Common Misconceptions in Mathematics (produced for a tda project coordinated by Teresa Robinson www.robinsoneducation.co.uk)

Number

Misconception CorrectionWhen multiplying two numbers, the answer is always bigger than both original numbers.

True for whole numbers.Investigate what happens when one or both of the numbers is a fraction less than 1.Investigate what happens when negative numbers are introduced.Frequently use the word ‘of’ instead of ‘times’.

0.1 x 0.1 is incorrectly given as 0.1 0.1 x 0.1 = 1/10 x 1/10 = 1/100 or 0.01.1/10 of 1/10 = 1/100 to justify answer.

3 ÷ ¼ is incorrectly seen as equivalent to 3 ÷ 4 and has a value of 0.75

3 ÷1/2 is 1 ½. Divide 3 apples into halves. How many pieces? 3 ÷ ¼ means how many quarters are there in 3?There are 4 quarters in 1, so 3x 4 or 12 in 3.Use diagrams to justify answers.

x² = 25 therefore x = 5 (-5)(-5) = 25Relating percentages to fractions such that10% = 1/10 so 5% = 1/5

% means out of 100, so 5% means 5/100 which can be simplified

1 is a prime number Prime numbers have two factors¼ = 0.4 ¼ = 25/100 = 0.25Relating percentages to decimals such that 6% = 0.6

60% = 60/100 = 0.6 and 6% = 6/100 = 0.06

a² = 2 × a a² = a × aa-2 = -a² e.g. 5-2 = -25 Show students the pattern:-

5³ = 1255² = 2551 = 55° = 15-1 = 1/5 divide by 5

-2-3 = 5 Avoid the word ‘minus’ and the expression that a double minus is a positiveEmphasis on subtract and negativeTalk about the difference and explore a mixture of similar questions such as -2+3, -3+2, 3-2, 2-3,...Use a vertical number line to justify answerWhen two operations appear together ring them and replace them with the correct single operation e.g. 2 + -3 = 2-3 = -1

3293 → 3 to 1 significant figure Use money to explain about order of magnitude and that the number must stay at the approximately the same valueAddition of fractions E.g. ½ + ¼ = 2/6

Use diagrams to justify answer.

Shape and Space

Misconception CorrectionCommon shapes are not recognised unless they are ‘upright’

When appropriate draw shapes in a different position – at an angle, upside down, facing a different direction.Ask students to draw pairs of congruent shapes, where one is a reflection of the other, to justify answer.

The square drawn on the hypotenuse is frequently drawn as a rhombus.

Get students to draw the square using a set square, or use the corner of another piece of paper to ensure a right angle. Draw the triangle in different aspects.

The diagonal of a square is the same length as the side

Investigate by drawing and measuring. Can they estimate the length of the diagonal for a 5cm by 5cm square, giving answer to 1 dp? Draw it and measure how close they were.

a² + b² = c² used where c is not the hypotenuse Define the hypotenuse repeatedly.Write a² + b² = (hyp)².Write (area smallest square)²+(area middle square)²=(are largest square)².

Use the Sine and Cosine Rule with right-angled triangles

Give the class right-angled triangles when teaching the Sine and Cosine Rules. Ask if there is a quicker way if nobody suggests unnecessary work.

Converting m³ to cm³ and vice versa Picture m³ and cm³. Metre stick demonstration.1m³ measures 1m by 1m by 1m i.e. 100cm by 100 cm by 100cm. Use a drawing to justify.

The centre of rotation does not matter Give students one shape and ask them to rotate it about lots of different centres of enlargement.Bearings from A to B given as bearing from B to A Draw an arrow facing north on A (position bearing is from)Using the wrong trigonometric ratio Learn SOHCAHTOA or mnemonic such as ‘Silly Old Harry Can Always Have Tea On Aeroplanes’

List the side you have and the side you want. Then identify the correct ratio.A right angle triangle has to be scalene Challenge students to draw a right angled triangle which is not scalene. i.e. isosceles triangle with angles 45°, 45° and

90°Squares and rectangles are not parallelograms Ask students to list the properties of a parallelogram

Ask students if a rectangle fits these criteria. Repeat for a square.Identifying dimensions.Formula for length, area or volume?

Replace all variables with a tick or with one letter and analyse.

Ratio of shape Separate the shapesNot recognising irregular shapes Show examples at all opportunitiesUsing the wrong scale on a protractor Estimate the angle before measuring. Is it more or less than 90°?

Classify angle first – is it acute, obtuse or reflex?Look from zero and follow round to the line.

Putting the bottom edge of the protractor along one of the lines forming the angle

Put the cross on the intersection.

An enlargement makes things bigger Use scale factors of ½, 1/3, ¼, etc. Then discuss.Consider enlargements of scale factor -1, -2, -3 etc.What happens for a scale factor of -1/2? -1/3?

A rectangle and parallelogram have 4 lines of symmetry

Fold a rectangle. Fold a parallelogram.Use tracing paper and flip over along line of symmetry.

Algebra

Misconception Correctiona × a × a = 3a and a³ = a × 3 There are rules for algebra and the definition of a³ is

a × a × a. 3a means 3 lots of a or a + a + a. Use substitution to justify.

3a² = (3a)² (3a)² means 3a all squared i.e. (3a) × (3a) = 3a × 3a = 9a²Whereas 3a² means 3 × (a²) = 3 × a × a

( a + b )² = a² + b² Use a diagram to justify answer.(a + b )² = (a + b)(a + b) = a² +ab + ab + b² = a² + 2ab + b²

Trail and improvementx³ + 2x² = 100Answer to 2 dp is 100.14

Encourage students to check their answer

Expanding brackets5( x + 3 ) = 5x + 3

Justify using a diagram.Ask students to check their answer by substituting in a value for x.

Factorisingx² + 5x = ( )( )

Practice makes perfect. Provide a mixture of questions.

y = 2 is vertical When plotting/sketching y =2 ask students to firstly write down some coordinates where y = 2.Rules of indices a° = 0x-2 = -x²

Explore patterns such as5³ = 1255² = 2551 = 55° = 1 5-1 = 1/5 divide by 5

Locating the wrong region when sketching graphical regions for inequalities

Encourage students to pick a point in their region and check that it works for the inequality.

Gradient and intercept mixed up when equation of line not given in the form y = mx + c

Encourage students to always rearrange the equation of the line in the form y = mx + c and then compare to find the gradient and intercept.

√-4 = 2 (-2)² = 4 and √4 = 2. √-4 is not possible (at this stage of their learning)All algebraic expressions can be combinede.g. 11a + 3a – 3 = 11a 3x + x² = 4x or 4x²

Encourage students to check that they have correctly simplified by substituting numbers

If x =-3 than x² = -3² = -9 Encourage students to use brackets when substituting e.g. x² = (-3)²

x + 5 can be cancelled to get 1 + 5 x

Rewrite as x/x + 5/x

Data Handling

Misconception CorrectionThe location of the line of best fit on a scatter graph.

Instruct the students to hold a ruler at 90 degrees to the page so that they can see both above and below their line. Move the ruler over the graph to find the best position.

When asked to identify the modal class from a grouped frequency table students give the highest frequency

Continuously ask students to describe the information in the table before they begin each question.

The probability of it raining today is ½ because it either rains or does not rain.

Describe different situations i.e. The sun is out and there is not a cloud in the sky. What is the probability of it raining? There are big black clouds and it is really dark. What is the probability of it raining?

Distinction between mean, median and mode The mean is mean because it is nasty to work outMode begins mo which stands for most oftenMedian - Median looks like the work Middle. Get students to write middle underneath medianMiddle

Range is a type of averageRange is written as 6-9 and not 3

Range is a measure of the spread of the data and is given as a number.

Finding the mean from a frequency distribution students divide by the number of classes

Encourage students to discuss what information the table is showing before they begin the question.Initially students could write out the data in the long form to calculate the mean.

Finding the mean from a grouped frequency distribution students divide by the number of classes

Ask students to read the information in the table.Ensure that they check that their answer is reasonable.

When drawing pie charts students do not move the protractor round to draw the next sector

Emphasise that the angles must add up to 360°

Using the mid points of the class intervals to plot a cumulative frequency curve

Discussion concerning what a cumulative frequency graph is illustrating

Reading cumulative frequency graphs as less than and not more than

Discussion concerning what a cumulative frequency graph is illustrating

Probability can be greater than 1 Complete a probability number line where 0 is impossible and 1 is certain.Ask students to write events in appropriate locations on the number line.

A bar chart and a histogram are the same Get students to draw a bar chart to illustrate a carefully chosen grouped frequency distribution.Discuss concerns and any misleading information.

That you add the fractions on the branches of a tree diagram

Explain that the branches are multiplying out to form more branches.For many questions if you add the branches the answer will be greater than 1!

When finding the median from a stem and leaf diagram students only write down the part of the number on the leaf and not on the stem.

Is your answer sensible?