The picture below is of a collection of blue chips that have fallen on the floor

Question

The picture below is of a collection of blue chips that have fallen on the floor. Determine how many chips there are expressed first as a Base 5 numeral then as a Base 8 numeral assuming that you will not first count them in the usual way (in Base 10) and then convert that quantity to each base. Instead, try to arrive at your answers by working withing each base (5 and 8). Include in your thread, responses for each prompt below i.) How may Base 5 chips are there? How many Base 8 chips are there? ii.) Describe your approach in detail. That is, if you were able to count the chips remaining entirely in each base, what was your approach? (Hint: Think about how you'd count a large number of items that may not be organized in a systematic way in Base 10. Do you group the objects to help you count them? You may find inspiration thinking about this and the relationship to the place values in each base. ) ii.) Discuss how you think this exercise relates to understanding place value, a key foundational learning objective in elementary school mathematics

Explanation / Answer

Solution:

Since childhood, we are taught to do our calculations in base 10. Base 10 uses numerals from 0 to 9. The place determines the value.Each place on the left is 10 times its value on the right.

When we count numbers in base n, we use numbers from 0 to n-1. For eg a number in base 2 can be represented in the form of 0's and 1's only. Similarly a number in base 5 can be represented using 5 numerals from 0 to 4 only. Tjhe place of the digit on the left is 5 times the place on the right.

The sequence of the numbers in base 5 are 0,1,2,3,4,10,11,12,13,14,20.....So the important point to note down is after 10 immediately comes after 4 in base 5.

Similarly the numbers in base 8 are in the order as 0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20 and so on

a) By Counting, the no of base 5 chips are 323 and the no of base 8 chips are 130. So the essence of the topic is adding the chips in different bases without counting in base 10 and then doing connversions accordingly. If the concept is clear, it is possible to directly do the calculations in base besides 10.

b) The approach is simple to use numbers from 0 to n-1 in doing calculations in base n. So while counting number s in base 5, the sequence will be 0,1,2,3,4,10,11,12,13,14,20.... the point to note is that numbers greater than or equal to 5 in base 10 will never appear in base 5 (i.e. 5,6,7,8 & 9 will never be used in base 5 digits).

I hope the soultion is clear!

c) The counting exercise relates to understanding that the place of the value on the left is 'n' times the place on the left. eg: 21 in base 5 means 2 fives and 1 one is analogous to 100 in base 10 means 1 hundred, 0 tens and 0 ones.

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