Entrance Slip: Market Scales Puzzle

     The values of the four weights must be 1, 3, 9 and 27 grams. These are the weight I chose because these numbers can be represented using base 3. That is

1=3^0

3=3^1

9=3^2

27=3^3.

Using an one span scale and following the similar procedure above, the five weights will be

2^0= 1

2^1= 2

2^2= 4

2^3= 8

2^4= 16

    In order to expand on this exercise, as a future teacher, I would assign each student to work in small groups to create an overview that illustrates the connection between measuring weights and number theory. Hopefully, this sparks an idea to create a game based on their research and use graphics to show how different pairings can produce a full number. After the students have finished these exercises, having a group chat to examine the concepts of number theory would be my next step. Using applications of mathematics in everyday life to connect challenging concepts is the goal here. This invites my students to ask any new questions or offer any fresh perspectives.

   This problem explores how weights can symbolize numbers using specific bases and binary systems with powers of two and three. Also, the merging of the topic number theory is also examined also. Students immediately see how any whole numbers can be described, which reflects the behaviour of binary numbers. As students go through the word problem, they will also learn about basic combinatorics concepts as they identify different ways to combine different elements to reach certain totals. In the end, it increases their understanding of the fundamental links between mathematics and quantitative concepts.

*EDIT*

Using powers of three: The weights' values should be that every number between 1 and 40 grams is represented.

So because we've learning about base theory, I decided to use the algebraic approach. Thinking how to represent each number between 1 and 40. Weights may be placed on both pans of a two-pan scale, not just one. It also suggests that a weight on the same pan as the herbs minus the overall weight, while a weight on the other pan increases it.

Step 1: To balance the scale, the weights are divided between the two pans and the herbs are put on one pan.

Step 2: The herbs' weight is efficiently reduced if a weight is placed on the same pan as them.

It is also important to note these principles usedL

The ternary (base-3) system, which uses the digits -1, 0, and +1, allows for each weight to either:

• Add (if placed on the opposite pan),
• Subtract (if placed on the same pan as the herbs),
• Not be used (if the weight is not on either pan).

For example: What happens if I need 7 g of herbs

Arrange the herbs in a single pan.
Put the three grams of weight (minus three grams) on the same pan
To scale for 7 grams, (7 - 3 = 4).

So, we test this out to weigh 40,

to add to 40=27+9+3+1

then using the principle of base theorem, I created a blueprint to my answer. 

so, 40=(3^3)+(3^2)+(3^1)+(3^0) then use the same theorem and mindset/approach for the base 2 theorem. 

(a picture exemplifying brainstorming using technology)