reminder Flashcards
reminder
Squaring Numbers Ending in 0
1) Drop the 0 at the end of the number
2) Multiply the remaining number by itself
3) Put “00” on the end of the number
20:
2 * 2 +”00” –> 400
120:
12*12 + “00” –> 14 400
reminder
Squaring Numbers Ending in 5
1) Drop the 5 at the end of the number
2) Multiply the remaining number by a number 1 higher than itself OR
square the number and then add one more of the number
3) Put “25” on the end of the number.
15 –> 1 * 2 + “25” –> 225
85 –> 8 * 9 +”25* –> 7225
805 –> 80^2 + 80 + “25” –> 648 025
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Squaring Numbers 1 Greater Than Numbers With Known Squares
(N+1)^2^ = N^2^+2N+1 = N^2^+N+(N+1)
121^2 = 120^2 + 120 + 121 =14,641
351^2 = 350^2 + 350 + 351 =
= (34100+25)*100 + 701 =
= 122500 + 701 =
= 123,201
reminder
Squaring Numbers 1 Less Than Numbers With Known Squares
As in the previous example, (N-1)^2^ is N^2^-2N+1 or N^2^-N-(N-1).
So 349^2^ is 350^2^ (122,500) minus 350 minus 349 (or twice 350 minus one) 699 which taken from 122,500 gives 121,801 which is 349^2^. Note that it is often simpler to subtract twice N (700) and then add one for the final result.
reminder
Squaring Numbers 2 Greater Than Numbers With Known Squares
As in the previous two examples, (N+2)^2^ is N^2^+4N+4 or N^2^+4(N+1).
So 352^2^ is 350^2^ (122,500) plus four times 351 (1404) which added to 122,500 gives 123,904 which is 352^2^. Note that it is often simplest to double N twice (double 350 is 700 and double that is 1400) added to 122,500 give 123,900 and then add four for the final result of 123,904.
reminder
Squaring Numbers 2 Less Than Numbers With Known Squares
As in the previous example, (N-2)^2^ is N^2^-4N+4 or N^2^-4(N-1).
So 348^2^ is 350^2^ (122,500) minus four times 349 (easily calculated as the double double of 350 minus four: 1396) which taken from 122,500 gives 121,104 which is 348^2^. Note that it is often simplest to subtract double double of N (122,500 minus 1400 is 121,100) and then add four for the final result of 121,104.
reminder
Here are the steps to calculate squares of numbers from 25 to 50. They may seem tricky at first, but they’re easily done in your head after regular practice:
1) Figure out the difference between your number and 50, and call that difference N.
2) Subtract 2500 - (100 * N).
3) Add N^2^ the previous total. You should know N^2^ automatically from section 2.
The result will be the square of the number from 25 to 50!
Let’s try 44^2^. The difference between 44 and 50 is 6, so we mentally do 2500-600, giving us 1900. 6^2^, as we already know, is 36. Adding this to our previous total, we get 1,936!
As another example, farther away from 50, we’ll try to figure out 32^2^. The difference between 32 and 50 is 18, so we subtract 2500-1800, giving us 700. 18^2^, as you know, is 324. 700+324=1024, which is 32^2^!
reminder
Squaring Numbers from 50 to 75
1) Figure out the difference between your number and 50, and call that difference N.
2) Add 2500 + (100 * N).
3) Add N^2^ the previous total. Again, you should know N^2^ automatically from section 2.
The process is very similar, but note that we add in step 2, instead of subtracting. That’s the only difference.
Let’s use this process to figure out the answer for 57^2^. The difference between 57 and 50 is 7.
2500+700=3200
7^2^=49
3200+49=3249
Trying this with 72, we see the difference between 72 and 50 is 22.
2500+2200=4700
22^2^=484
4700+484=5184
reminder
Squaring Numbers from 75 to 100
With numbers from 25 to 75, we’ve focused on their difference from 50. As we’re getting close to 100, instead. Here’s the process to use for numbers from 75 to 100:
1) Figure out the difference between your number and 100, and call that difference N.
2) Subtract N from the number to be squared.
3) Multiply that number by 100 (just add two zeros at the end)
4) Add N^2^ to that number.
To help clarify things, let’s try to figure out 97^2^, using this process:
1) 100-97=3, so N=3
2) 97-3=94
3) 94*100=9400
4) 9400+(3^2^)=9400+9=9409
Trying this with a number farther away, like 81^2^ gives us:
1) 100-81=19, so N=19
2) 81-19=62
3) 62*100=6200
4) 6200+(19^2^)=6200+361=6561
reminder
Squaring Numbers from 100 to 125
1) Figure out the difference between your number and 100, and call that difference N.
2) Add N to the number to be squared.
3) Multiply that number by 100 (just add two zeros at the end)
4) Add N^2^ to that number.
Once again, step two simply changes from subtracting to adding.
Let’s try and figure 106^2^ using this method:
1) 106-100=6, so N=6
2) 106+6=112
3) 112*100=11,200
4) 11,200+(6^2^)=11,200+36=11,236
How about 124^2^?
1) 124-100=24, so N=24
2) 124+24=148
3) 148*100=14,800
4) 14,800+(24^2^)=14,800+576=15,376
Just to be complete, what about 125^2^?
1) 125-100=25, so N=25
2) 125+25=150
3) 150*100=15,000
4) 15,000+(25^2^)=15,000+625=15,625
Don’t forget, though, that there is a much easier way to do 125^2^! Remember?
1) Dropping the 5, we get 12
2) 12+1=13
3) 12*13=156
4) Tacking the 25 on the end, we get 15,625.
reminder
Another method for squaring two digit numbers
This is best shown through an example.
Let us say that we wanted to square 43.
The first step is separate it into a problem where we are multiplying two numbers as such. 43 * 43
The next step is to round down to the nearest multiple of ten on one of the numbers, then add up on the other one the same amount we rounded down. In this case we are rounding down to 40, subtracting 3, and adding 3 to the other side, giving us 40 * 46. We must also remember the value that we rounded by.
We now multiply 40 times 46 in our head. The simple way to do this is to drop the zero, then multiply from left to right. 4 * 4 = 16, plus a zero on the end equals 160, and 4 * 6 = 24, and now add the two for 184, and finally add the zero back for 1840.
We now take the number we rounded by earlier, square it, and add it to the total. In this case 3^2 = 9, so our answer is 1840 + 9, or 1849.
With practice this can be done very quickly. Lets go over one more to make sure the process is clear.
56 ^ 2 - Initial Problem
56 * 56 - Split the problem
60 * 52 rounding 4 - Round the numbers
3120 rounded 4 - Multiply the rounded numbers
3120 + 16 - Square the number we rounded by.
3136 - Add the two numbers for the result.
The only thing that might slow you down on this method is multiplying the rounded numbers. But with practice you can do it nearly instantaneously.
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Every square of any number ending with n has the same last digit as n^2^.
For example, 123456789^2^ and 98789^2^ must both end with 1 because 9^2^ is 81. Likewise, 1234567^2^ and 797^2^ must both end with 9 because 7^2^ is 49.
