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The Real Number SystemNaturals inside integers inside rationals inside reals, with…8th GradeREALevery number on the line−3, 0, ½, √2, πRATIONALcan be written as a fraction−3, 0.25, ⅔, 7INTEGERSwhole numbers, positive or negative… −2, −1, 0, 1, 2 …WHOLEcounting numbers plus zero0, 1, 2, 3 …NATURALthe counting numbers1, 2, 3 …IRRATIONAL — real, but never a fraction√2 = 1.41421356…π = 3.14159265…e = 2.71828…Decimal never ends and never repeats. Any non-perfect square root is irrational.0.333… repeats, so it IS rational: it equals ⅓.−2−10123√2e−3/2π sits just past 3 — every real number has exactly one home on this line.The Real Number Systemlittleactivity.com

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Math · 8th Grade

The Real Number System

Naturals inside integers inside rationals inside reals, with irrationals beside them.

How to use it

Every 'which set does it belong to?' question is answered by this diagram.

Free · print-ready PDF · US Letter and A4

Grade
8th Grade
Ages
13–14
Subject
Math
Pages
1

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Linear FunctionsSlope, intercepts, the three forms of a line, and what each…8th GradeSLOPE — rise over runrunrisem = (y₂ − y₁) / (x₂ − x₁)Positive → uphill. Negative → downhill.Zero → horizontal. Undefined → vertical.Always subtract in the same order, top and bottom.THE THREE FORMSSlope-intercepty = mx + bbest for graphing — b is where it crosses the y-axisPoint-slopey − y₁ = m(x − x₁)best when you have a point and the slopeStandardAx + By = Cbest for intercepts — set x = 0, then y = 0GRAPHING FROM y = mx + b1Plot b on the y-axis — that is your starting point.2From there, count the rise, then the run, and plot a second point.3Repeat once more, then join the three points with a straight line.Parallel lines: same m. Perpendicular lines: slopes multiply to −1.Linear Functionslittleactivity.com
Math

Linear Functions

Slope, intercepts, the three forms of a line, and what each one is good for.

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Exponent & Radical RulesEvery exponent law with a worked example, plus roots and…8th GradeEXPONENT LAWSProductaᵐ × aⁿ = aᵐ⁺ⁿx³ · x⁴ = x⁷Quotientaᵐ ÷ aⁿ = aᵐ⁻ⁿx⁷ / x³ = x⁴Power of a power(aᵐ)ⁿ = aᵐⁿ(x³)² = x⁶Power of a product(ab)ⁿ = aⁿbⁿ(2x)³ = 8x³Zeroa⁰ = 1any base but 0Negativea⁻ⁿ = 1 / aⁿ2⁻³ = 1/8Fractionala^(1/n) = ⁿ√a9^(1/2) = 3SCIENTIFIC NOTATIONa × 10ⁿ, where 1 ≤ a < 104 800 000 = 4.8 × 10⁶  ·  0.00032 = 3.2 × 10⁻⁴Big number → positive power. Small number → negative power. Count the decimal moves.ROOTSPerfect squares: 1 4 9 16 25 36 49 64 81 100 121 144Perfect cubes: 1 8 27 64 125 216√50 = √(25 × 2) = 5√2 — pull out the largest perfect square.Exponent & Radical Ruleslittleactivity.com
Math

Exponent & Radical Rules

Every exponent law with a worked example, plus roots and scientific notation.

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Pythagoras & TransformationsThe theorem, its converse, distance formula, and the four…8th GradeTHE PYTHAGOREAN THEOREMbaca² + b² = c²c is the hypotenuse — opposite the right angle.Find the hypotenuse: c = √(a² + b²)Find a leg: a = √(c² − b²)Triples worth knowing: 3-4-5 · 5-12-13 · 8-15-17 · 7-24-25THE CONVERSE & THE DISTANCE FORMULAIf a² + b² = c², the triangle is right-angled. If a² + b² > c² it is acute; if less, obtuse.d = √[(x₂ − x₁)² + (y₂ − y₁)²]— Pythagoras on the gridTRANSFORMATIONSTranslationslide(x, y) → (x + a, y + b)same size, same shapeReflectionflipover x-axis: (x, −y) · over y-axis: (−x, y)same size, mirroredRotationturn90° about origin: (x, y) → (−y, x)same size, turnedDilationresize(x, y) → (kx, ky)similar, not congruentThe first three keep congruence. Only dilation changes size.Pythagoras & Transformationslittleactivity.com
Math

Pythagoras & Transformations

The theorem, its converse, distance formula, and the four transformations.

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Literary Analysis EssayThesis, topic sentence, quote integration, analysis — with the…8th GradeTHE QUOTE SANDWICH (I·C·E)INTRODUCESet the scene — who is speaking, and when.CITEThe quotation, kept short, with the reference.EXPLAINTwo sentences minimum on what it proves.PARAGRAPH SHAPETopic sentenceOne idea, linked to the thesis. No quotation yet.ContextWhat happens just before the quotation.EvidenceThe quotation, embedded in your own sentence.AnalysisWord-level: why this word, this image, this order.LinkBack to the thesis, and forward to the next idea.EMBEDDING A QUOTATIONWeak:Dickens describes fog. “Fog everywhere.” This shows fog.Strong:Dickens opens with “fog everywhere”, the repetitionsmothering the city before a single character speaks.Never end a paragraph on a quotation — the last word must be your analysis.THESIS TESTIf nobody could disagree with it, it is a summary, not a thesis.Literary Analysis Essaylittleactivity.com
Literacy

Literary Analysis Essay

Thesis, topic sentence, quote integration, analysis — with the stems for each.

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Academic Word RootsThirty roots that carry academic and test vocabulary, with…8th GradeOne root unlocks a dozen words. Learn the root, then collect the family.ambi / amphibothambiguous, amphibiananthrophumananthropologybenegood, wellbenefit, benevolentcede / cessgo, yieldprecede, recessionchromcolourmonochromecognknowrecognise, cognitivecontraagainstcontradict, contrastcorpbodycorporation, corpsedemopeopledemocracy, demographicequiequalequator, equilibriumfidfaith, trustconfide, fidelitygenbirth, kindgenerate, geneticshydrwaterhydrate, dehydrationlog / logueword, studydialogue, biologyluc / lumlightlucid, illuminatemagngreatmagnify, magnitudemalbadmalfunction, malicemortdeathmortal, mortalitynovnewnovel, innovateomniallomnivore, omnipotentpathfeeling, diseaseempathy, pathologyphillovephilosophypon / posplace, putcomponent, depositscopelook atmicroscopesens / sentfeelsensory, sentimentsolalone, sunsolitude, solartang / tacttouchtangible, contacttemptimetemporary, tempovertruthverify, verdictvit / vivlifevital, reviveON A TESTSplit the unknown word, translate each part, then check it against the sentence.Academic Word Rootslittleactivity.com
Literacy

Academic Word Roots

Thirty roots that carry academic and test vocabulary, with examples.

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The Periodic TableThe full table by group, with metals, non-metals and the…8th Grade1H2He3Li4Be5B6C7N8O9F10Ne11Na12Mg13Al14Si15P16S17Cl18Ar19K20Ca21Sc22Ti23V24Cr25Mn26Fe27Co28Ni29Cu30Zn31Ga32Ge33As34Se35Br36Kr37Rb38Sr39Y40Zr41Nb42Mo43Tc44Ru45Rh46Pd47Ag48Cd49In50Sn51Sb52Te53I54Xe55Cs56Ba57La58Ce59Pr60Nd61Pm62Sm63Eu64Gd65Tb66Dy67Ho68Er69Tm70Yb71Lu72Hf73Ta74W75Re76Os77Ir78Pt79Au80Hg81Tl82Pb83Bi84Po85At86Rn87Fr88Ra89Ac90Th91Pa92U93Np94Pu95Am96Cm97Bk98Cf99Es100Fm101Md102No103Lr104Rf105Db106Sg107Bh108Hs109Mt110Ds111Rg112Cn113Nh114Fl115Mc116Lv117Ts118Og123456789101112131415161718118 elements. Columns are groups (same outer electrons); rows are periods (same shells).FAMILIESAlkali metalsAlkaline earthTransition metalsPost-transitionMetalloidsNon-metalsHalogensNoble gasesLanthanidesActinidesWHAT THE COLUMNS TELL YOUGroup 1 metals react violently with water · Group 17 halogens are the most reactivenon-metals · Group 18 noble gases have full outer shells and barely react at all.The Periodic Tablelittleactivity.com
Science

The Periodic Table

The full table by group, with metals, non-metals and the families named.

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Lab Report GuideWhat goes in each section of a lab report, and how to write a…8th GradeTitleWhat was tested, in one specific line — not “Experiment 3”.QuestionThe exact question the experiment answers.HypothesisIf … then … because … — a prediction with a reason.VariablesIndependent (changed), dependent (measured), controlled (kept the same).MaterialsEverything, with quantities. Another class should be able to repeat it.ProcedureNumbered steps, past tense, no explanation — just what was done.DataTable first, then a graph. Label every axis with units.AnalysisWhat the numbers show. Trends, not opinions.ConclusionAnswer the hypothesis using the data. Say whether it was supported.Error & next stepsWhat could have skewed it, and what you would change.A CONCLUSION THAT SCORES"The hypothesis was supported. Plants in full light grew 6.2 cm on average,compared with 2.1 cm in shade — a difference of 4.1 cm over 14 days."State the verdict · cite the numbers · never introduce new claims.Lab Report Guidelittleactivity.com
Science

Lab Report Guide

What goes in each section of a lab report, and how to write a conclusion that scores.

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