Multiply using duplication and mediation
8 x 169
I suspect you want something like this:
8 x 169
= 4 x 338
= 2 x 676
= 1 x 1352
= 1352
I halved the first factor and doubled the second factor until the first factor becomes 1
Most people can double a number and half a number in their heads.
You would obviously start halving the smaller number.
The problem arises when the "halved" number is odd.
I used to teach this method as a novelty and called it "Newfie" multiplication, (a non-politically correct reference to my fellow countrymen from Newfoundland)
Here is how you would handle a problem like
34 x 56
= 17 x 112
= 8 x 224 , should have been 8.5, but oh well , just drop the .5
= 4 x 448
= 2 x 896
= 1 x 1792
Now drop all the rows where the first factor ( the one you halved) is even, leaving
17 --- 112
1 --- 1792
add the numbers in the right column
112+1792 = 1904
so 34 x 56 = 1904
Using this method, one does not have to know the multiplication table, only being able to double a number and to take half of a number.
My challenge to the students was to explain why this worked.
To multiply using duplication and mediation, we need to utilize the property of multiplication: multiplying a number by 2 is the same as doubling it. Here's how we can approach this problem step by step:
Step 1: Write down the given numbers.
8 x 169
Step 2: Start with the first number and multiply by 2, duplicating it until it reaches a power of 2 higher than the second number.
8 x 2 = 16
16 x 2 = 32
32 x 2 = 64
64 x 2 = 128
128 x 2 = 256
Step 3: Take the second number (169) and write it as a sum of powers of 2 using mediation. Start with the largest power of 2 less than 169, then repeat the process with the remainder until you reach 1.
169 = 128 + 32 + 8 + 1
Step 4: Now, write down the duplicated numbers corresponding to the powers of 2 from step 2.
256, 128, 64, 32, 16
Step 5: Cross out any duplicated numbers that are not needed in the mediation step. In this case, we can cross out 256.
Step 6: Add the remaining duplicated numbers, corresponding to the powers of 2 that were not crossed out, using mediation.
128 + 32 + 8 + 16 = 184
Step 7: The final step is to add up all the results from step 6 to get the final answer.
184 + 1 = 185
Therefore, 8 multiplied by 169 using duplication and mediation is equal to 185.