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# Summer School Insights: Week 6 FDP equivalence Craig Barton

Hello everyone,

This is the sixth post in our Eedi Summer School series, where I provide insights from the cohort of Year 6 students who are taking part in our seven week programme. My hope is that this may give you an indication of where our Year 7 students might struggle come September.

Last week’s insights on fractions caused quite a stir on Twitter, as the following thread shows.

Let’s see if we can be just as controversial this week as we turn our attention to Fraction, Decimal and Percentage equivalence.

Our group of 10 and 11 year olds did well at this quiz, scoring an average of 69%. But it is worth noting that this is the lowest average score seen across any of the quizzes so far - one percent lower than last week.

Here are the proportion of students getting each question correct in reverse order from the most poorly answered to the best answered:

We can see that Question 6 is the one that caused our group the most problems. What is Question 6, I hear you ask? It is this…

Take a moment to consider where you predict students might go wrong with this question.

Here are the results from our Summer School students:

Why did almost half of the students get this question wrong? And what was so alluring about the most common wrong answer of “none of these”?

Let’s have a look at some examples of student explanations to see if we can get to the bottom of it:

because if the denominator turns into 100 you'll have to do the same thing to the numerator and it won't be the same number so it cant be any of these decimals
8 cant make 100 but can make 200 by multiplying it by 25 so 3 times 25 is 75 and since it is a decimal divide by 100

i think it is none of these because it can't have a 7 in it and it can't be 0.3 and it can't be 0.38

As with all these misconceptions, I find it useful to ask myself two questions:

• Why are students going wrong?
• What am I going to do about it?

The data surprised me here. I would have expected the most common wrong answer to be B, 0.38. I have seen lots of students believe that fractions can be converted into decimals by essentially taking the numerator and denominator and just squidging them together. This is a fundamental misconception about the relationship between fractions and decimals. But as we can see from our young students’ responses, they are aware this answer is not correct.

However, the majority of students appear to have gone wrong here because they only seem to have one tool in their armoury for converting fractions to decimals - to make the denominator into 100 by multiplying. As the first two explanations above clearly explain, that is difficult to do when you have an 8.

One possible solution is to focus on transforming numbers into 100 via multiplication AND division separately. So, students could be presented with a sequence of numbers such as the following and asked to come up with combinations of divisions and multiplications that would transform each into 100:

2
4
8
40
20
80
120
125
150

Not having to worry about also converting the numerators, together with the related nature of the questions, should enable students to focus more on different ways to get to 100. This could lead to some interesting discussions about different approaches for dealing with certain numbers - i.e. is it best to deal with 8 by first dividing by 2, or maybe dividing by 4, or maybe multiplying by 25 and then halving the answer? All of this thinking, discussion and resulting strategies will stand students in good stead when we move to finding equivalent fractions with denominators over 100, and then finally to converting to decimals. Again we see my interpretation of Atomisation - carefully sequencing the building blocks to enable students to cope with a more complex idea.

I had a look on my Variation Theory website, and no such sequence currently exists for converting fractions to decimals that includes denominators like eighths or fortieths - in other words, those denominators that it is not quite so easy to multiply up to 100. If any one fancies creating one, please be my guest! There are templates available here.

Let me show you the second and third worst answered questions from this quiz, together with the proportion of students opting for each answer:

Again, it might be useful to ask yourselves the questions:

• Why are students going wrong?
• What am I going to do about it?

You can access the whole quiz here, and the Insights page that will give you access to the percentages and student explanations here.

Next week we have Ratio, and I will be back with some more insights based on students’ responses.

Take care of yourselves, thanks for reading and stay safe
Craig and the Eedi Team Written by
Craig Barton