How to Use This Book

Grade 8 Mathematics · 2023 Virginia Standards of Learning

This book covers every Grade 8 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.
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. A few 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.

How the problems are numbered

In this book, item numbers run straight through a whole chapter. Chapter 3's guided practice starts at item 1, and the numbers keep climbing to the last item of the chapter review — they do not restart at each lesson. So "item 47" names exactly one problem in the chapter, which is what lets the answer key and the coverage table point at specific problems without ambiguity.

About the calculator

Grade 8 is different from Grade 7 here, and the difference is worth knowing. In Grades 6 and 7, some standards are assessed without a calculator. In Grade 8, none are. VDOE provides the Desmos Virginia Scientific Calculator in the toolbar for the entire Grade 8 test, and states that the Grade 8 assessment will not contain items assessed without a calculator.

That is not permission to stop thinking. Three chapters in this book still ask you to work by hand, because in those chapters a calculator has nothing to offer:

• Chapter 1 (subsets of the real numbers). No calculator can tell you which subset a number belongs to. $\sqrt{9}$ and $\sqrt{10}$ look equally decimal on a screen; only one of them is rational.
• Chapter 2 (comparing and ordering). The standard asks you to name the two consecutive natural numbers a square root lies between and justify which is the better approximation. A calculator hands you $\sqrt{40} \approx 6.32$ with no justification attached, and the justification is the graded part.
• Chapter 4 (equivalent expressions). Deciding whether $3(x+4)$ and $3x+12$ are the same expression is not a computation. There is no number to enter.

Everywhere else, use the calculator — and use it the way a mathematician does: estimate first, compute second, then ask whether the answer is the size you expected. An answer you cannot judge for reasonableness is an answer you do not really have.

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 the real number system first, then percent applications, then expressions, equations, and inequalities, then functions, then geometry and measurement, then probability and data. Chapters are self-contained enough to reorder, with five dependencies worth respecting:

• Real-number work (Chapters 1–2) comes before Chapter 10, where the Pythagorean Theorem produces irrational lengths that students must estimate and place on a number line.
• Expressions (Chapter 4) come before equations (5), which come before inequalities (6) — inequalities are the equation procedure plus one rule that has no equation analogue.
• Relations and functions (Chapter 7) come before linear functions (8): domain, range, and the definition of a function are the vocabulary Chapter 8 reasons in.
• Translations (Chapter 13) come before reflections and compositions (14), since a composition is built from the single transformations.
• Boxplots (Chapter 16) come before scatterplots (17). Both run the data cycle, but univariate distribution comes before bivariate association.

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 items. The volume's SOL coverage index lists all sixteen standards and where each is addressed.

Where a standard is split. One standard is taught across two chapters, because its Knowledge and Skills bundle distinct skills:

• 8.MG.3 — translations in Chapter 13 (parts a and d); reflections, compositions, and the properties preserved by each in Chapter 14 (parts b, c, e, f, g)

Every other standard maps to exactly one chapter.

Modeling is required, not optional. The Grade 8 standards name specific representations: concrete and pictorial models in 8.PFA.1a and 8.PFA.4a — algebra tiles, in practice — number-line graphs of solution sets in 8.PFA.5, subset representations in 8.NS.2a, right triangles "in various orientations" in 8.MG.4c, nets and concrete objects for pyramids and cones in 8.MG.2, and boxplots and scatterplots in 8.PS.2 and 8.PS.3. Where the standard names a model, this book teaches with that model rather than jumping to the procedure. Lessons that derive a formula derive it the way the standard asks, then use it.

Three conventions this book fixes explicitly, because the mathematics is ambiguous without them and different textbooks — and different calculators — choose differently:

• Quartiles (Chapter 16): order the data; the median splits it into a lower and an upper half; when the count is odd the median belongs to neither half; each quartile is the median of its half. Every figure, answer, and IQR in the volume follows this rule. A graphing calculator or spreadsheet may use a different interpolation rule and report a slightly different quartile. That is a difference in convention, not an error — but since Grade 8 students now have a calculator on the entire test, expect the discrepancy to surface in class, and name it when it does.
• Pi (Chapter 11): answers containing $\pi$ are given exactly in terms of $\pi$, then approximated with $\pi \approx 3.14$. The answer keys were computed with $3.14$, not a calculator's stored $\pi$, and the two disagree in the last decimal place.
• Line of best fit (Chapter 17): 8.PS.3 asks students to sketch the line, so this book sketches a plausible fit by hand and never presents a regression line as the one correct answer. Any line that reasonably follows the trend is acceptable; the reasoning about direction and strength is the graded content.

Assessment. Each lesson has an exit ticket. 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 8, and what Grade 8 builds toward

Grade 8 is the bridge year. It is where the number system finally closes, where linear relationships become fully general, and where the habits of Algebra 1 are either built or missed. Four shifts define it:

• The number system closes. Grade 7 worked in the rationals. Grade 8 admits irrational numbers — $\sqrt{2}$, $\pi$ — and asks students to place them, order them, and say why they cannot be written as fractions. Chapter 10 then makes irrational numbers unavoidable rather than exotic: most right triangles have one.
• Linear becomes general. Grade 7 introduced $y = mx$, proportional and through the origin. Grade 8 adds the intercept to reach $y = mx + b$, and with it the idea that a rate of change and a starting value are different pieces of information. Algebra 1 then treats this form as a given.
• Function arrives as a definition. Students have graphed relationships for years. In Grade 8 "function" becomes a claim you can test — one output per input — with domain and range as the vocabulary for saying what a function accepts and produces.
• Data reasoning becomes bivariate. Grade 7 read the shape of one distribution from a histogram. Grade 8 keeps that work in boxplots and then adds a second variable, where the question shifts from what is typical to do these two things move together — and to the discipline of not calling that causation.

By the end of the year, a student should be able to take one linear situation and move fluently among its description, table, equation, and graph; find a missing side of a right triangle and say whether the answer is rational; and run the data cycle end to end on a question of their own.

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. The Grade 8 calculator policy described above is drawn from VDOE's Desmos Online Calculator guidance and its published table of online testing tools.

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.