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Mathematics Achievement: Grade 5-6 School Mathematics

Thirty school-maths questions in thirty minutes across six strands, where a wrong answer means you worked it out wrongly, not thought about it wrongly.

44 topics · 165 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Place Value and Reading Large Numbers

  1. How it worksHow does each step left along a number multiply a digit's worth by ten?
  2. Worked exampleFind the value of the digit 4 in 2,458,013 and name the place it sits in.
  3. Common mistakeWhy isn't the largest digit in a number always the digit worth the most?
Your turneasy1 of 3
What is the value of the digit 7 in the number 758,013?

The Four Operations with Whole Numbers

  1. How it worksHow does carrying a ten in addition move a whole bundle into the next column?
  2. How it worksWalk me through 6,042 minus 1,875 and show where every borrow comes from.
  3. Worked exampleFind 48 x 26 by splitting it into 48 x 20 plus 48 x 6.
  4. Common mistakeWhy doesn't a remainder mean you have done the division wrong?
Your turneasy1 of 3
Find 48 times 27 by splitting it into 48 times 20 plus 48 times what is left.

Rounding and Making a Reasonable Estimate

  1. How it worksHow does the digit just to the right decide which way a number rounds?
  2. Worked exampleShow me 3,486 rounded to the nearest hundred and then to the nearest thousand.
  3. Common mistakeWhy isn't 449 rounded to the nearest hundred by rounding the 49 up first?
  4. Going furtherWhat happens to an estimated sum when every number gets rounded the same direction?
Your turneasy1 of 3
Round 7,486 to the nearest hundred.

Order of Operations

  1. How it worksHow does an agreed order of operations stop one expression from having two meanings?
  2. Worked exampleFind the value of 20 - 2 x (3 + 4) and show each step in the order it must happen.
  3. Worked exampleFind the value of 6 + 18 / 3 - 2 x 4 by working left to right where the rules allow.
  4. Common mistakeWhy isn't multiplication always done before division when both show up?
Your turneasy1 of 3
Find the value of 20 - 2 x (3 + 5).

Factors, Multiples, Greatest Common Factor and Least Common Multiple

  1. How it worksHow does listing factors in pairs make sure you never miss one?
  2. How it worksHow do two lists of multiples show you the first number both counts land on?
  3. Worked exampleFind the greatest common factor of 18 and 24, then the least common multiple of 4 and 6.
  4. Common mistakeWhy doesn't a greatest common factor ever come out bigger than the smaller number?
  5. Ties togetherHow does a least common multiple turn into the common denominator that fractions need?
Your turneasy1 of 3
What is the greatest common factor of 16 and twice 16?

Prime Numbers and Divisibility Rules

  1. How it worksHow does adding up a number's digits tell you whether 3 or 9 divides it?
  2. Worked exampleFind which of 2,001, 2,003, 2,005 and 2,007 divides by 9 with nothing left over.
  3. Common mistakeWhy isn't 1 counted as a prime when nothing except itself divides it?
  4. Going furtherWhat happens when you break 60 all the way down into a product of primes?
Your turneasy1 of 3
What is the sum of the digits of the number 4,706?

Exponents as Repeated Multiplication

  1. How it worksHow does a small raised number count how many times a base gets multiplied?
  2. Worked exampleFind the value of 3 to the fourth power and of 5 squared, then say which wins.
  3. Common mistakeWhy isn't 4 cubed the same thing as 4 times 3?
Your turneasy1 of 3
Find the value of 4 to the fourth power.

Equivalent Fractions and Simplest Form

  1. How it worksHow does re-dividing one shaded strip turn 1/2 into 4/8 without changing anything?
  2. Worked exampleFind the simplest form of 18/24 by pulling out the factor they share.
  3. Common mistakeWhy doesn't dividing only the top of a fraction leave its value alone?
  4. Going furtherWhat happens to a fraction's picture when top and bottom are both multiplied by 3?
Your turneasy1 of 3
One half is rewritten as an equivalent fraction with a bottom number of 70. What is the new top number?

Comparing and Ordering Fractions and Decimals

  1. How it worksHow does cutting two bars into matching pieces let you compare them at a glance?
  2. Worked exampleCompare 2/3, 5/8, 0.7 and 3/5 and put them in order from smallest to largest.
  3. Common mistakeWhy isn't 0.5 smaller than 0.45 even though 45 is bigger than 5?
  4. Going furtherWhat changes about comparing two fractions when their top numbers match instead?
Your turneasy1 of 3
Write the fraction 5/8 as a decimal.

Adding and Subtracting Fractions and Mixed Numbers

  1. How it worksHow does a common denominator re-cut two bars until their pieces finally match?
  2. Worked exampleFind 3/4 plus 5/6 and write the answer as a mixed number.
  3. Worked exampleFind 5 1/3 minus 2 3/4 by borrowing one whole and turning it into twelfths.
  4. Common mistakeWhy doesn't adding fractions mean adding the tops and adding the bottoms?
Your turneasy1 of 3
Find 3/4 plus 8/12. Give your answer as a decimal.

Multiplying and Dividing Fractions

  1. How it worksHow does taking 2/3 of 3/4 of a square show up as a shaded overlap?
  2. How it worksHow does flipping the second fraction turn a division into a multiplication?
  3. Worked exampleFind 2/3 times 9/10, and then find 3/4 divided by 1/2.
  4. Common mistakeWhy isn't dividing by 1/4 the same as dividing by 4?
Your turneasy1 of 3
Find 2/3 times 7/10. Give your answer as a decimal.

Adding, Subtracting, Multiplying and Dividing Decimals

  1. How it worksHow does counting decimal places tell you where the point lands in a product?
  2. Worked exampleFind 4.7 times 0.6 and show exactly why the point ends up where it does.
  3. Worked exampleFind 12.6 divided by 0.4 by sliding both points the same number of places.
  4. Common mistakeWhy doesn't multiplying two decimals always give something bigger than both of them?
Your turneasy1 of 3
Find 4.7 times 0.7.

Converting Between Fractions, Decimals and Percents

  1. How it worksHow does a hundred-square turn any fraction into a percent you can see?
  2. Worked exampleShow me how 3/8 becomes a decimal and then becomes a percent.
  3. Worked exampleFind 0.35 written as a fraction in simplest form and also as a percent.
  4. Common mistakeWhy isn't 0.2 the same thing as 2 percent?
Your turneasy1 of 3
Write the fraction 5/8 as a percent. Give just the number.

Finding a Percent of a Number

  1. How it worksHow does shading part of a hundred-square show what a percent of a number means?
  2. Worked exampleFind 25 percent of 64, and then find 15 percent of 80.
  3. Going furtherWhat happens to a 40 dollar price after a 25 percent cut and then a 10 percent cut?
  4. Common mistakeWhy doesn't a 20 percent rise followed by a 20 percent fall bring you back where you started?
Your turneasy1 of 3
Find 25 percent of 700.

Ratio and Simple Proportion

  1. How it worksHow does a ratio of 3 to 2 draw as two towers that grow together?
  2. Worked exampleFind how many boys sit in a class of 30 when boys to girls run 3 to 2.
  3. Common mistakeWhy isn't a ratio of 3 to 2 the same as saying 3 out of every 2 students are boys?
Your turneasy1 of 3
In a class of 50 students the ratio of boys to girls is 3 to 2. How many boys are in the class?

Number Patterns and What Comes Next

  1. How it worksHow do you test a guessed rule against every term before you trust it?
  2. Worked exampleFind the next two numbers in 3, 7, 11, 15 and name the rule that builds them.
  3. Worked exampleFind the missing number in 1, 4, 9, 16, blank, 36 and say what the pattern is doing.
  4. Going furtherWhat changes about a pattern when each term doubles instead of gaining a fixed amount?
Your turneasy1 of 3
A pattern starts at 3 and adds 4 at every step: 3, 7, 11, 15 and so on. What is term number 15?

Writing an Expression from Words

  1. How it worksHow do you turn each phrase of a sentence into one piece of an expression?
  2. Worked exampleBuild an expression for 3 tickets at t dollars each plus a 2 dollar booking fee.
  3. Common mistakeWhy isn't twice the sum of x and 4 written as 2x + 4?
Your turneasy1 of 3
A problem says 9 less than a number. A student writes 9 - n. Why is that the wrong way round?

Evaluating an Expression by Substitution

  1. How it worksHow does dropping one number into every copy of the letter finish the job?
  2. Worked exampleFind the value of 4n - 7 when n is 6, and then again when n is 2.
  3. Worked exampleFind the value of the expression 3(a + b) when a equals 5 and b equals 4.
  4. Common mistakeWhy isn't 2x equal to 25 when x is 5?
Your turneasy1 of 3
Find the value of 4n - 7 when n is 11.

One-Step and Two-Step Equations

  1. How it worksHow does taking the same amount off both pans leave a balance still even?
  2. Worked exampleSolve 3x + 5 = 20 by undoing the plus 5 first and the times 3 second.
  3. Worked exampleSolve n / 4 = 9, and then solve 2n - 7 = 11.
  4. Common mistakeWhy isn't 3x + 5 = 20 solved by dividing only the 3x part by 3?
  5. Ties togetherHow does a two-step equation undo the very same two steps a number machine did?
Your turneasy1 of 3
Solve n / 4 = 15. What is n?

Input-Output Tables and the Rule That Makes Them

  1. How it worksHow do you check a guessed rule on every row of a table at once?
  2. Worked exampleFind the rule sending 20 to 11, 16 to 9, 10 to 6 and 4 to 3, then find the input giving 15.
  3. Going furtherWhat happens to a table when its rule adds first and multiplies second instead?
Your turneasy1 of 3
A rule sends 20 to 11, 16 to 9, 10 to 6 and 4 to 3. Which input gives an output of 16?

Plotting a Point on a Coordinate Grid

  1. How it worksHow do you walk across first and up second to land on (3, 5)?
  2. Worked exampleGraph the points (3,5), (6,5), (8,2) and (2,2) and find which quadrilateral they make.
  3. Common mistakeWhy isn't (2, 5) the very same point as (5, 2)?
  4. Going furtherWhat happens to a plotted shape when 3 is added to every across coordinate?
Your turneasy1 of 3
A rectangle has corners at (2,2), (8,2), (8,8) and (2,8) on a grid. What is its area, in square units?

Points, Lines, Rays and Naming Angles

  1. How it worksHow do you name an angle with three letters and keep the corner in the middle?
  2. Worked exampleDraw a right angle, an acute angle and an obtuse angle, then sort them by size.
  3. Common mistakeWhy doesn't drawing an angle's arms longer make the angle itself bigger?
Your turnmedium1 of 3
6 rays all start from the same point and spread out in different directions. How many different angles can be named at that point?

Measuring and Classifying Angles

  1. How it worksHow does a protractor turn the opening between two rays into a number?
  2. Worked exampleFind the missing angle when two angles sitting on a straight line are 125 and x.
  3. Worked exampleFind the angle between a clock's hands at 3 o'clock and then at 4 o'clock.
  4. Going furtherWhat happens to the other three angles when two lines cross and one of them is 40 degrees?
Your turneasy1 of 3
Two angles sit side by side on a straight line. One measures 142 degrees. How many degrees is the other?

Classifying Triangles and Quadrilaterals

  1. How it worksHow do sides and angles hand a triangle two different family names?
  2. How it worksHow does a family tree of quadrilaterals tuck squares inside rectangles?
  3. Worked exampleFind the name for a quadrilateral with exactly one pair of parallel sides.
  4. Common mistakeWhy isn't every rectangle a square when every square is a rectangle?
Your turneasy1 of 3
An isosceles triangle has two equal angles of 58 degrees each. How many degrees is the third angle?

The Angle Sum of a Triangle

  1. How it worksHow does tearing off three corners and fitting them together prove the 180 degrees?
  2. Worked exampleFind the third angle of a triangle whose other two measure 47 and 68 degrees.
  3. Common mistakeWhy isn't a huge triangle's corner total any bigger than a tiny triangle's?
  4. Going furtherWhat happens to that total when you go from a triangle to a four sided shape?
Your turneasy1 of 3
Two angles of a triangle measure 58 degrees and 66 degrees. How many degrees is the third?

Polygons, Circles and Their Parts

  1. How it worksHow do the radius, the diameter and the way around one circle depend on each other?
  2. How it worksHow does counting sides name a polygon from triangle all the way up to octagon?
  3. Worked exampleFind the diameter of a circle with a radius of 7 centimetres and compare the two.
  4. Common mistakeWhy isn't a shape with a curved side ever called a polygon?
Your turneasy1 of 3
A circle has a radius of 11 centimetres. How many centimetres is its diameter?

Perimeter and Area of Rectangles and Triangles

  1. How it worksHow does walking all the way around a shape add up to its perimeter?
  2. How it worksHow do unit squares tiling the inside of a rectangle turn into length times width?
  3. Worked exampleFind the perimeter and the area of a rectangle 12 feet long and 8 feet wide.
  4. Worked exampleFind the area of a triangle with a base of 6 feet and a height of 8 feet.
  5. Common mistakeWhy isn't a triangle's area just base times height with no halving?
Your turneasy1 of 3
A rectangle is 16 feet long and 9 feet wide. How many feet of fence would go all the way around it?

Area of Composite Shapes

  1. How it worksHow do you split an odd shape into pieces you already know how to measure?
  2. Worked exampleFind the area of a patio made from a 12 by 8 rectangle joined to a triangle 6 wide and 8 tall.
  3. Going furtherWhat happens when you get that same odd shape's area by subtracting rather than adding?
Your turneasy1 of 3
A patio is a 12 by 8 rectangle with a triangle joined to one end, 8 feet wide and 8 feet tall. What is the total area, in square feet?

Symmetry, Congruence and Slides, Flips and Turns

  1. How it worksHow does sliding, flipping or turning a shape leave its size and shape untouched?
  2. Worked exampleFind how many lines of symmetry a square has and how many a long rectangle has.
  3. Common mistakeWhy doesn't turning a triangle upside down make it a different triangle?
  4. Going furtherWhat changes about a shape when you stretch it twice as wide but keep the height?
Your turneasy1 of 3
A rectangle measures 7 units by 6 units. It is stretched to twice as wide while its height stays the same. What is the new area, in square units?

Solids and Their Faces, Edges and Vertices

  1. How it worksHow does unfolding a box into a flat net show every face at once?
  2. Worked exampleFind the faces, edges and vertices on a rectangular box and on a square pyramid.
  3. Common mistakeWhy isn't a cylinder called a prism when both have two matching ends?
Your turneasy1 of 3
A prism has ends shaped like a polygon with 8 sides. How many faces does the whole solid have?

Units and Tools for Length, Weight, Capacity, Time and Temperature

  1. How it worksHow do you pick the unit that makes a measurement easy to say out loud?
  2. Worked exampleFind the best unit and the right tool for weighing a letter, a person and a truck.
  3. Common mistakeWhy isn't the bigger number always the bigger amount when the units are different?
Your turneasy1 of 3
A letter, a person of 75 kilograms and a loaded truck all have to be weighed. Which set of units fits the three of them best?

Converting Units Within One System

  1. How it worksHow does multiplying or dividing by the conversion number pick your direction?
  2. Worked exampleFind how many inches make up 4 feet 7 inches, using 12 inches in one foot.
  3. Worked exampleFind how many centimetres are in 2.5 metres when nobody hands you the conversion.
  4. Common mistakeWhy doesn't changing to a smaller unit mean you should divide?
Your turneasy1 of 3
How many inches are in 7 feet 8 inches? Use 12 inches in 1 foot.

Reading a Ruler, a Scale and a Thermometer

  1. How it worksHow do you work out what one small tick mark is worth on any scale?
  2. Worked exampleFind the length of a pencil ending three ticks past 5 when each inch splits into quarters.
  3. Going furtherWhat happens to your reading when the same dial counts by 2 rather than by 1?
Your turneasy1 of 3
On a ruler where each inch is split into quarters, a pencil ends 2 ticks past the 5 inch mark. How long is the pencil, in inches?

Elapsed Time and Money Problems

  1. How it worksHow does jumping to the next whole hour make elapsed time easy to add up?
  2. Worked exampleFind how long a film runs if it starts at 2:45 and ends at 4:20.
  3. Worked exampleFind the change from a 20 dollar bill after 3 notebooks at 2.85 dollars each.
  4. Common mistakeWhy doesn't 1.5 hours mean one hour and fifty minutes?
Your turneasy1 of 3
A film starts at 2:45 in the afternoon and ends at 7:20. How many minutes does it run?

Volume of a Rectangular Solid by Counting Cubes

  1. How it worksHow does one layer of cubes times the number of layers give the whole volume?
  2. Worked exampleFind the volume of a box 5 cubes long, 3 cubes wide and 4 cubes tall.
  3. Common mistakeWhy isn't volume measured in square units the way area is?
  4. Going furtherWhat happens to a box's volume when only its height gets doubled?
Your turneasy1 of 3
A box is 6 cubes long, 5 cubes wide and 6 cubes tall. How many unit cubes fill it?

Mean, Median, Mode and Range

  1. How it worksHow do the mean, the middle value and the most common value answer different questions?
  2. Worked exampleFind the mean, median, mode and range of 4, 7, 7, 9 and 13.
  3. Common mistakeWhy isn't the median found before the numbers have been put in order?
  4. Going furtherWhat happens to both the mean and the median when one number jumps from 13 to 130?
Your turneasy1 of 3
Find the mean of 4, 7, 7, 9 and 18.

Bar Graphs, Line Graphs and Pictographs

  1. How it worksHow does a pictograph key turn half a little symbol into a real number?
  2. Worked exampleFind the total when a pictograph shows 3 and a half book symbols worth 4 books each.
  3. Worked exampleBuild a bar graph from votes of 12, 8, 15 and 5 and find which bar ends up tallest.
  4. Going furtherWhat changes about a pictograph total when its key says 10 books per symbol instead of 4?
Your turneasy1 of 3
A pictograph shows 7 and a half book symbols, and its key says each symbol is worth 4 books. How many books is that?

Circle Graphs and Tally Charts

  1. How it worksHow does a tally chart's bundle of five make a long count fast to read?
  2. Worked exampleFind how many of 60 students picked soccer when its slice fills a quarter of the circle.
  3. Common mistakeWhy doesn't a fat slice on one circle graph mean more people than a thin slice on another?
Your turneasy1 of 3
A circle graph covers 400 students, and the soccer slice fills exactly a quarter of the circle. How many students picked soccer?

Probability of a Simple Event as a Fraction

  1. How it worksHow does counting the winning outcomes over all outcomes build a probability fraction?
  2. Worked exampleFind the probability of drawing the letter C from cards labelled A through K.
  3. Worked exampleFind the probability of rolling higher than 4 on one six sided number cube.
  4. Common mistakeWhy isn't a coin due to land heads after five tails in a row?
Your turneasy1 of 3
A set of cards is labelled 1 through 16, one label each. What is the probability of drawing one particular card? Give a decimal.

Certain, Likely, Unlikely and Impossible

  1. How it worksHow do you place an event on a line running from impossible to certain?
  2. Worked exampleFind where a chance of 7 out of 8 sits on a line from impossible to certain.
  3. Common mistakeWhy isn't an unlikely event the same thing as an impossible one?
Your turneasy1 of 3
A chance of 5 out of 8 is marked on a line running from impossible to certain. What decimal is it?

Listing All the Possible Outcomes

  1. How it worksHow does a tree diagram grow one branch for every choice you could make?
  2. Worked exampleFind how many outfits come from 3 shirts and 4 pairs of shorts, then list them all.
  3. Worked exampleFind every outcome of tossing two coins and the chance of getting one of each.
  4. Going furtherWhat happens to the number of outcomes when a third coin joins the toss?
Your turneasy1 of 3
A drawer holds 5 shirts and 6 pairs of shorts. How many different outfits can be made from them?

No Calculator: Mental Math and Checking Your Work

  1. How it worksHow does breaking a number into friendly parts let you multiply inside your head?
  2. Worked exampleFind 25 x 16 in your head by doubling and halving instead of writing it out.
  3. Going furtherWhat happens to your speed when you check an answer by estimating rather than redoing it?
Your turneasy1 of 3
Find 25 times 20 in your head by halving 20 and doubling the 25 instead of writing it out.

Process Errors Hiding in the Answer Choices

  1. How it worksHow does each wrong choice match one exact mistake like a dropped decimal point?
  2. Worked exampleFind the wrong answers that appear when 15 x (8 + 17) is done three mistaken ways.
  3. Common mistakeWhy isn't an answer correct just because it turned up in the list of four?
  4. Going furtherWhat happens when you finish the problem before ever looking at the four choices?
Your turneasy1 of 3
Work out 15 x (8 + 20) correctly.

Pacing 30 Questions in 30 Minutes

  1. How it worksHow does a section ordered easy to hard change which questions you skip?
  2. Worked exampleFind how much time is left for the last 10 questions when the first 20 took 22 minutes.
  3. Common mistakeWhy isn't skipping a hard question the same as giving up on it?
Your turneasy1 of 3
Mathematics Achievement gives 30 minutes for 30 questions. The first 20 questions took 24 minutes. How many minutes are left for the last 10?

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