Two-color counters are typically flat discs that are red on one side and yellow on the other. Despite their simple nature, two-color counters can be used to model a large number of math topics such as addition, subtraction, multiplication, division, place value, comparing numbers, integer operations. Some two-color counters are bean-shaped instead of rounded. Some two-color counters use red and white instead of red and yellow.
Virtual manipulatives, such as Brainingcamp's Two-Color Counters Manipulative, include flip, copy, snap and drag capabilities to make work fast and easy. The virtual two-color counters come with tens frames, grouping circles, and a part-part-whole mat. The two-color counters can be turned into integer chips so that students can also see "+" and "-" symbols on the chips. Dragging opposite signed integer chips on top of each another creates a zero-pair. The zero-pairs are dimmed so that students can visualize that these pairs sum to zero. Finally, Brainingcamp's two-color counters include group selection, dragging, and copying.
To add means to put together and find the total. To add 2 + 5, place 2 yellow counters and 5 red counters. There are 7 counters in total, so 2 + 5 = 7. Using a Part-Part-Whole grid, place 7 counters in the Whole. Copy the 7 counters and place some of the copies in the one Part and some of the copies in the other Part. Move counters between the two parts to discover all the pairs of numbers that sum to 7.
To subtract means to take apart and find the difference. Subtraction is the opposite of addition. To subtract 5 – 3, place 5 yellow counters then flip 3 of the counters red. There are 2 yellow counters remaining, so 5 – 3 = 2. Using a Part-Part-Whole grid, place 5 counters in the Whole section. Copy the 5 counters, move 3 of the copies to one Part section, and the remainder of the copies to the other Part section. There are 2 counters in the other Part section, so 5 – 3 = 2.
Using a Ten Frame and 12 counters, fill the Ten Frame with as many counters as possible. The model shows that 12 is made of 1 ten and 2 ones. Remove the counters and place 7 yellow counters and 8 red counters to model the addition 7 + 8. Fill the Ten frame to show that 7 + 8 equals 1 tens and 3 ones, or 13.
To compare the numbers 6 and 4, place 6 yellow counters in a row and 4 red counters in a row below the yellow counters. The yellow row is 2 counters greater than the red row, so there are 2 more yellow counters than red counters.
Multiplication can be thought of as the first number telling how many groups to create of the second number. To multiply 3 × 2, place 2 counters into a group. Then make two more copies to create 3 groups of 2. There are 6 counters in total, so 3 × 2 = 6. Multiplication can also be modeled using arrays. The multiplication 3 × 2 can be thought of as taking 2 rows of 3 columns. There are a total of 6 counters in the array, so 3 × 2 = 6.
Division can be thought of as separating the first number into a number of groups equal in size to the second number, or into a number of groups equal in count to the second number. Dividing 12 ÷ 3 can be modeled as separating 12 counters into 3 equal groups. There are 4 counters in each group, so 12 ÷ 3 = 4. Dividing 12 ÷ 3 can also be modeled as separating 12 counters into equal groups of 3 counters. There are 4 groups in total, also showing that 12 ÷ 3 = 4. Notice that division is the inverse of multiplication. The counters also show the multiplication 3 × 4 = 12 and 4 × 3 = 12. So 12 ÷ 3 is the same as finding 3 × ? = 12.
A positive and a negative counter form a "zero pair" because their sum is zero. To add integers, create as many zero pairs as possible and then sum the remaining counters. To sum -5 + 3, place 5 red negative counters and 3 yellow positive counters. Next combine red and yellow counters to create as many zero pairs as possible. There are 2 red counters remaining, so -5 + 3 = -2.
Subtraction can be thought of as removing or taking away. To subtract 5 – (-3), start by modeling the first number with 5 yellow positive counters. Next remove 3 red negative counters. Since there are not enough red negative counters, add 3 zero pairs and then remove 3 red negative counters. There are 8 yellow positive counters remaining, so 5 – (-3) = 8.
When multiplying, the first number tells how many groups to make of the second number. To multiply 3 × (-2) make 3 groups of 2 red negative counters. There are 6 red negative counters in total, so 3 × (-2) equals -6. Clear the counters. To multiply -3 × 2, remove 3 groups of 2 yellow positive counters. Since 3 × 2 = 6, remove 6 positive yellow counters. Since there are not enough positive counters, add 6 zero pairs and then remove 6 positive counters. There are 6 red negative counters remaining, so -3 × 2 equals -6.
Division can be thought of as separating the first number into a number of groups equal in size to the second number, or into a number of groups equal in count to the second number. To divide -12 ÷ (-3), separate 12 red negative counters into groups of 3 red negative counters. There are 4 groups in total, so -12 ÷ (-3) = 4. Clear the counters. To divide -12 ÷ 3, separate12 red negative counters into 3 groups of equal size. There are 4 red negative counters in each group, so -12 ÷ 3 = 4.
Ratios and Proportions
A ratio is a comparison of two quantities. Place 2 yellow counters and 3 red counters. The ratio of yellow to red counters can be written using words (2 to 3), using a colon (2:3), or as a fraction (2/3). A proportion is an equation stating that two ratios are equivalent. Make a copy of the 2 yellow counters and 3 red counters. The proportion 2/3 = 4/6 shows that the ratios represent equivalent comparisons.
A fraction is a number that represents a part of a whole or a set. Place 4 yellow counters. Flip the color of one of the counters red. The red counter represents the fraction 1/4 because there is 1 red counter out of 4 total counters in the set. Make a copy of all the counters. The red counters now represent 2/8 of the set. 1/4 and 2/8 are called "equivalent fractions" because they represent the same parts of a whole. Clear the workspace and then add 15 yellow counters. To find how many counters are in 1/5 of the set, divide the 15 counters into 5 equal groups. Each groups has 3 counters, so 1/5 of 15 is 3.