Ki Teitzei- Weights & Measures, Balanced Scales

"You shall not have in your pouch a stone and a stone- a large one and a small one. You shall not have in your house a measure and a measure- a large one and a small one. A perfect and honest stone shall you have, a perfect and honest measure shall you have, so that your days shall be lengthened on the land that Hashem, your G-d, gives you. For an abomination of Hashem, your G-d, are all who do this, all who act fraudulently." ~Devarim 25;13-16

Rashi on these passages explains that these stones refer to weights and measures. He also clarifies that it's not saying that you're not allowed to use different size weights. Rather, it means that you may not use two weights that are different weights but look to be the same, which would enable you to trick someone else into thinking that you are using the heavier weight when you're really using the lighter one.

Weights & Measures- Balanced Scales:
Classic scales, ones that were used before analog and digital scales with internal weight mechanisms were developed, worked by balancing two sides with each other. With an item of weight on either side, if the right side dips lower, then the item(s) on the right side are heavier; if the left side dips lower, then the item(s) on the left side are heavier; if the sides are even with each other, then the items on the two sides weigh the same amount. 

The concept of balancing a scale is also one that is commonly used now when teaching pre-algebraic and algebraic concepts. In this format, the idea of balancing equivalent combinations of numbers and variables is compared to balancing weights on a scale. If you know that the two sides of an equation are balanced, then you can perform the same operation to both sides of the balance- similar to adding or subtracting the same amount of weights on both sides of a scale- in order to isolate a variable on one side while keeping the equation balanced so the other side tells you the value of the variable.

Some examples:

If you know that-

x + 3 = 15

the "=" tells us that x + 3 is the same as (balances with) 15. Imagine that the "x" is one weight with an unknown value and the "3" is another weight with a value of 3. The "15" is a single weight with a value of 15. For younger children who need to physically manipulate to help them work through the problem, it might be an unknown weight, 3 weights with a value of 1, and 15 weights with a value of one. This set-up means that if you take away 3 from both sides, the scale (so to speak) will remain balanced. This leaves us with just our unknown, "X" weight on one side and [15 - 3 =] 12 on the other side. So, now we know that the unknown weight has a value of 12.

A more complex example:

If you know that-

5 x Y = 20

again, the "=" tells us that the 5 x Y is the same as (balances with) 20. Here, we imagine that we have 5 weights which all have the same unknown value of "Y" on one side of the scale and a weight with value of 20 (or 20 weights with a value of 1) on the other side. This set-up means that if you divide both sides into 5 equal groups, you can match-up groups of equivalent values. When we divide the "5 x Y" side by 5, we will have the 5 weights separated into 5 groups of one weight in each. With the more simplistic set-up, we can divide the 20 weights into 5 groups, and we'll have 4 weights with a value of 1 in each group. This means that 1 weight "Y" is the same as 4 weights. So, the unknown weight has a value of 4.

Parsha Connection:
In this week's parsha, we are warned against not using two weights that have the same shape and size, but have different weights. If you think about labeled weights that we use nowadays, does that mean that we're not allowed to have more than one weight? I have to choose if I'm going to measure everything with a 1 lb weight or a 5 lb weight? Rashi explains that it means that I can have weights of different weight, but I can't have, for example, a 1 lb weight and a 5 lb weight that look to be the same shape and size. You need to have weights that are clearly distinguishable from one another so that when you are weighing out items of value, it will be clear that you are in no way cheating regarding the value of the items being weighed.

Everyday Connection:
Practicing with balancing scales gives children an opportunity to manipulate the concept of equivalence. Building an understanding of being able to manipulate both sides of a scale in the same way and still maintain equivalence is a critical skill for developing algebraic thinking.

Have you every tried playing on a see-saw? If you have two people who are closer in weight, then they will balance each other and can have fun bouncing each end up and down. Sometimes, if you have one heavier person and two lighter people, you can put the two lighter people together on one side and they will approximately balance the one heavier person. If the heavier person tried to balance with just one lighter person on the other side, the heavier person will be stuck down on the ground, while his friend is stuck up in the air- not heavy enough to weigh himself down against his friend.