# Drawing Cumulative Frequency Diagrams – GCSE Maths Insight of the Week 4

“What misconceptions do GCSE Maths students have when it comes to plotting cumulative frequency diagrams from grouped data?”

The number of students around the country taking these weekly quizzes is growing and growing, and as it does, their teachers are learning more and more about the key holes in their knowledge. If you haven’t got involved yet, now is the perfect time. It is completely free, and always will be.

Each week we set all our Year 11 Higher GCSE students a GCSE Essential Skills Quiz from my Diagnostic Questions website. These quizzes are completely free and are designed to test the content that appears on the majority of GCSE higher maths papers. These are the kind of questions that students should get right, but which time and time again cost our Year 11s the valuable marks that prevent them getting the grade they want.

We set our quizzes on a Monday, the students complete them for Friday, and then I sit down with a Mellow Birds coffee and a yoghurt (it is the weekend, after all) to analyse the results. I try to find one key insight, and then share it with staff in our weekly Departmental Meeting in a 5 minute slot called “Insight of the Week”.

Quiz 4 is below, followed by the key insight from this week.

How our students performed compared to the rest of the world: We had issues with percentage multipliers and probability tree diagrams, but by far our biggest issue was with plotting cumulative frequency diagrams:

A measly 18% of our Year 11s got this question correct, with Options B and C tempting many of them. Plotting cumulative frequency diagrams is one of those topics people can take for granted. Students have drawn 100s of them so assume they are dead easy. Likewise, I know I have been guilty of assuming that our students will be absolutely fine, and instead focussing on more tricky topics. However, there is no hiding from the fact that every time our students encounter a cumulative frequency diagram on an exam, the same classic mistakes with plotting the points occur again and again. That was clearly evident in this question.

Incorrect Answer B (45% of our students chose this)

This was the most popular choice amongst our lot. As you can see from the explanation, they have correctly identified that cumulative frequency needs to be plotted against the upper boundary of the interval. However, they have fallen into the trap of plotting frquency, and not cumulative frequency

In cumulative frequency you use the highest point which is 40 and the frequency is also plotted which is 12.

You have to use the upper bound and the 12

Incorrect Answer B (31% of our students chose this)

Coming in at Number 2 on the popular choices list is an answer that reveals two misconceptions. These students have also plotted frequency instead of cumulative frequency, but have also plotted against the midpoint of the interval instead of the upper bound:

30 is the midpoint in between 20 and 40 and c is at 30 and 12

The middle is £30 and it shows the pound sign on the x axis, then frequency is on the y axis and then you just plot where it’s telling you to plot

Incorrect Answer C (5% of our students chose this)

Not many of our students were tempted by this, but those who did revealed that they remembered how to calculate cumulative frequency, but were still tempted by that all too inviting midpoint:

it shows the correct point on the graph between £20 and £40 as you add 8 and 20 to get the cumulative frequency

Find in between 20 and 40 for 30, then find the cumulative frequency for the coordinates of (30,20)

One of my motivations for developing Diagnostic Questions was so that students all around the world could learn from each other. When your students finish a quiz, please encourage them to review their answers, reading through other students’ explanations, until they find the magic one that makes sense to them. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

Cumulative frequency is adding the frequencies one by one and must always be plotted on the upper bound

you plot it at the upper bound (40) and the cumulative frequency is 20 because 8+12 = 20

because you would plot at the upper bound and as cumulative frequency means running total you would do 8 + 12=20

Upper bound and the running total

Tackling the Misconception in Class

So, when we discussed this insight in our Departmental Meeting, what ideas did we come up with for resolving the misconceptions in lessons this coming week with our students:

• Emphasise where the word cumulative comes from (to accumulate) and ensure students know it means a ‘running total’
• I feel it is important then when teaching any stats lesson involving a table of values, whether it be grouped frequency on non-grouped frequency, students should be challenged to say what each number means. So, if you have a cumulative frequency column, put a circle around one of the numbers, and ask the students to write down what it means in terms of the question. This gets students away from automatically processing numbers without thinking, and hopefully will make them make less mistakes.
• One colleague mentioned she does a zigzag method of continuous adding which means students cannot move on to the next row until they have added the previous one.
• Another colleague mentioned teach the students to add an extra column to all exam questions whenever they see cumulative frequency and know that they have to do something with the data before plotting it.
• We mentioned doing a ‘Frequency polygon vs. cumulative frequency’ lesson where for example, they are given 4 tables (2 polygon 2 cumulative) and just the question where it says draw the graph. Highlight the differences between the questions that way
• Another colleague said “Stop them plotting midpoints!”……

The series of GCSE Essential Skills Quizzes are available hereand will always be completely free.

Quiz 5 is available here

I hope these quizzes will prove useful to help your students develop the essential skills necessary for success at GCSE, and aid your teachers gain a deeper understanding of how your students learn.

# Squaring Negative Numbers – GCSE Maths Insight of the Week 3

“What misconceptions do GCSE students have when squaring negative numbers in the context of a table of values for a quadratic graph?”

The number of students around the country taking these weekly quizzes is growing and growing, and as it does, their teachers are learning more and more about the key holes in their knowledge. If you haven’t got invovled yet, now is the perfect time. It is completely free, and always will be.

Each week we set all our Year 11 Higher GCSE students a GCSE Essential Skills Quizfrom my Diagnostic Questions website. These quizzes are completely free and are designed to test the content that appears on the majority of GCSE higher maths papers. These are the kind of questions that students should get right, but which time and time again cost our Year 11s the valuable marks that prevent them getting the grade they want.

We set our quizzes on a Monday, the students complete them for Friday, and then I sit down with a Mellow Birds coffee and a yoghurt (it is the weekend, after all) to analyse the results. I try to find one key insight, and then share it with staff in our weekly Departmental Meeting in a 5 minute slot called “Insight of the Week”.

Quiz 3 is below, followed by the key insight from this week.

How our students performed compared to the rest of the world: We had issues with interpreting pie charts and area in context, but our Achilles’ heel this week was all about negative numbers:

Only 43% of our Year 11s got the question correct, with option C tempting almost one third of them: Negative numbers is a tricky issue for many students. Confusion about when to apply the famous “two minuses make a plus” rule (a rule, incidentally, that is illegal to say in my classroom), together with issues around squaring numbers, which direction to move on number lines, and the context of the table of values all seem to have contributed to the problems students found with this question. Let’s dive in and have a look at their answers and explanations.

Incorrect Answer A (16% of our students chose this)

A classic error, with students believing that -4 multiplied by -4 equals minus 16. Crucially, this group of students have an additional misconception when it comes to subtracting numbers from negative numbers as they believe that -16 – 1 = -15.

-4 squared = -16 – 1 = -15

Incorrect Answer B (30% of our students chose this)

Our most popular incorrect answer, catching the eye of almost 1 in 3 of our Year 11s. The misconception here is the same as the one above, but with a correct final step:

x = -4 to find y… y = -4 squared – 1 -16 -1 = -17 y = -17

-4^2 equals -16. – 1 off that puts it to -17

Incorrect Answer C (11% of our students chose this)

C is a different kettle of fish. Here students have remembered that when you square negative numbers they become positive, but have made the classic mistake of doubling instead of squaring:

-4^2 = 8 -1 = 7

One of my motivations for developing Diagnostic Questions was so that students all around the world could learn from each other. When your students finish a quiz, please encourage them to review their answers, reading through other students’ explanations, until they find the magic one that makes sense to them. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

-4 x -4 = +16 (two minuses make a positive) 16 – 1 = 15

because -4 squared =16 and then take one =15

Tackling the Misconception in Class

So, when we discussed this insight in our Departmental Meeting, what ideas did we come up with for resolving the misconceptions in lessons this coming week with our students:

• I have always taught negative numbers using the bowl of soup method. I wrote an article about it back in 2011here. There are lots of other ways teachers have of providing some sort of context for negative numbers that avoids potentially dangerous rules, including Harry Potter style potions, good and bad deeds, trains going into and out of tunnels, linking it to speed, and more! Perhaps this is the way forward.
• Write out -4 squared in full as -4 x -4
• A few of us suspected out students were using a calculator, and falling into the deadly trap of typing in -4^2 and getting -16 out. So, it would be a case of reinforcing the need to add brackets around all negative numbers when using a calculator, and showing a few examples of how the trusty calculator can in fact let you down!
• We thought relating the question to the actual quadratic graph it produces was important. If students are expecting the U-shape of a positive quadratic, and go ahead and plot their co-ordinates, then they are likely to notice there is a problem when one of their points does not fit the pattern.

The series of GCSE Essential Skills Quizzes are available hereand will always be completely free.

Quiz 4 is available here

I hope these quizzes will prove useful to help your students develop the essential skills necessary for success at GCSE, and aid your teachers gain a deeper understanding of how your students learn.

# Expanding Double Brackets – GCSE Maths Insight of the Week 2

Each week we set all our Year 11 Higher GCSE students a GCSE Essential Skills Quiz from my Diagnostic Questions website. These quizzes are completely free and are designed to test the content that appears on the majority of GCSE higher maths papers. These are the kind of questions that students should get right, but which time and time again cost our Year 11s the valuable marks that prevent them getting the grade they want.

We set our quizzes on a Monday, the students complete them for Friday, and then I sit down with a Mellow Birds coffee and a yoghurt (it is the weekend, after all) to analyse the results. I try to find one key insight, and then share it with staff in our weekly Departmental Meeting in a 5 minute slot called “Insight of the Week”.

Why not join in too by setting these quizzes to your Year 11s? The more the merrier!

Quiz 2 is below, followed by the key insight from this week.

How our students performed compared to the rest of the world: We had big issues with properties of kites and rearranging formula, but the question that caused our students the most problems was the following classic fusing together algebra and geometry:

Only 27% of our Year 11 students got this question correct, with A and C luring in over half of them. Now, expanding double brackets is an essential skill, and we would hope our students would have more success in this area. But I suspect it was the added context of the question that caught our lot out. This becomes more apparent was we read some of their explanations:

Incorrect Answer A (29% of our students chose this)

The classic incorrect way to multiply out squared brackets. I think I’ve told my Year 11s approximately 6,373 times to write the brackets out next to each other and either use FOIL or the smiley-face method. 5 of my students went for this option, so it seems I am going to have to try something else.

X * X = X squared 5*5 = 25

“All sides in a square are equal x * x = x2 5 * 5 = 25

Incorrect Answer C (33% of our students chose this)

Our most popular incorrect answer, catching the eye of 1 in 3 of our Year 11s. This is a completely different misconception to the one above. Here, students are displaying algebraic proficiency, but are muddling up area with perimeter. Would they also do this in a non-algebraic context? We will need to test our Year 11s to find that out.

I think the answer is C because it is a square so all the sides are equal. so to work out the formula of the area you just times x+5 by 4

As it is a square and all sides are of equal length so: 4 (×+5)=4×+20

Incorrect Answer D (11% of our students chose this)

I find this answer fascinating, but when you think about it, it makes total sense. Students, for years, have been used to working out areas that have actual number answers, and also solving equations that have actual number answers. Now they are asked to find an area, and yet they do not know the value of x. We need to ensure our Year 11s are comfortable leaving answers to questions in terms of x, otherwise they could come a cropper!

We need to know the length and width, but we don’t know the value of x

its D because you don’t know the value of x

One of my motivations for developing Diagnostic Questions was so that students all around the world could learn from each other. When your students finish a quiz, please encourage them to review their answers, reading through other students’ explanations, until they find the magic one that makes sense to them. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

(x+5)^2 = x^2 + 10x +25 x * x = x^2 5 * 5 = 25 5 * 1 = 5 *2 = 10x

(x + 5)(x + 5) x times x is x squared 5 times x equals 5x 5 times x equals 5x again 5 times 5 equals 25 5x plus 5x =10x

Tackling the Misconception in Class

So, when we discussed this insight in our Departmental Meeting, what ideas did we come up with for resolving the misconceptions in lessons this coming week with our students:

• Getting students to mark on the base and height of the square
• Ensuring the write out the formula for area
• Putting brackets around the (x + 5). This led on to a big discussion about how it could well be good practise to encourage students to put brackets around all algebraic expressions when they appear in a geometrical context. This will obviously help with multiplying, but may also be beneficial when faced with things like: (x + 4) – (x – 2), and the classic missing of the double-negative.
• Similar to last week, substituting values in for x and seeing which of the expressions give the same answer

The series of GCSE Essential Skills Quizzes are available hereand will always be completely free.

Quiz 3 is available here

I hope this quizzes will prove useful to help your students develop the essential skills necessary for success at GCSE, and aid your teachers gain a deeper understanding of how your students learn.

# Simplifying Expressions – GCSE Maths Insight of the Week 1

Each week we set all our Year 11 Higher GCSE students a GCSE Essential Skills Quiz from my Diagnostic Questions website. These quizzes are completely free and are designed to test the content that appears on the majority of GCSE higher maths papers. These are the kind of questions that students should get right, but which time and time again cost our Year 11s the valuable marks that prevent them getting the grade they want.

We set our quizzes on a Monday, the students complete them for Friday, and then I sit down with a Mellow Birds coffee and a yoghurt (it is the weekend, after all) to analyse the results. I try to find one key insight, and then share it with staff in our weekly Departmental Meeting in a 5 minute slot called “Insight of the Week”.

Why not join in too by setting these quizzes to your Year 11s? The more the merrier!

Quiz 1 is below, followed by the key insight from this week.

How our students performed compared to the rest of the world: The Insight of the Week comes from the question that caused our students the most problems, which was on simplifying algebraic expressions:

Only 37% of our students got this question correct, with C being the most common answer chosen, luring in 47% of them. Let’s look at some of our students’ explanations to see if we can understand why:

Incorrect Answer A (10% of our students chose this)

This may be students mistaking the rules for multiplying with the rules for addition. Many student explanations for this choice of answer also highlight their compulsion to combine the powers of d together.

2×3=6 DxD=D squared D squared add D = D cubed

Because d is 1 so if there’s 2 ds and one spare then 3ds

Incorrect Answer B (6% of our students chose this)

Students opting for this answer believe you can add the numbers together and add the powers together, but interestingly also may have some notion of the need to keep numbers and letters separate

3+2=5 (d squared)

d2 and d =d3

Incorrect Answer C (47% of our students chose this)

By far the most popular choice. Students are determined to combine the terms in a way that seems the most sensible: add the numbers and add the powers.

i think this because 2 and 3 is 5 and if you add the d’s on you get in a total of 3 d’s

3 plus 2 is 5 and d to the power of 2 plus another d is d to the power of 3

3 + 2 = 5 d squared + d = d cubed d cubed + 5 = 5d3 (d cubed)

One of my motivations for developing Diagnostic Questions was so that students all around the world cold learn from each other. So, what are our Year 11 students’ favourite correct explanations to help them resolve their own misconceptions?

because d squared and d are different things so you can’t put them together because it wouldn’t make sense

you cannot group these terms together – d and d squared are two completely different things.”

because you can only collect like terms and these aren’t like terms

Tackling the Misconception in Class

So, when we discussed this insight in our Departmental Meeting, what ideas did we come up with for resolving the misconceptions in lessons this coming week with our students:

• Substitute numbers in for d to show that the expressions are not equal
• Explicitly name the terms different things – “d” and “d squared”
• Write the expressions out in full, so 2d^2 would be 2 x d x d
• Write the expressions in the form d^2 + d^2 and d + d + d, and compare this to a + a and b + b + b