Math and the Mind's Eye Components
The Math and the Mind's Eye collection is made up of fourteen units each of which contains 4-12 activities. All Units except XIII require manipulatives - details are available on the product catalog pages.
The Units
Each activity within the units contains a quick synopsis, suggested prerequisites, and a materials list. Step-by-step instructions are included along with extensive diagrams and a comments section that offers background information, suggested teaching strategies, and possible student responses. See Sample Materials for examples.
For additional details select View Activities or click the Unit Title for the Product Catalog page.
Unit I: Seeing Mathematical Relationships
Introduction to using manipulatives and models when representing mathematical relationships.
The Handshake Problem—The handshake problem is used to illustrate the use of visual thinking in mathematical problem solving. In Part I, an expression is obtained for the number of handshakes if everyone in the classroom shakes hands with one another. This expression is evaluated in Part II.
Cube Patterns—The beginning building in a sequence of cube patterns are constructed. Students are asked to use visual observations and mental images to describe other buildings in the sequence and determine the number of cubes needed to construct them.
Pattern Block Trains & Perimeters—‘Trains’ of pattern blocks exhibiting certain geometric patterns are constructed. Students are asked to describe other trains that exhibit the same patterns. These descriptions are then used as a basis for determining the perimeters of trains.
Diagrams & Sketches—Students are asked to create mental images of situations described in story problems. They are then asked to draw sketches or diagrams, based on their images that lead to solutions of the problems.
Unit II: Visualizing Number Concepts
Exploration of number concepts, from basic operations to greatest common divisors and least common multiples, using tiles and grid paper.
Basic Operations—Students use tile to express their perceptions of the four basic arithmetic operations. Models for these operations used in subsequent activities are introduced. (Available in Sample Materials section).
Odd & Even Numbers—A tile pattern is used to visually represent the concepts of even and odd numbers.
Factors & Primes—Counting numbers are represented by rectangular arrays of tile. The arrays are used to determine the factors of a number and introduce the concepts of prime and composite numbers.
Averaging—The average of two or more numbers is related to the process of evening off columns of cubes. Methods for arriving at the average of a set of numbers are developed.
Greatest Common Divisors—The greatest common divisor of two numbers is obtained by finding the largest square that tiles a rectangle that has those numbers as dimensions.
Least Common Multiples—The least common multiple of two numbers is obtained by finding the smallest square composed of rectangles that has those numbers as dimensions.
Unit III: Modeling Whole Numbers
Use of Base 5 and Base 10 number pieces to model whole numbers.
Prerequisite activities found in Unit II.
Grouping & Numeration—Base 5 number pieces are used to examine the role of grouping and place value in recording numbers. The results are extended to other bases.
Linear Measure & Dimension—Base 5 number pieces are used to introduce linear measure. The relationship between the dimensions and the area of rectangular regions is discussed.
Arithmetic and Number Pieces—Base 5 number pieces are used to perform arithmetic operations. Emphasis is placed on modeling arithmetical operations rather than developing paper-and-pencil processes.
Base 10 Numeration—Base 10 number pieces are used to examine the roles of grouping and place value in a base 10 numeration system.
Base 10 Addition & Subtraction—Base 10 number pieces are used to portray methods for adding and subtracting multi-digit numbers.
Number Piece Rectangles—Base 10 number pieces are used to find the area and dimensions of rectangles as a preliminary to developing models for multiplication and division.
Base 10 Multiplication—Base 10 number pieces and base 10 grid paper are used to portray methods of multiplying whole numbers.
Base 10 Division—Base 10 number pieces and base 10 grid paper are used to portray methods of dividing whole numbers.
Unit IV: Modeling Rationals
Egg cartons, fraction bars, and base 10 pieces are used to form understandings about fractions and decimals.
Prerequisite activities found in Unit III.
Egg Carton Fractions—Egg carton diagrams are used as visual models to introduce fractions and fraction equivalence.
Fractions on a Line—Line segments are divided into equal parts as a means of introducing the division model for fractions.
Fraction Bars—The Fraction Bar model for fractions is introduced and used to discuss fraction equality and inequality.
Addition & Subtraction with Fraction Bars—Fraction bars are used to illustrate processes for adding and subtracting fractions.
Multiplication & Division with Fraction Bars—Fraction bars are used to illustrate processes for multiplying and dividing fractions.
Introduction to Decimals—With the aid of base 10 number pieces, the concept of a decimal is introduced and decimal notation is discussed.
Decimal Addition & Subtraction—Base 10 number pieces are used to develop processes for adding and subtracting decimals.
Decimal Length & Area—The dimensions and areas of rectangles are found and the distinction between linear measure and area measure is discussed.
Decimal Multiplication & Division—Base 10 number piece rectangles are used to find the product and quotient of decimals.
Fraction Operations via Area: Addition/Subtraction—Fractions are represented by areas of rectangular regions, and fraction sums and differences found by finding the sums and differences of areas.
Fraction Operations via Area: Multiplication—Two fractions are multiplied by viewing them as the dimensions of a rectangle and their product as the rectangle’s area.
Fraction Operations via Area: Division—The quotient of two fractions is found by constructing a rectangle for which the area and one dimension are given.
Unit V: Looking at Geometry
Geoboards are used to explore basic concepts in geometry such as geometric figures, area, perimeter, and volume.
Geoboard Figures—The geoboard is used as a means of representing geometric figures and as a medium for geometric explorations.
Geoboard Areas—Regions are formed on a geoboard and their areas are determined using formula-free methods.
Areas of Silhouettes—The area of a region is found by determining the number of unit squares needed to cover it.
Geoboard Triangles—The size, shape, and area of a certain geoboard triangles are investigated.
Geoboard Squares—The lengths of geoboard segments are determined by viewing them as he sides of squares.
Pythagorean Theorem—The Pythagorean relationship is developed visually with the use of dot paper drawings.
Geoboard Perimeters—The perimeters of geoboard polygons are determined and the relationship between area and perimeter is explored.
An Introduction to Surface Area & Volume—Solids of a given volume are formed with cubes and their surface areas determined by constructing grid paper coverings.
Shape and Surface Area—The effect of shape on surface area is investigated.
Areas of Irregular Shapes—Basic area concepts are used to estimate the areas of irregularly shaped regions.
Unit VI: Modeling Integers
Bicolored counting pieces are used to introduce and explore signed numbers.
Prerequisite activities found in Units II and V.
Counting Piece Collections—Bicolored counting pieces are used to introduce signed numbers and provide a model for the integers.
Adding & Subtracting Signed Numbers—Counting pieces are used to find sums and differences of signed numbers.
Counting Piece Arrays—A rectangular array of counting pieces is formed. Rows and/or columns of the array are turned over and the effect on the net value of the array is noted. Edge pieces are introduced and the relationship between the net values of an array and the net values of its edges is investigated.
Multiplication & Division of Signed Numbers—Counting piece arrays, with edge pieces, are used to model multiplication and division of signed numbers.
Unit VII: Modeling Percentages and Ratios
Use of counting pieces as well as diagrams and sketches to explore percentages and ratios.
Prerequisite activities found in Units I, IV, and V.
Introduction to Percentages—Areas and lengths are used to introduce the meaning of percentage. Given the meaning of 100%, students develop their own procedures for carrying out computations involving percentages.
Fractions, Decimals & Percentages—A portion of a quantity can be described in terms of a fraction, a decimal, or a percentage. Ways of doing this, and of converting from one way to another are investigated.
Ratios—The concept of ratio is introduced as a way of comparing the number of black pieces to the number of red pieces in collections of counting pieces.
Percentages & Ratio Problems—Diagrams and sketches are used to solve story problems involving percentages and ratios.
Unit VIII: Visual Encounters with Chance
Examination of probability, data collection, and graphing through simulation games.
Prerequisite activities found in Unit IV.
Sampling, Confidence & Probability—Samples are drawn from Hidden Sack in order to predict likely vs. unlikely proportions. Students’ confidence in their predictions is examined. An area model for representing the results of a probability experiment is introduced. Comparisons between guesses, experimental probabilities and theoretical probabilities are made.
Identifying Like Traits by Sampling—This activity builds upon the experience of making decisions based on random samples begun in the Sampling, Confidence and Probability activity. In addition, histograms are used to represent and to compare data samples. Histograms provide another convenient visual representation of data.
Experimental & Theoretical Evidence—The distribution of sums for rolling two dice is investigated using both experimental and theoretical evidence. The context is set within a game in which players attempt to find an optimal strategy to win.
Checker-A Game—Results from a binomial experiment with equally likely outcomes (odd and even rolls on a regular die) are compared to theoretical probabilities determined by counting the possible sequences of 6 tosses of the die. This activity is a precursor for work on counting strategies, Pascal’s triangle and the binomial distribution.
Checker-B Game—Results from a binomial experiment with unequally likely outcomes (odd and even products of faces of two regular dice) are compared to theoretical probabilities. Comparisons are made between the Checker-A and Checker-B games to point out the differences between equally likely and unequally likely binomial experiments.
Cereal Boxes—Simulation by a probability experiment is a tool often used when a direct theoretical approach to a probability problem is inaccessible. The cereal box problem uses the “sample until” technique that frequently occurs in problems involving chance. Visual representation of data, such as median marks, line-plots and box-plots are introduced to get at the concepts of central tendency, range and variation.
Monty’s Dilemma—A probability simulation in which this game can be played many times very quickly proves to be a powerful mechanism for understanding what the best strategy is—to stick or to switch.
Unit IX: Picturing Algebra
Exploration of formulas, algebraic notation, equivalent expressions, and equations using toothpicks and counting pieces.
Prerequisite activities found in Units IV and VI.
Toothpick Squares: An Introduction to Formulas—Rows of squares are formed with toothpicks. The relationship between the number of squares in a row and the number of toothpicks needed to form them is investigated, leading to the introduction of algebraic notation and the use of formulas. (Available in Sample Materials section).
Tile Patterns, Part I—Tile patterns are used to generate equivalent expressions and formulate equations.
Tile Patterns, Part II—Algebraic expressions are represented as sequences of tile arrangements. Examining the properties of these arrangements leads to solving equations.
Counting Piece Patterns, Part I—The net values of arrangements in counting piece patterns are determined. Functional notation for net values is introduced.
Counting Piece Patterns, Part II—Counting piece patterns are used to introduce equations involving negative integers.
Counting Piece Patterns, Part III—Counting piece patterns are extended to include arrangements corresponding to non-positive, as well as positive, integers.
Counting Piece Patterns, Part IV—Counting piece patterns are used to introduce quadratic equations.
Unit X: Seeing Symmetry
Development of an awareness of symmetry, congruence, and angle using various manipulatives.
Paper Folding—Students predict and describe the results of several paper-folding and cutting problems. The accompanying discussion develops geometric language and an awareness of concepts such as congruence, angle, and symmetry.
Mirrors & Shapes—Students investigate possible shapes that can be seen as the mirror is moved about on various geometric figures.
Shapes & Symmetries—The students begin by drawing “frames” around shapes and exploring different ways of fitting them in. This is followed by identification of lines of reflection and centers of rotation. The students are encouraged to make generalizations, to clarify, to make conjectures, and to pose and solve problems.
Strip Patterns—Students examine strip patterns and classify them according to their symmetries.
Combining Shapes—To review and extend ideas about symmetry and develop problem-solving strategies, students explore ways of producing symmetrical figures by joining given shapes together.
Symmetries of Polygons—Students classify hexagons and other polygons according to their symmetries.
Polyominoes & Polyamonds—The students consider shapes made by joining together squares or equilateral triangles, and some tessellations based on these shapes. They classify the shapes by symmetry and extend symmetry concepts to tessellations.
Unit XI: Graphic Algebraic Relationships
Graphs and graphing calculators are used to describe algebraic relationships.
Introduction to Graphs, Part I—The values of the arrangements in extended sequences are graphed. The graphs are examined for information about these values.
Introduction to Graphs, Part II—Extended sequences of arrangements are augmented so their graphs become continuous.
Introduction to Graphs, Part III—Further investigations with continua of arrangements are explored.
Introduction to Graphing Calculators, Part I—Graphing calculators are introduced to provide an alternative to the “by-hand” method of plotting graphs, and as a way to represent continua of arrangements in more detail. Connections between Algebra Piece, graphing, and symbolic representations of patterns are developed.
Introduction to Graphing Calculators, Part II—Continued explorations with graphing calculators, and investigating systems of equations are present in this activity. Connections between the Algebra Piece, graphing, tabular, and symbolic representations of equations are reinforced.
Unit XII: Modeling Real and Complex Numbers
Introduction to real and complex numbers.
Prerequisite activities found in Units IV, XI. Unit V also helpful.
Heximals & Fractions—The relationship between heximals and fractions is investigated with the help of base six pieces.
Decimals & Fractions—The relationship between decimals and fractions is investigated.
Fraction Sums & Differences—Procedures are developed for finding the sums and differences of algebraic fractions, based on area properties of rectangles.
Fraction Products & Quotients—Procedures are developed for finding the products and quotients of algebraic fractions, based on area properties of rectangles.
Squares & Square Roots—Methods of constructing squares of integral area are introduced. Properties of squares and square roots, including the Pythagorean theorem, are developed.
Complex Numbers—Green and yellow bicolored counting pieces are used to introduce complex numbers and their arithmetical operations.
Unit XIII: Sketching Solutions to Algebraic Equations
Use of sketches to help solve algebraic equations.
Prerequisites found in Units IX, XI, and XII.
Sketching Solutions—Sketches are used to solve standard algebra problems.
Sketching Quadratics, Part I—Sketches are used to solve problems involving quadratic relationships.
Sketching Quadratics, Part II—Further ways of using sketches to solve quadratics are discussed.
Equations Involving Rational Expressions—Sketches are used to solve equations involving rational functions.
Irrational Roots—The irrationality of the square root of 3 is established. The method is extended to other roots.
Unit XIV: Looking at Linear Algebra
A visual geometric introduction to linear transformations, matrices, determinants, and systems of equations.
Transformations—The effects of transforming the basis vectors of a coordinate grid are depicted.
Coordinate & Vector Arithmetic—Coordinates of grids are introduced. Vector arithmetic is developed and used to obtain formulas for linear transformations.
Matrices & Determinants—The matrix and the determinant of a linear transformation are defined.
Products of Transformation—The product, or composition, of two transformations is defined and the matrix of the resulting transformation determined.
Scaling & Shearing—Scaling and shearing transformations are introduced. Later we shall see that all linear transformations are compositions of these two types of transformations.
Inverses—The inverse of a linear transformation is defined and methods of finding inverses are developed.
Systems of Equations—Methods of solving systems of two linear equations in two unknowns are developed.
Invariance—An invariant set of points is defined and examples given. Eigenvectors and eigenvalues are introduced.
Space Transformations—Concepts introduced in previous activities are extended to three-dimensional settings.
