Explorer
Every studio, from easiest to hardest.
Number and place value
547
- counting and comparing to 10Match apples one to one, count them and compare. Then meet the signs <, > and =.Play and learn with the hoopoe
- Numbers to 20: a ten and some onesRead and build the numbers 10 to 20 with two ten-frames. One more, the number between, and first, second, third.Play and learn with the hoopoe
- Number lines, counting in steps, odd and evenFind numbers on a number line, count in 2s, 5s and 10s, and discover odd and even numbers.Play and learn with the hoopoe
- Tens and ones: numbers to 100Bundle ten ones into a ten. Build and read numbers to 100 with tens and ones.Play and learn with the hoopoe
- Hundreds and thousands: place value and expanded formBuild, read and write numbers to 10 000 with base-ten blocks, write them in expanded form, compare them from the highest place, and meet Roman numerals.
- Large numbers, rounding and estimatingRead and build numbers in the millions in groups of three digits, round to the nearest 10, 100 or 1000 by looking at the next place, and estimate a sum before you work it out.
- Estimating and rounding to tens and hundredsRound numbers to the nearest ten or hundred and estimate sums and differences without an exact calculation.
- Negative numbers and integersCarry the number line on to the left of zero. Show temperatures below zero and depths below sea level, then add and subtract with jumps.
- Rounding decimalsZoom in between a number's two neighbours and see which is nearer: to the nearest whole number, to 1 and 2 decimal places, and to significant figures. Then find every number that rounds to the same value.
- Factors, multiples and primesArrange counters into rectangles to find a number's factors. See why some numbers make only one rectangle.
- Divisibility, HCF, LCM and powersShade multiples on a hundred square to discover divisibility rules. Then common multiples and factors, square numbers and powers.
- Multiplying and dividing integers; absolute valueMultiply and divide negative numbers with hops on the number line, find the pattern behind the signs, and measure distance from zero.
- Rational numbers: ordering and the four operationsPut negative fractions and decimals on one number line, compare them, then add and multiply them as moves and as scalings.
- The sieve for prime numbersWrite the numbers 1 to 50 and cross out multiples until only the primes are left, then find out where to stop.
- Subtracting a negative numberRewrite every subtraction as an addition, see subtracting a negative on the line, and work two-digit sums in tens and ones.
- Powers of negative numbers and doubling growthFollow the order of operations with powers, understand the difference between (−3)² and −3², and see why repeated doubling gets large so fast.
- Prime factors, HCF × LCM, and exact rootsSplit two numbers into primes with factor trees. Build the HCF and LCM from the primes, and find square and cube roots by pairing and grouping the primes in threes.
- Laws of indices, zero and negative powersWrite powers out as factors on a fraction bar: multiply by putting the factors together, divide by cancelling equal factors above and below, and see why a⁰ = 1 and why a negative index means a fraction.
- Dividing rational numbersFind the reciprocal of a rational number and turn divisions of signed numbers into multiplications. See why dividing by zero makes no sense.
- nth roots and rational powersLearn nth roots through powers: even and odd roots, multiplying roots, then fractional powers like 2713 and 823.
- Standard form (scientific notation)Slide the digits with ÷10 and ×10 until one non-zero digit is left of the decimal point, and watch the power of ten count the slides.
- Estimating square roots; irrational numbers on the lineTrap a square root between whole numbers, zoom in to tenths and hundredths by squaring, and sort rational from irrational numbers on the real line.
- Surds and simplifying themSplit the number under a root into factors and take each pair out: √12 = 2√3. Multiply roots, add like surds, and rationalise a denominator.
- Euclid's method for the highest common factor (HCF)Find the highest common factor (HCF) of two numbers with a chain of divisions, without listing any factors.
- Absolute value of real numbers and approximate calculationSee absolute value as distance from the origin, meet the three cases, compare |a + b| with |a| + |b|, and calculate with approximate roots.
- Root properties, rational powers and irrational equationsFind the root of a quotient and the root of a root, compare roots of different degrees, and solve an irrational equation by squaring, checking the condition and substituting. Section: Grade 10, nth roots, rational powers, irrational equations; the Grade 8 square root is in another studio.
- Numbers in words: ten thousands, hundred thousands and billionsBuild and read big numbers: five-digit numbers, a hundred thousand, and numbers past the million.
- Roots: square and cube roots and their signsCount the numbers that square to a positive number, see why a negative has no real square root but does have a cube root, and why the root of a square is the absolute value.
- Rounding on the number lineFind the number on the number line and see which ten, hundred or thousand it is nearest to. A number exactly halfway rounds up.
- Numbers in wordsBuild two- and three-digit numbers from tens and ones, and read and write them in words.Play and learn with the hoopoe
- Divisibility in the integersThe division algorithm, properties of divisibility, and the gcd written as a combination of the two numbers.
- Number table and skip countingSkip count in 3s, 4s, 6s and 7s on a number table, and see repeated addition with rods.
- Congruence in the integersNumbers with the same remainder on division by m are congruent. See how to calculate with them.
- Place value in decimalsWrite and read decimals in a place-value table with ones, tenths and hundredths columns, build them in expanded form and compare them.
- Approximation: rounding and truncatingApproximate a number to the nearest 10, 100, 1 or 0.1 with two methods, rounding and truncating, tell which one is rounded, approximate a fraction and estimate a product.
- Approximate values and errorFind the absolute and relative error, compare the accuracy of measurements and bound the error of a sum.
- Composition of 6–10 and zeroSplit the numbers 6, 7, 8, 9 into two parts with counters. Then find out what happens when all, or nothing, is taken away.
- Composition card and stripsWrite the two parts that make 6, and build a strip of 7 cells.
- Properties of the arithmetic square rootFind roots of squares, products, quotients and powers with the properties of the arithmetic square root, as in the Grade 8 Ukrainian book. Section: Grade 8, arithmetic square root; nth roots are in the Grade 10 studio.
- The natural series and the coordinate rayFind the number before a given one, find out why there is no largest natural number, mark K(5) on a coordinate ray and read a scale.
- Division with remainder, divisibility by 9 and 3Find out why the remainder is smaller than the divisor, build the dividend with a = bq + r and test divisibility by 9 and 3 by the digit sum.
- Nested roots: √(a ± 2√b)Get rid of a root under a root: write the expression as the square of a sum or a difference and find the numbers from their sum and product.
- Comparing and ordering big numbersCompare numbers up to a million place by place, put them in order and add whole thousands.
- Binary numbers and Roman numeralsRead and write binary numbers with the place values 1, 2, 4, 8, 16 and compare Roman numerals with our place value system.
- Distance between points and equations with a modulusFind the distance between two points of the number line as |a - b| and solve equations |x - a| = b by marking points.
- Divisibility of sums, differences and products; tests for 4, 8, 6 and 11Decide whether a sum, difference or product is divisible without calculating it, and use the tests for 4, 8, 6 and 11.
- Heron's method and irrational numbersApproximate a square root with the Heron method (a rectangle that becomes a square), then see why √2 is not a fraction.
Calculation
430
- Adding and taking away within 10Put apples together, take some away and compare. Then make 10 from two numbers.Play and learn with the hoopoe
- Within 20: bridging ten and fact familiesFill the ten first, then put the rest. Add three numbers, swap the numbers round and find the four facts in a family.Play and learn with the hoopoe
- Mental adding and subtracting with jumpsShow each addition and subtraction as jumps on a number line: tens then ones, or to the next ten first.
- Equal groups: multiplying and dividingPut apples into equal groups, build arrays and see how multiplying and dividing fit together.Play and learn with the hoopoe
- Times tablesLearn the times tables with pictures and practise them in a short game: 2, 5 and 10 first, then 0 and 1, then the rest up to 12 × 12.
- Multiplying and dividing by 10, 100 and 1000When you multiply by 10, every digit moves one place to the left; when you divide by 10, one place to the right. See it on the place-value chart, first with whole numbers, then with decimals.
- Column addition and subtractionWrite numbers in columns and add or subtract one column at a time: carry a ten to the next column, exchange from the column next door, and check your answer.
- Column multiplicationMultiply in columns: short multiplication by a 1-digit number with carried digits, long multiplication with two rows and a placeholder 0, and then decimals.
- Division with remainders and long divisionWrite a division digit by digit, see what a remainder is and what to do with it in a problem, and check a division by multiplying.
- Extra: adding and subtracting on the abacusAdd and subtract 2-digit numbers by moving beads on an abacus. Exchange ten ones beads for one tens bead, and back.
- Multiplying round numbers and calculating with decimalsMultiply numbers that end in zeros (40 × 30, 650 × 300), add and subtract decimals in columns, and multiply two decimals by counting decimal places.
- Written multiplication by multi-digit and round numbersMultiply in a column by a three-digit number, a number with a zero and a round number, using partial products.
- Multiplying with blocks: split into partsMultiples of ten first, then a multi-digit number times a one-digit number with hundred, ten and one blocks: multiply each part and add.
- Four-digit addition and subtraction with thousand blocksAdd and subtract four-digit numbers in the place-value table with ones, tens, hundreds and thousands blocks, exchanging ten for one of the next place.
- Counting in groups and multiplication storiesCounting in steps of 7 and 200, "times as many" stories, changing dollars into toman and speed problems with a multiplying machine.
- Parts of an equality: addend, sum, differenceAdd and subtract by parts on a number line, name the parts of an equality and subtract through addition.
- Unknown component and change of the sumFind the unknown addend, minuend and subtrahend by rule, compare expressions without calculating and watch how the sum changes.
- Two-digit numbers: by digits, by parts, the rounding trickAdd and subtract two-digit numbers by digits and by parts, and work out 38 + 9 through 10, giving back the extra one. Section: Grade 2; Grade 3 oral methods are in another studio.Play and learn with the hoopoe
- The Pythagoras table and how results depend on the numbersExplore the multiplication table, multiply and divide in sequence, see how a product and a quotient depend on their components.
- Oral methods: by digits, rounding, dividing in partsAdd and subtract by digits and by rounding, divide numbers by splitting them and divide round numbers in place-value units. Section: Grade 3; Grade 2 two-digit numbers are in another studio.
- Smart ways to multiply and divideMultiply and divide by 5, 25, 50, 11, 101, 9 and 99 in handy ways, as the book does.
- Laws of operations and convenient calculationUse the commutative, associative and distributive laws to calculate conveniently, and simplify expressions with rational numbers.
- Expression, equality, inequalityTell expressions, equalities and inequalities apart, cross out false equalities and compare expressions without calculating. Ukrainian grade 2: numerical expressions, signs =, < and >.
- Division with a remainder by 10 and 100Divide with a remainder by 10 and 100 by covering digits on the right, and divide numbers ending in zeros by striking out the same number of zeros. Ukrainian grade 4.
- Chains of arithmetic actionsFollow the arrows: a number in the circle, an action on the flag. Find the number at the end of the chain and restore a rubbed-out flag. Ukrainian grade 1.
- Addition tablesRead the addition tables of 1, 2, 5 and 7, cover the results and write them in. Then the sums of the second ten. Ukrainian grades 1-2.
- Subtracting 6, 7, 8, 9Replace the minuend by a sum of convenient addends, one of which equals the subtrahend. Then the other addend is left. Ukrainian grade 1.
- Adding and subtracting a sumAdd a sum to a number, subtract a sum from a number and a number from a sum, writing equalities with brackets as in the Ukrainian Grade 2 book.
- Multiplying and dividing by partsMultiply and divide two-digit numbers by one-digit numbers by splitting the number into easy parts, and check with the inverse operation.
- Checking calculations: estimate and inverseEstimate the answer before you calculate, work in a column, check with the inverse operation, and decide whether a result (a calculator's too) is reasonable.
Fractions
118
- What is a fraction?Split one whole into equal parts and use a fraction to say how many parts you took.Play and learn with the hoopoe
- Fractions on a number lineFractions are numbers too, and each one has its own place on the number line. Split the space from 0 to 1 into equal parts and place points.
- Equivalent fractionsThe same amount can be written as different fractions. See how the fraction changes when the parts get smaller or bigger, but the amount stays the same.
- Comparing fractionsWhich fraction is bigger? Line the bars up and see whose shading goes further.
- Adding and subtracting fractionsPut fractions side by side on bars, then add and subtract them. When the denominators are different, make the parts the same size first.
- Multiplying and dividing fractionsUse bars to see that multiplying by a fraction means "how many lots" or "what fraction of", and dividing by a fraction means "how many fit".
- DecimalsFind tenths and hundredths on the number line, and see how decimals are written and read.
- Division and fractionsDo a division digit by digit, and see that every fraction is really a division you can turn into a decimal.
- A fraction of an amountShare apples equally and take some of the parts. Then scale a recipe up and down.
- Percentages and fraction–decimal–percent conversionPer cent means "out of a hundred". Use a hundred square and a 100% bar to see that a fraction, a decimal and a percentage are three names for one amount.
- Multiplying and dividing mixed numbersUse bars to see how multiplying and dividing mixed numbers works, and which operation comes first in a calculation.
- Decimal division and recurring decimalsTurn a decimal division into a division of whole numbers, keep dividing until you see digits repeat, and turn a recurring decimal back into a fraction.
- Complex fractionsA fraction whose top or bottom is a fraction: share a fraction on a bar, count fractions inside fractions, and divide by flipping the lower fraction.
- Subtracting mixed numbers and calculating with fractionsSubtract mixed numbers, open up a whole when needed, solve expressions with complex fractions and scale up a recipe.
- A part of a number and the number from its partFind a part of a number and find the number from its part: first one part, then the whole or several parts. Section: Grade 3 (part of a number and the number from its part); Grade 5-6 fractions are in another studio.
- Improper fractions, number from its fraction, percentage ratio, decimal approximationShow an improper fraction on bars, find the whole from the value of a part or of a percent, work out the percentage ratio of two numbers and round a fraction to hundredths. Section: Grades 5-6; the Grade 3 part-of-a-number studio is separate.
- Fractions of units of measureWhat part of a metre is 6 cm, and what part of a year is 3 months? Change a fraction of a unit into smaller units and compare it with a named number. Ukrainian grade 4, the section «We study fractions».
- A fifth and a tenthSplit a whole into 5 and into 10 equal parts, compare 15 and 110, and find a fifth and a tenth of an amount.
Ratio and proportion
17
- Ratio and sharing in a ratioUse two-colour bars to see what 2 : 3 means, simplify ratios, and share an amount in a given ratio.
- Proportion and the unitary methodUse a proportion table to find a missing amount: first the amount for one, then for any number. Inverse proportion comes last.
- Scale factors and similar shapesEnlarge a shape by a scale factor, spot similar shapes, find a scale factor, and draw a similar shape from one side.
- Percentage change, multipliers and reverse percentagesThe starting amount is a 100% bar. An increase makes the bar longer (120%), a decrease shorter (85%). Use the bars to find multipliers, percentage changes and original prices.
- Rates, unit prices and densityUse a ratio table to find the amount "for each one": the price per kilogram, the litres per pound, and the mass of each cubic centimetre.
- Scale drawings and mapsMeasure on a map and a room plan with a ruler, and use the scale to find the real sizes.
- Graphs of direct proportion and inverse proportionPlot a direct proportion from a table, find k from one point, fit y = k/x to a point with a slider, spot the direct-proportion graph and plot an inverse proportion.
- Extra: compound proportionIn compound proportion three quantities change together. In the table change one quantity at a time and decide whether each change is direct or inverse.
- Distance, speed, time: s = v · tFind distance, speed and time from a table and from the formulas s = v · t, v = s : t, t = s : v.
- Two bodies moving and joint workFind the distance between two bodies moving at once, and the time for joint work, from a table and a solving plan. Section: Grade 4, motion of two bodies and joint work; the fourth proportional is in another studio.
- Proportional division and the same quantitySolve problems with the short record price, count, cost: proportional division and finding the same quantity from two differences or two sums.
- Fourth proportional and double reduction to unitySolve problems by the method of ratios and by reduction to unity, and problems with two changes. Section: Grade 4, fourth proportional and double reduction to unity; motion and joint work are in another studio.
- Similar figures: length, area and volumeWhen a figure is enlarged by scale factor k, lengths grow k times, areas k squared times and volumes k cubed times. Use a ratio table to find the missing size.
- Work: productivity, time, outputSolve work problems: productivity · working time = total output, with a table and with the triangle scheme.
- Fourth proportional: the same quantitySolve problems with a quantity that stays the same (the mass of one box, a price, a total mass): find it first, then answer the question.
- Processes: motion and workSolve problems of simultaneous motion and of joint work with one plan: speed · time = distance, productivity · time = output.
- Percent: base value, percentage value, rateFind the percentage value W, the rate p % or the base value G with the percent strip, the rule of three and the formula W = G · p %.
Money
7
- Coins and notesGet to know coins and notes, make an amount of money in different ways, and see that more coins is not always more money.
- Shopping and changeAdd up the prices in a basket, pay, and work out the change by counting up.
- Percent in money: discounts, profit and lossUse a 100% bar to see how much a discount takes off a price, and what percentage of the cost a seller gains or loses.
- Simple and compound interestFollow savings year by year: with simple interest the same amount is added every year, with compound interest the money is multiplied by the same number. Then see how something loses value each year.
- VAT, cash discount, selling price and per milleNet and gross prices with 19 % and 7 % VAT, a trade discount followed by a cash discount, the selling-price schedule, and per mille.
- Percent and percentage pointsHow "rose by 5 %" differs from "rose by 5 percentage points": a rate of 20 % to 25 %, the share of top pupils, a bank problem. This follows the Ukrainian Year 9 algebra book, the page "How to avoid ambiguity in percent problems".
- Loans: differentiated and annuity paymentsTake a loan and watch the debt grow and shrink: repaid in equal shares of the debt, or in equal payments. Compare what the borrower pays in the end.
Measurement
333
- Comparing: longer, heavier, fullerWhich is longer? Which is heavier? Measure with paper clips, hand spans and cups.Play and learn with the hoopoe
- Measuring length, mass and capacityMeasure with a ruler, scales and a measuring jug. Estimate first, then measure.Play and learn with the hoopoe
- Clocks and calendarsTurn the hands of a clock, write an afternoon time in 24-hour time, and find days on a calendar.Play and learn with the hoopoe
- perimeter and areaWalk round a shape to find its perimeter, cover it with unit squares to find its area, and see that shapes with the same area can have different perimeters.
- Quarter hours and order of timeShow quarter past and quarter to on the clock, write a time two ways and decide which job took longer.
- Converting unitsConvert metres to centimetres, grams to kilograms and litres to millilitres. The conversion strip helps you.
- Area of triangles, parallelograms, trapezia and circlesMove pieces to see why the area of a parallelogram is base × height, why a triangle is half of that, complete a trapezium with a copy of itself, and estimate the area of a circle.
- volume of cuboidsBuild boxes from unit cubes, count cubes layer by layer, and see why the volume of a cuboid is length × width × height.
- Extra: thermometers and imperial unitsRead a thermometer in °C, then meet inches, feet, pounds and pints and their metric equivalents.
- Volume and surface area of prisms and cylindersTurn prisms and cylinders, find the base (of a prism, cylinder, pyramid or cone), fill them layer by layer to see why volume = base area × height, and open the net (of a solid) to work out the lateral surface area and the total surface area.
- Measuring more exactly and a part of a unitMeasure with your own unit, count the small parts of a centimetre and find the middle of a line.
- Pyramids, cones, spheres and composite solidsPour a pyramid into its prism and a sphere into its cylinder to see where the 13 and the 23 come from. Open a cone into a sector, find its slant height with Pythagoras' theorem, and work out the volume and surface area of a tower.
- Tube, tin and box: volume and surface problemsRoll a sheet into a tube and give its capacity in litres, compare the sheet needed for a tin and a box, and find the volume of an open box.
- Limits of accuracy: upper and lower boundsA rounded measurement is really somewhere in a range. Draw the range, then use the bounds to find the biggest and smallest answers.
- Measuring time: minutes and secondsFind an arrival time, work out the time between two clocks with time arrows, and convert between seconds, minutes and hours.
- Kilometres, capacity and the mass of waterConvert metres and kilometres, tell a container's capacity from the volume of liquid in it, and see that one litre of water is 1000 cubic centimetres and weighs 1 kg.
- Area of a rhombus and a circleFind the area of a rhombus with four triangles and with its diagonals, see why a rhombus is half the rectangle round it, and work out the area of a circle from its sectors and with the formula pi × radius × radius.
- Metres with centimetres, kilograms with gramsRead and write two-unit measures (325 cm = 3 m 25 cm), choose a suitable unit and read half a centimetre on the ruler.
- The decimetre and actions with quantitiesLearn that 1 dm = 10 cm, measure ribbons in decimetres and add lengths given in decimetres and centimetres.
- Named numbers: arithmetic with themAdd, subtract, multiply and divide named numbers: write them in one unit, do the sum and write the result in two units.
- Area of a rectangle and units of areaFind the area of a rectangle with S = a · b, find an unknown side and learn the units of area: cm², dm², m², are, hectare. Section: Grade 4, area of a rectangle and units of area; equal figures are in the Grade 5 studio.
- Time: adding and subtracting named numbersChange larger units of time into smaller ones, add or subtract, and write the answer in two units. Find how long an event lasted from its start and end.
- Prisms, parallelepipeds and pyramids: properties, sections, frustumFind a box diagonal, the volume of an oblique prism, the lateral area of a regular pyramid, the volume of a frustum, and a section of a pyramid.
- Sections of a cylinder, cone and ball, the sphere equationFind the diagonal of an axial section, the radius of a section of a cone and of a ball, and read the centre and radius from a sphere equation.
- Prisms: problems with angles and diagonalsFind the height and the lateral edge of a right prism from a diagonal and an angle, work out the lateral area and give answers with a root.
- Pyramids: where the apex projectsFrom equal lateral edges or equal dihedral angles, decide which centre of the base the apex projects to, then find the height and the lateral area.
- Volumes of solids: problemsFind the volume of a part from the rise of the water, the radius of a melted ball and the volume of a solid made by turning a triangle or a trapezium. Keep π in the answer.
- Perimeter of a polygonFind the perimeter of triangle ABC and of rectangle DMPK as the sum of the lengths of all sides.
- Arc length, sector area and segment areaFind the length of an arc of n°, the area of a circular sector and of a segment, with l = πRn/180 and S = πR²n/360.
- Sector, arc and segmentA sector is α/360° of a circle: find its arc length and area, find the angle from the arc length, and find the area of a circular segment.
- Parts of a ball and the oblique prismFind the area of a spherical zone, the volume of a segment and of a sector of a ball, and the volume of an oblique prism. Type answers with π: 20π is entered as 20π.
- Units of mass, speed and areaConvert tonnes, centners and kilograms, km/h to m/s and hectares to square kilometres, and choose a sensible unit.
- Centners, tonnes and centuriesChange tonnes and centners into kilograms and find which century a year belongs to.
Shape and geometry
282
- Flat shapes and symmetryMake a triangle and a square on the pinboard, find the triangles, and draw a line of symmetry.Play and learn with the hoopoe
- Corners, circles, rectangles and symmetryCount corners, find the shape with no corner, draw a rectangle, fold a shape and cut one into equal parts.
- 3-D shapesRecognise the cube, cuboid, cylinder, cone, sphere and pyramid, count faces and edges, and make a net of a cube.Play and learn with the hoopoe
- Lines, rays, segments and anglesFind the rays, draw parallel and perpendicular lines on dotty paper, and compare angles with a right-angle corner.
- Triangles and quadrilateralsSort triangles by their sides and angles, meet the quadrilateral family, and draw the diagonals of a rhombus.
- Polygons and angle sumsFind the regular polygons, discover the angle sum of a triangle, and split polygons into triangles to find their angle sums.
- Measuring angles with a protractorRead and draw angles with a protractor. Find angles on a straight line, vertically opposite angles and angle bisectors.
- circles: radius, diameter and πMeasure the diameter and radius of a circle. Wrap a tape round a circle and find how many diameters make the circumference.
- Ruler-and-compass constructionsUse compasses (pair of compasses) and a ruler to meet chords and arcs, construct a perpendicular bisector and an angle bisector, and build triangles from their sides.
- Angles with parallel linesDrag and turn two parallel lines and a transversal to see which angles always stay equal, which add up to 180°, and what happens when the lines are not parallel. Then use a parallel line to show why a triangle's angles add up to 180°.
- Triangle inequality; medians, altitudes and bisectorsBuild a triangle from three sticks and find out which three lengths make a triangle and which don't. Then test a median, an altitude and a perpendicular bisector on a triangle you reshape yourself, and see where the three medians always meet.
- Congruent shapes and congruent trianglesSlide, turn and flip a shape onto another. Then match the corresponding vertices of two triangles, choose the congruence condition, write a proof line by line, and drag a counterexample that shows SSA is not a condition.
- Pythagoras' theorem and its converseBuild squares on the sides of a right-angled triangle and compare their areas, see why the theorem is true with four copies of the triangle, find a missing side, use the converse to test for a right angle, and mark √2 on the number line with compasses.
- Reasoning and proof in geometryDrag an isosceles triangle and see that its base angles always stay equal. Then prove it with reasons, use it in a new proof, and show that another claim is false with a counterexample.
- The intercept theorem and similar trianglesSlide a line parallel to one side of a triangle and see that the ratio it cuts on the other two sides always stays equal. Then try the converse, similar triangles and scale factors.
- LociFind every point at a given distance from a point, from a line segment, from two points or from the two arms of an angle, and see what shape they make. Then use compasses to shade a region that meets two conditions at once.
- Lines and circles, angles at the centre and circumference, tangentsSlide a line across a circle and see when it cuts the circle, when it just touches it and when it misses. Bring two circles together, turn a tangent round a circle, and see that the angle at the circumference is always half the angle at the centre.
- Cyclic polygons, incircles and lengths in a circleBuild a triangle's circumcircle and incircle and see where their centres are. Then drag the corners of a cyclic quadrilateral round a circle and test two crossing chords: which relationship always stays true?
- Plans, elevations and Pythagoras in 3DTurn a tower of cubes, recognise its front, side and top views and draw them on a grid. Then find the space diagonal of a box with Pythagoras' theorem used twice.
- Extra: right-triangle theoremsIn a right-angled triangle with a 30° angle, the side opposite it is half the hypotenuse; and the median to the hypotenuse is half of it too. See both, prove one and use them.
- Axioms, segments and theoremsSee that one line passes through two points and that AB = AC + CB, find the midpoint of a segment and tell a theorem from its converse.
- Extra: tessellationsFit copies of a regular polygon round a point and see which fit with no gaps and no overlaps, why the regular pentagon leaves a gap, and tile with two kinds of polygon.
- Extra: regular polyhedra and cross-sectionsTurn the regular polyhedrons, count their faces, vertices and edges and discover V − E + F = 2. Then cut a cube with a knife and see which shapes a cross-section can be.
- The exterior angle, sides and angles, right trianglesSee that an exterior angle equals the sum of two interior angles, that the larger angle lies opposite the longer side, and use the congruence criteria of right triangles.
- Chords, loci and the four steps of a constructionSee that the segment from the centre to the midpoint of a chord is perpendicular to it, find the distance to a chord and count the solutions of a locus construction.
- Conic sections: slicing a coneCut a cone with different planes and see where the circle, ellipse, parabola and hyperbola come from. Then see what the ratio of the distance to a focus and to a directrix builds.
- Regular polygons and the excircle of a triangleFind the side and apothem of a regular polygon from the radius of the circumscribed or inscribed circle, and the radius of the excircle of a triangle.
- Lines and planes in spaceUse a cuboid to find how two lines can sit in space: parallel, intersecting or skew. Then look at a line and a plane, a line perpendicular to a plane, and the angle between two skew lines.
- Axioms of solid geometry and sectionsPass a plane through three points, find the line where two planes meet, recognise a line parallel to a plane, and name the shape of a section of a cube.
- Parallel planes, projection and perpendicularsTest two planes for parallelism, compare segments between parallel planes, find the length of a parallel projection and the distance from a point to a plane.
- Symmetry of polygons, interior and exterior anglesCount the lines of symmetry of regular polygons, find the angle at each corner and see why the exterior angles always add up to 360 degrees.
- Circle theorems and the angle bisector theoremProve that two tangents from a point are equal, and learn that the bisector of an angle of a triangle cuts the opposite side in the ratio of the other two sides.
- Perpendicular and parallel through a point, and constructing from dataFrom a point outside a line, construct a perpendicular and a parallel, then build a rhombus and an isosceles triangle with the compasses by analysing the problem first.
- Constructing an angle and triangles with compasses and rulerConstruct an angle equal to a given one, a triangle from two sides and the angle between them, from a side and two adjacent angles, and a right triangle from a leg and the hypotenuse.
- Loci in construction problems: a point, a circle, a triangleFind the wanted point as the crossing of two loci. Three tasks from the Ukrainian book: a point on a line at a given distance from another line, a circle tangent to a line at a given point, and a triangle from a side and the circumradius.
- Proof by contradictionLearn to prove theorems by contradiction: assume the conclusion is wrong, reach a contradiction, and conclude.
- Ceva, Menelaus, Ptolemy and the Euler lineOptional stories from Grade 8 geometry: when three cevians meet at a point, when three points are collinear, the product of the diagonals of a cyclic quadrilateral, and the Euler line.
- Rays, segments and diagonalsName points with letters, rays and segments, draw a segment of a given length and count the diagonals of a polygon by solving a simpler problem.
- Symmetry and folding paperCount symmetric figures with a line of symmetry, draw the mirror image and predict the shape after folding and cutting paper.
- Geometry problem solving: given, claim and planSeparate the given from the claim, plan before writing a proof, find the shortest road with a perpendicular and Pythagoras, and see circle applications.
- The house of quadrilateralsSort trapeziums, kites, parallelograms, rhombuses, rectangles and squares by symmetry and diagonals, and construct a kite.
- Metric relations in the right triangle and properties of a proportionDraw the altitude to the hypotenuse of a right triangle and see that AH² = BH×HC; find a side with it, use the adding property of a proportion, and build Pythagoras from the same figure.
- Thales, circumcircle and tangentProve the Satz des Thales and its converse, find the circumcircle of a triangle, and construct a tangent with the Thaleskreis.
- Circle and discTell a circle from a disc, find the centre and radii and mark the points that lie on the circle.
- Constructing triangles with a planPlan figure, numbered description and number of solutions for triangle constructions, also with a height.
- Statement, converse and proofSplit a theorem into Voraussetzung and Behauptung, form the Kehrsatz and refute it with a Gegenbeispiel, and prove and use the Satz des Pythagoras.
- Oblique drawings of solidsShorten depth edges with the scale q, spot faulty oblique drawings and read true sizes from a Schrägbild.
- Planes of symmetry and sectionsCount planes of symmetry of solids, find axial sections of cylinder and cone, and use 3-D coordinates for the space diagonal.
- Stretching in the coordinate systemCompute images of points and lines under a Zentrische Streckung with the vector rule, find k, and find the Schwerpunkt of a triangle.
- Perpendicular and slant, line–plane angle, dihedral angleFind a slant from the perpendicular and its projection, measure the angle between a line and a plane and a dihedral angle by its linear angle, then show that two planes are perpendicular.
- The parallelogram, rectangle, rhombus and squareLearn the elements of a quadrilateral, the properties and criteria of a parallelogram, and the diagonals of a rectangle and a rhombus, as in the Grade 8 Ukrainian geometry book.
- The trapezium, its midline, a circumscribed quadrilateral and the area of a trapeziumLearn the properties of an isosceles trapezium, the midline, the criterion for a quadrilateral with an inscribed circle and the area of a trapezium.
- The second and third tests for similar trianglesFind similar triangles by two sides and the angle between them, or by three sides, work out an unknown side, and avoid the trap of an angle that is not between the sides.
- The segment, its length, and the lineCompare segments by laying one on another, find the length of AB from two parts, measure the distance between points and find how many lines pass through two points.
- Polygons, equal figures, area, are and hectareName a polygon by its vertices, find an equal figure by laying it on another, learn how equal figures differ from figures with equal area, and convert a field's area to hectares. Section: Grade 5, equal figures, area, are and hectare; the formula S = a · b is in the Grade 4 studio.
- The cylinder and the cone: bases and netFind the bases of a cylinder and a cone and work out the lateral area of a cylinder from its net.
- Point, straight line, curve, ray, segment, broken lineFind points, straight lines and curves in a picture. Name the parts of a line that points make and measure the length of a broken line. Ukrainian grade 1.
- Secants and the tangentFrom a point P outside a circle two secants, or a secant and a tangent, are drawn. The products of the segments agree: PA · PB = PC · PD and PT² = PA · PB.
- Parallel lines meeting a plane and parallel projectionFind lengths of segments of parallel lines meeting a plane, and ratios of segments under parallel projection.
- Doubling a median and Varignon's theoremExtend a median to get a parallelogram and prove a bound for the median; join the midpoints of the sides of any quadrilateral and see what shape appears.
- Ratios of areas of trianglesCompare areas without calculating them: triangles with a common height have areas in the ratio of their bases, a median halves a triangle, and a vertex moving on a parallel line keeps the area.
- Similarity in practice: height from a shadow, an inaccessible distanceUse similar triangles for measuring in the field: the height of a tree from its shadow, a distance across a river, and the ratio of areas of similar figures. Lay solutions out as Given, Find, Solution, Answer.
- Angles between chords, secants and a tangentAn angle whose vertex is not on the circle is measured by arcs: between two crossing chords, half the sum of the arcs; between two secants, half the difference; between a tangent and a chord, half the arc.
- Two circles and their common tangentsThe position of two circles depends on the distance between centres d and the radii R and r: move a circle and see how many common points and common tangents there are.
- Review of Grade 10 solid geometryChoose the right next step of a proof about skew lines, then solve problems on the angle between lines, parallel projection and a perpendicular to a plane.
- Check yourself: polyhedra, solids of revolution, volumesMultiple-choice tests with options A, B, C, D on the lateral area of a prism, a cylinder and a cone, and on the volume of a ball.
- Polyhedra: faces, edges, verticesFind the bases, side faces and edges of a pyramid, a prism and a cuboid, and count their faces and edges.
- Area of the orthogonal projection of a polygonUse the theorem on the area of a projection, S′ = S · cos α, to find the area of a projection or of the polygon itself.
- Where a line meets a plane, where two planes meetConstruct the point where a line meets the base plane of a cube and the line where two planes of a tetrahedron meet, using the axioms of solid geometry.
- A point equidistant from the vertices or the sides of a polygonIf M is equally far from the vertices of a triangle, its projection is the circumcentre; if equally far from the sides, the incentre. Find the distance from M to the plane.
- Angle, right angle, rectangle and squareName an angle by letters, find a right angle with the model, tell a rectangle from other quadrilaterals and a square from other rectangles.
- Regular polygons: angle, number of sides, constructionFind the number of sides of a regular polygon from its angle, test whether it exists, and build a hexagon by chords R and a 2n-gon.
- Symmetry and enlargement in spaceFind the mirror image of a point in a plane, work out how a volume grows under an enlargement, and find all the planes of symmetry of a cube and a ball.
- Polyhedra, nets and shortest pathsDecide which polyhedra are convex, use Euler's formula, unfold a prism and find the spider's shortest path over a brick.
- Inscribed and circumscribed solidsFind the radius of a ball inscribed in a cube or a cone, and of a ball round a cuboid, and the volume of a cylinder round a prism.
- Drawing on squared paperDraw on squared paper as in an exercise book: a segment of a given length, a square, a rectangle and a shape with a given perimeter.
- The set square and figures of rectanglesDraw a right angle and a perpendicular with a set square and find the area of a figure made of rectangles.
- Cube buildings: building plans and viewsRead a building plan with heights, draw the front and top views and work out how many cubes a building has at most.
- Grid paper: building, routes and area by cellsBuild a rectangle on a grid, measure the length of a route, find the area of a figure by cells and estimate a round shape with a palette.
- Set square, ruler and protractorDraw a perpendicular and a parallel with a set square, measure and draw angles with a protractor, and draw the height of a triangle.
- Tilings and border patternsCount the shapes around a point in a tiling, continue a border pattern by sliding and by reflecting, and find mirror lines.
- Maps in a rowApply two maps one after the other, find the carrier graph of a moving point and watch the order.
Position and direction
16
- Position, direction and turnsFind left and right, above, below and between. Then turn an arrow a quarter, a half and three quarters of a turn.Play and learn with the hoopoe
- coordinates and translationPlot points from their coordinates, find the missing vertex of a shape, and translate shapes on a grid.
- Reflection, rotation and rotational symmetryPlot points in all four quadrants, reflect shapes in an axis, rotate them about a point, and find shapes with a centre of symmetry.
- Transformations and enlargementSee which transformations keep lengths, angles and orientation, enlarge a shape from a centre by 2 and by 12, find the centre of an enlargement, and see what happens to the area.
- Reflection and rotationReflect a shape in a vertical line, a slanted line and the x-axis, complete a shape from its half, rotate shapes about different points and find the coordinates of an image.
- Composing moves, symmetry and the reflection trickMeet two reflections in a row, the glide reflection, the axes of symmetry of a polygon and the shortest path found with a reflection.
Patterns and sequences
8
- Repeating and number patternsShapes and numbers that follow a rule. Find the pattern, continue it and fix it.
- Input–output machines and sequence rulesA number goes into a machine and another comes out. Find what the machine does, then find the rule of a number sequence.
- The nth term, and geometric sequencesBuild a table from a matchstick pattern, find its nth term and use it to count figure 20 without drawing it. Then meet sequences that multiply each time.
- Sequences, nth term and recurrenceUse aₙ notation to find terms, build a sequence from a recurrence relation, spot a periodic sequence and count triangular numbers.
- Sigma and sums of seriesRead Σ notation, find sums of arithmetic and geometric series, and see why an infinite sum can still be a number.
- Special sums, differences and modellingBuild the sum of squares, add a series by the method of differences, find the nth term of a quadratic sequence and solve a real problem with an arithmetic series.
- Graph of a sequence, monotonic and boundedDraw a sequence as points (n, a_n): an arithmetic one lies on a line, a geometric one on a curve. Then decide whether a sequence always grows and whether it stays below a limit.
- Number patterns: Päckchen, ANNA numbers and number wallsContinue structured sums, find the rule, build ANNA numbers and work out a missing brick in a number wall.
Algebra
47
- Order of operations, letters and formulaeWork out expressions step by step in the right order. Then let a letter stand for a number and use formulae.
- Expressions, like terms and substitutionBuild expressions with algebra tiles, collect like terms, cancel zero pairs, then substitute a value for x.
- Solving linear equations on the balanceAn equation is a balance. Try values, then solve equations with x on one or both sides, fractions and brackets by doing the same to both pans, and change the subject of a formula.
- Forming and solving equations from word problemsLet a letter stand for the unknown, turn the story into an equation, solve it, then check the answer in the story.
- Expanding a bracket and factorisingSee a bracket as a rectangle: expand by painting its area with tiles, then work backwards and find the sides of a rectangle to factorise.
- Multiplying binomials and identitiesMultiply two brackets as a rectangle of tiles and on a grid, see why (a + b)² has two ab strips, and tell an identity from an equation.
- Inequalities, intervals and solving themShow the solutions of an inequality on a number line: filled or hollow circle, segment or arrow. Then solve inequalities and see why multiplying by a negative number turns the sign round.
- Polynomials: standard form, subtracting, multiplyingWrite a polynomial in standard form, subtract with brackets, multiply three terms by two on a grid and raise a product to a power.
- Simultaneous linear equationsFind the point that fits two equations at once: read it where the lines cross, draw the second line, eliminate a letter, meet parallel lines and get an exact answer where the graph only gives an estimate.
- Factorising quadratics and further identitiesArrange quadratic tiles into a rectangle to find its two brackets, use zero pairs for negatives, take out a common factor first, and meet the sum of two cubes.
- Equations in two variables and systemsFind the pairs that satisfy an equation in two variables, draw its graph, build a system from a story and find its solution.
- Algebraic fractionsFactorise the top and the bottom of an algebraic fraction and cancel whole factors. Find the values x cannot take, then add, subtract and divide algebraic fractions over a common denominator.
- Grouping and cubes: factorisingFactorise a four-term polynomial by grouping, factorise sums and differences of cubes and choose where to start when several methods fit.
- Quadratic equations: factorising, completing the square, the formulaSolve quadratic equations by factorising, complete the square with tiles as al-Khwarizmi did, and use the discriminant to see how many solutions there are.
- Polynomials and polynomial divisionExtension: the degree and standard order of a polynomial, then dividing polynomials by filling a multiplication grid backwards, with a remainder and with missing powers.
- Horner's scheme and integer rootsDivide a polynomial by x − a with Horner's table, find the remainder and the integer roots among the divisors of the constant term.
- Sign tables and fractional inequalitiesFind where an expression is positive and where it is negative: mark the zeros, fill in a sign table under the number line, and read off the solution of an inequality such as x − 1x + 2 ≥ 0.
- Inequalities in two variables and regionsDraw a boundary line, make it solid or dashed, test a point and shade the region; then find the region that fits two or three inequalities at once.
- Numerical inequalities and their propertiesCompare numbers by the sign of the difference a − b, prove inequalities, add inequalities term by term and estimate the value of an expression.
- Systems of linear inequalities in one variableSolve each inequality of a system, show the solutions on a line and find their common part: a segment, a single number or the empty set.
- Systems of equations in two variablesFind the solutions of a system with an equation of degree 2 as common points of graphs, solve it by substitution and form a system from a word problem.
- Simultaneous equations: substitutionPut the expression for one unknown into the other equation, keep the bracket, isolate an unknown first, and check the answer on the graph and in both equations.
- Rational and radical equationsSolve equations with fractions and equations with square roots, and throw out the extra roots by checking.
- Sum and product of roots, hidden quadratics, line meets parabolaRead the sum and product of the roots from the coefficients, solve an x to the fourth equation by a change of letter, and find where a line meets a parabola.
- Remainder theorem, factor theorem and roots of polynomialsFind a remainder without dividing, spot a factor with the factor theorem, find the roots of a cubic and read coefficients from roots.
- Quadratic inequalities and parametersSolve quadratic inequalities by the signs of their factors, and see how a letter in an equation turns 'how many roots?' into an inequality.
- Rearranging formulaeGet one letter on its own in a formula: use the same moves as on the balance, with letters.
- Binomial expansion for any power: approximationsUse the first terms of the binomial expansion to estimate roots and reciprocals of numbers near 1.
- Parabola through three pointsSet up and solve a system of three linear equations for a, b, c and check the parabola on its graph.
- Equations: the rule for the unknown componentCompose an equation from a story, solve it with the rule for the unknown component and check by substitution. Section: Grades 3-4, equations with an unknown component; formulas and the root of an equation for Grades 5-7 are in another studio.
- Operations on rational fractions: subtraction, opposite bottoms, powers, identitiesSubtract rational fractions with the sign change, find extra factors, handle opposite bottoms, raise a fraction to a power and prove identities, as the Grade 8 Ukrainian book does.
- Equivalent equations, rational equations and word problemsLearn what equivalent equations and an extraneous root are, solve a rational equation with a check, and form equations for motion and work problems.
- Incomplete quadratic equations and the quadratic trinomialSolve the three kinds of incomplete quadratic equation, find the roots of a quadratic trinomial and factorise it with its leading coefficient.
- Formulas, equations and the root of an equationUse the distance formula, find the unknown part of an operation, check a root by substitution, move terms across with the opposite sign and learn when ax = b has no root. Section: Grades 5-7, formulas, the root of an equation, ax = b; Grade 3-4 equations are in another studio.
- Systems of equations as models of word problemsTranslate problems about motion, work, digits and rectangles into equations: name the unknowns x and y, write the system, solve it and check that the answer fits the story. This follows the Ukrainian Year 9 algebra book, section 14.
- Expressions with a variableA letter instead of a box gives an expression with a variable. Put in the value of the variable and work out the value of the expression. Ukrainian grade 2.
- Inequalities with a variable: how a component affects the resultSolve inequalities with a variable in two ways: by selection and by reducing to an equation. Write the answer with an ellipsis. Ukrainian grades 3–4.
- The interval methodMark the roots on the number line, put a sign on each gap and read the solution of an inequality such as (x + 1)x − 4x − 2 ≤ 0.
- Equations with a parameterA parameter is a number that is not given but fixed. Look at the graph and split into cases: for which a are there roots, and how many?
- Multiplying and dividing algebraic fractionsMultiply and divide algebraic fractions: factor, cancel, turn division into multiplication by the reciprocal, and say which values of the letter are not allowed.
- Whole part of a fraction and equations in whole numbersTake the whole part of an algebraic fraction, find last digits and remainders of powers by their cycle, and solve equations in whole numbers.
- Quadratic equation with a parameterSlide a and watch the parabola go up and down: the number of roots follows the discriminant, the signs of the roots follow Vieta's theorem.
- Irrational inequalities and equivalent systemsSolve inequalities with a square root by turning them into an equivalent system, with the domain and the sign of the right-hand side in mind.
- Systems and sets of inequalities; exponential and logarithmic systemsTell "and" (a system, the common part) from "or" (a set, the union), solve mixed systems and systems of exponential and logarithmic equations.
- Word problems: mixtures, alloys and percentsSet up a before-and-after table for a solution, an alloy and a price, choose x so that the condition holds, and check that the answer suits the story.
- Structure of terms and order of operationsName the main operation of a term, do the steps in the right order and put brackets to change the result.
- Placeholder equations and inverse operationsWrite a story as an equation with a placeholder, find the number with the inverse operation, solve riddles backwards and check by substitution.
Coordinate geometry
12
- Gradient, intercept and y = mx + cMeasure how steep a line is with a step triangle, set m and c on sliders, plot a line from a table of values and find the equation of a line through two points.
- Line forms, parallel and perpendicular lines, distance (between two points) and midpointRearrange the general form of a line into y = mx + c, draw parallel and perpendicular lines, find the distance between two points with Pythagoras and the midpoint as an average.
- Dividing a segment in a ratio, and the distance from a point to a lineFind the point that divides a segment in a given ratio, and the distance from a point to a line with the formula.
- Area of a triangle from coordinates, and geometry problemsFind the area of a triangle from the coordinates of its three vertices, and use coordinates to solve geometry problems.
- The circle: equation, line and circle, tangentBuild and read the equation of a circle, find where a line meets a circle, and work with the tangent and its length.
- Conic sections and the parabolaRecognise the parabola as the set of points equally far from a focus and a directrix, and read its equation with the vertex at the origin and at any point.
- The ellipse, the hyperbola and a line against themFind the foci of an ellipse and a hyperbola from the equation, build an ellipse with its centre anywhere, draw the asymptote of a hyperbola and test a line against a parabola.
- Parametric equations and polar coordinatesPlot a curve from parametric equations point by point, eliminate the parameter, model a motion and convert polar coordinates to Cartesian.
- The gradient of a line and collinear vectorsFind the gradient k from the angle of inclination of a line (k = tan α, including an obtuse α), perpendicular lines and a vector collinear with a given one.
- A line and the axes: triangle, perimeter and areaFind where a line ax + by = c meets the axes, the perimeter and area of the triangle it cuts off, and where two lines meet.
- Area with a determinantCompute the area of parallelograms and triangles from vectors and coordinates with the two-row determinant.
- Coordinates of a point in spaceDecide where a point lies, then find the distance between two points, the midpoint of a segment and a symmetric point from their coordinates.
Functions and graphs
32
- relations, ordered pairs and functionsShow a relation as a mapping diagram, plot its ordered pairs, decide which relations are functions and try the vertical-line test on a graph.
- Real-life graphs, speed and growthRead a distance–time graph for stops and speeds, sketch the graph of a walk, match a filling bottle to its graph and compare doubling with steady growth.
- Graphs of quadratic functionsFill in a table for y = x2 − 2x − 3 with negative x and plot it, see why the bottom of a parabola is round, read the root (of an equation)s, y-intercept and turning point off the graph, and see what a does in y = ax2.
- Function notation, domain and rangeWork out values of f(x) from a rule, find the input that gives an output, write the domain of a fraction and a square root, and read the range from a graph.
- Modulus and integer-part functionsSee the modulus in a table and on a graph, solve a modulus equation as two crossings with a line, read the steps of the integer-part (floor) function and plot a taxi fare that rises in steps.
- Increasing, decreasing, even and odd functionsFind the intervals where a graph rises or falls, reflect points in the y-axis and through the origin, and use f(-x) to test whether a function is even, odd or neither.
- Sum, composition and inverse of functionsAdd two functions, keep the order in a composite function, find the inverse of a machine by undoing its steps, build the inverse graph by reflecting in y = x, and use a horizontal line to see which functions have an inverse.
- Translating, reflecting and stretching graphsMove the graph of y = (x − h)² + k with sliders, read the turning point of a parabola, see where f(2x) sends a point, and combine several transformations on one parabola.
- Cubic graphs, sketching polynomials and asymptotesFill in and plot a cubic table, build a polynomial graph from its roots and its y-intercept, and draw the vertical and horizontal asymptotes of a rational function as lines.
- The linear function: formula, table, graphDefine a function by a formula, a table and a graph, draw a line from k and b, find the zero of a function and where it is positive.
- The quadratic function and properties of functionsFind the vertex of a parabola by the formula, read the properties of a function from its graph, solve a quadratic inequality graphically and draw the graphs of y = |f(x)| and y = f(|x|).
- The number e, growth and decaySee why (1 + 1n)n settles at e, lay the graph of y = a×ex on a given curve, and build growth and decay models.
- Exponential functions with any base, and exponential equationsFill in a table of 2^x, see that a base above 1 gives a rising curve and a base between 0 and 1 a falling one, and solve an exponential equation by writing both sides with one base.
- Logarithms and log tablesSee that a logarithm is the power the base is raised to, build the log graph as the mirror image of the exponential graph, and use characteristic, mantissa and antilogarithm with a log table.
- Laws of logarithms and equationsSee the product and power laws, solve a logarithmic equation, and straighten data with logarithms.
- Piecewise, sign and modulus propertiesRead a piecewise function, plot the sign function, build y = |x − 1| + 2 by moving the V, and read modulus inequalities off a graph.
- Maximum and minimum values, power functionsFind the greatest and least value of a function on a set, compare the graphs of y = xⁿ for even and odd n, and meet the graph of y = x⁻ⁿ.
- Shifted hyperbolasMove the hyperbola with b and c, read the asymptotes, D and W, and calculate the y-axis intercept.
- Families of lines and parabolasSlide the parameter of a family, find the carrier graph of the vertices and solve the tangent problem with the discriminant.
- Exponential equations and inequalities: methodsTake out a common power, substitute t = 2ˣ, read a root off a graph, and decide when an inequality keeps its sign.
- The functions y = k/x and y = √x and their graphsPlot a hyperbola from a table, find which quadrants its branches lie in, find k from a point and plot y = √x.
- Logarithmic identity, change of base, the logarithmic functionUse the basic logarithmic identity, change base, and compare logarithms with a base between 0 and 1. Section: Grade 11, logarithm properties and the logarithmic function; logarithmic equations are in another studio.
- Logarithmic equations and inequalitiesSolve equations with a sum of logarithms, check roots against the domain, substitute t = lg x, find a variable base, and solve inequalities with a base between 0 and 1.
- The function y = x² and its graphMake a table of values for y = x², plot the points, learn the properties of the graph and solve x² = 4 and x² = x + 2 from the graph.
- Properties of a function from its graphFrom the graph of a function on [−4; 7], find its zeros, the intervals where it keeps one sign, an interval of increase and its greatest value. This follows the Ukrainian Year 9 algebra book, section 8 "Properties of a function".
- Graphs of y = ⁿ√x and powers with a rational exponentCompare the graphs of ³√x, ⁴√x and √x, find the domain of an nth root and of a power with a rational exponent, and compare numbers by monotony.
- Growth and decay: B(t) = B0 · q^tTell linear, quadratic and exponential tables apart, find a doubling time and a half-life with the logarithm, and see where an exponential model stops being realistic.
- Polynomial functions: ends, zeros and termsRead the behaviour at the ends from degree and leading coefficient, tell simple from double zeros, solve a biquadratic equation and find a term from a graph.
- Greatest value problems without derivativesSet up a target function for a fenced rectangle, a rectangle in a triangle and a box, and read the greatest value from the vertex or from a table. State the domain of the variable.
- Function families and their parametersShift power and exponential functions with sliders, read domain, range and asymptote, find the inverse function as a reflection and match a graph with its term.
- Fractional rational functionsTell a pole from a removable gap, read limits, find asymptotes and build a term from properties.
- A function term from conditionsTurn properties into equations for f, f′ and f″, find parameters and the locus of the extreme points of a family.
Vectors
16
- Directed segments, vectors and translation vectorsDraw vectors from tail to head and read them as column vectors, find equal and opposite vectors, and translate a shape by a vector.
- Adding vectors, scalar multiples and unit vectorsAdd vectors head to tail and with a parallelogram, subtract one vector from another, stretch and flip a vector with a number, and write any vector with i and j.
- Extra: vectors in motionAdd up a robot's moves into one move, send it home with the opposite vector, and see how a river's current changes where a swimmer goes.
- Vectors in space: length, distance and direction cosinesA vector in space is the diagonal of a cuboid. Find the length of a vector, the distance between two points, a midpoint and the direction cosines from their coordinates.
- The scalar product and the vector productThe scalar (dot) product of two vectors is a number that tells us the angle between them; the vector (cross) product is a vector perpendicular to both. Work out both and use them.
- Area and volume with the cross productThe length of the cross product of two vectors is the area of their parallelogram. With three vectors, a determinant gives the volume of the parallelepiped.
- Equations of a line and a plane in spaceA line in space is fixed by a point and a direction vector, and a plane by a point and a normal vector. Write their equations and check points against them.
- Angles, intersections and distances in spaceFind the angle between a line and a plane, where a line meets a plane, the distance of a point from a plane and the angle between two planes, using scalar products and equations.
- Vectors in proof and problem solvingSolve the balance point of weighted points, a geometric proof with vectors (the line joining the midpoints of two sides of a triangle) and problems about speed and parallelograms by drawing vectors.
- A vector in the plane: coordinates, length, dot productFind the coordinates and length of a vector, the dot product by coordinates and by angle, the condition for perpendicular vectors and the angle between vectors.
- The dot product of vectors: lengths, angle, coordinatesFind the dot product from lengths and the angle, square a sum of vectors, find the cosine of the angle between vectors and the angle of a triangle from its vertices.
- Vectors in space: sum, scaling, collinear and coplanarAdd vectors with the parallelepiped rule and in coordinates, multiply a vector by a number, and find out when three vectors lie in one plane.
- Vectors on a prism and a cuboid: equal, sums, collinearTell equal, codirectional and opposite vectors apart on a prism, add vectors by the triangle rule, express one vector through others and find collinear vectors by coordinates.
- Writing a vector in terms of two vectorsAny vector in the plane can be built from two vectors that do not lie on one line: p = xa + yb. Draw the parts on the grid, find the coefficients, write AM in a parallelogram and find the length of a projection.
- Position of lines and planes in spacePoint test for a line, normal vector by the cross product, the position of two lines and the intersection of a line and a plane.
- Distances in spaceDistance point–plane with the Hessian normal form, foot of the perpendicular, parallel planes and skew lines.
Trigonometry
20
- Trigonometric ratios in right-angled trianglesGrow and shrink a right-angled triangle with a 35° angle and see that the sides change but their ratio doesn't. Then name the opposite, adjacent and hypotenuse for each angle, and find a side and an angle from a table of values, without a calculator.
- Exact values for 30°, 45° and 60°Halve an equilateral triangle and a square to find exact values of sine, cosine and tangent (trigonometric ratio) for 30°, 45° and 60°, link tan to the gradient of a line, and solve an equation such as sin x = 12.
- Angles of any size, the unit circle and radiansTurn a point round the unit circle by angles bigger than 90°, negative ones included, read sine and cosine from its position, measure the angle in radians and find exact values such as sin7π6.
- Related angles, arcs and sectorsMirror a point on the unit circle to see how the sine and cosine of θ and 180° − θ are linked, use the reference angle to find cos 120°, then work out the length of an arc and the area of a sector with the angle in radians.
- Graphs of sine and cosine, period and amplitudeRead the graphs of sine and cosine: where cosine is zero, how the period and the greatest value change in y = sin 2x and y = 2sinx + 1, and how secant and inverse sine are read from the same curves.
- Trigonometric identities and proofUse sin2θ + cos2θ = 1 and tanθ = sinθcosθ to find one ratio from another, spot the faulty line in a ‘proof’, and learn what counts as a proof of an identity.
- Compound, double and triple angles, sums and productsUse the addition formulae with exact values to find sin 75°, then work with double and triple angles, turn a sum of sines into a product, and write √3sinx + cos x as a single sine.
- Trigonometric equations: quadratic and systemsSolve trigonometric equations by finding every angle from 0° to 360°: tan x = −1, quadratic equations in sin x or cos x (one needs an identity first), and a system of two equations in x and y.
- Sine rule, cosine rule, tangent rule and 3DSolve triangles drawn to scale with the sine rule, the cosine rule and the tangent rule, then use right-angled triangles to find an angle in a cuboid.
- Triangle area: sine, Heron and the circlesFind the area of a triangle from two sides and the angle between them, or from three sides with Heron’s formula, then use the area to find the radii of the circle inside and the circle round a triangle.
- Tangent: graph, domain and periodRead y = tan x: its sign in each quadrant, where it is undefined, its period and the solutions of tan x = 1.
- Parity and reduction formulasFind out that sine, tangent and cotangent are odd and cosine is even, then use the reduction rule: when the function changes its name and which sign it gets.
- Properties of trigonometric functionsRead the properties of sine, cosine and cotangent from their graphs: period, parity, zeros, intervals of decrease, domain and range.
- General solutions of trigonometric equationsThe line y = b meets the graph of sine, cosine or tangent in infinitely many points. Write all solutions as a formula with n ∈ ℤ, and in an equation that reduces to a quadratic reject the root that cannot occur.
- Sine, cosine and tangent of an angle from 0° to 180°Define sine and cosine by the coordinates of a point on the unit semicircle, find the values for obtuse angles and use sin²α + cos²α = 1.
- Solving trianglesSolve a triangle in four cases: a side and two angles, two sides and the angle between them, three sides, and two sides with an angle opposite one of them. Section: Grade 9, solving any triangle; right triangles are in the Grade 8 studio.
- Consequences of the cosine ruleThe type of a triangle from its longest side (acute, right or obtuse), the sum of the squares of the diagonals of a parallelogram and a problem about sides AB and BC. This follows the Ukrainian Year 9 geometry book, section 2 "The cosine rule".
- Cotangent, identities and solving right trianglesLearn the cotangent and the main trigonometric identity, solve right triangles from a hypotenuse and an angle or from two legs, and solve practical problems. Section: Grade 8, right triangles; any triangle is in the Grade 9 studio.
- Trigonometric inequalities and choosing rootsUse a graph to see where a sine or cosine is above or below a number, and pick the roots of an equation on a given segment.
- f(x) = a · sin(b · (x + c)) + dChange amplitude, midline, period and shift of the sine curve, read the shift inside the bracket correctly and use the model for a tide.
Statistics
132
- Collecting data, tallies and pictogramsAsk the class, count with tally marks and draw a pictogram.
- Bar charts and line graphsDraw bar charts and line graphs from a table, read them, and choose the right chart for a question.
- Tallies, sums with tallies and tens pictogramsSort and count with tally marks, add and take away with tallies, and draw a pictogram where one picture is 10.
- Pie charts and the meanColour and read pie charts, level bars to find the mean, and tell numerical data from data in words.
- Mode, median, mean and rangeLine up data cards to find the median, read the mode and range from a dot plot, watch the mean move when one value changes, and choose the best average.
- Outliers and comparing two data setsCompare two classes' standing long jumps using an average and the range, drag one value far out and watch the mean, median and range react, and decide whether an outlier is a mistake or real.
- Grouped data, histograms and frequency polygonsSort heights into equal classes, build a frequency table with relative frequencies, draw a histogram whose bars touch and join the class midpoints into a frequency polygon.
- Mean of grouped data and cumulative frequencyEstimate the mean of grouped data with a midpoint × frequency column, combine two classes' averages properly, build a cumulative frequency column and graph, and read the median off the graph.
- Bivariate data and scatter graphsPlot pairs of data, name the kind of data, describe the correlation, drag a line of best fit, predict from it and tell correlation from cause and effect.
- populations, samples and collecting dataTake random samples of 10 and of 50 from a school of 600 pupils and compare them with the whole school. Spot a biased sample, and choose the right way to collect data for each question.
- Extra: mean absolute deviationSee each dot's distance from the mean, average those distances, find out why signed differences always add up to zero, and compare two data sets by their spread.
- Quartiles and box plotsCut ordered data into four equal parts, find Q₁ and Q₃, mark the five numbers of a box plot and measure the middle half with the interquartile range.
- Variance, standard deviation and coefficient of variationMeasure spread with the squared distances from the mean: variance, standard deviation, and the coefficient of variation for comparing spreads in different units.
- The shape of data and the normal curveRead the shape of a histogram (symmetric or skewed), see how skew pulls the mean away from the median, and use the 68–95–99.7 rule for bell-shaped data.
- Correlation and the regression lineRead the strength of a correlation from r, find the mean point, set the regression line with its gradient and intercept, and predict inside the data.
- Stem-and-leaf diagrams and time seriesWrite data in a stem-and-leaf diagram and read it, then draw a time series as a line graph and smooth it with a moving average.
- Unequal class widths and two-variable dataDraw a histogram with classes of different widths using frequency density, read the frequency back from a bar, and read two variables of the same people on one graph.
- Poverty line, price index and inflationFind a poverty line from the median income, work out a price index, and see what inflation does.
- Trend, interpolation and extrapolationRead the pattern and trend of a time series, estimate a value between recorded ones (interpolation) and beyond them (extrapolation).
- Collecting data, scale and line graphsTally the answers of a survey, choose a good scale for a graph axis and read a 12-month temperature line graph.
- Percentages in statistics and probabilityWrite the result of a survey as a percentage, find a percentage from the angle of a pie sector, draw a percentage pie and write a probability as a percentage.
- Types of variable, collecting data and estimationTell qualitative from quantitative and discrete from continuous variables, take a stratified sample, find the fair questionnaire question and estimate a population mean from a sample.
- Data tables: reading and fillingRead a simple table, find the biggest value, work out the difference and fill a table from the data.Play and learn with the hoopoe
- Sorting by a signSort things into groups by colour and by shape, count each group and find the biggest.
- Timetables: lessons and trainsRead school and train timetables: when does a lesson end, how long is a break or a trip, which train to take.
- Testing simple hypotheses, covariance and least squaresDecide how many heads make a fair-coin hypothesis implausible, compute covariance and the correlation coefficient for five points, and find the line with the smallest sum of squared errors.
- Common features and classifyingFind the common feature of a set, sort shapes into groups by two features at once and find the odd one out.
- Reading data criticallyCompare groups, spot the Simpson effect, judge a sample, tell percent from percentage points and correlation from cause.
- Absolute and relative frequencyMake a tally chart of a survey, work out relative frequencies as fractions, decimals and percent and compare groups of different size.
- Misleading chartsSpot charts that give a wrong impression, such as a cut axis or pictures of different size, draw them honestly and choose headlines that the data support.
- Misleading charts and samplesSee how a cut axis changes a graph, choose a chart for a statement, judge whether a survey sample is representative, and see that the same mean can hide very different spreads.
- Normal distribution and the σ rulesSet μ and σ of the Gauss curve and use the σ rules 68.3 %, 95.4 % and 99.7 %.
Probability
18
- Chance: certain, possible, impossiblePick counters from a bag, spin a spinner and roll a dice. Say whether each event is certain, possible or impossible.
- experimental probability and theoretical probabilitySpin a spinner, throw a bottle cap and roll dice: 10 times, 100 times, 1000 times. Watch the relative frequency settle near the probability.
- sample spaces, combined events and complementsList every outcome of a coin and a spinner, two coins and two dice. Find the favourable cells in a 6 × 6 grid, and get the probability that something doesn't happen from the probability that it does.
- Venn diagrams, events and two-way tablesSort a class of 30 by football and chess in a Venn diagram, flip the same data into a two-way table and back, see events as sets and work out "or" probabilities with the addition rule.
- Tree diagrams, the product rule and independent eventsGrow tree diagrams to count outfits and race results, write probabilities on the branches, multiply along a path, add paths, and see what changes when the bead is not put back.
- The rules of probability, continuous outcomes and expected countsAdd the probabilities of outcomes that cannot happen together, find which lists can be probabilities, work out a probability as a part of a length, and predict how many times something happens.
- Conditional probabilityUse a table, a Venn diagram and a tree to work out the probability of one event when you already know another, see why the order of "given" matters, and meet dependent events.
- Independent events and total probabilityTest whether two events are independent with a table, see why events that cannot happen together are not independent, and add the paths of a tree to find the probability of an outcome that can arise in several ways.
- Random variables and expected valueGive every outcome of two dice a number, find the probability of values of that number, work out the average of a die, and decide how much a game is worth per throw.
- Unequal chances and Bayes' lawGive each outcome its own weight and add them. Then read a tree backwards: a bad bolt is found, which machine made it?
- Binomial and Poisson distributionsCount the paths of a tree that give exactly k successes, see where the binomial formula comes from, and use the Poisson formula for events that arrive at random.
- The normal distribution and z-scoresMark distances from the mean in standard deviations on a number line, use the 68–95–99.7 rule, work out a z-score, and find the probability in a tail.
- Sample means, sample proportions and the central limit theoremSee that the means of samples are far less spread out than single values, work out the standard error of a mean and of a proportion, and use the central limit theorem to find a probability.
- Variance of a random variable, the geometric distribution and Chebyshev's inequalityCompute a variance from a distribution table, find the variance of a sum of independent variables, find a probability of 'until the first success' and bound a deviation with Chebyshev's inequality.
- Geometric probability, density and continuous random variablesFind the probability that a random point lands in part of a figure, segment or arc as a ratio of measures, understand a density, and meet the uniform and exponential distributions.
- Probability and combinationsFind the probability of an event with P(A) = m : n when the outcomes of a choice are counted with combinations Cₙᵏ and arrangements Aₙᵏ.
- Estimating π with random points and stripsCount random points in a quarter circle to estimate π, see how the scatter shrinks with more points, and compare with the strip method.
- One-sided significance testWrite H₀, find the rejection region at the 5 % level, judge the type 1 error and read the result correctly.
Sets and logic
23
- Sets: elements, subsets, union, intersection and complementSort number cards into the ovals of a Venn diagram, write sets in braces, shade unions, intersections and differences, and count with the diagram.
- Arrangements and groups: permutations and combinationsGrow the tree of arrangements to count who can win gold and silver, line friends up in a row, then see why a group of people is counted fewer times than an arrangement.
- The binomial theorem and Pascal's triangleSee (a + b)³ as three brackets and count the paths that give each term, read the coefficients from Pascal's triangle, and use them for expansions with numbers.
- Counting without double counting, and the pigeonhole principleCount the pupils in two clubs without counting the ones in both twice, count the multiples of 2 or 3, then use the pigeonhole principle to be sure that two things share a box.
- Open sentences, subsets and the laws of setsTell statements from open sentences, build the solution set of an open sentence, count subsets, and check the distributive and absorption laws on a Venn diagram.
- Statements, and, or, not and if…thenDecide which sentences are statements, use and, or and not correctly, test a promise "if … then …" and find the contrapositive.
- Quantifiers, valid arguments and De MorganNegate "all", draw "every multiple of 6 is a multiple of 3", tell valid arguments from invalid ones and use De Morgan's laws on sets.
- Proof: direct, cases, contradiction and inductionProve that odd + odd is even with algebra, check a claim by splitting it into cases, prove that there is no largest even number by contradiction and take the induction step for the odd numbers.
- Ordered pairs, relations and equivalenceMark A × B as points, pick out a relation, reverse it to get the inverse and see why "same remainder" is an equivalence relation.
- Counting with repeats: arrangements, groups and stars and barsArrange marbles of the same colour in a row, count codes with repeated digits, split students into rooms and count flower bunches with stars and bars.
- Set-builder notation and matching infinite setsKnow ℕ, ℤ, ℚ and ℝ, list the members of a set from its condition, write the condition for a set, and see why the evens and the naturals are "the same size".
- Binomial coefficients, Pascal's triangle and proof by countingSplit groups of friends into two kinds, prove Pascal's rule by counting, and see why each row adds up to a power of 2.
- Graphs: vertices, edges and degreeRead a graph: find the degree of each vertex, compare the sum of the degrees with the number of edges, and see why some graphs cannot exist.
- Graphs: dominating setsPlace stations so that every point of the map is covered, and find the fewest that will do: the domination number of the graph.
- Rules of sum and product, Pₙ, Aₙᵏ and CₙᵏTell the rule of sum from the rule of product, use the textbook formulas for permutations, arrangements and combinations, and solve an equation with factorials.
- Combinatorial problems: listing, the tree and the product ruleFind all the ways to place three girls on floors, grow a tree of options and find out why the number of ways is found by multiplying.
- The rule of sum and the rule of productIn how many ways can we choose "a or b" and "a and b": the sum rule, the product rule, pairs (a; b), a road scheme and problems about digits. This follows the Ukrainian Year 9 algebra book, section 20 "Basic rules of combinatorics".
- True or falseRead statements about the fruit with the words "all", "every" and "some" and decide whether they are true.
- If ... then ...Cross out numbers using clues, draw conclusions from "if ... then ..." rules and build chains of reasoning.
- Directed graphs and problems solved with graphsRead directed graphs, count handshakes and games in a league and find the shortest route by turning the problem into a graph.
- Connectedness, paths, cycles and tracing a graphCount connected components, find the vertices of odd degree and decide whether a graph can be traced without lifting the pencil.
- Necessary and sufficient conditionsTell when a condition is necessary, when it is sufficient and when it is both, and refute a false claim by a counter-example.
- Trees and plane graphsRecognise trees, count the edges of a tree and of a spanning tree, and count the faces of a plane graph with Euler's formula.
Complex numbers
4
- Complex numbers: i, adding, multiplying, dividingMeet the number i with i² = −1, add complex numbers as points on the plane, multiply them like brackets and divide with the conjugate.
- Quadratics and polynomials with complex rootsSee why x² + 4x + 13 = 0 has no real roots, solve equations whose roots are complex, plot them on the plane and find the missing root of a cubic.
- The Argand diagram: modulus, argument and turningRead complex numbers as points on the Argand diagram, find their modulus and argument, reflect them in the real axis and multiply by turning and stretching.
- Powers, roots and circles in the Argand diagramUse e^{iθ} as a point on the unit circle, raise numbers to powers with De Moivre, find the four roots of a number as a square and draw a circle with |z − c| = r.
Matrices
6
- Matrices: order, entries and special typesRead a matrix like a table: find an entry by its row and column, state its order and build the zero, identity and diagonal matrices.
- Adding, scaling, multiplying and transposing matricesAdd and subtract matrices of the same order, multiply by a scalar, multiply row by column, and turn rows into columns.
- DeterminantsWork out 2 × 2 and 3 × 3 determinants, see what swapping rows does, and see how the determinant scales areas.
- Inverse matrices and solving systemsFind the inverse of a 2 × 2 matrix from its determinant, find a 3 × 3 inverse by row operations, and solve a system with it.
- Cramer's rule and Gauss eliminationSolve a system with determinants (Cramer), then by row operations (Gauss), and see how a zero determinant gives parallel lines.
- Matrices as transformationsA matrix moves the points of the plane: apply it to a point, build it from the images of i and j and find the points it leaves in place.
Calculus
21
- Limits: getting close to a numberFill in a table of values close to a point where a function is undefined, read the limit from the graph, and see why the left-hand and right-hand limits must agree.
- Working out limits: laws, 0 over 0 and sineWork out limits of sums and products from graphs, open up the 0 over 0 form by factorising and cancelling, and meet the sine limit.
- Infinite limits and limits at infinitySee why 1x2 grows huge near zero, draw a horizontal asymptote, and open up the forms ∞ ÷ ∞ and ∞ − ∞.
- Continuity and rate of changeSee when a graph can be drawn without lifting the pen, find the k that makes a function continuous, and squeeze an average rate of change down to an instantaneous one.
- From secant to tangent: what a derivative isSlide a point Q along the graph towards P, watch the gradient of the secant in a table settle on the gradient of the tangent, draw the tangent, and see why a corner has no derivative.
- The rules of differentiationTry the power, product, quotient and chain rules on a number at one point, and check them against tangents and normals on a graph.
- Derivatives of sine, exponentials, logs and inversesTest the derivative of sin x near zero, draw the tangent to ex, use ln x as the mirror image of ex to find its derivative, and carry the inverse-function rule over to arctan.
- Implicit, parametric and second derivatives, and connected ratesFind the gradient of a circle at a point by implicit differentiation, work out the gradient of a parametric curve, take a second derivative, and see how a changing side makes the area change.
- Tangent equation and greatest and least value on an intervalWrite the tangent equation with y = f(x₀) + f′(x₀)(x − x₀), find a tangent parallel to a given line and one through a point off the curve, then find the greatest and least value of a function on [a;b].
- The sign of the derivative and the shape of the graphUse the sign of f′(x) to find where a graph rises and falls, place the stationary points, find the point of inflection with f″(x) and work out a maximum value.
- Rolle, l'Hôpital and Maclaurin seriesFind where the tangent is horizontal, find the point of the mean value theorem, take a 0/0 limit with derivatives and find a coefficient of the Maclaurin series of cosh x.
- Estimating area, Riemann sums and the trapezium ruleEstimate the area under y = x² with thin rectangles and with trapezia, and see which number the estimates head for as the strips get narrower.
- From antiderivative to definite integralFind an antiderivative and see why +C is needed, evaluate a definite integral with F(b) − F(a), find the mean value of a function and meet the integral of 1x.
- Methods of integrationChange the variable and the limits together, integrate by parts, split a fraction into partial fractions, pick a trigonometric substitution and decide whether an improper integral has a value.
- Areas, volumes and arc lengths by integrationUse the definite integral to find the area under a curve and between two curves, a volume of revolution, an arc length and a polar area.
- Differential equationsWrite a rate of change as a differential equation, find the solution y = Cekx on a graph and solve a separable equation with a starting condition.
- Curve sketching and optimisationFind the asymptotes and key points of a graph, then use a maximum to solve problems such as the largest area and the largest volume.
- A research plan for sketching a graphStudy a function by a plan: domain, parity, zeros, increasing and decreasing, extreme points, then mark the key points of the graph.
- The monotony schemeFrom the signs of the derivative on every gap, mark with arrows where the function rises ↗ and falls ↘, and find the extreme points.
- From the graph of f to the graph of f′Find where f′ is zero, read f′ from the graph of f, locate the point of inflection as an extremum of f′ and the moment of fastest growth.
- Integral function and the fundamental theoremBalance areas up to x, find extrema of the integral function and calculate with J(x) = F(x) − F(a).
Problem solving
217
- Problem solving: draw a pictureRead the problem. Say what you know and what you need to find. Draw it, then solve it.Play and learn with the hoopoe
- Problem solving: number sentencesTurn a story into a number sentence. Put a box □ where the number you don't know goes, then find it.Play and learn with the hoopoe
- Problem solving: model, pattern, eliminate, guess and checkFour ways to solve a problem: list every case in order, find the pattern, cross out what can't be, and guess, check and improve.
- Problem solving: sub-problems, simpler cases and systematic listsBreak a big problem into steps, solve a simpler version first, list every case in order, and choose a strategy yourself.
- problem solving: grade 7 strategiesPractise systematic listing, sub-problems and simpler problems with money problems and larger numbers.
- problem solving: profit and enough informationFind a shopkeeper's profit with a table, and decide which numbers a problem needs, which it does not, or when one is missing.
- Problem: short record and solutionWrite the condition briefly, make an equality, solve the problem in one or two steps and write the answer.
- Problems: times as many, how many more, the inverse problemMake a number several times bigger, compare by «how many times», solve indirect problems and inverse problems. Section: Grades 2-3, "times as many", "how many more", inverse problem; price/quantity/cost tables are in another studio.
- Table problems: price, quantity, costWrite the short record as a «price, quantity, cost» table and solve problems on the sum, difference and ratio of two products. Section: Grade 3, table problems price/quantity/cost; "times as many" problems are in another studio.
- Compound problemsTell simple from compound problems, then write the solution by actions and as one expression with brackets.
- Compound problems with multiplication and divisionSolve compound problems: the remainder, sums and differences of products, and several times more or fewer.
- Compare the problems: what changed?Compare two related problems, predict how the solution changes, and solve the changed one.
- Given, solution, answerWrite a problem the way it is written in an exercise book: Given, Find, Solution (by questions, by actions, as one expression) and Answer.
- Logic puzzlesSolve brain-teasers: fill a table from clues, find knights and liars, pour water between buckets and find who did it.
- Instructions and algorithmsFollow instructions step by step, run an algorithm machine and put the steps of a plan in the right order.
- Use and changeSolve problems about use per day ("per day × days") and about change ("before - after") in steps and write the answer.
- Fermi problems: estimate or exact?Estimate big numbers in steps with reference sizes, round for a quick estimate and decide when an exact answer is needed.