Math Department Endeavour Primary School Purpose of workshop Misconceptions and mistakes by topic: ◦ Whole Numbers (P5) ◦ Fractions (P5 and P6) ◦ Ratio (P5 and P6) ◦ Percentage (P5 and P6) ◦ Geometry (P5 and P6) ◦ Mensuration (P5 and P6) Equip parents with knowledge of common misconceptions and mistakes pupils make. Help parents to help pupils better. Consistent between what is taught in school and support from home. Provides a good support structure that can reduce stress in pupils. If pupils can overcome these misconceptions, they can fare much better. What is a misconception? It is defined as – ‘a view or opinion that is incorrect because it is based on wrong thinking or understanding.’ Misconception 1: When you divide a number, the answer will be smaller and when you multiply a number, the answer becomes greater. Misconception 2: When you see the word ‘more’ in the sentence, you always add. 1. Writing in numerals or in words Write in numerals: Three hundred and four thousand and sixtyfive Correct answer : 304 065 Common mistakes: 300465, 4365 300 + 4000 + 65 Three hundred and four thousand and sixtyfive 304 065 Write in words: 217 389 vs 217 089 Misconception: can only use ‘and’ once! 2. Order of Operations Do the following sum: a) 8 + 3 – 1 = ? Ans: 10 and 6 Did you get them right? b) 8 – 3 + 1 = ? Now do these: c) 5 x (8 – 4 ÷ 2) ÷10 = Is your answer 1 or 3? Or do you get a totally different answer? Why do pupils get the wrong answer? BODMAS It is not true that you have to follow the order as shown, e.g. D before M and A before S BODMAS is a misconception In Primary school, we seldom encounter ‘of’ in the order of operations. Pupils generally only see B, D, M, A, S Use hierarchy structure First priority B Then, D M or M D Lastly, A S or S A from left to right from left to right Let’s try again: 5 x (8 – 4 ÷ 2) ÷10 = 5 x (8 – 2) ÷10 = 5 x 6 ÷10 = 30 ÷10 = 3 1. Fraction as division 5 =? 8 Pupils work out 8 ÷ 5 instead of 5 ÷ 8. Why? Misconception: Only a larger number can be divided by a smaller number. Show counter example: 2 pizzas can be shared with 4 people. 2. Answering ‘Fraction of …’ qns: John has 7 books, Mary has 5 books. Express the number of Mary’s books as a fraction of the number of John’s books. 5 7 Express the number of John’s books as a fraction of the number of Mary’s books. 7 5 Pupils may give the first answer as they believe the numerator has to be smaller than the denominator Misconception 1: First number is the numerator, second number is the denominator, Misconception 2: Larger number is the denominator. Consider another question: John has 7 books, Mary has 5 books. What fraction of the number of Mary’s books is the number of John’s books? 5 7 7 or 5 Rule: the first number after ‘fraction of’ is the denominator 3. Multiplication of fractions (cancellation) 22 x 14 x 1 7 1 2 What are the common mistakes usually found here? a) Cancellation between 2 numerators or 2 denominators b) Double cancellation: 1 denominator with 2 numerators 4. Calculator error Use your calculator to do this: 1 1 x2=? 2 Pupils sometimes did not use mixed number key but used fraction key instead. Pupils who pressed ‘1’ first, then the fraction key to enter half gets a wrong answer. Try doing this? 1) Press the shift button 2) Press the fraction button 5. Dealing with remainder Compare the two questions: 1) 1 1 Sarah spent 4 of her money on a bag and 4 2) 1 1 Sarah spent of her money on a bag and 4 4 of it on a purse. of the remainder on a purse. Tendency to missed out the ‘remainder’. 1 7 6. Mrs Tan had kg of flour. She used 2 8 of it to make some cookies. How much flour had she left? 7 1 8 2 or 7 1 x 8 2 Pupils need to be alert on the presence or absence of units. 1. Simplest form All ratio answers need to be in the simplest form unless specified. Do not leave ratio in decimal notation. e.g. 3.5 : 4 : 2 = 7 : 8 : 4 2. Not alert on ratio requested. Sam has $34 and Frank has $35. What is the ratio of Frank’s money to Sam’s money? What is the ratio of Frank’s money to the total amount of money? Tendency to give the ratio answer based on order the numbers appear. 3. Answering in the wrong format The ratio of the number of boys to the number of girls in the school in 2 : 3. What fraction of the total number of pupils are girls? Pupils answer in ratio instead of fraction. 4. Unable to identify standard ratio type 1) 2) 3) The ratio of Ali’s stamps to John’s stamps is 3 : 1. Ali uses 12 stamps and the ratio 1 Quantity unchanged - John becomes 3 : 2. The ratio of Ali’s stamps to John’s stamps is 3 : 1. After Ali gave John 12 stamps, the ratio becomes 2 : 1. Total remain unchanged The ratio of Ali’s stamps to John’s stamps is 3 : 1. If they both buy 12 stamps each, the ratio becomes 2 : 1. Difference is unchanged 5. Using wrong original ratio E.g. The ratio of Ali’s stamps to John’s stamps is 3 : 1. Ali uses 12 stamps and the ratio becomes 3 : 2. How many stamps do they have altogether in the beginning? 1 Quantity unchanged - John Before Ali : John 3 : 1 After Ali : John 3 : 2 Before Ali : John 3 : 1 6 : 2 Ali – 12 After Ali : John 3 : 2 1 Quantity unchanged - John 3u ---- 12 1u ---- 4 4u ---- 4 x 4 = 16 Should have solved for 8u instead of 4u. Good practice: Cancel out the original ratio. 1. Wrong mathematical sentence Do this: Change to percentage 12 a) 25 7 b) 8 Method 1 is to convert denominator to 100. Method 2 is to multiply fraction with 100% Method 2 is to multiply fraction with 100% 12 x 100 = 48% (incorrect statement) 25 12 x 100% = 48% (Correct statement) 25 However, try doing both using the calculator. What do you notice? When using calculator, do not press the % key. 2. Using wrong base Mr Jahan’s mass was 70kg in 2004. His mass increased to 84 kg in 2014 and reduced after much dieting to 67.2kg in 2016. a) b) What is the percentage increase in his mass in 2014? What is the percentage decrease in his mass in 2016? a) Increase: 84 – 70 = 14 14 x 100% = 20% 70 b) Decrease: 84 – 67.2 = 16.8 16.8 x 100% = 20% 84 Common mistake: use wrong denominator Mr Jahan’s mass was 70kg in 2004. His mass increased to 84 kg in 2014 and reduced after much dieting to 67.2kg in 2016. a) What is the percentage increase in his mass in 2014? b) What is the percentage decrease in his mass in 2016? 1. Unable to identify parallel lines and angles within 2. Not labeling angles 180° – 132° = 48° 48° – 23° = 25° 180° – 90° – 25° = 65° 132° – 65° = 67° No way of knowing what the working done are for. 3. Unable to see isosceles triangles within a rhombus By adding the equal lines, pupils can see the isosceles triangles better. 4. Wrong naming of type of angles 180° - 23° - 132° = 25° (alternate angle) (vertically opposite angle) 1. Forgetting to multiply the half 2. Not able to identify the correct base and height pair 2. Not able to identify the correct base and height pair 3. Difference of areas mistake 1 x 42 x 3 2 1. Wrong use of formula Area of circle π x r x r (encouraged) π x r2 Circumference of circle 2xπxr π x d (encouraged) Encouraged to distinguish the formulas better. 2. Calculate for a circle instead of part of a circle 3. Identifying the correct radius and diameter Small semicircle: Radius = 5cm Diameter = 10cm Big quadrant: Radius = 10cm Diameter = 20cm 4. Incomplete sides when finding perimeter Pupils failed to add the two straight lines. 4. Incomplete sides when finding perimeter Pupils failed to add the two straight lines. 5. Mathematically wrong statement Correct: 2u 60 or 2u = 60 Incorrect: 2 60 2 = 60 Statement says area of circle but pupils calculated for a quadrant instead. Missing π in the working answer 6. Doing things the hard way Find area of quadrant Answer x 2 Find area of square – Area of quadrant Answer x 2 Add both answers Shortcut: Area of square x 2 6. Doing things the hard way Shortcut: Area of square – Area of semicircle Can you see it? 7. The value of pi Usually, we use 4 values of pi 1) 2) 3) 4) 3.14 In terms of π Calculator value 22 7 Pupils must use the value of π as mentioned in the question. Some pupils used the wrong value. What is the value of π in the calculator? 3.141592654 So how do pupils use this value? Imagine needing to find perimeter of quadrant Pupils tend to round off first and that will result in inaccurate answer. Pupils are advised to use the symbol π until the last step, then press the value of π into the final answer. When pupils do not get the marks they are supposed to get, there are usually 3 reasons: a) Do not understand the question. b) I understand but I do not know how to do. c) I understand and I know how to do but I am not alert and careless. Common mistakes made in exams may cause pupils to lose enough marks to cause a drop of 1 grade and in some cases, even two grades.