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Systems of EquationsGraphing, substitution and elimination side by side, plus what…9th GradeGRAPHING — where the lines crossThe solution is the point that satisfies bothequations — where the two graphs meet.Good for a check; poor for exact answers.SUBSTITUTION — when one variable is already aloney = 2x + 1  and  3x + y = 111. Replace y: 3x + (2x + 1) = 112. Solve: 5x + 1 = 11 → x = 23. Back-substitute: y = 2(2) + 1 = 5Solution: (2, 5)ELIMINATION — when the terms line up2x + 3y = 12  and  2x − y = 41. Subtract the equations: 4y = 8 → y = 22. Substitute back: 2x − 2 = 4 → x = 3If nothing cancels, multiply one equation through first.Solution: (3, 2)HOW MANY SOLUTIONS?Onelines crossdifferent slopesNoneparallel linessame slope, different interceptInfinitethe same linesame slope, same interceptSystems of Equationslittleactivity.com

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Math · 9th Grade

Systems of Equations

Graphing, substitution and elimination side by side, plus what 'no solution' looks like.

How to use it

Pick the method from the shape of the equations, not from habit.

Free · print-ready PDF · US Letter and A4

Grade
9th Grade
Ages
14–15
Subject
Math
Pages
1

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Function ToolkitDomain, range, notation and the parent functions with their…9th GradeFUNCTION NOTATIONf(x) = 2x + 1 → f(3) = 7f(x) is NOT f times x. It means "the output of f when the input is x".A relation is a function when every input has exactly one output — the vertical line test.DOMAIN & RANGEDomainevery input x is allowed — watch for ÷0 and √(negative).Rangeevery output y the function can actually produce.PARENT FUNCTIONSLinearf(x) = xstraight lineQuadraticf(x) = x²parabola, opens upAbsolute valuef(x) = |x|V shapeSquare rootf(x) = √xhalf a sideways parabolaExponentialf(x) = 2ˣflat then steepCubicf(x) = x³S through the originREADING A GRAPHx-intercepts are the zeros. The y-intercept is f(0).Function Toolkitlittleactivity.com
Math

Function Toolkit

Domain, range, notation and the parent functions with their graphs.

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QuadraticsFactoring, the quadratic formula, the discriminant and…9th GradePARABOLA ANATOMYvertexrootrootaxis of symmetryy = ax² + bx + ca > 0 → opens up · a < 0 → opens downAxis of symmetry: x = −b / 2ay-intercept is c. Roots are where y = 0.Roots, zeros, solutions and x-intercepts all mean the same thing.SOLVE BY FACTORING — try this firstx² + 7x + 12 = 0 → (x + 3)(x + 4) = 0 → x = −3 or −4Find two numbers that multiply to c and add to b. Here: 3 × 4 = 12 and 3 + 4 = 7.Zero product property: if two things multiply to zero, one of them must be zero.THE QUADRATIC FORMULA — always worksx = ( −b ± √(b² − 4ac) ) / 2aWrite a, b and c down first, with their signs. Most errors are a lost minus sign.x² + 7x + 12: a = 1, b = 7, c = 12.THE DISCRIMINANT b² − 4acpositive → 2 real rootszero → 1 repeated rootnegative → noneCheck it before you factor — it tells you whether factoring will work at all.Quadraticslittleactivity.com
Math

Quadratics

Factoring, the quadratic formula, the discriminant and parabola anatomy.

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Geometry ReferenceAngle relationships, triangle rules, congruence shortcuts and…9th GradeANGLE RELATIONSHIPSLinear pairsupplementary — add to 180°Vertical anglescongruentCorresponding (parallel lines)congruentAlternate interiorcongruentSame-side interiorsupplementaryTRIANGLE RULESAngle sumthe three angles add to 180°Exterior angleequals the two remote interior angles addedIsoscelesbase angles opposite equal sides are equalTriangle inequalityany two sides must add to more than the thirdMidsegmentparallel to the third side and half its lengthPROVING TRIANGLESCongruent:SSS · SAS · ASA · AAS · HL (right triangles)Similar:AA · SSS (ratios) · SAS (ratio + included angle)Not a shortcut: SSA and AAA.AAA proves similar, never congruent.Mark the given information on the diagram before writing a single line of proof.Geometry Referencelittleactivity.com
Math

Geometry Reference

Angle relationships, triangle rules, congruence shortcuts and the proof vocabulary.

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Literary TermsThe terms high-school essays are expected to use, grouped and…9th GradeThe vocabulary a high-school essay is expected to use — precisely, not decoratively.DictionThe writer's word choice, and its connotations.SyntaxSentence construction — length, order, fragments.ToneThe writer's attitude towards the subject.MoodThe atmosphere the reader is put into.MotifA recurring image or idea that builds meaning.AllusionA reference to another text, myth or event.JuxtapositionTwo contrasting things placed side by side.ParadoxA statement that contradicts itself yet holds true.OxymoronTwo contradictory words joined: deafening silence.SatireRidicule used to expose a flaw or folly.Verbal ironySaying the opposite of what is meant.Situational ironyThe outcome is the opposite of what is expected.Dramatic ironyThe reader knows what a character does not.AnaphoraThe same phrase repeated to open successive lines.UnderstatementDeliberately making something sound smaller.CharacterisationDirect (told) or indirect (shown through action).Two terms used exactly beat ten used loosely.Literary Termslittleactivity.com
Literacy

Literary Terms

The terms high-school essays are expected to use, grouped and defined.

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Thesis & EvidenceHow to build a defensible thesis and the evidence sandwich…9th GradeTHE THESIS FORMULAAlthough [counter-view], [your claim]because [reason 1] and [reason 2].“Although Macbeth is warned repeatedly, his fall is self-authored,because he reads prophecy as permission and silences every doubt.”ARGUABLE vs NOTNot a thesis“This essay is about the water shortage.”Not a thesis“Pollution is bad for rivers.”A thesis“River charges, not household bans, are the only lever that has cut usage.”THE EVIDENCE SANDWICH1Set upGive the context the quotation needs.2QuoteShort. Embedded in your own sentence.3AnalyseTwo sentences on the choice of words.4LinkTie it back to the thesis, explicitly.CHECKCould a reasonable person write the opposite essay? If not, sharpen the claim.Thesis & Evidencelittleactivity.com
Literacy

Thesis & Evidence

How to build a defensible thesis and the evidence sandwich that supports it.

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Cell Structure & FunctionEvery organelle with its job, plus transport across the…9th GradePLANTANIMALORGANELLES BY JOBNucleusControl centre — holds the DNA.Cell membraneDecides what gets in and out.CytoplasmJelly where the reactions happen.MitochondriaReleases energy from sugar.RibosomesBuild proteins.Endoplasmic reticulumTransport network for proteins and fats.Golgi bodyPackages and ships what the cell makes.VacuoleStorage. Large and permanent in plants.Cell wallRigid outer layer — gives shape and strength.ChloroplastMakes food using sunlight (photosynthesis).Transport: diffusion (high → low) · osmosis (water) · active transport (needs energy).Cell Structure & Functionlittleactivity.com
Science

Cell Structure & Function

Every organelle with its job, plus transport across the membrane.

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Periodic TrendsAtomic radius, ionisation energy and electronegativity, with…9th GradeHHeLiBeBCNOFNeNaMgAlSiPSClArKCaScTiVCrMnFeCoNiCuZnGaGeAsSeBrKrRbSrYZrNbMoTcRuRhPdAgCdInSnSbTeIXeCsBaLaCePrNdPmSmEuGdTbDyHoErTmYbLuHfTaWReOsIrPtAuHgTlPbBiPoAtRnFrRaAcThPaUNpPuAmCmBkCfEsFmMdNoLrRfDbSgBhHsMtDsRgCnNhFlMcLvTsOg123456789101112131415161718across a period →THE FOUR TRENDSAtomic radiusDECREASES →INCREASES ↓more protons pull the same shell in tighterIonisation energyINCREASES →DECREASES ↓harder to remove an electron held more tightlyElectronegativityINCREASES →DECREASES ↓F is the highest; noble gases are excludedMetallic characterDECREASES →INCREASES ↓metals give electrons away easilySHORTCUTRadius grows down and left. Everything reactive-with-electrons grows up and right.Periodic Trendslittleactivity.com
Science

Periodic Trends

Atomic radius, ionisation energy and electronegativity, with arrows for each trend.

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