# Color Tiles

Color Tiles are a versatile collection of 1-inch square tiles that come in four colors—red, green, yellow, and blue. They are pleasant to handle and easy to manipulate. Students can use the tiles to act out story problems involving all sorts of everyday objects. Learning to use small colored squares to represent such objects is a significant step in the process of learning to abstract.

Although Color Tiles are simple in concept, they can be used to develop a wide variety of mathematical ideas at many different levels of complexity. Young children who start using Color Tiles to make patterns may likely talk about numbers of different colored tiles. Some children may even spontaneously begin to count and compare numbers. The fact that the tiles are squares means that they fit naturally into a grid pattern, and when Color Tiles are used on top of a printed grid—for example, a number chart—the tiles can be used to discover many number patterns. As they record their patterns, students are also using their spatial skills and strategies to locate positions of particular tiles. When making patterns, students often provide the best inspiration for one another. Given sufficient time, some student will come up with an idea that excites the imagination of other students. It is preferable that new ideas arise in this way, because then students develop confidence in their own abilities to be creative. Though students need to explore patterns freely, some students may also appreciate challenges, such as being asked to make patterns with certain types of symmetry or patterns with certain characteristics, such as specific colors that represent different fractional parts. Logical thinking is always involved when students investigate Color Tile patterns, because, in order to recognize and continue a visual pattern, students must form conjectures, verify them, and then apply them.

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.

Color Tiles are wonderful tools for assessing students’ mathematical thinking. Watching students work with their Color Tiles gives you a sense of how they approach a mathematical problem. Their thinking can be “seen”, in that thinking is expressed through the way they construct, recognize, and continue spatial patterns. When a class breaks up into small working groups, you are able to circulate, listen, and raise questions, all the while focusing on how individuals are thinking. Here is a perfect opportunity for authentic assessment.

Having students describe their designs and share their strategies and thinking with the whole class gives you another opportunity for observational assessment. Furthermore, you may want to gather students’ recorded work or invite them to choose pieces to add to their math portfolios. ## Lessons

Early learners can use Color Tiles to gain experience with counting, estimating, comparing area measurement, and place value. Color Tiles are also useful to represent objects in word problems as students act out various real-world situations described by operations.

Here are two activities for getting to know Color Tiles and to begin using them to learn and understand the math that they represent.

 Activity Curriculum Strands Topics Half and HalfChildren predict whether or not the outlines of various shapes can be filled with an equal number of Color Tiles of two different colors. They use Color Tiles to check their predictions, then create addition sentences to describe their results. Number & Logic CountingComparingEstimation & AreaFractions Last SurvivorIn this game for two players, children take turns removing one or two Color Tiles from a group of 13 tiles in an effort to be the player who takes the last tile. NumberLogic CountingMental mathGame strategies

The intermediate grades can use Color Tiles to explore factors and multiples from a geometric perspective. Students can also use the tiles to apply fraction concepts to a problem-solving situation. Rather than relying solely on the mathematical symbols, students can use Color Tiles to test their thinking.

Color Tiles lend themselves handily to geometry lessons, facilitating logical reasoning, as well as developing spatial skills.

Here are two activities using Color Tiles in different contexts.

 Activity Curriculum Strands Topics Logic RiddlesStudents create riddles that provide clues about Color Tiles that they have hidden in a paper bag. Then they try to solve one another’s riddles. NumberLogic CountingFractionsDeductive reasoning Sides And AnglesStudents try to build as many arrangements of eight Color Tiles as they can. Then they determine which arrangements have the fewest and the greatest numbers of sides and angles. GeometryNumberPatterns/Functions CountingComparingPolygonsOrganizing dataInterpreting data

Color Tiles are a useful tool at these grade levels for helping students gain a deeper understanding of area and perimeter, sequencing, symmetry, and probability. For example, students can use the manipulatives to create and identify patterns seen in real life and use these patterns to make predictions.

Here are two activities using Color Tiles in different of contexts.

 Activity Curriculum Strands Topics Border TilesStudents use Color Tiles to model a series of squares and then figure out the number of border tiles and intereior tiles in each square. NumberPatterns & FunctionsGeometry Pattern recognitionSquare numbersPredicting Patterns and FunctionsStudents figure out how to determine the number of Color Tiles needed for designs in a sequence without actually creating the designs. NumberPatterns & FunctionsGeometry SequencesPredictingPatterns

### Skills Taught

• Area
• Chance
• Classifying
• Combinations
• Combined Probability
• Comparing
• Computation
• Congruence
• Counting
• Creating Patterns
• Deductive Reasoning
• Equivalence
• Estimation
• Experimental Probability
• Following Directions
• Game Strategies
• Geometry
• Graphing
• Growth Patterns
• Interpreting Data
• Line Symmetry
• Logic
• Looking for Patterns
• Making Patterns
• Making Predictions
• Measurement
• Mental Math
• Multiples
• Number
• Number Sequences
• Organizing &Analyzing Data
• Organizing &Interpreting Data
• Organizing Data
• Pattern Recognition
• Patterns/Functions
• Perimeter
• Permutations
• Polygons
• Predicting
• Probability/Statistics
• Properties of Numbers
• Proportions
• Ratios & Proportions
• Rectangles
• Rotational Symmetry
• Sampling
• Scale Drawing
• Sequence of Perfect Squares
• Sequences
• Shape Recognition
• Similarity
• Sorting
• Sorting and Classifying
• Spatial Reasoning
• Spatial Visualization
• Square Numbers
• Symmetry
• Theoretical Probability
• Transformational Geometry
• Transformations
• Using Patterns
• Using Patterns to Make Predictions
• Visual Sequences
• Volume
• Writing Algebraic Expressions

### Subjects Taught

• Decimals
• Division
• Fractions
• Geometry
• Measurement
• Multiplication
• Percents
• Probability & Statistics

### Essential Manipulatives

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### See the Research

Read a summary of the research in the white paper Why Teach Mathematics with Manipulatives?