I’m thinking of four numbers. The mode of the numbers is 7; the median is 7.5; the mean is 9. What are the numbers?
First, you need definitions:
MODE: The most popular, or most frequently occurring item of data
MEDIAN: The middle number when the data set is ordered. If there are four numbers (as in this case) the median is exactly half way between the second and third values.
MEAN: This is calculated by working out the total of your numbers, and then dividing by how many numbers you have.
Here are the steps to solve the puzzle:
1) If 7 is the mode, then two of our numbers will be 7. We cannot have three 7′s, because this would affect the median. Our numbers are therefore 7, 7, x, x
2) If the median is 7.5, our third number is 8, because 7.5 is exactly half-way between 7 and 8. Our numbers are now 7, 7, 8, x
3) If the mean is 9, then the total of our numbers is 36, because 9 = 36/4. At the moment, our numbers total 7+7+8=22. This means that our last remaining number is 14, making our numbers 7, 7, 8, 14.
If you enjoyed solving that problem, then try this:
I am thinking of five numbers. The smallest number is 10. The range is 9. The mode is 11. The mean is 13. What are the five numbers?
NB RANGE: The difference between the highest and lowest values.
If you’re feeling brave, post your solution below!
Rote learning gets a bad press. The term is associated with good old-fashioned chalk-and-talk teaching methodologies… “Sit down and shut up! Open your books at page 359 and complete questions 1 to 59 before the bell…” But as any teacher knows, it remains the best way to learn a lot of what we need to make us good at maths.Fortunately, times have changed, and technology has opened up new ways to teach by rote. Here are some of the techniques we employ in our tuition centre, and within our app, to liven things up:
- Introduce a time constraint on a question: “You have five seconds to answer each of these questions before you get it wrong…” (useful in forcing children to guess – not always a bad thing when it comes to learning facts by heart!)
- Introduce a time constraint on an exercise: “How many of these times-tables can you do in one minute?”
- Compete – against others, either in the classroom or across the globe on tutpup.com or similar
- Compete against last week’s score. Children almost always improve their score, and this spurs them on more.
- Instant feedback. Rote learning is deathly if nothing’s marked until the end of the lesson – and then it’s all wrong. IT allows children to see instantly if they are on the right track.
- Muscle memory: remember this can play an important part in learning some tasks by rote. For example, it is vital when completing sums by column addition to always show working (this is the purpose of the overlay on these questions in DoodleMaths). The brain remembers the sequence of muscle movements as much as the principle itself.
- Set the bar at the right height: in other words, ensure the balance of success/challenge is correct for the individual.
Here are some areas of Key Stage 2 maths that lend themselves well to such techniques:
- Doubles and halves
- Number bonds to 10, 20, 100 and number facts up to 10 and 20
- Times Tables (leading to associated division facts)
- Column addition, subtraction, multiplication and division
- Naming shapes and solids
- Units for measurement
- Adding and subtracting 9, 19, 99 etc.
- Compass points
- Decimal and fractional bonds to 1
- Calculating the fraction of a quantity
- Fraction/decimal equivalents
This is the tip of the iceberg. But what are we doing when we learn these things by rote? Simple: we are committing a mathematical fact to our long-term memory. Storing facts in our long-term memory frees up our working memory to perform more complex calculations. For example, 80 – (4 x 7) becomes a lot easier if you know 4 x 7 = 28 as a fact immediately.
I’m not saying all maths needs to be learned in a rote fashion. There are equally plenty of areas of maths where a deeper conceptual understanding is required. But most of the children that we meet find maths difficult because they are frustrated by their lack of knowledge, rather than their lack of understanding.
You are allowed to use any mathematical operator including exponents and factorials. Scroll down for my five ways. I had to get creative for the last one, and I must admit to spending a good part of my evening on it!
1. 5 x 5 x (5 – 5/5)
2. 5 x (5 + 5 + 5 + 5)
3. 5 x 5 x 5 – 5 x 5
4. (5 + 5) ^ ((5+5)/5)
5. 5! – (5 + 5 + 5 + 5)
If you found any other ways, please please please submit below!
The $1 billion dollar iPad roll-out in Los Angeles is on hold for the time-being since three students hacked their district-issued iPads within days.
Far more disturbing than this is that there appears to have been little or no discussion with LA teachers regarding the educational content on these devices. Pearson appear to have this sewn-up.
Teachers are all different. They have their own individual styles, strengths, habits and foibles; it is getting the right blend of these characteristics that makes a school great. Children are all different, too, and this means that each child will always find a teacher in an educational institution that they admire, respect, and see as a role model. It’s this unique blend of individuals that makes every single school develop its own culture and ethos, and thus allows (some) parents to have a choice in the education of their child.
Given this, it should come as no surprise that teachers like to choose resources which supplement their own good work – resources that complement their own style of teaching and will work with the blend of individuals that are in their charge. I have taught in four very different schools and had to adapt my teaching style every time. “One-size fits all” just does not work. Teachers have been prescribed enough with the Common Core Standards (or the National Curriculum in the UK) without having to be told exactly how to deliver them.
So that’s my first problem with Pearson providing the content for these iPads. My second is this: I’m afraid that for all their vast financial investment, the Pearson Common Core System of Courses is uninspiring, and in no way matches the innovative nature of the iPad itself. They do not delight. As I have stated in previous blogs, subject-specific apps need to offer something new: technology alone does not raise standards. The simple rule is this: educational apps need to enhance existing provision, rather than simply replace existing teachers, worksheets and textbooks if they are to have an impact. Innovation of this nature tends to come from individuals finding solutions to their own problems, rather than corporate, salaried employees being paid to search for them.
I’m sure educators in Los Angeles will be able to introduce and use other resources apart from Pearson’s. If not, it won’t be the students hacking their devices – it will be the teachers.
On the assumption that your five-year old has a grasp of addition, these are the most important numerical facts a child can learn at this age.
Children who are good at maths have committed a lot of what they know to heart. By this, I mean that important number facts have been learnt and committed to long-term memory. Note that it is widely accepted that there are two ways to commit something to long-term memory: either learn by understanding (perhaps you’d learn the events leading to the start of WW1 in this way) or rote learning (times-tables must be learnt this way in my view.)
I digress… back to the point in hand. The more number facts a child has committed to long-term memory, the more they free up their working memory to perform more complex calculations. A child who can recall the doubles of numbers below ten can then also learn the following without much more effort:
- Near doubles: if you know that 6+6=12, you can instantly work out that 6+7=13.
- Adjusted doubles: to work out 6+8, change it to 7+7 and use your doubles. Doubles, near doubles and adjusted doubles account for the majority of addition facts to 20.
- Double 10, 20, 30 etc. and 100, 200, 300 etc. This innately teaches children place value, and excites them because they are using big numbers! You’ll get them doubling 1000, 20,000 before you know it.
- 2x table: same as your doubles!
- Halving: the reverse of your doubles. But you have to learn them off by heart: if you choose to teach doubles by adding a number to itself, whilst this is sensible in the short term, in the long term many children learning how to halve will attempt some kind of subtraction. Better to get them to learn off-by-heart early on.
- If they can halve, they can quarter. Teach it by halving, and halving again.
- And you can even lead in to percentages, because 50% is one half.
- Partitioning: if they know their doubles confidently off-by-heart, they can double any number by partitioning. Double 24? Well, double 20, double 4, then put it back together.
Of course, the other by-product of doing this is it gets children into the habit of committing numerical facts to their long term memory from an early age. Because maths is never duller than when you are still continually counting on your fingers at 9 or 10 years of age…
In the third part of our real-life insight into the way DoodleMaths works, our two guest bloggers give their view on the app’s benefits. Following the start of a new school year, six-year-old pupil Tabitha and her mum, Sophie, consider what impact the app has made to Tabitha’s maths over the summer holiday.
DoodleMaths has been fabulous even throughout the very, very end of the summer holidays. Going back to school it has helped me do amazing at maths!
At school, I got a sticker that said “You did it”!!! because I got all my divide sums correct!!!
I am better at amounts of money and three digit number sums. Everything on Doodle Maths has helped me do better at school because it is an awesome app.
It’s great because there is so much to do on it like: New This Week, Seven- A- Day, Games Pages, Topic Index and My Back Pages. There’s a Parents section for mum and dad.
I’ve given my pet cat on the app a name, she’s called Molly. I have reached 350 Doodle Stars, which means I can choose different things for her to wear.
I’ve started doing algebra with letters like n or x or p. And the 14 times table, it’s brilliant.
On the Topic Index I’ve done things like Word Solving, plus, minus, times tables and division.
It is perfect and amazing doing DoodleMaths most days and peaceful too.
The Parent’s View – Sophie, 33-and-a-half
It’s been just over two months since we downloaded DoodleMaths as part of a plan to keep Tabitha’s maths ticking over during the summer holiday. Has her enthusiasm for the app dipped during that time? No. Does she still ask me every morning whether she can do an exercise or two on it? Yes.
But I suppose the most important question is whether the time she has spent on the app has improved her mathematics. And, judging by the response from Tabitha and her teacher in the first couple of weeks back at school, the answer is a resounding Yes.
Many schools are introducing iPads, or other tablets, into the classroom. They’re popular with parents (and especially prospective parents). There’s undoubtedly a media buzz around what is still considered new technology. The potential benefits are becoming more and more apparent, both educationally and arguably financially, with ESSA Academy reporting an annual saving of £65K on photocopying budget following their introduction. The US is leading the way, with 4.5 million iPads sold into education institutions so far, and Los Angeles recently penning a deal with Apple to provide iPads for every one of 600,000+ students within the next year. But if they’re not introduced effectively, it can be a huge waste of money.
We have worked with a number of schools in the last year who have introduced, or are at the point of introducing, iPads. So in the spirit of sharing best practice, here’s a summary of what we believe works and doesn’t work:
- Appoint the right person to oversee the introduction to iPads: someone who will embrace the opportunity and become ambassadors for the technology. The success of their use at Cedar’s School of Excellence, Greenock can be largely put down to the commitment of Fraser Speirs, Head of IT, in his commitment to their implementation. Without the right person, you may find them gathering dust.
- Pilot with a few iPads, perhaps a one-third class set or a half set. A one-to-one program is a nice ideal, but perhaps bring-your-own-device will be more realistic by the time you are at that stage.
- Are you sure you want iPads? The Kindle Fire is a reasonably-priced alternative, with the Amazon App Store working hard to develop educational content. Little fingers can work equally-well on an iPad Mini, too.
- Think about how to use them across the curriculum. There are many good subject-specific apps, but equally focus on non-subject specific and be creative in their use. The iPad is a Swiss-army knife of tools waiting to be unlocked. The camera can be used to snap artwork at various stages in the creative process. Experiments can be videoed. And these can be imported into a document in an instant. Combine with Apple TV: teachers or students can display their work, annotate it, work through step-by-step, and display to the rest of the class. Communicate: apps like Socrative allow teachers to quiz their students; it automatically collates their responses. Research the Internet – use a safe internet browser such as Browser for Kids to monitor content.
- Of the good subject-specific apps, view these as supplementing and enhancing your teaching practice, and don’t slip into replacing your teaching with an app. Avoid apps which replicate worksheets – an expensive and inferior solution in many cases. Seek out subject-specific apps that can offer something beyond your usual teaching, be it a vast amount of content, (Life and Death in Pompeii, for example), high levels of interactivity (try Solar Walk), or real-time monitoring of each individuals’ progress (like our very own DoodleMaths for Schools).
- And apps are not just for students, either: great teacher planning apps – such as iDoceo – can enhance record-keeping, too.
- Practicalities: you’ll need school-wide wi-fi, internet security, iPad cases and a secure stowage unit. Number each device. Each device should have its own Apple ID. And tablets are designed for personal use – sharing an iPad between two for an activity just doesn’t work.
- Finally, don’t see iPads as entirely replacing PCs. They still have their role in the classroom. But for integrating technology directly into everyday lessons, the tablet is going to play a huge role.
I’m interested in any comments on this one!