2x2 criss cross order of operation

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#1 24 May, 2016 - 12:24
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2x2 criss cross order of operation


I am trying to develop my skill at 2x2 multplication. I had been working from left to right but I found that if I do the cross multiplications (tens x units) first, then the units and finally the tens it seems much easier. In particular I find it much easier to keep a 3 digit partial product in my head. and I'm much less likely to confuse the 10's and 100's.

e.g. 47x84

first step: (40x4) + (80x7) = 720
2nd step: 720+7x4 = 748
3rd step: 4x8x100 + 748 = 3948

If I do it the traditional way and start with 40x80, I have to hold onto a 4 digit partial product for several steps. I find that challenging.

24 May, 2016 - 16:55
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There isn't a right way. You can learn to accumulate right to left (although I wouldn't recommend it). As you practice 2x2 multiplication over about the next 16 weeks, if you don't quit... You will "discover" all sorts of exciting short cuts and tricks... Then as you continue to improve your numeracy you will add new facts that make 2digit multiplication easier. Learning your first 99 squares fluently is a huge advantage in this area and a good intermediate challenge. Nailing 2digit multiplication then opens up 3 an 4 digit squares and multiplication... What you find as you become comfortable with the numbers is that you usually have 3 or 4 equally strong methods and that you can cross check your own work mentally almost as fast as resolving the initial solution... So just keep practicing for 3 or 4 months and you should see what I mean :)

25 May, 2016 - 07:06
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Robert,
Thank you for your encouragement. A few months seems quite doable. I have the squares up to 100 but retrieval is slow. That's another problem, how to speed up random access. Perhaps part of it is just developing fluency with the Major system to the point where it becomes "transparent" and doesn't need decoding.

25 May, 2016 - 09:15
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Learning your squares through calculation is the important part. Fast difference of squares gives you a whole other layer to work multiplication with. It also gets you working addition and subtraction nicely.

Confidence calculating squares carries over nicely into developing confidence calculating 2 digits. There is a similar progression for 3 digit squares. You will likely become distracted by other things along the way but if you keep returning to daily practice progress can be made.

26 May, 2016 - 03:05
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Hi Zvuv!

I wholeheartedly concur with Robert and his advice.
Your way of doing the 2x2 is fine. Keep on doing this.

Play with the difference of squares like Robert says. This will speed up your 2x2's.
Example: 17X23 = 20^2 - 3^2 = 400 - 9 = 391.
This is a completely different way of doing a 2x2 compared to the criss cross method.

In order to get proficient with multiplication, do a calculation using different methods. Also, do a guesstimate first. At least get a feel for the ball park the answer is in.

Do a search on this part of the forum. In the last couple of years there has been a lot of advice for multiplication, especially 2X2's.

26 May, 2016 - 18:55
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Thank you kinma.

I have been scanning the forum. I have a little notebook in which I write tricks for 2x2. I practice with a spread sheet that generates random 2x2's. Right now my goal is to get perfect accuracy. Next step will be to do the problem while holding it in my head. Currently, I stare at the spread sheet. Progress is slow but I am improving.

27 May, 2016 - 13:03
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Quote:

So just keep practicing for 3 or 4 months and you should see what I mean :)

How much time per day would you recommend spending on practice?

27 May, 2016 - 14:02
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zvuv wrote:

e.g. 47x84

first step: (40x4) + (80x7) = 720
2nd step: 720+7x4 = 748
3rd step: 4x8x100 + 748 = 3948

Let's whet your appetite for some different ways of calculating this. How does this sound?

again: 47x84

first step: 50 X 81 = 4050 (+3, -3)
2nd step: 3 X 34 = 102 (difference 47 and 50 = 3, 84 and 50 = 34)
3rd step: 4050 - 102 = 3948

2 June, 2016 - 16:20
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Thank you kinma. I'll add that to my little note book of calculating tricks.

3 June, 2016 - 08:58
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Once you get comfortable with 2x2. 89-90% you will find that working squares and 3x3 as well will help your 2x2. "overreaching". You will almost certainly discover weaknesses in your addition and subtraction. Identify and practice these tootoo. "remedial". Check out some Kinma''s posts on logarithms, approximation and modulo 11 and 9. “motivation"....

At some point you may become interested in math. It is quite a bit more fun than sodoku, Rubik's Cube or cross word puzzles but sadly or happily you always feel stupid no matter how much of it you manage to master.

On the plus side mathematicians tend to humble and helpful.

The other nice thing is math can be fun even if you don't have any talent. There are all level of mental challenges available.

4 June, 2016 - 06:37
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Another imho important one is this.
You can see 89 as 80 plus 9.
It certainly is pronounced that way.
However, you can see also this as 90 minus 1.

Use this in criss cross multiplication.

If you do 89 x 89 using the criss cross you get:
80 x 80
9x 80 + 9 x 80
9 x 9
And a lot of carrying.

Now try this:
90 -1
90 -1
_____ x
90 x 90
-1 x 90 + -1 x 90
-1 x -1
For me this is easier. Less carry and easier to remember numbers.
Of course now you are subtracting the middle part instead of adding.

4 June, 2016 - 06:55
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The 89 is 90-1 carries forward into all addition and subtraction. Being able to rapidly look at 267 and see 300 - 33 increases your options tremendously. Hard subtraction can be converted to easy addition. And the other way around almost without fail. Acquiring the mental flexibility to do this in your head while doing a mental calculation is a skill that seems to improve with daily practice and number of sleeps. I have wondered at what point practice improvements tail off and natural talent becomes a limiter but I would have to ask someone else as it take more practice than the off and on year or so that I have been playing with it. - or alternatively I have no natural talent and my small daily improvements are already the tail ;)

Youth clearly helps the speed of initial learning but brute force perseverance seems to be able to get average folk the first 80% of the way. (and 80% seems pretty magical to people at 0%). When you occasionally meet someone gifted, or worse gifted and practiced it gives you an appreciation of how sharp the brain can be. They usually have a few tips as well if you are willing to listen.

4 June, 2016 - 12:39
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What he ^ said!

Once you start to see numbers like this you will see things like why the difference of squares 'work'.
Try for example doing 89 X 91 using the above method:

90 -1
90 +1
_____ x

90^2
-1 x 90 + 1 x 90 (=0)
-1 x +1 = -(1^2)
or
90^2 - 1^2

The whole point in mental calculation is making what looks difficult easy.
For that there are a lot of 'tricks'.
The rest is plain addition and subtraction.

Quote:

Hard subtraction can be converted to easy addition

This is so true!
A lot of people find 124 - 89 difficult. Probably because of the double carry.
However; they can easily do 89-24 (=65).
The complement of 65 is 35 and that is your answer to 124 - 89.
Think in complements. If you see 54 try to also see its complement (46).
If you see 89 try to also see 11.
Subtracting 89 from 124 is the same as adding 11 (its complement) to 24.

As Robert said:

Quote:

Hard subtraction can be converted to easy addition

8 June, 2016 - 09:13
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I'm not great at maths but that compliment thing is amazing, thanks

9 June, 2016 - 09:28
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I am practicing via the method on this site....

http://freetestprep.net/quantitative/multiplication/speed-multiplication-2/

Seems complicated at first but even though i'm bad at math, it's quite easy to multiply two digits mentaly via this method.

Watch the video and read the point (try a different order).

9 June, 2016 - 13:39
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Observations on practice on a sample size of 1 with no objectivity, terrible bias, no recorded facts and a complete lack of scientific training ( you have been warned)...

In terms of practice time the answer like everything else is 'it depends'. Occasional massed practice can be helpful. (reaching)

If you were smart about it you could probably productively improve over time with something like 3 - 15-20 minute sessions per day. With my recent McJob (I recently found a more interesting job) I tended to use commute time. Currently'ish... Review of recently learned material (in this case 2-digit squares - about 70% known as facts) is about 20 minutes. Then practice with 3 digit square in the 100's which stretches my 2 digit squares for 20 minutes. Then work 2x2, 3x3, or 4x4 multiplication for 20.

If I'm tired/brain dead I may not take on calculation that is hard 'for me' and simply work on improving the gaps in my 2x2 squares (currently not terribly difficult at all). ....

An extra session on my commute back may help if I'm not tired but may or may not be productive. Number of sleeps often seems to be as important as actual study time in terms of integrating things in a way that becomes a normal part of your thinking patterns. At my age it may never or it may take a solid 10 years (alternately the 10,000 hour rule may be reasonable for expertise - if expertise is integrating a skill to the point that is as natural as language or walking). Gifted folk appear to be able to bypass this volume of effort. In some cases correct training reduces the total effort hours. Incorrect training seems to have the opposite effects.

Some days are sharper than others and I feel like I can stretch a bit further with normal calculation.

I also take days and sometimes a week or more off. I either get stale with practice or side tracked with life. Each time I return there is some back tracking but there is also some new semi-permanent facts that let me move forward a little more quickly and further (not a lot).

There is another aspect to this in that being able to 'think'/concentrate for longer periods of time is a skill in of itself that appears to take practice. Being a bit obsessive obviously helps but once you reach a point of distraction you are no longer being productive trying to focus narrowly and obsession does not help a bit.

If there were a nice package that monitored learning, calculation speed, strong and weak areas and self-tailored problem solving to the individual I think that learning could be a great deal more efficient. The challenge is that while I could almost certainly get someone else to the point I have reached much faster than I reached it I have no idea how to efficiently move forward other than to use what has worked well so far and read the posts on this forum carefully at the right time, then attempt to extrapolate.

None of the Math-games/trainers that I have found so far do a good job of building a student's abilities. Most are extremely limited, very few keep track of the student's abilities, and a lot focus on tricks rather than thinking and integrating facts and methods effectively. At some point I will likely try building an application but in order for it to be really good I think I would/will need a fairly big database of people's learning statistics in arithmetic calculation that could be leveraged to proactively help individuals improve.

In the interim I keep playing with numbers and trying to get my math skills to a solid undergraduate level while my wife tells me how silly that is.

12 June, 2016 - 11:53
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I have the 1st 100 squares memorized but I am much faster at the crisscross method than I am using the difference of squares. I think this is just a matter of training.

Thank you Robert for your description of your practice methods. I too find that taking a break can help things solidify.

About myself. I am 65 yrs old. For most of my career I worked in the defense industry doing applied math. I have some talent as a mathematician but despite years of training, I never learned to do arithmetic accurately. I was pretty good at estimating, a skill most people working in engineering develop, but I could never carry out calculations accurately by hand and this always bothered me. Applied math is more about the properties of sets of number than it is about actual, specific numbers and an interest in one does not always carry over to interest or skill in the other. Anyway, finally, at this advanced age, I've decided to overcome this lifelong limitation. It's been really interesting to confront the nature of the errors that I tend to make. Did you know that 4x8 is sometimes 48 ? And more likely to be so when I'm tired? :)

13 June, 2016 - 11:12
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Quote:

I am much faster at the crisscross method than I am using the difference of squares. I think this is just a matter of training.

It is.

I advocate trying to multiply using many systems.
Try doing the 78 x 83 using criss cross first, then using squares (using 78 x 82 + 78).
Then using the anchor method (80 x 81-2x3).

Mental plasticity is what it breeds.

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