As counters, Color Tiles are very important early number models. Eventually students will develop more abstract concepts of numbers and will not be dependent on manipulation of objects. Color Tiles can help students build such abstract structures. The tiles fall naturally into certain patterns, as shown below, and enable students to visualize the relationships as represented by the tiles.
Since there are large numbers of them, Color Tiles are useful for estimation and developing number sense. Students can take a handful, estimate how many, then separate the tiles into rows of 10 to identify how many “tens” and how many “ones” there are. The colors of the tiles also make them useful in developing the concept of place value. For example, students can play exchange games in which each color tile represents a place value—ones, tens, hundreds, and thousands. Exchange games can work for subtraction as well as addition, and can also refer to decimals, where tile colors would represent units, tenths, hundredths, and thousandths.
Color Tiles are very suitable for developing an understanding of the meaning of addition. The sum 2 + 3 can be modeled by taking two tiles of one color and three of another, and then counting them. Subtraction problems can also be modeled either traditionally–put up five tiles and take two away–or by taking five tiles of one color, then covering two with a different color so that it is obvious that three tiles of the original color are left. Either of these methods of modeling makes the connection between addition and subtraction apparent.
Color Tiles are also ideal for developing the concept of multiplication, both as grouping and as an array. To show 3 × 4, students can make three groups with four tiles in each group and then arrange them in a rectangular array of three rows of four tiles. The advantage of the array is that by turning it students can see that 3 × 4 = 4 × 3. The array model also leads naturally into the development of the formula for the area of the rectangle. In fact, Color Tiles are especially suitable for exploring all area and perimeter relations.
Color Tiles can be used to explore all the different ways that squares can be arranged, subject to certain constraints. One classic investigation is to find all tetraminoes, pentominoes, and hexominoes, that is, all ways to arrange either four, five, or six tiles respectively, so that one complete side of each tile touches at least one complete side of another tile. Color Tiles can be used to investigate how many different rectangular arrays a given number of tiles can have. This helps students to discover that for some numbers—prime numbers—the only possible rectangular arrays are one-tile wide. At an upper-grade level, the colors of tiles can represent prime numbers, and a set of tiles can be used to represent the prime factorization of a number. For example, if a red tile represents 2 and a green tile represents 3, the number 24 might be represented by three red tiles and one green tile, since 24 = 2 × 2 × 2 × 3. This representation of numbers in terms of factors can help students to understand procedures for finding greatest common divisors and least common multiples. Since Color Tiles all feel exactly the same, they can be used to provide hands-on experience with sampling. By using a collection of tiles in a bag, students can investigate how repeated sampling, with replacement, can be used to predict the contents of the bag. Since the tiles are square, they can also be used to represent entries in a bar graph drawn on 1-inch grid paper. For example, class opinion polls can be quickly conducted by having each student place a tile in the column on a graph that corresponds to his or her choice.
To stimulate algebraic thinking, number sentences can be introduced in which each number is covered with tiles. The challenge for students is to figure out what is under each tile. Students will learn that sometimes they can be sure of the number covered, as in 4 + __ = 6, while at other times they cannot, as in __ + __ = 6. This use of tiles lays the groundwork for introducing a variable.