MUL 06

 In this week's tasks we take a sideways look at place value by considering divisibility and performing calculations in the context of these British imperial units of weight:
the hundredweight (cwt.), stone (st.) and pound (lb.). 

The smallest unit that we use in these tasks is pounds, so when we say a quantity is divisible by a given number, we mean there are no pounds left over when it is divided by that number.

Monday: Here we start by asking students to convert a given quantity into pounds, to familiarise them with the units that we are using. We then look at the divisibility of various quantities.

In Question 1,
3 cwt. is 24 st.,
24 st. + 1 st. is 350 lb.,
350 lb. + 10 lb. is 360 lb.,
360 lb. is divisible by 6.

In part 2a) we can argue that 5 cwt. 4 st. is equal to a whole number of stones, which is equal to 14 times that number of pounds and so is divisible by 7.

In part 2b) we can argue that 5 cwt. is divisible by 8 as it is 5 × 8 ×14 lb., and 4st. is divisible by 8 as it is 4 × 14 lb. = 4 × 2 × 7 lb. = 8 × 7 lb.

Tuesday: A single, simple task. Short and sweet....

9 cwt. 5 st. is a whole number of stones, which is 14 times that number of pounds and so it is divisible by 14. When 1 lb. is added to that amount, it is no longer divisible by 14. 

Wednesday: Part b) will be quite demanding for some students, but it offers nice parallels with the long division algorithm. It is more complex in the sense that the exchange from one unit to the next does not involve a constant factor of 10. At the same time, this 'irregularity' makes the issue of exchanging or carrying more explicit.

Part a):

5 cwt. is 40 st.; 40 st. + 4 st. is 44 st., which is 616 lb.; 616 lb. + 2 lb. is 618 lb.

618 lb. is divisible by 3: 618 lb. ÷ 3 = 206 lb.

 

Part b):

The neatest way to perform the division is to start with the cwt. and then convert any remainder into st., and so on. Some students might start with the lb. and ‘borrow’ some st. and then some cwt. if necessary, but it’s messy.

Starting with cwt., we get this:

5 cwt. ÷ 3 = 1 cwt. with 2 cwt. or 16 st. left over.
16 st. + 4 st. = 20 st.
20 st. ÷ 3 = 6 st. with 2 st. or 28 lb. left over.
28 lb. + 2 lb. = 30 lb.
30 lb. ÷ 3 = 10 lb.
So 5 cwt. 4 st. 2 lb. ÷ 3 = 1 cwt. 6 st. 10 lb. Check: 1 cwt. 6 st. 10 lb. = 206 lb., the result in part a).

Thursday: In Q1 we are given a quantity that is divisible 3. We add a multiple of 3 lb. to the quantity in various ways, so the result is still divisible by 3 each time.

A similar thing happens in Q2 where we add multiples of 4 lb. each time, though the effect may not be quite so obvious in part b) where we add 2 st.

Friday: In Wednesday's division calculation it pays to start with cwt., then st., then lb. Here it is easier to keep track by first multiplying the pounds, then the stones, then the hundredweights.

1.  3×10 lb. = 30 lb. = 2 st. 2 lb. Check-off 2 lb.    
    2 st. + 3×4 st. = 14 st. = 1 cwt. + 6 st. Check-off 6 st.
    1 cwt. + 3×5 cwt. = 16 cwt. Check.
Some students might multiply the cwt. first, then the st., then the lb. This works but it entails carrying more information in one’s head.
Q2 brings out some parallels with multiplying by 10 in our normal (base 10) number system, where each number is shifted one column to the left.
In part a) we multiply by 14, which simply converts the given number of pounds into that number of stones, and as it happens (!) we can exchange these for one hundredweight. So we just have to calculate 14 × 3 cwt. which is is 42 cwt.
In part b) we multiply by 8, which simply converts any given number of stones into that number of hundredweight. It doesn’t convert pounds to stones in the same simple way, but closer inspection might reveal that 8 × 7 lb. is 4 × 14 lb. which is 4st. So we then have to calculate 8 × (2+5+4) cwt. which is is 88 cwt.