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Beast Academy - Perfect Squares


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Can someone explain how these are helpful? I can see where changing 49 x 51 to 50 x 50 makes for easy mental math, but what about changing 64 x 66 to 65 x 65 or 104 x 106 to 105 x 105? Are these problems all meant to be done mentally? I'm specifically asking aout page 55 in book 3B.

 

I've used a range of programs with my dd, but mostly traditional, so it's likely I'm just missing something here.

 

Thanks so much!

Lisa

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They're mental math tricks. They're helpful in everyday math (like keeping a running total at the grocery store, math competitions where speed matters (I have a DD who loves that sort of thing) and in just plain understanding how math works-that you really ARE still doing the same problem, which comes in handy when you move on to algebra and higher math.

 

SM spends a lot of time on this, as does BA. I hadn't realized just how much difference it made until my DD started doing competitions and I started regularly seeing really, really good math students, often several grades ahead of her, falter on fairly simple questions that she could solve mentally because she was used to the kinds of manipulations that SM was asking her to do (BA has ended up being a review for her-it came out about a year too late, but I think it would have had the same effect).

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They're mental math tricks. They're helpful in everyday math (like keeping a running total at the grocery store, math competitions where speed matters (I have a DD who loves that sort of thing) and in just plain understanding how math works-that you really ARE still doing the same problem, which comes in handy when you move on to algebra and higher math.

 

SM spends a lot of time on this, as does BA. I hadn't realized just how much difference it made until my DD started doing competitions and I started regularly seeing really, really good math students, often several grades ahead of her, falter on fairly simple questions that she could solve mentally because she was used to the kinds of manipulations that SM was asking her to do (BA has ended up being a review for her-it came out about a year too late, but I think it would have had the same effect).

 

So if the idea is to show that you are doing the same problem minus 1, I kind of get that. But, how is mental math any easier with a problem like 64 x 66 vs 65 x 65? How would you tackle this as a mental math problem?

 

Thanks!

 

Lisa

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So if the idea is to show that you are doing the same problem minus 1, I kind of get that. But, how is mental math any easier with a problem like 64 x 66 vs 65 x 65? How would you tackle this as a mental math problem? (I'm not even sure these are supposed to be mental math problems, to be honest.)

 

Thanks!

 

Lisa

 

65x65 = (60+5)(60+5) = 60x60 + 2x60x5 + 5x5. If you know your squares well, you can compute that very rapidly -- more rapidly than by having to remember what to carry with 64x66.

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65x65 = (60+5)(60+5) = 60x60 + 2x60x5 + 5x5. If you know your squares well, you can compute that very rapidly -- more rapidly than by having to remember what to carry with 64x66.

 

Okay, I would have done 60x60 + 60x5 + 60x5 + 5x5, but I wouldn't have done it as a mental math problem.

 

I guess I don't get it. I've got to carry with both problems. Once with 65x65 and twice with 64x66.

 

I always did well in math in school, but some of this kind of stuff is so different, I'm just not sure whether I want to pursue it or not, especially because I can't really explain the point of it to my dd.

 

Thanks for your response.

 

Lisa

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65x65 = (60+5)(60+5) = 60x60 + 2x60x5 + 5x5. If you know your squares well, you can compute that very rapidly -- more rapidly than by having to remember what to carry with 64x66.

:iagree:

 

I think doing 64x66 mentally without paper and pencil would be rather difficult otherwise... or for that matter, doing 65*65. The above method does not involve any "carrying".

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I don't know anything about beast academy, but I have used Singapore Math. Mental math strategies are sometimes only really applicable under certain circumstances, and you would not apply them under all circumstances. For example, adding when a number is close to 10 or 100, e.g. 98 + 487 = 100 + 487 - 2, you use when the number is close to 10 or 100. The Singapore Math also taught some strategies for multiplication, but again when the number was close to a ten or 100. Is Beast Academy recommending changing 66 x 64 to 65 x 65, or is it rather suggesting this strategy for when both factors are close to a ten? By the same difference?

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Do a lot of people really do this mentally for their jobs? Aren't most people at a computer where they have access to a calculator?

 

I mean, sure, it is a great technique to teach kids who need more material to cover in math, and some people think it is fun and for competitions and such. But for a kid with serious memorization and processing speed issues, I'm trying to determine if spending some of our precious slow time on something like this is worth it.

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But for a kid with serious memorization and processing speed issues, I'm trying to determine if spending some of our precious slow time on something like this is worth it.

 

For the student you describe, I would not consider any of the AoPS products, including BA, to be the best fit for math.

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Okay, I would have done 60x60 + 60x5 + 60x5 + 5x5, but I wouldn't have done it as a mental math problem.

 

I guess I don't get it. I've got to carry with both problems. Once with 65x65 and twice with 64x66.

 

 

Doing this mentally, it's easier to start from the larger numbers and keep running totals rather do all the multiplications first and then all the additions (the way you'd do this if you were writing things down). Moving it to be a square reduces significantly the number of different operations you have to perform and the amount of things you have to keep track of during the operation.

 

60^2 is 36 100s.

5*60 is 3 100s, but there are 2, so that's 42 100s

5^2 is 25.

 

so 4225.

 

but you moved the two numbers 'in' by 1, so the result is off by 1^2.

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60^2 is 36 100s.

5*60 is 3 100s, but there are 2, so that's 42 100s

5^2 is 25.

 

so 4225.

 

but you moved the two numbers 'in' by 1, so the result is off by 1^2.

 

I'd also add that this is why squaring numbers that end in 5 is easy, because your 'middle term' will always be 2xsomethingx5 = 10xsomething.

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There is a "trick" for 65 x 65 that BA teaches, but it's based on a conceptual understanding of factoring. So if you get the problem of 64 x 66, then it really only takes 5 seconds to solve it. Good for parties...

 

DH is good at math, but never learned the above BA way, and it was fun for DD to teach it to him. Many of the concepts that BA teaches can be solved traditionally, but it's awfully fun to present it in a way that may spark a love for math.

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I don't know anything about beast academy, but I have used Singapore Math. Mental math strategies are sometimes only really applicable under certain circumstances, and you would not apply them under all circumstances. For example, adding when a number is close to 10 or 100, e.g. 98 + 487 = 100 + 487 - 2, you use when the number is close to 10 or 100. The Singapore Math also taught some strategies for multiplication, but again when the number was close to a ten or 100. Is Beast Academy recommending changing 66 x 64 to 65 x 65, or is it rather suggesting this strategy for when both factors are close to a ten? By the same difference?

 

It wasn't clear to me whether these problems were meant to be done mentally, but I got the feeling they were. The examples they gave in the lesson portion were all very simple and could easily be done mentally, which is what made me think that.

 

We don't use Singapore, but my daughter is easily able to implement the type of strategies you describe above. To answer your questions, some of the problems brought the number close to a 10, which made them simple to do mentally. Others were like the example I gave - changing 64 x 66 to 65 x 65. Multiplying 65 x 65 mentally was not something she had ever done before and I'm not sure whether or not it was expected? Still not sure.

 

Lisa

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:iagree:

 

I think doing 64x66 mentally without paper and pencil would be rather difficult otherwise... or for that matter, doing 65*65. The above method does not involve any "carrying".

 

The Beast Academy lesson really just left you with the 65 x 65 without any further guidance on how to work it mentally. So, perhaps their intention was to do the problem with pen and paper. The solution in the answer section simply showed "65 x 65 = 4225 - 1".

 

Lisa

Edited by LisaTheresa
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There is a "trick" for 65 x 65 that BA teaches, but it's based on a conceptual understanding of factoring. So if you get the problem of 64 x 66, then it really only takes 5 seconds to solve it. Good for parties...

 

DH is good at math, but never learned the above BA way, and it was fun for DD to teach it to him. Many of the concepts that BA teaches can be solved traditionally, but it's awfully fun to present it in a way that may spark a love for math.

 

Okay, I will have to go back through my book and find the "trick". Somehow I missed it. Thanks!

 

Lisa

 

ETA: Would it be to multiply 70 x 70 and then 60 x 60 and then take half of that and add it to 60? We were talking about that in the car. That seemed easier to me than trying to do 65 x 65, but still wasn't as quick and easy and getting out a pencil and paper and just multiplying 65 x 65 (at least for us!).

 

ETA (again): Okay, I give up. I thought I had it figured out, but realized that was wrong too. I still stand by this statement --> I think Beast Academy needs a Teacher's Manual for lame math teachers (Me!).

Edited by LisaTheresa
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Okay, I will have to go back through my book and find the "trick". Somehow I missed it. Thanks!

 

Lisa

 

ETA: Would it be to multiply 70 x 70 and then 60 x 60 and then take half of that and add it to 60? We were talking about that in the car. That seemed easier to me than trying to do 65 x 65, but still wasn't as quick and easy and getting out a pencil and paper and just multiplying 65 x 65 (at least for us!).

 

ETA (again): Okay, I give up. I thought I had it figured out, but realized that was wrong too. I still stand by this statement --> I think Beast Academy needs a Teacher's Manual for lame math teachers (Me!).

 

Sorry, I was away from home, but I have the textbook with me now. It's on pages 54 and 55 of 3B. I wouldn't have the child memorize the trick until he understands why it's done that way (eg. 6 x 6 +6 with an ending of 25). This particular trick is common knowledge, apparently, but not 64 x 66 using the middle number squared minus 1. I noticed that the common trick may not actually be understood completely by some people, so Beast took the effort of factoring with pictures, just to make sure that the kids are understanding this trick.

 

I would use manipulatives with a lower number like 15 x 15 or draw it out, like what they did on page 56 of the textbook. And I would also stack another set of manipulatives for 14x16 to see the shortage of 1 unit. Or if you don't have enough c-rods, base 10, then use an example such as 5x5 and 6x4.

 

Take a look at page 56 to see that it's really 60 x 60 + 600.

Edited by crazyforlatin
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So if the idea is to show that you are doing the same problem minus 1, I kind of get that. But, how is mental math any easier with a problem like 64 x 66 vs 65 x 65? How would you tackle this as a mental math problem?

 

Well, did you see the part a little earlier in the chapter where they show a quick shortcut for squaring any number that ends in 5? 65 x 65 = 60 x 70 + 25, so 4225. You multiply the number before the 5 by the next larger number and then tack a 25 on the end. Changing 64 x 66 into 65 x 65 -1 only makes sense if you've already done the other part first.

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Sorry, I was away from home, but I have the textbook with me now. It's on pages 54 and 55 of 3B. I wouldn't have the child memorize the trick until he understands why it's done that way (eg. 6 x 6 +6 with an ending of 25). This particular trick is common knowledge, apparently, but not 64 x 66 using the middle number squared minus 1. I noticed that the common trick may not actually be understood completely by some people, so Beast took the effort of factoring with pictures, just to make sure that the kids are understanding this trick.

 

I would use manipulatives with a lower number like 15 x 15 or draw it out, like what they did on page 56 of the textbook. And I would also stack another set of manipulatives for 14x16 to see the shortage of 1 unit. Or if you don't have enough c-rods, base 10, then use an example such as 5x5 and 6x4.

 

Take a look at page 56 to see that it's really 60 x 60 + 600.

 

Thanks, Rivka. I did go back earlier this afternoon and take another look at the textbook. We are using BA as a supplement and squeeze it in when we have time. There had been too much time between when we read the text and when we did the problems in the workbook and my short term memory isn't what it was.

 

I found the "trick". It took me a bit to understand what they were doing. Hopefully, my dd got something out of it.

 

Thanks again!

Lisa

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Well, did you see the part a little earlier in the chapter where they show a quick shortcut for squaring any number that ends in 5? 65 x 65 = 60 x 70 + 25, so 4225. You multiply the number before the 5 by the next larger number and then tack a 25 on the end. Changing 64 x 66 into 65 x 65 -1 only makes sense if you've already done the other part first.

 

Yes, I totally forgot about that part of the textbook. We follow BA's recommended sequence and wind up reading a large part of the text and then doing a chunk of the workbook. When we originally read that section, it didn't make a ton of sense to me, so I didn't focus on it too much. Now I get what they were doing.

 

I will be interested to see if this method actually sticks with her. I don't have any cuisinaire rods, but I may have to invest in some.

 

Lisa

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And for anyone that's interested, if you have a square that is 65 x 65, you take one row from each end and add it to one side of your square. Then you have 60 x 70 with a 5 x 5 square left over. So, it's 4200 + 25. Then, since your actual problem is 64 x 66, you have to subtract one, so your final answer is 4,224.

 

After today, I don't think I will ever forget this method. I'm just not so sure about dd.

 

Lisa

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Well, did you see the part a little earlier in the chapter where they show a quick shortcut for squaring any number that ends in 5? 65 x 65 = 60 x 70 + 25, so 4225. You multiply the number before the 5 by the next larger number and then tack a 25 on the end. Changing 64 x 66 into 65 x 65 -1 only makes sense if you've already done the other part first.

 

Yup, this is what I was remembering as I thought about it during the day. This ds could do easily.

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For the student you describe, I would not consider any of the AoPS products, including BA, to be the best fit for math.

 

Yeah, but for a statistical outlier 2E kid, I'm not sure there is a best fit. Something will always require tweaking. At least BA is very visual, even if it is designed for fast processors. What else is there that introduces advanced concepts for younger kids?

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Thanks redleaf now I see what they all talking about. :-)

 

It seems a neat trick to teach mathy people but not something for everyday use.

 

 

ahh...but it is, because the process is more generally applicable than just to numbers that differ by 2.

 

 

63 x 67 = (65-2) x (65+2)=65^2 - 2^2

 

seeing as you can move the numbers 'in' by whatever amount is convenient and still just have to handle squares means you can apply this to a much larger set of numbers. For any pair of numbers, then, you can square the average and then subtract 1/2 the distance squared. That's usually pretty quick.

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This was a chapter where DD stumbled too. Definitely a teachers guide would have come in handy, or a better explanation in the Guide. It was easy to miss the concept b/c the explanation was written "backwards" for comical expression...but just made it too complex!

 

I ended up using base then baseten 100 squares, 10 rods and units to demonstrate the concept (I.e. 15 x 15 = 10 x 20 +25 ) or (14x16=15x15-1).

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