Jefferson County Public Schools

Compass Montessori School Wheat Ridge, Colorado

June 2000

MATHEMATICS CONTENT STANDARDS

MATHEMATICS CONTENT STANDARDS

INTRODUCTION

The Jefferson County School District content standards for mathematics are intended to help schools prepare students for the demands of work and citizenship in the twenty-first century. The standards on the following pages establish expectations for ALL students. It is believed that these standards represent an achievable level of competence for all children. They are not intended to be a ceiling; many students’ performances will significantly exceed the standards. In no way will these standards inhibit the progress of high-achieving students.

In compliance with the Jefferson County School District standards, the educators at Compass Montessori School recognize that the standards can be successfully met by all students, yet the paths individual students take to reach the standards will be different. The differences among unique students, individual teachers, and diverse communities should be perceived as strengths which can be harnessed to create an instructional program which suits the needs of all participants. The purpose of this document is to demonstrate the correspondence and alignment of the mathematics curriculum at Compass Montessori School with the Jefferson County School District standards for mathematics.

This document explains the alignment of Montessori materials and lessons to

the Jefferson County Mathematics Standards. Each standard is divided into several components which further specify the content, and these are reflected in one or more benchmarks. Montessori classes are multi-age and grade. The three levels in elementary consist of Pre-Kindergarten/Kindergarten, Lower elementary (grades 1-2-3), and Upper elementary (grades 4-5-6). The Montessori examples in the standards reflect how the curriculum satisfies the standards for the first two levels in the Montessori program. Specific terminology is referenced in the glossary at the end of this document.

MATHEMATICS CONTENT STANDARDS 1. Number Sense - Students develop number sense and use numbers and number relationships in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems. 2. Algebra - Students use algebraic methods to explore, model, and describe patterns and functions involving numbers, shapes, data, and graphs in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems. 3. Data Analysis – Students use date collection and analysis, statistics, and probability in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems. 4. Geometry - Students use geometric concepts, properties, patterns, and relationships in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems. 5. Measurement - Students use a variety of tools and techniques to measure and apply the results in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems. 6. Computation - Students link concepts and procedures as they develop, use, and explain computational techniques, including estimation, mental arithmetic, paper-and-pencil, calculators, and computers in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems.

Standard 1: Students develop number sense and use

numbers and number relationships in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems. Key components In order to meet this standard, the students will

1.1 construct and interpret number meaning through real-world experiences and the use of hands-on materials; 1.2 represent and use numbers in a variety of equivalent forms (for example, fractions, decimals, percents, exponents, scientific notation); 1.3 know the structure and properties of the real number system (for example, primes, factors, multiples, relationships among sets of numbers); and 1.4 use number sense, including estimation and mental arithmetic, to determine the reasonableness of solutions. Benchmarks - Grades K- 3 In grades K-3, students will A. demonstrate the meaning and structure of whole numbers, fractions, and decimals (including money), using models drawings, calculators, and computers in real-world situations;

Montessori Examples: K- Short chains, 100 board, numerical symbols, Cards and counters, fraction skittles, fraction

insets. 1-2 Sort and count golden beads.

Label and record short and long number chains. 3 Identify and state the value of coins with

money exercises. Explain the fractional parts of the fraction insets B. develop, test, and explain conjectures (hypotheses) about properties they apply to number sense;

Montessori Examples: K- addition strip board, finger board, making sets,

Nine tray. 1-2 predict what happens when number order is

reversed in addition and subtraction. After exploration, explain why the numbers can or cannot be reversed, using addition and subtraction strip boards.

3 Predict what happens when number order is reversed in multiplication and division. After exploration, explain why the numbers can or cannot be reversed. Multiplication layout with the colored bead bars - division board.

C. estimate, determine, and justify the reasonableness of solutions;

Montessori Examples: K- addition strip board, finger board, small bead stair, Cards and counters.

1-2 Estimate the number of beans in a jar, count the exact number, and discuss the reasonableness of the estimates. 3 Determine and explain strategies for estimation.

D. use ordering and grouping to demonstrate place value concepts;

Montessori Examples: K- Making tens, cards and counters, short chains,

Nine tray. 1-2 Sort objects into ones, tens, and hundreds.

Golden beads - stamp game - dot board 3 Sort objects into ones, tens, hundreds and thousands. Golden beads - stamp game - dot board E. use place value concepts to read and write whole numbers; and

Montessori Examples: K- sandpaper numerals with symbols, numerical rods with symbol cards.

1-2 read and write three digit numbers. Golden beads- composition of numbers with golden beads 3 Read and write six and seven digit numbers. Bead frame F. use numbers to count, to measure, to label, and to indicate location in problem solving situations.

Montessori Examples:

K- small bead stair, bead frame, red and blue rods addition strip board, golden bead materials. 1-2 Use a number line or chart to solve addition and subtraction problems; strip boards or Use ordinal numbers as one way to indicate location. 3 Use a number line or chart to solve multiplication and division problems. Multiplication Bingo chart

Standard 2: Students use algebraic methods to explore model, and describe patterns and functions involving numbers, shapes, data, and graphs in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems. Key Components In order to meet this standard, students will

2.1 identify, describe, analyze, extend, and create a wide variety of patterns in numbers, shapes, and data; 2.2 describe patterns using mathematical language; 2.3 solve problems and represent real-world situations mathematically; 2.4 compare and contrast different types of functions, and 2.5 describe the connections among representations of patterns and functions, including words, tables, graphs, and symbols.

Benchmarks --K- 3

In grades K-3, students will A reproduce, extend, create, and describe patterns and sequences using a variety of materials.

Montessori examples: K- Tower of Cubes, Broad Stair, Red Rods, Color Boxes

I, II, and III, Constructive Triangle Boxes. 1-2 Use manipulatives to form simple repeating patterns. Colored bead bars - Pythagoras board. Find patterns in the real world 3 Increase difficulty of patterns B. represent patterns and other relationships using tables, graphs, and open sentences and recognize connections among these representations;

Montessori examples: K- Small bead stair on graph paper, 45 bead layout

1-2 Develop and explain a pictograph showing the changes in an apple tree throughout the seasons. 3 Develop and explain a graph depicting the phases of the moon. C. recognize when a pattern exists, identify a rule that generates the pattern, and

use that information to solve a problem; and Montessori examples: K- Positive/negative snake game, red and blue rods,

table rods, die rolling, 100 board. 1-2 Determine the total number of eyes in a group of five children. Explain the reasoning. 3 Determine the next two numbers in the pattern 1,4,7,10... Explain the reasoning. D. observe and explain how a change in one quantity can produce a change in another.

Montessori examples: K- Graphing numbers of boys vs. girls, voting on

issues, show representation on graph paper. 1-2 Show how a change in the number of hands produces a change in the number of fingers.

Show how a change in the number of tricycles produces a change in the number of wheels.

3 Extend above examples to include higher numbers and various operations.

Standard 3: Students use data collection and analysis, statistics, and probability in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems (Data Analysis). Key Components

In order to meet this standard, students will

3.1 solve problems by systematically collecting, organizing, describing, and

analyzing data using surveys, tables, charts, and graphs; 3.2 make valid inferences, decisions, and arguments based on data analysis;

and 3.3 use counting techniques, experimental probability, or theoretical

probability, as appropriate, to represent and solve problems involving uncertainty. Benchmarks--Grades K- 3

In grades K-3, students will A. collect and display data using surveys, tables, charts, and graphs;

Montessori examples: K- Calendar work, weather charts.

1-2 Survey class to find total number of pets owned by students. Display data.

3 Measure how far each student throws a ball. Display data. B. interpret data using the concepts of largest, smallest, most often, and middle;

Montessori examples: K- Tower of Cubes, Knobbed Cylinders, graph height of

children. 1-2 Use the pet graph from Benchmark A to determine the largest and smallest number of pets or; make and interpret their own graphs. 3 Analyze data from Benchmark A to determine longest and shortest throw, most frequent distance thrown and the mid-point; or make their own graphs. C. make predictions based on data obtained from surveys;

Montessori examples: K- Casting dice as a means of determining probability.

1-2 Survey one class of students on favorite dinosaurs. Make predictions for another class based on this

information. 3 Survey one class to determine the students' favorite lunches. Determine which lunches should be brought most often by students. D. generate, analyze, and make predictions based on data obtained from chance devices

Montessori examples: K- Use previous data and add variables to change outcomes.

1-2 Draw colored objects out of a jar one at a time, record color and return object to the jar. Analyze data and make predictions of what the color of the next object drawn would be. 3 Roll two number cubes ten times and record data. Analyze data and predict which total will come up most often in the next thirty rolls. E. solve problems using various strategies for making combinations.

Montessori examples: K- Color box III, shading of like colors.

1-2 Use manipulatives or actual models to make combinations of white sneakers with red and/or blue socks. 3 Extend 1-2 example to include additional colors of sneakers and socks.

Standard 4: Students use geometric concepts, properties, patterns, and relationships in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems (Geometry). Key Components

In order to meet this standard, students will

4.1 connect various physical objects with their geometric representations; 4.2 connect mathematical concepts from across the standards with their geometric representations; 4.3 recognize, draw, describe, and analyze geometric shapes in one, two, and

three dimensions; 4.4 make, investigate, and test conjectures about geometric shapes and patterns, 4.5 solve problems and model real-world situations using geometric

concepts. Benchmarks--Grades K-3

In grades K-3, students will A. recognize, identify, describe, compare, and classify geometric shapes and patterns in the real world;

Montessori examples;

K- Constructive Triangle Boxes, Geometric Cabinet Geometric Solids, Metal Insets.

1-2 Take a walk, look for, and identify shapes and patterns. Discuss likenesses and differences. 3 Identify some of the basic shapes that occur repeatedly in more complex structures. Discuss likenesses and differences.

B. recognize the relationships (symmetry and congruence) among shapes and patterns using hands-on materials

Montessori examples: K- Metal Insets, Geometric Cabinet, use of paper cutouts,

Blue rectangle box, Constructive Triangles 1-2 Box of triangles - Cut out paper shapes 3 Make a pattern by rotating, flipping, and sliding a single shape. Blue rectangle box. C. draw and build physical models of geometric figures;

Montessori examples: K- Constructive Triangle Boxes, Metal Insets, Geometric

Cabinet, toothpick and sugar cube activity, Geo boards. 1-2 Sort shapes by categories. Rectangle Box, Blue rectangle box, Box of sticks. 3 Construct two and three-dimensional geometric figures. Box of sticks. D. relate geometric ideas to measurement and number sense; Montessori examples:

K- Constructive Triangles, represent height by using yarn, or other items in classroom. Measure on wall height from beginning of year to mid year.

1-2 Explore properties of shapes, such as measurement of sides and the number of sides. Geometric cabinet, constructive triangles. 3 Explore properties of perimeter and area by covering a space with blocks and examining the relationship between perimeter and area. E. investigate and predict the results of combining, subdividing, and changing shapes;

Montessori examples: K- Constructive Triangles, Blue rectangle box, paper

tangrams, art work with recycled paper shapes. 1-2 Combine two shapes to make a new shape. Rectangle box. Blue rectangle box.

3 Hold a long loop of yarn with classmates so that each hand serves as a vertex, and explore the effect of

changing the size of an angle, or increasing the number of sides while the perimeter is unchanged.

F. solve problems using geometric relationships and spatial reasoning. Montessori examples: K- Blue triangle box, Broad Stair, Red Rod Maze

1-2 Explore with puzzles, calendar type grids and tangrams. 3 Explore what shape results when a cylinder is cut down the side and spread flat.

Standard 5: Students uses a variety of tools and techniques to measure and apply the results in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems (Measurement). Key Components

In order to meet this standard, students will

5.1 determine the degree of accuracy needed in a given situation and select

appropriate tools and units of measure; 5.2 understand and apply the attributes of length, capacity, weight, mass,

time, temperature, perimeter, area, volume, and angle measurement in problem-solving situations;

5.3 make and use direct and indirect measurements to describe and compare real- world phenomena;

5.4 describe and use rates of change-(for example, temperature as it changes throughout the day, or velocity as the rate of change of distance over time) and other derived measures; and

5.5 understand the structure and use of metric and U.S. customary systems of measurement. Benchmarks--grades K- 3

In grades K-3, students will A. select and use appropriate standard and non-standard units of measurement in problem-solving situations; Montessori examples: K- Clock work, digital and analog to determine time and a ruler, standard and metric to determine distances of

measurement.. 1-2 Use a ruler (standard) to measure length.

3 Use metric and U.S. customary units to measure to a more precise degree of accuracy.

(for example, nearest 1/4 inch or millimeter). B. estimate and use measures of length, perimeter, capacity, volume, weight, time,

temperature, distance, and area; Montessori examples: K- Use of manipulative clock, use paper clips, pencils and

other objects to show length. Use measurements of cups and quarts to determine weight of liquids (heavier/lighter). Use of Baric Tablets for weight of solids. 1-2 Estimate and measure the various attributes of an object (for example length, weight, capacity,), using standard and non-standard measurement tools. 3 Expand above example to include U.S. customary and metric units. C. compare and order objects according to measurable attributes;

Montessori examples: K- Red Rods, Baric Tablets, Small bead stair, sequence cards

of times of day 1-3 Use golden beads to order and compare measurable attributes. D. describe and explain the concepts related to units of measurement (for example, noting the differences among the units of measure of length, area, and volume);

Montessori examples: K- Use of sweep hand on analog clock or watch

1-2 Use straws to measure the length of a desk and use squares to measure the area of the desk, Note the difference in units of measure of length and area. 3 Expand the 1-2 example to include standard units of measure. E. demonstrate a sense of measurement by using the approximate measure of familiar objects; and

Montessori examples: K- Use of Red rods to determine height, and length.

1-2 Use hands to measure the width of a desk, the length of the chalkboard, and the height of a table. 3 Measure the length of their own pace and use that information to find the length of the room.

F. demonstrate the relationships between units within a system of measurement. Montessori examples: K- Use of clock to explore seconds in a minute, minutes in

an hour, hours in a day. Use of calendar to explore days in a week, weeks in a month, months in a year. Time line of years of the child. 1-2 Explore the relationships between centimeters and meters, ounces and pounds, seconds and minutes. 3 Determine the number of centimeters in a meter, cups in a quart, ounces in a pound.

Standard 6: Students link concepts and procedures as they develop, use, and explain computational techniques, including estimation, mental arithmetic, paper-and-pencil, calculators, and computers in problem-solving situations and communicate with appropriate mathematical language the reasoning used in solving these problems (Computation). Key Components

In order to meet this standard, students will

6.1 model, explain, and use the four basic operations--addition, subtraction,

multiplication, and division--in problem-solving situations; 6.2 develop, use, and analyze algorithms; and 6.3 select and apply appropriate computational techniques

(for example, approximate or exact) to solve a variety of problems and determine whether the results are reasonable. Benchmarks--Grades K- 3 In grades K-3, students will A. demonstrate conceptual meanings for the four basic arithmetic operations of addition, subtraction, multiplication, and division. Montessori examples:

K- Negative Snake game, positive snake game, stamp game, cards and counters, small bead frame.

1-2 Show and explain operations with manipulatives or in picture/number form for addition and subtraction. Strip boards, golden beads, stamp game, snake games. 3 Extend K-2 example to include multiplication and division. Multiplication board, division board, bead frame, checkerboard, division tubes. B. demonstrate and explain how the four basic arithmetic operations are related to

one another; Montessori examples: K- Cards and counters, small bead stairs, Making tens,

addition strip board, subtraction strip board, stamp game with static and dynamic problem situations, 45-layout. 1-2 Use golden beads, addition strip board, and subtraction strip board to illustrate related addition and subtraction problems. 3 Use repeated addition to demonstrate and explain multiplication. Golden beads, stamp game. Model and explain inverse of multiplication and division. multiplication and division boards. C. use physical models to add and subtract commonly used fractions with common denominators and decimals including money;

Montessori examples: K- Fraction insets, fraction skittles (up to 1/10)

1-2 Explore fractions using Fraction Insets. 3 Add and subtract using fraction Insets. D. demonstrate understanding of and automatic recall of basic facts;

Montessori examples: K- Finger board, bead stair, addition strip board, problem ticket activity with tiles, small table rods. 1-2 Demonstrate 8+7 using addition strip board. Use addition and subtraction facts accurately in oral and written responses. Addition Bingo and subtraction charts. 3 Demonstrate 5x3 using colored bead bars. Use multiplication and division facts accurately in oral and written responses. E. construct, use, and explain procedures to estimate and compute with whole numbers;

Montessori examples: K- Small table rods, Addition Charts 3, 4, 5 and 6, subtraction

charts, number or object stamping, showing a 1-1 correlation. 1-2 Use addition strip board to show all the ways to make 7. 3 Use mental arithmetic strategies: 8+7=(8+8) -1 F. select and use appropriate methods from among mental arithmetic, estimation,

paper- and-pencil, calculator, and computer for computing with whole numbers in problem-solving situations; and

Montessori examples: K- Using verbal and visual cues explore what you need to

make more/less. 1-2 Given the talk of buying tires for 3 bikes, use mental arithmetic, estimation, paper-and-pencil, a calculator, or a computer to compute the total cost.

3 Given the talk of purchasing 5 pounds of apples for $.99 a pound, use mental arithmetic, estimation, paper-and- pencil, a calculator, or a computer to compute the total cost.

G. explain whether an estimate is acceptable in place of an exact answer in certain situations.

Montessori examples: K- Explore how many blocks it may take to construct a large

structure and ask the child to build from his estimation, is it what he/she envisioned? 1-2 Discuss real-life situations (lunch count, tickets on an airplane, money, popcorn) in which estimation is appropriate or not. 3 Extend K-2 examples to include situations that involve larger numbers.

The materials defined in this glossary are specific to the Montessori Method and are used to isolate individual mathematical skills. They are designed to be revisited with increased complexity as students master and build on their skills. Montessori teachers receive specific training and are certified to instruct students in the use of these materials and how they are applicable to each level of elementary education.

GLOSSARY addition strip board - a manipulative on which the children perform addition. Wooden strips represent the addends and are placed on a wooden grid chart to calculate the sum. baric tablets - three sets of wooden tablets (6 or 9 in each set depending on the manufacturer) in a box divided into three compartments. Each set is of a different wood and, therefore, differ in weight: heavy, medium and light. Tablets are of identical size. bead frame – manipulative representing the passage from the real quantity to representative hierarchical quantities. Ten is represented by one bead, as is 100 and 1000. The position of the bead indicates the value. blue rectangle box - box of manipulative blue triangles which can form triangles, rhombus, square, trapezoid, rectangles, parallelograms, quadrilaterals. box of sticks - a box of manipulatives consisting of various sizes of sticks with which a student may create triangles, quadrilaterals, and angles. A measuring right-angle is available and a curved half-circle. broad stair –ten solid wooden rectangular prisms. They are consistent in length at 20 cm. and vary in width and height by one cm. cards and counters – a box or container with a set of cards with symbols 1 to 10 printed or mounted on them. A collection of 55 counters or discs. To show visual representation of odd and even, the discs are placed in order 1 and two is represented by discs side by side. Three (and all odd numbers) are represented by two discs side by side and the third or odd disc being placed to the far left.

checkerboard – a manipulative checkerboard-style material with numerals placed along the bottom and the side to represent the multiplicand and the multiplier. The color of the squares in the checkerboard represent units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions and billions. Colored bead bars placed on the squares enable the students to multiply and discover products. color box I – a box containing six tablets, 2 red, 2 yellow and 2 blue. color box II – a box containing two of the following colors in tablets, orange, green, purple, brown, black, white, gray. color box III – a box containing 63 tablets which represent 7 shades of nine colors, red, blue, yellow, orange, green, purple, brown, pink and gray. In the lessons of color boxes I and II the student uses visual discrimination to match colors. In the lessons of color box III the student uses visual discrimination to shade the various colors in their darkest to lightest shading order.

colored bead bars - color-coded bars of fused beads in sets of one, twos, threes, fours, etc. through ten. constructive triangle boxes – a progressive series of angled shapes marked with solid black lines indicating the point of connection. Each angled manipulative shape is color-coded. In this series are triangle, hexagon, octagon, parallelogram, trapezoid, rhombus, square, and rectangle. division board - a manipulative for distributive division. It consists of a pitted wooden board that holds individual beads to represent the dividend and has a place for skittles to represent the divisor. division tubes - a manipulative designed for division, consisting of tubes of beads color-coded to represent place value. Pitted boards representing place value are used for the distribution of beads to discover quotients in division problems. dot board – a board with a matrix pattern of six columns and four rows. Columns are labeled from left to right: 10,000-1,000-100-10-1 in green, blue, red color-coding of the decimal system. The board may be used for addition and multiplication. finger board – a control chart showing the facts from 1 to 9. Child places index finger on the sums to be added and intersects fingers to get answer. fraction insets – a manipulative that has twelve circles, beginning with one whole and ending with a circle divided into twelfths. fraction skittles – a series of 4 segmented skittles, each one representing a whole, a

half, a third and a fourth. They are color coded on the inside segments, so that when color- coded they represent the proper fraction. They are the same height and diameter. geo-board – an 8” square wooden apparatus with projected nail heads on it. It is used with rubber bands to create geometric shapes. geometric cabinet - a cabinet with a series of drawers containing wooden geometric shapes. Included are triangles, curvilinear figures, and quadrilaterals. geometric solids – a set of solid wooden blue representations of a sphere, cylinder, cube, ellipsoid, ovoid, square-base pyramid, triangular shaped pyramid, rectangular prism, triangular-based prism and cone. golden beads - gold beads representing place value, consisting of unit beads, ten bars (ten beads fused together), hundred square (100 beads fused in a square), thousand cubes (1000 beads fused together). long chains - a manipulative of color-coded hanging chains of fused beads representing the cubes of numbers one through ten. making tens – a quantity of ten bars, colored bead bars to equal the number of tens and a counter. Alternating from each individual bead stair (top of one and bottom of the other) make tens. (1 plus 9 equals 10, 2 plus 8 equals 10 and so on) multiplication board - a manipulative that consists of a wooden board pitted to hold the product made of red beads. Cards representing the multiplicand and a red disc representing the multiplier are used to find products. nine tray - rectangular wooden tray containing a small rectangular piece of wood with indentations to hold 9 unit beads. A square box with low sides containing nine ten bars, two square boxes with higher sides, one containing nine hundred squares and a second box containing a thousand cube. A small wooden box containing symbols, unit symbols from 1 – 9 in green, tens symbols 10 – 90 in blue, hundreds symbols 100 – 900 in red and thousand symbol 1,000 in green. Four pieces of felt in corresponding colors green, blue, red and green. numerical symbols – a series of individual wooden squares with black manuscript symbols from one to 10 printed on each individual square. one hundred board - a wooden board with a natural wooden frame. The board is lined to form 100 squares. A wooden box with a divider in the middle to hold 100 tiles each representing numerals 1 – 100. Pythagoras board - a manipulative on which students perform multiplication. It is a

square wooden board with the multiplier and multiplicand on perpendicular axis and the products represented with one-inch plastic tiles. Rectangle box - manipulative consisting of two yellow equilateral triangles, equal in size and each having a black line along one side. Four isosceles right triangles, equal in size; two green ones, each with a black line on the hypotenuse, two yellow ones, each with a black line on one of the equal sides which inscribe the right angle. Seven scalene right triangles, equal in size: two gray triangles, each with a black line on the hypotenuse, two green triangles, each with a black line on the longer of the two sides which inscribe the right angle. Two yellow triangle, each with a black line on the shorter of the two sides which inscribe the right angle. A red right triangle with a black line on the hypotenuse. A small red scalene obtuse triangle with a black line on the side which is opposite the obtuse angle. Control chart of each quadrilateral formed with this material and labeled. red rods - ten solid wooden rods, usually red in color, differing in length; the shortest rod is 10 cms. long, each succeeding rod is 10 cms. longer than the preceding one, the longest is 100 cms. (1 meter). sandpaper numerals - a sandpaper or tactile representation of a number in symbol form from 1 to 10 short chains - a manipulative of color-coded hanging chains of fused beads representing the squares of numbers one through ten. snake games – a box containing color bead bars for the numerals one to nine. A box with a set of black and white bead bars. All bead bars through five are black. From six through nine the bead bars are black for the first five beads and white for successive beads; a box of ten bead bars an a box of golden bead bars. Addition is performed using these materials. The subtraction snake game contains the same items with the addition of a box of several sets of gray bead bars to denote the subtrahend. spindle boxes – two separate compartments (1-4 and 5-9) with numeral symbols one through nine printed on each individual slot. 45 wooden spindle rods for placement by quantity into the corresponding numbered slot. stamp game - a manipulative with which students can perform all four operations. It consists of one-inch wooden tiles that are coded by color and numerals representing place value up to and including thousands. subtraction strip board – a board labeled with numbers from one to eighteen at the

top. Tan and blue strip boards to represent subtrahend and the minuend. A booklet of printed equations, and a control chart for self-correcting are included. tangram – a Chinese puzzle consisting of a square cut into five triangles, a square, and a rhomboid, to be reassembled into different figures. tower of cubes –a set of ten solid wooden cubes (usually painted pink): cubes vary in size from one cubic cm. to ten cubic cms. (1 cubic cm., 2 cubic cms., ….to 9 cms., 10 cubic cms.): 1,000 of the 1 cubic cm. cubes equal one of the 10 cubic cm. cubes.