Numbers up to Sixty on a Counting Board
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Prerequisite Students already know how to represent numbers up to twenty. Task 1 Students work in groups of two, three or four. The teacher gives to each group the task of representing one or two numbers within the range from one to sixy with the smallest possible number of counters. The instructions may be very general or quite specific. Examples 1. "Represent the numbers forty-seven and forty-nine with the smallest possible number of counters." 2. "Represent the number forty-seven with three counters in two different ways. Also represent the number forty-nine with three counters. Can you represent it in two different ways? Explain why not." Answers should be shared with the whole class. Students may use either white or red tokens, but they are not allowed to mix them on the same board. This activity should be carried out several times until most students feel very comfortable with the task. Example Answers 
The leftmost and middle images show the number 47 represented on the board with three counters. These images depict the two ways to represent 47 on the board. The rightmost image shows the number 49 represented on the board with three counters. The number 49 cannot be represented in two ways with three counters because one counter must be placed on 40, while the other two counters must be placed on 4 and 5. All other representations of 49 on the board require more tokens. Task 2 Find all ways of representing the numbers one through five with one, two, or three counters. Again you may use only one color of tokens during this task. Example Answers One Counter Answers 
One counter can be used to represent the numbers one through five by placing the counter on the desired number. Two Counter Answers 
Two counters placed on the 1 square represent the number 2. 
One counter placed on the 1 square and one counter placed on the 2 square represent the number 3. 
In the leftmost image, one counter placed on the 1 square and one counter placed on the 3 square represent the number 4. In the rightmost image, two counters placed on the 2 square also represent the number 4. 
In the leftmost image, one counter placed on the 1 square and one counter placed on the 4 square represent the number 5. In the rightmost image, one counter placed on the 2 square and one counter placed on the 3 square also represent the number 5. Three Counter Answers 
Three counters placed on the 1 square represent the number 3. 
Two counters placed on the 1 square and one counter placed on the 2 square represent the number 4. 
In the leftmost image, two counters placed on the 1 square and one counter placed on the 3 square represent the number 5. In the rightmost image, one counter placed on the 1 square and two counters placed on the 2 square also represent the number 5. Information for Teachers 1. Representation of the numbers six to nine with two counters.
6 = 5+1 = 4+2 = 3+3
7 = 5+2 = 4+3 8 = 5+3 = 4+4 9 = 5+4 
In the leftmost image, one counter on the 5 square and one counter on the 1 square represent the number 6. (5+1 = 6) In the center image, one counter on the 4 square and one counter on the 2 square represent the number 6. (4+2 = 6) In the rightmost image, two counters on the 3 square represent the number 6. (3+3 = 6) 
In the leftmost image, one counter on the 5 square and one counter on the 2 square represent the number 7. (5+2 = 7) In the rightmost image, one counter on the 4 square and one counter on the 3 square represent the number 7. (4+3 = 7) 
In the leftmost image, one counter on the 5 square and one counter on the 3 square represent the number 8. (5+3 = 8) In the rightmost image, two counters on the 4 square represent the number 8. (4+4 = 8) 
One counter on the 5 square and one counter on the 4 square represent the number 9. (5+4 = 9) 2. The smallest whole number that requires four tokens on this board is 66. 
One counter on the 50 square, one counter on the 10 square, one counter on the 5 square, and one counter on the 1 square is one example of the number 66 represented on the board. 3. Representation of the numbers 1 to 5 with two or three counters.
1 none
2 = 1+1 3 = 2+1 = 1+1+1 4 = 3+1 = 2+2 = 2+1+1 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1
Two counters on the 1 square represent the number 2. (1+1 = 2)
In the leftmost image, one counter on the 2 square and one counter on the 1 square represent the number 3. (2+1 = 3) In the rightmost image, three counters on the 1 square represent the number 3. (1+1+1 = 3)
In the leftmost image, one counter on the 3 square and one counter on the 1 square represent the number 4. (3+1 = 4) In the center image, two counters on the 2 square represent the number 4. (2+2 = 4) In the rightmost image, one counter on the 2 square and two counters on the 1 square represent the number 4. (2+1+1 = 4)
In the leftmost image, one counter on the 4 square and one counter on the 1 square represent the number 5. (4+1 = 5) In the second image from the left, one counter on the 3 square and one counter on the 2 square represent the number 5. (3+2 = 5) In the second image from the right, one counter on the 3 square and two counters on the 1 square represent the number 5. (3+1+1 = 5) In the rightmost image, two counters on the 2 square and one counter on the 1 square represent the number 5. (2+2+1 = 5) Number Board index |