons that derive a formula — the area of a parallelogram, the area of a triangle, the circumference of a circle — derive it the way the standard asks, then use it.

Assessment. Each lesson has an exit ticket. Chapters of five or more lessons have a mid-chapter check. Every chapter has an end-of-chapter review organized by Knowledge and Skill, so a review score points at a specific standard rather than a vague topic.

What builds into Grade 6, and what Grade 6 builds toward

Grade 6 is the transition from whole-number arithmetic to the reasoning that Algebra 1 depends on. Three shifts define the year:

• Numbers extend in both directions. Students meet negative numbers as quantities, not just as subtraction results, and learn to compare and operate with them.
• Comparison becomes multiplicative. Ratios, rates, and proportional relationships replace "how many more" with "how many times as many."
• Relationships become symbolic. Students write equations and inequalities to represent situations, which is the skill Grade 7 and Algebra 1 build on directly.

By the end of the year, a student should be able to move fluently among a context, a table, a graph, and an equation describing the same relationship.

Sources and independence

Standards text is drawn from the Mathematics Standards of Learning for Virginia Public Schools, 2023, approved by the Virginia Board of Education on August 31, 2023 and fully implemented in the 2024–2025 school year.

This book is an independent open educational resource. It is not published, reviewed, or endorsed by the Virginia Department of Education. Where this book says a standard requires something, the requirement comes from the standard's own Knowledge and Skills text; the teaching sequence, examples, exercises, and explanations are original to this book.