The game of numerical representations.
We are given a series of connected boxes and a quantity of dots. In order to represent the given quantity we must follow a simple fixed rule.
For our first game; 1←2, the rule says that whenever two dots are present in a box they must explode to produce one dot in the left adjacent box and all dots start out in the right most box. The quantities remaining in the boxes in the final state is the code for the given quantity of dots we started with.
Q1: What is the code for the number 2 in the 1←2 game? ______
What is the code for the number 3 in the 1←3 game? ______
What is the code for the number 5 in the 1←5 game? What is the code for the number
90? 5 = ______90 = ______
Q2: What is the value of a single dot when it appears in each of the first four boxes for each of the three games just mentioned?
1←2 System
10 =
1←3 system
10 =
1←5 system Code for 5 =
9(10)=18(5)
90 =
Q3: What is the code for 10 in the 2←3 game? What is the code for the quantity 17?
Code for 10 = ______Code for 17 = ______
Q4: Express the code for the quantity 206 dots in the 1←16 game. What is special about this game?
Code for 206 = ______
Q5: What do we equate with the operation of exploding dots? ______
Adding Codes: Consider the 1←10 rule as that is our familiar game to play.
Q1: Add the two codes 163 and 489 using boxes and dots. What is their sum?
Q2: What do you think of the addition from left to right or from right to left and leaving carry to the end?
Multiplying Codes: Stay with base 10.
Q1: Multiply the number 56243 by 7 using boxes and dots to arrive at the product.
Q2: Can you suggest a method of multiplying a three digit by a three digit number such as 354 x 672?
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Subtraction: Again stay in base 10.
Here we introduce a new character in this game. It is the hole or anti-dot. The anti-dot represents a negative quantity or the opposite of a dot. An anti-dot cancels one dot.
In the process of subtraction we are combining dots and anti-dots.
Q1: Subtract: 423-254.
Q2: Subtract: 19-45.
Q3: Subtract: 456-632.
Q4: Add: -12+3; Subtract: 3-12.
Does anyone else feel “combine” is a better term than subtraction?
Division: 1←10 Rule
Q1: Any suggestions on how we go about the division process in terms of the box and dots game?
Q2: Start with very simple problems: 3906÷3 and 20486÷2.
Q3: Moving forward one small step consider 402÷3.
Q4: See what you can do with 156÷12 and 214506 ÷102.
Q5: Just two more for the drill: 44793÷21 and 637824÷302.
Q6: Show the boxes and dots for the division problems: 23÷12, 5210÷4, 4857÷23, and 31533÷101.
Decimals: 1←10
Q1: Express 18 as a decimal by treating the fraction as a division problem using boxes and dots.
Q2: Show that 13 is a repeating decimal using boxes and dots.
Q3: Perform the following division problems: 19÷8 and 37÷6.
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Can we apply the game to negative numbers in general?
Addition
Subtraction
Multiplication
Division
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