# Mount Vernon Field Trip, T

MONTESSORI UPPER ELEMENTARY

MATH CURRICULUM ALIGNMENT

Based on the Maryland Voluntary State Curriculum

Montessori Mathematics

Grades 4 to 6

June 2006

Prince George’s County Public Schools

[pic]

PGIN 7690-3472

BOARD OF EDUCATION

OF

PRINCE GEORGE’S COUNTY, MARYLAND

Beatrice P. Tignor, Ed.D., Chair

Howard W Stone, Jr., Vice Chair

John R. Bailer, Member

Abby L. W. Crowley, Ed.D., Member

Charlene M. Dukes, Ed.D., Member

Robert O. Duncan, Member

Jose R. Morales, Member

Judy G. Mickens- Murray, Member

Dean Sirjue, Member

Leslie Hall, Student Board Member

John E. Deasy, Ph. D., Chief Executive Officer

Shelley Jallow, Chief Academic Officer

Patricia Miller, Director of Curriculum and Instruction

Gladys Whitehead, Ph.D., Coordinating Supervisor, Academic Programs

Pamela Shetley, Ph.D., Director of the FOCUS Office

ACKNOWLEDGEMENTS

We wish to acknowledge the following teachers for their hard work and dedication to the creation of this outstanding document:

John Feeley Susan Holmes

Laure Fleming Marion Lebensbaum

Janet Goodspeed Cynthia Peil

Gwendolyn C. Harris Kimberly Strayhorn

We also wish to thank all of the Curriculum Writing Production Center staff for their assistance.

Table of Contents

Introduction……………………………………………………………………………………………………………………………………………………………5

Montessori Math Materials List…………………………………………………………………………………………………………………………………….6

Voluntary State Curriculum Chart (VSC) ………………………………………………………………………………………………………………………..8

Quarterly Guidelines ……………………………………………………………………………………………………………………………………………….14

Montessori Alignment with VSC………………………………………………………………………………………………………………………………….15

Montessori Great Lessons ………………………………………………………………………………………………………………………………………..93

Montessori Sample Lessons ……………………………………………………………………………………………………………………………………104

Appendix ……………………………………………………………………………………………………………………………………………………………123

History of Math Command Cards

Rounding and Estimating Lessons

Calculator Lessons

Problem Solving Sequence

Problem Solving Activity Cards

Measurement Activity Cards

Money Lessons

Probability Activity Cards

Statistics Activity Cards

Introduction to Mathematics in the Montessori Elementary Classroom

The human mind is naturally mathematical. Mathematics is another language of communications. Mathematical inventions are a reflection of the culture. The child at the second plane of development (6-12 years old) is interested in exploring how and why mathematics was developed and its usefulness to humanity. To assist the child’s inner development, the elementary teacher offers experiences that lead towards mathematical abstraction. First, the Montessori Great Lessons are introduced. Next, concrete experiences are presented, using hands-on materials manipulative practice with individual concepts. Then the teacher gives math nomenclature as the first step leading to abstraction. Repetition through a variety of parallel, interrelated work is given to keep the child's interest. Also, interest is maintained through work on problem solving skills, as the child creates his/her own problems. After extensive work with the sensorial (manipulative) materials, the child naturally moves toward abstraction until only paper and pencil remain.

The child's work follows five steps:

1. The teacher presents "The Story of Numbers,” giving a vision of the whole and inviting children to begin

the adventure of mastering mathematics.

2. Children work individually with math materials, acquiring precision.

3. The teacher presents mathematical vocabulary.

4. Children are encouraged through questioning to begin to make generalizations.

5. Children begin to work on paper (abstract level).

The mathematics program at the 6-12 level is designed to awaken the exploratory imagination of the reasoning mind. The child, from his work in the Montessori preschool, already has as a strong base for understanding math concepts. Correlation between quantity and symbol, base-ten place value to thousands, an introduction to the four operations using whole numbers, and memorization of math facts have been introduced and practiced. The major domains of study in the elementary class are:

1. Algebra, patterns, and functions

2. Geometry

3. Measurement

4. Statistics

5. Probability

6. Number relationships and computation

7. Processes of mathematics

Description of Montessori Math Materials

The 100 Board has 100 tiles numbered from 1-100, designed to be laid out in rows of ten on a 10 x 10 grid.

The Bead Bars are colored bead bars representing different quantities, used to explore patterns of numbers.

The Chain Cabinet holds ten cubes 1-10, the squares of the cubes, the chains of the cubes and the chains of the squares. Colors are the same as for the Bead Bars.

The Strip Boards are used to learn addition and subtraction facts and to lead children to discover numerical patterns. They are printed with a grid of squares, and use strips of wood cut in sizes from 1-9, and 1-18 to represent quantities.

The Multiplication Board is a wooden board with ten holes vertically and ten holes horizontally totaling 100 holes. Red beads are used to create the product.

The Division Board has nine holes vertically and nine holes horizontally totaling 81 holes. Green beads are used to create the quotient. Skittles represent the divisor.

The Finger Charts are series of charts with the basic math facts printed on them. They lead students to discover numerical patterns.

The Golden Bead Material is a three dimensional representation of place value. It uses single beads to represent units, ten beads on a wire to represent tens, 10 ten bars wired together in a square to represent hundreds, and 10 hundred squares stacked and held together in a cube to represent thousands.

The Stamp Game uses color-keyed "stamps" to represent the decimal system. Green stamps represent units and thousands, blue stamps represent tens, and red stamps represent hundreds. Place value is written on each stamp.

The Dot Game uses the same color coding system on a board divided into 5 columns. Each column has 25 rows of 10 small squares. The dot game reinforces the concept of exchanging tens in addition.

The Small and Large Bead Frames are color-coded abacuses.

The Racks and Tubes are used for division. They contain 700 color-coded beads sorted in tubes with ten each. The beads are used to represent the dividend. Color-coded boards with indentation to hold the beads are used to hold the quotient. The divisor is created with color coded skittles.

The Wooden Hierarchical Material is a large three-dimensional representation of place value to one million. It uses the same color coding system.

The Checkerboard is a rectangular board with four horizontal rows, each row containing color-coded nine squares.

The Bank Game is a set of cards to 9,000,000 in the hierarchical colors.

The Golden Bead Frame is an abacus is made of golden beads.

The Peg Board is a board with thirty holes vertically and thirty holes horizontally. Colored pegs use the same color coding system- green for units, blue for tens, red for 100. Color-coded skittles are also used with the pegboard.

The Fraction Insets, Divided Square Material, and Equivalency Insets are used in the study of fraction equivalence and operations on fractions.

The Decimal Material is laid out in columns to represent place values less than one unit.

The Constructive Triangle Boxes are a set of boxes holding various triangles color-coded to encourage exploration of geometric forms created by combining triangles.

The Geometry Cabinet has six drawers containing wooden figures to represent plane-closed figures.

The Geometry Sticks consist of sticks of several sizes and colors that can be fixed to a cork board to aid in the study of geometric figures such as lines and angles.

The Geometric Solids consist of nine different wooden forms painted blue.

The Yellow Area Material uses flat yellow quadrilaterals to encourage exploration of the concept of area.

The Volume Material is a series of containers and cubes used to explore volume.

The Binomial, Trinomial and Arithmetic Cubes contain colored cubes and prisms used in the preschool to refine the visual sense and discrimination of form. In the elementary a series of exercises leads students to discovery of the algebraic formulas.

The Numerical Decanomial helps in the memorization of multiplication tables, and is a preparation for squares, cubes and square and cube roots .

VSC and Montessori Upper Elementary

Math Curriculum Alignment

Table of Contents

|Grade 4 |Grade 5 |Grade 6 |

|VSC |Algebra, Patterns, and Functions |VSC |Algebra, Patterns, and Functions |VSC |Algebra, Patterns, and Functions |

|1.0 | |1.0 | |1.0 | |

|1.A.1.a |Page 15 |1.A.1.a |Page 15 |1.A.1.a |Page 15 |

|1.A.1.b |15 |1.A.1.b |15 |1.A.1.b |15 |

|1.A.1.c |16 |1.A.1.c |16 |1.A.1.c |16 |

|1.A.1.d |16 |1.A.1.d |16 |1.A.1.d |16 |

|1.A.2.a |17 | | | | |

|1.A.2.b |17 | | | | |

|1.A.2.c |18 | | | | |

|1.B.1.a |18 |1.B.1.a |18 |1.B.1.a |18 |

|1.B.1.b |18 |1.B.1.b |18 |1.B.1.b |18 |

| | |1.B.1.c |19 |1.B.1.c |19 |

| | | | |1.B.1.d |19 |

|1.B.2.a |19 |1.B.2.a |19 |1.B.2.a |19 |

|1.B.2.b |21 |1.B.2.b |21 |1.B.2.b |21 |

| | | | |1.B.2.c |21 |

| | | | |1.B.2.d |21 |

| | | | |1.B.2.e |22 |

|1.C.1.a |23 |1.C.1.a |23 |1.C.1.a |23 |

|1.C.1.b |24 |1.C.1.b |24 |1.C.1.b |24 |

| | | | |1.C.1.c |26 |

| | | | |1.C.2.a |25 |

| | | | |1.C.2.b |25 |

|Grade 4 |Grade 5 |Grade 6 |

|VSC |Geometry |VSC |Geometry |VSC |Geometry |

|2.0 | |2.0 | |2.0 | |

|2.A.1.a |26 |2.A.1.a |26 |2.A.1.a |26 |

|2.A.1.b |26 |2.A.1.b |26 |2.A.1.b |26 |

|2.A.1.c |27 |2.A.1.c |27 |2.A.1.c |27 |

| | |2.A.2.a |28 |2.A.2.a |28 |

| | |2.A.2.b |28 |2.A.2.b |28 |

| | | | |2.A.2.c |28 |

| | | | |2.A.2.d |29 |

|2.B.1.a |29 |2.B.1.a |29 | | |

|2.B.1.b |30 |2.B.1.b |30 | | |

|2.B.2.a |31 |2.B.2.a |31 | | |

|2.C.1.a | |2.C.1.a |32 |2.C.1.a |32 |

| | | | |2.C.1.b |33 |

| | | | |2.C.1.c |33 |

|2.D.1.a |34 |2.D.1.a |34 |2.D.1.a |34 |

|2.E.1.a |35 |2.E.1.a |35 |2.E.1.a |35 |

|Grade 4 |Grade 5 |Grade 6 |

|VSC |Measurement |VSC |Measurement |VSC |Measurement |

|3.0 | |3.0 | |3.0 | |

|3.A.1.a |36 |3.A.1.a |36 | | |

|3.A.1.b |37 |3.A.1.b |37 | | |

|3.A.1.c |37 | | | | |

|3.B.1.a |38 |3.B.1.a |38 |3.B.1.a |38 |

|3.B.2 |39 |3.B.2.a |39 |3.B.2 |39 |

|3.C.1.a |40 |3.C.1.a |40 |3.C.1.a |40 |

|3.C.1.b |41 |3.C.1.b |41 |3.C.1.b |41 |

|3.C.1.c |41 |3.C.1.c |41 |3.C.1.c |41 |

| | |3.C.1.d |42 |3.C.1.d |42 |

| | | | |3.C.1.e |42 |

|3.C.2.a |42 |3.C.2.a |42 | | |

|3.C.2.b |43 |3.C.2.b |43 | | |

|3.C.2.c |44 | | | | |

|Grade 4 |Grade 5 |Grade 6 |

|VSC |Statistics |VSC |Statistics |VSC |Statistics |

|4.0 | |4.0 | |4.0 | |

|4.A.1.a |45 |4.A.1.a |45 |4.A.1.a |45 |

|4.A.1.b |45 |4.A.1.b |45 |4.A.1.b |45 |

| | |4.A.1.c |46 |4.A.1.c |46 |

| | |4.A.1.d |46 | | |

| | |4.A.1.e |47 | | |

| | |4.A.1.f |48 | | |

|4.B.1.a |48 |4.B.1.a |48 |4.B.1.a |48 |

|4.B.1.b |49 |4.B.1.b |49 |4.B.1.b |49 |

| | |4.B.1.c |50 |4.B.1.c |50 |

| | |4.B.1.d |50 | | |

| | |4.B.1.e |51 | | |

|4.B.2.a |52 |4.B.2.a |52 |4.B.2.a |52 |

|4.B.2.b |52 |4.B.2.b |52 | | |

| | | | | | |

| | | | | | |

|Grade 4 |Grade 5 |Grade 6 |

|VSC |Probability |VSC |Probability |VSC |Probability |

|5.0 | |5.0 | |5.0 | |

| | |5.A.1.a |53 | | |

|5.B.1.a |54 |5.B.1.a |54 |5.B.1.a |54 |

| | | | |5.B.1.b |54 |

| | | | |5.B.1.c |55 |

| | | | |5.C.1.a |56 |

| | | | |5.C.2 |56 |

| | | | |5.C.3 |56 |

| | | | |5.C.4 |56 |

|Grade 4 |Grade 5 |Grade 6 |

|VSC |Number Relationships and Computation |VSC |Number Relationships and Computation |VSC |Number Relationships and Computation |

|6.0 | |6.0 | |6.0 | |

|6.A.1.a |58 |6.A.1.a |58 |6.A.1.a |58 |

|6.A.1.b |59 |6.A.1.b |59 |6.A.1.b |59 |

|6.A.1.c |60 |6.A.1.c |60 |6.A.1.c |60 |

|6.A.1.d |61 |6.A.1.d |61 |6.A.1.d |61 |

| | |6.A.1.e |61 |6.A.1.e |61 |

|6.A.2.a |62 | | | | |

|6.A.2.b |62 | | | | |

|6.A.2.c |63 | | | | |

|6.A.2.d |63 | | | | |

|6.A.2.e |64 | | | | |

|6.A.2.f |64 | | | | |

|6.A.2.g |65 | | | | |

|6.A.2.h |66 | | | | |

|6.A.3.a |67 | | | | |

|6.A.3.b |67 | | | | |

|6.B.1.a |68 |6.B.1.a |68 |6.B.1.a |68 |

|6.B.1.b |69 |6.B.1.b |69 | | |

|6.B.1.c |69 |6.B.1.c |69 | | |

| | |6.B.1.d |70 | | |

|6.C.1.a |71 |6.C.1.a |71 |6.C.1.a |71 |

|6.C.1.b |72 |6.C.1.b |72 |6.C.1.b |72 |

|6.C.1.c |73 |6.C.1.c |73 |6.C.1.c |73 |

|6.C.1.d |74 |6.C.1.d |74 |6.C.1.d |74 |

|6.C.1.e |75 |6.C.1.e |75 |6.C.1.e |75 |

|6.C.1.f |76 |6.C.1.f |76 |6.C.1.f |76 |

|6.C.1.g |77 |6.C.1.g |77 | | |

| | |6.C.1.h |78 | | |

|6.C.2.a |79 |6.C.2.a |79 |6.C.2.a |79 |

|6.C.2.b |79 |6.C.2.b |79 | | |

| | |6.C.2.c |80 | | |

| | | | |6.C.3.a |80 |

| | | | |6.C.3.b |82 |

|Grade 4 |Grade 5 |Grade 6 |

|VSC |Processes of Mathematics |VSC |Processes of Mathematics |VSC |Processes of Mathematics |

|7.0 | |7.0 | |7.0 | |

|7.A.1.a |82 |7.A.1.a |82 |7.A.1.a |82 |

|7.A.1.b |82 |7.A.1.b |82 |7.A.1.b |82 |

|7.A.1.c |82 |7.A.1.c |82 |7.A.1.c |82 |

|7.A.1.d |83 |7.A.1.d |83 |7.A.1.d |83 |

|7.A.1.e |83 |7.A.1.e |83 |7.A.1.e |83 |

|7.A.1.f |83 |7.A.1.f |83 |7.A.1.f |83 |

|7.A.1.g |84 |7.A.1.g |84 |7.A.1.g |84 |

|7.A.1.h |84 |7.A.1.h |84 |7.A.1.h |84 |

|7.B.1.a |85 |7.B.1.a |85 |7.B.1.a |85 |

|7.B.1.b |85 |7.B.1.b |85 |7.B.1.b |85 |

|7.B.1.c |85 |7.B.1.c |85 |7.B.1.c |85 |

|7.B.1.d |86 |7.B.1.d |86 |7.B.1.d |86 |

|7.C.1.a |87 |7.C.1.a |87 |7.C.1.a |87 |

|7.C.1.b |87 |7.C.1.b |87 |7.C.1.b |87 |

|7.C.1.c |87 |7.C.1.c |87 |7.C.1.c |87 |

|7.C.1.d |88 |7.C.1.d |88 |7.C.1.d |88 |

|7.C.1.e |89 |7.C.1.e |89 |7.C.1.e |89 |

|7.C.1.f |89 |7.C.1.f |89 |7.C.1.f |89 |

|7.C.1.g |89 |7.C.1.g |89 |7.C.1.g |89 |

|7.C.1.h |89 |7.C.1.h |89 |7.C.1.h |89 |

|7.D.1.a |90 |7.D.1.a |90 |7.D.1.a |90 |

|7.D.1.b |90 |7.D.1.b |90 |7.D.1.b |90 |

|7.D.1.c |91 |7.D.1.c |91 |7.D.1.c |91 |

|7.D.1.d |92 |7.D.1.d |92 |7.D.1.d |92 |

Montessori Upper Elementary Mathematics

Quarterly Overview for 4th, 5th, and 6th Grade Students

|First Quarter |Second Quarter |Third Quarter |Fourth Quarter |

| | | | |

|Statistics (All Indicators) |Algebra: |Probability (All Indicators) |Accelerated Curriculum with Extended Assessment |

| |Equations and Inequalities | |Limits: |

|Algebra: | |Measurement: | |

|Patterns and Functions |Geometry: |Weight and Capacity (Standard and Metric) |Number Concepts: |

|Expressions / Order of Operations |Nomenclature (lines, angles, and polygons) |Applications of Formulas (Area, Perimeter, |Whole Number Operations |

|Coordinate Grids |Congruence and Similarity |Volume) |Fraction Operations |

| |Transformations |Composite Figures |Decimal Operations |

|Number Concepts: |Analyzing Quadrilaterals, Triangles, and Circles|Time |Integers Operations |

|Whole Number Place Value |Measurement: | |Ratios and Scale Models |

|Whole Number Operations |Length (Standard and Metric) |Number Concepts: | |

|Integers and Exponents (6th Grade) |Degrees of an Angle |Decimal Place Value |Geometry: |

|Divisibility |Numbers: |Decimal Operations |Analyze Circles |

|Factors and Multiples |Analyze Fractions |Equivalent Forms | |

| |Equivalent Forms |Money |Measurement: |

| |Fraction Operations (+,-,x) |Percent |Use Measurement Tools |

| | | |Apply Measurement Formulas (Surface Area, |

|Processes: All Indicators |Processes: All Indicators | |Volume) |

| | | | |

| | |Processes: All Indicators |Processes: All Indicators |

Montessori Math and the Maryland Voluntary State Curriculum

This sequence has been kept as closely aligned to the Montessori math curriculum as possible, but adjustments have been made based on the indicators tested quarterly in Prince George’s County Public Schools. If teachers systematically incorporate these indicators into their presentations and discussions, students will be comfortable with quarterly benchmark tests and the Maryland School Assessment (MSA). The Montessori Math Lessons are designed to be presented individually or in small groups. Each presentation isolates one learning objective. Where students begin in the sequence, and how quickly they progress, depends on the developmental needs of each individual student. The teacher observes and responds to each individual child’s learning needs. Grade level expectations are intended to be used only as an aid to planning. Teachers will use their Montessori curriculum albums as their primary guides, and each student will progress at his or her own unique and appropriate pace.

Montessori Upper Elementary Math Curriculum Alignment

VSC-Mathematics-Standard 1.0 Knowledge of Algebra, Patterns, and Functions: Students will algebraically represent, model, analyze, or solve mathematical or real-world problems involving patterns or functional relationships.

|Grade 4 |Grade 5 |Grade 6 |

|Topic 1.A. Patterns and Functions |Topic 1.A. Patterns and Functions |Topic 1.A. Patterns and Functions |

|1.A.1. Identify, describe, |Quarter |1.A.1. Identify, describe, |Quarter |1.A.1 Identify, describe, |Quarter |

|extend, and create numeric |1 |extend, and create numeric |1 |extend, and create numeric |1 |

|patterns and functions | |patterns and functions | |patterns and functions | |

|a. Represent and analyze |Montessori Lessons: |a. Interpret and write a rule |Montessori Materials: |a. Identify and describe |Montessori Lessons: |

|numeric patterns using skip |Short and Long Bead Chains, Bead|for a one-operation (+, -, x, ÷|Command Cards |sequences represented by a |Short and Long Bead Chains, |

|counting |Bars, Flash Cards, |with no remainders) function | |physical model or in a function|Bead Bars; Student Created |

|Assessment limit: Use patterns|Multiplication Charts |table | |table |Models |

|of 3, 4, 6, 7, 8, or 9 starting| | | | | |

|with any whole number (0 – 100)|Albanesi Cards: | | | | |

| |9.A -K |Assessment limit: Use whole | | | |

| |10.A –G |numbers or decimals with no |SFAW: 2-14; TE 1 | | |

| | |more than 2 decimal places (0 –|DIS: J-8; p. 15-16, 90 | |Glencoe: 7-6, 7-6a |

| |POW (06-07): Week 1 |1000) | | | |

| | | |MSA Finish Line: | | |

| |SFAW 2-9; TE 1 | |Pages 8-11 | | |

| |DIS: M-35; p. 69-70, 127 | | | | |

| |SFAW 2-10 / 2-11; TE 1 | | | | |

| |DIS: M-15; p. 29-30, 107 | | | | |

| |DIS: J-18; p. 35-36, 100 | | | | |

| | | | | | |

| |MSA Finish Line: | | | | |

| |Pages 8-11 | | | | |

|b. Create a one-operation |Montessori Lessons: |b. Create a one-operation |Montessori Lessons: |b. Interpret and write a rule |Montessori Lessons: |

|(+ or -) function table to |Word Problem Cards |(x, ÷ with no remainders) |Word Problem Cards |for one-operation (+, -, x, ÷ |Word Problem Cards |

|solve a real world problem | |function table to solve a real | |) function table | |

| | |world problem |SFAW: 3-15; TE 1 |Assessment limit: Use whole |GLencoe: 9-6a, 9-6 |

| | | | |numbers or decimals with no | |

| | | | |more than two decimal places (0| |

| | | | |– 10,000) | |

|Grade 4 |Grade 5 |Grade 6 |

|c. Complete a function table |Montessori Materials: |c. Complete a one-operation |Montessori Materials: |c. Complete a function table |Montessori Materials: |

|using a one operation (+, -, ×,|Command Cards |function table |Command Cards |with a given two-operation rule|Command Cards |

|÷ with no remainders) rule | |Assessment limit: Use whole | | | |

| |SFAW 3-13; TE 1 |numbers with +, -, x, ÷ (with |POW (06-07): Week 3 |Assessment limit: Use the |MSA Finish Line: |

|Assessment Limit: Use |DIS: J-13; p. 25-26, 95 |no remainders) or use decimals | |operations of (+, -, x), |Pages 8-11 |

|operational symbols (+, -, x) | |with no more than two decimal |SFAW: 2-14; TE 1 |numbers no more than 10 in the | |

|and whole numbers (0-200) | |places with +, - (0 – 200) |DIS: J-8; p. 15-16, 90 |rule, and whole numbers (0 - | |

| | | | |50) | |

|d. Describe the relationship |Montessori Materials: |d. Apply a given two operation |Montessori Materials: | | |

|that generates a one-operation |Command Cards |rule for a pattern |Command Cards | | |

|rule | | | | | |

| | |Assessment limit: Use two |POW (06-07): Week 4 | | |

| | |operations (+, -, x) and whole | | | |

| | |numbers (0 – 100) |SFAW: 3-15; TE 1 | | |

| | | | | | |

| | | |MSA Finish Line: | | |

| | | |Pages 12-15 | | |

|1.A. 2. Identify, describe, |Quarter | | | | |

|extend, analyze, and create a |1 | | | | |

|non-numeric growing or | | | | | |

|repeating pattern | | | | | |

|a. Generate a rule for the next|Montessori Materials: | | | | |

|level of the growing pattern |Command Cards; Student Created | | | | |

|Assessment limit: Use at least |Models | | | | |

|3 levels but no more than 5 | | | | | |

|levels |POW (06-07): Week 2 | | | | |

| | | | | | |

| |MSA Finish Line: | | | | |

| |Pages 12-15 | | | | |

| | | | | | |

| | | | | | |

| | | | | | |

| | | | | | |

|Grade 4 |Grade 5 |Grade 6 |

|b. Generate a rule for a |Montessori Materials: | | | | |

|repeating pattern |Command Cards; Student | | | | |

|Assessment limit: Use no more |Created Models | | | | |

|than 4 objects in the core of | | | | | |

|the pattern |MSA Finish Line: | | | | |

| |Pages 12-15 | | | | |

|c. Create a non-numeric growing|Montessori Materials: | | | | |

|or repeating pattern |Command Cards; Student Created | | | | |

| |Models | | | | |

|Grade 4 |Grade 5 |Grade 6 |

|Topic 1.B. Expressions, Equations, and Inequalities |Topic 1. B. Expressions, Equations, and Inequalities |Topic 1. B. Expressions, Equations, and Inequalities |

|1.B.1 Write and identify |Quarter |1.B.1 Write and identify |Quarter |1.B.1 Write and evaluate |Quarter |

|expressions |1 |expressions |1 |expressions |1 |

|a. Represent numeric quantities|Montessori Lessons: |a. Represent unknown quantities|Montessori Lessons: |a. Write an algebraic |Montessori Lessons: |

|using operational symbols (+, |Word Problem Cards; Math |with one unknown and one |Word Problem Cards; Math |expression to represent unknown|Word Problem Cards; Math |

|-, ×, ÷ with no remainders) |Vocabulary Card Set; Chart for |operation (+, -, ×, ÷ with no |Vocabulary Card Set; Chart for |quantities |Vocabulary Card Set; Chart for|

| |four basic operations; |remainders) |four basic operations; Problem | |four basic operations; Problem |

|Assessment limit: Use whole |Problem Solving Sequence | |Solving Sequence |Assessment limit: Use one |Solving Sequence |

|numbers (0 – 100) | |Assessment limit: Use whole | |unknown and one operation (+, |Glencoe: 1-1 / 1-6 |

| |SFAW 2-12; TE 1 |numbers (0 – 100) or money ($0 |SFAW: 2-12; TE 1 |-) with whole numbers, | |

| |DIS: J-18; p. 35-36, 100 |- $100) | |fractions with denominators as |PGCPS 6th CFPG: Supplemental |

| | | | |factors of 24, or decimals with|Lesson 1 |

| |SFAW 3-13 / 3-14; TE 1 | | |no more than two decimal places| |

| |DIS: J-20; p. 39-40, 102 | | |(0-200) | |

| |DIS: J-21; p. 41-42, 103 | | | | |

|b. Determine equivalent |Montessori Materials: |b. Determine the value of |Montessori Materials: |b. Evaluate an algebraic |Montessori Materials: |

|expressions |Math Fact Families; Bead Bars |algebraic expressions with one |Math Fact Families; Bead Bars |expression |Math Fact Families; Bead Bars |

| | |unknown and one operation | | | |

|Assessment limit: Use whole |SFAW 2-13; TE 1 | |POW (06-07): Week 22 |Assessment limit: Use one |Albanesi Cards: |

|numbers (0 – 100) |DIS: J-19; 37-38, 101 |Assessment limit: Use +, - with| |unknown and one operation (+, |72.A-G |

| | |whole numbers (0-1000) or ×, ÷ |SFAW: 2-13; TE 1 |-) with whole numbers (0 – | |

| |MSA Finish Line: |(with no remainders) with whole| |200), fractions with |Glencoe: 1-6, 1-7 |

| |Pages 16-19 |numbers (0-100) and the number |MSA Finish Line: |denominators as factors of 24 | |

| | |for the unknown is no more than|Pages 16-19 |(0 – 50), or decimals with no |MSA Finish Line: |

| | |9 | |more than two decimal places (0|Pages12-15 |

| | | | |– 50) | |

| | | | | | |

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| | | | | | |

VSC-Mathematics-Standard 1.0 Knowledge of Algebra, Patterns, and Functions: Students will algebraically represent, model, analyze, or solve mathematical or real-world problems involving patterns or functional relationships.

|Grade 4 |Grade 5 |Grade 6 |

| | |c. Use parenthesis to evaluate |Fourth Quarter |c. Evaluate numeric expressions|Montessori Lessons: |

| | |a numeric expression |Accelerated Curriculum |using the order of operations |Integers |

| | | |Montessori Materials: | | |

| | | |Command Cards |Assessment limit: Use no more |POW (06-07) Week 2 |

| | | |POW (06-07) Week 35 |than 4 operations | |

| | | |SFAW: 3-13 and 3-16; TE 1 |(+, -, x, ÷ with no remainders)|Glencoe: 1-5, 9-1 |

| | | |DIS: J-23; p. 45-46,105 |with or without 1 set of | |

| | | | |parentheses or a division bar |MSA Finish Line: |

| | | | |and whole numbers (0-100) |Pages 16-19 |

| | | | |d. Represent algebraic |Montessori Materials: |

| | | | |expressions using physical |Command Cards, Student Created |

| | | | |models, manipulatives, and |Models |

| | | | |drawings | |

|1.B.2. Identify, write, solve, |Quarter |1.B.2. Identify, write, solve, |Quarter |1.B.2. Identify, write, solve, |Quarter |

|and apply equations and |2 |and apply equations and |2 |and apply equations and |2 |

|inequalities | |inequalities | |inequalities | |

|Grade 4 |Grade 5 |Grade 6 |

|a. Represent relationships |Montessori Lessons: |a. Represent relationships by |Montessori Lessons: |a. Identify and write equations|Montessori Lessons: |

|using relational symbols |Word Problem Cards; Math |using the appropriate |Word Problem Cards; Math |and inequalities to represent |Word Problem Cards; Math |

|(>, , , half of ten). Three is less than half of ten. Label (3half of ten).

Repeat this exercise with 10 ten bars and detachable numeral cards for tens to establish that 5 tens equal one-half of 10 tens, or one hundred. Repeat this exercise with 10 hundred squares and detachable numeral cards for hundreds to establish that 5 hundreds equal half of 10 hundreds, or one thousand. Via the child's imagination and logical reasoning, extend this concept throughout the decimal system, so that she understands that 5 in any place represents one-half of the next larger place.

Direct Aim: Student will gain experience with concrete materials and corresponding symbols of the concept of half of ten.

Presentation Three - Rounding

Lay out 10 units, 10 tens, and 10 hundreds in their appropriate places on the mat. Ask the child if she remembers the process for exchanging in the decimal system. Have her verbalize that every time we find 10 of any order (units, tens, hundreds, etc.), we exchange - the search for ten. Allow the child to complete the exchanging process with the golden beads you placed on the mat.

Explain that today we will learn a technique called rounding. The purpose of rounding is to make it easier for us to work with large numbers mentally. The rules of rounding are much like the rules of exchanging, with some variations.

Ask the child to lay out 4,681 on the mat with the golden beads. Explain that we will now experience what rounding means by following some simple rules. The basic rule for rounding states:

When rounding to a specific place, examine the quantity in the previous place. If this quantity is one-half of ten or more, we exchange it for one more in the rounded place, returning any quantities smaller than that place to the bank. We call this rounding up.

If the quantity in the previous place is less than one-half of ten, we leave the value of the rounded place as it is, returning any quantities smaller than that place to the bank. We call this rounding down.

We will now round 4,681 to the tens place. What is the place before the tens? (units) What is one-half of ten? (5) Which values are one-half of ten or more? (5, 6, 7, 8, 9) Which values are less than one-half of ten? (4, 3, 2, 1) What is the value in the units place? (1) Is this five or more? (no) Round down!

Put the unit back in the bank. What is our number now? (4,680) We have just rounded it to the tens place. We can say (and write) that 4,681 is about 4,680. We rounded down!

Compose another number (352) to round to the hundreds place. Ask the child, What is the place before the hundreds place? (tens) Is the number in the tens place 5 or more? Yes! Round the hundreds place up to 400 and return the tens and units to the bank. We can now write and say that 352 rounded to the hundreds place is 400. This is called rounding up.

Front-end rounding is the name given to rounding the largest place of a numeral.

Ask the older children to compare and contrast the process of rounding with that of exchanging in the decimal system in their math journals, using correct nomenclature for each.

Extensions - Estimating:

When estimating length, height, perimeter, or area, standard units of English and metric linear and square measurement must be previously experienced and understood.

Digital and analog clocks and the basic units of time measurement of must be understood to estimate time.

Units for measuring weight and liquid must be explored and their relationships established in order for estimation in these areas to be meaningful.

Degrees Celsius and Fahrenheit and their correlation with temperature must be previously known before estimating temperature.

Fractions and their decimal equivalents are the key domains preceding the estimation of mixed numbers and money.

Direct Aim: Student will understand the process of rounding and how it compares to exchanging.

Presentation Four - Rounding Decimal Numbers

When decimal concepts and numerals are understood by the students, rounding work may begin. Use the decimal board or the decimal golden mat and the decimal numeral cards to create quantity and symbol representing decimals. Apply the same rules for whole numbers to round decimals to various places.

Calculators

Materials: Calculator diagram, white board and markers, calculators (enough for a small group)

Presentation:

Place calculator diagram on lesson mat. Calculators are wonderful tools. We can use them to quickly complete equations. When do you think it would be useful to use a calculator? On white board write addition equation: 5 + 2 = 7. Using the diagram, identify the keys used to compute the equation. Distribute calculators. Invite children to compute the equation. Continue lesson with equations for all previously introduced operations: +, , x, ˜

Direct Aim: To familiarize children with the calculator.

Extensions:

Represent decimals to hundredths on a calculator.

Use the calculator to do multiplication with more than two digits in the multiplier.

Use the calculator to do or verify division and interpret results.

Montessori Problem Solving Sequence

Reading Skills in Problem Solving

Story problems should be part of the daily reading program of the elementary child. Familiarity with mathematical nomenclature and thinking, modeled on a daily basis by the teacher and classmates, will allow each child to move naturally into practical application of mathematical knowledge. There are many children’s books available from the library, which enhance mathematical thinking skills while making reading about math a joy.

Prerequisite Skills

The domain of problem solving concerns the application of mathematical knowledge to real life situations.

There are five sub skills required for children to experience success in this area:

1. Language: understanding the language used in identifying the four basic operations (addition, subtraction, multiplication, and division);

2. Modeling: the ability to picture in concrete ways the interactions and relationships described in the situation;

3. Calculating: calculating skills, including use of calculators, for performing operations with whole numbers, fractions, and integers;

4. Symbolizing: understanding the symbols and meanings of number sentences (equations and inequalities expressing relationships between quantities using symbols);

5. Writing Skills: explaining the process of solving the problem.

Skill One: Language of the Four Basic Operations

Materials

Write the words listed below on cards with the symbol for the operation on the back. Have the children sort the words into columns with the operational symbol at the top of each column. Make a chart for control of error.

Addition/

Multiplication Key

Words |Subtraction

Key Words

|Multiplication

Key Words

|Division

Key Words

| |combine

in all

altogether

total

added

buy

add

collect

plus

bring

put together

gather

how many

how much

how often

increase

include

more

|difference

greater than

more than

less than

fewer than

older than

years ago

younger than

remaining

left

give away

sell

went away

take away

missing

left over

extra

enough

|same

each

bundle of

times

groups of

teams of

sets of

groups in all

groups

altogether

squared

cubed

in each herd

on each team

in each box

laps in each

crayons per

race

box

|sharing

dividing

taking apart

separating

equal groups

equal shares

equal parts

in each

how many groups

how many per

equal sections

shared equally

equal amounts

splitting apart

how many in

each get

laps per race

each group

divided

each group

how many

on each

teams

boxes

team | |

Skill Two: Modeling the Problem

In order for the child to manipulate the information in a story problem, she must first understand the situation. This can be facilitated in a variety of ways:

1. Choose children to act out the parts of the story. Encourage them to use objects in the environment to make the situation concrete.

2. Encourage the child to use the golden beads and other Montessori materials in the classroom in a variety of ways to illustrate the dynamics of the situation. For example, if the story describes sharing a pizza, use the fraction insets for the dividing process.

3. Model for the children how to draw the characters and dynamics of the situation in a variety of ways.

4. Create the quantities described in the situation with Montessori materials or other objects in the environment. Label these quantities, telling what they represent.

5. Create a label for the information you want to know.

6. Use the real children and the real things in the environment as the basis of the story problems. Make up one story every day for the whole class to solve, using the names and interests of the children in the classroom.

Skill Three: Calculating

Children should use a wide repertoire of calculating tools in the classroom environment to solve story problems. The Golden Bead material is applicable to all situations. The stamp game, bead frames, the checkerboard, charts of simple equations, and all math materials can and should be used in reference to real life situations, both in initial presentations and in daily practice. The use of calculators is allowed in many testing situations. Using the calculator is an opportunity for the child to focus on the concept of number sentences.

Skill Four: Symbolizing the Problem

Overview:

A number sentence is an expression, written in mathematical symbols, of relationships between numerical or spatial quantities. Number sentences can be equations or inequalities. The most frequently used symbols in number sentences are as follows:

+ add < is less than

- subtract > is greater than

x multiply ≅ is congruent to

÷ divide ≈ is approximately equal to

= is equal to % percent

≠ is not equal to

In a number sentence, quantities and symbols are combined to show relationships. Number sentences represent the procedure used to solve the problem. A number sentence is also the expression of the active process of solving problems to establish these relationships.

Four birds sat on a tree branch. Two flew away. How many are left?

The number sentence reflects the thinking process:

4 birds (at first) − 2 birds (flew away) = 2 (left)

Sequence is an important factor in writing number sentences. (Commutative and associative properties of addition and multiplication are relevant here.) Using a calculator is a good way to focus on creating number sentences. The number sentence is exactly the sequence of buttons you push on the calculator to find the correct answer to the equation. Conversely, any number sentence may be solved with a calculator by entering its symbols correctly in sequence.

Materials:

1. paper slips with symbols: >, ................

................

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