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Math Lessons

Edited by: Jodie Unten

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By: Dr. Louise Bogart Revised 2009

Published by: The Christina Trudeau Education and Research Foundation 5862 Fountainhead St. SE Salem, OR 97306-9020 (503) 364-4362 http://www.cterf.org

All right reserved. Copyright Š2004 Louise Bogart No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocophying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. For information or permission contact: Dr. Louise Bogart louise@cterf.org www.cterf.org Revised: June 2009

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Math Rationale “In the practical application of the method it is helpful to know the sequence, or the various series, of exercises which must be presented to the child successively.” Maria Montessori The Montessori Method Montessori Math !

According to Dr. Bogart MSEB (Mathematical Sciences Education Board)

defines math as a science of pattern and order. Itʼs domain is numbers, chance, form, algorithms, and change (Math PPT, 7/27/10). !

Montessoriʼs 3-6 math curriculum consists or numeration to 10, base 10

place value, linear counting and operations. For each of the areas the first activity introduces the child to the quantity of the number, next is the introduction of the symbol, then the association of the quantity and symbol. Placement and sequencing of the materials is important, there is a purpose for the way the materials are arranged. Classrooms will have the same type of materials, but there may be a few variances in the sequence of the materials. For instance, during sensorial I learn to introduce the triangle box, small hexagon box, large hexagon box, rectangular box, last blue triangle. In Math I learned to present the shapes in a different way; blue triangle, rectangle box, triangle box, large hexagon box, small hexagon box. Every one has their own sequence and it is important to develop your own style and what works best for you.

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!

Sequence is important, it will help children develop from experiential to

abstract learning. It is important for children to explore, experience and experiment with the materials. After they have explored the materials they can begin to work with the materials. The first math activity is red and blue rods, this activity is an extension from a sensorial activity, red rods which teaches the child measurement. Red and blue rods help children to understand the quantities from one through ten. When the child has learned the quantities of the numbers from one through ten he/she will learn the names of the numerals (symbols) from one through ten (sandpaper numbers). Once the quantities and symbols have been mastered the association of the quantity and symbol will be presented together. The sequence of the activities will help children to develop their visual memory and physical internalization of the base ten. It will also develop order, concentration, coordination and independence. !

Spindle boxes helps to reinforce the association of quantities and

symbols, this activity also introduces children to the idea of zero. What is zero? Zero is nothing. Cards and counters is another way for the association of quantities and symbols. Introduction to the Short Bead Stair and search for ten are other numeration to 10 activities. !

Base 10 place value activities start with the introduction to golden bead

material, introduction to quantity; one, ten, hundred and thousand. Next is the introduction of the symbol, then the association of quantity and symbol. Crisis of nine teaches children that nine is the largest number that can be in written in any place, after counting to 9 you must change place. Other base 10 activities

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include Quantity Layout, Fetching Quantities, Symbol Layout and Fetching Symbols. !

Linear counting starts with teen counting 11-19 quantity, next is teens

symbols, the the association between the two. Tens quantity is followed by tens symbol, then the quantities and symbols are put together. Other activities include hundred board, hundred chain, and 1000 chain !

Operations use base 10 materials or golden beads to work with. The first

activity is static addition (no exchange), next is dynamic addition (with exchange). The next operation is static multiplication (without exchange), followed by dynamic multiplication (with exchange). Static subtraction (without exchange) is followed by dynamic subtraction (with exchange). The last operation is static division (without exchange), then dynamic division (with exchange). The sequence of addition followed by multiplication is done this way because multiplication is like addition, it始s just adding an amount more then one time. Subtraction follows multiplication because you start with a number then taking away an amount to find the difference. It is different thinking from addition or multiplication. Division is last, you start with an amount and divide it evenly (if possible) amongst X amount of people. Why Math? !

There is more than one way to solve a problem. Knowing that math has a

pattern and the relationship of the patterns will help children to grasp math and have an easier time of making sense of it all. A child needs to be able to explain his/her work and how she solved the problem or came to her answer.

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!

Materials and activities are used to meet the NCTM, HPCS and HCPS III

standards. Through the math materials children learn how to develop their number sense, numeration to 10, learn base 10 place value, linear counting, operations, graphing/data analysis, patterns, problem solving skills, measurement (non-standard, standard; English and metric), geometry, algebra and appropriate vocabulary. The math curriculum is developmentally appropriate for the 3-6 environment. !

Having manipulatives for children to work with will help them to see the

quantity of the number. The children will move from experiential to abstract learning through the use of manipulatives. !

Children also need to be introduced to “main stream/traditional schooling�

terminology of math. Through the materials and activities children learn and know how to do things, but sometimes it is called something else in Montessori. Teaching children all the vocabulary and terminology that has to deal with math is important. Having children be aware of the different vocabulary for example mass=weight. There are many ways to say one thing, it is important to expose the children to all the appropriate vocabularies. Where to use Math !

The prepared environment is very important to a childĘźs learning. Math is

every where, there are many activities that are in other area that prepare the child for math concepts. !

Math is integrated in all of the curriculum areas; language arts, music, art,

sensorial, children literature, cosmic and practical life. Language development is

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important. If there is more then one way to say something bring that to the childʼs attention. Giving the child a rich vocabulary will help them with language development in the future. In music there is a variety of songs that teach children about different math concepts. Art has a variety of different shapes that teach children about geometry. Children are able to use geometric solids to learn about shapes in sensorial. Many children literature books help to teach children about math concepts. Time concepts in cosmic will help children learn to learn how to tell time. Practical life uses measuring cups and other math tools to learn everyday life skills. !

Parents can help children learn about math concepts in their everyday

routines. Smith gives examples of things parents can let children do to practice matching “each buttonhole gets a button,” sets “animals that live on the land....animals that live in the water,” comparing “hot-cold” (2009). When to use Math !

!

Every child is different and he/she learns differently. It is important

as a teacher to learn the learning style of your children. Once you have learned your childrenʼs learning style you will be able to teach them to their fullest potential. In math there is more then one way to solve a problem. If a child is able to explain their reasoning behind their problem, it will help with the childʼs problem solving and reasoning skills. !

Math starts even before a child knows it. When they engage in sensorial

and practical life activities they are using math concepts; measurement, sequence, order, exactness (material placement) and calculation (water pouring).

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After children have worked with these materials they should be ready for the red and blue rod math activity. !

Math is all around us. Sequencing of materials is important. Knowledge

is build upon prior experiences. First a child learns about quantity from one to then, then symbol of the number from one to ten, last is the association of the quality and symbol together. Learning numeration to 10 is the base knowledge a child needs to further their math education. Next they will learn base ten place value, from there it is a pattern. Even though we may not always directly teach children about math they may be learning math in some way. Parents can help their child始s math education by attaching vocabulary to activities. For example, comparing hot and cold. Throw in appropriate vocabularies when teaching children things.

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References Bogart, L. (2010). Powerpoint. Montessori, M. (1967). The Discovery of the Child. New York, NY: Ballantine !

Book.

Montessori, M. (1966). The Secret of Childhood. New York, NY: Ballantine !

Book.

Smith, S. S. (2009). Early Childhood Mathematics. Boston, MA: Pearson !

Education, ! Inc.

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Table of Contents Early Childhood Materials Numeration to 10 Introduction to Base 10 Place Value Linear Counting Whole Number Operations with Golden Bead Rational Number Concept Development-Fractions

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NUMERATION TO 10 Quantity - Number Rods

Age: 3 1/2 - 6 Materials: Mat ten wooden rods measuring from one decimeter to one meter in length with alternate divisions of red and blue (each division is one decimeter) Presentation 1 (Lesson): 1. Invite the child(ren). 2. Prepare a long floor mat and bring rods to the mat beginning with the shortest rod. Place rods on the mat in mixed array. 3. Arrange rods in staircase fashion from shortest to longest placing rod #2 above rod #1. 4. Return rods to the shelf beginning with the longest rod. Continue so that all ten rods are in the stair. 5. Return mat. Presentation 2 (Lesson): 1. When child(ren) can correctly build the entire stair by themselves, do a three period lesson to teach the names of the quantities. Direct Aim (Objective): To teach the names for the quantities one through ten. Indirect Aims (Scaffolding): Preparation for higher math Preparation for science Preparation for language Preparation for art Develop visual memory, and physical internalization of the base ten Develop order, concentration, coordination, and independence Points of Interest (Motivation): Color pattern Handling the rods

Making stair Three period lesson

Control of Error (Self-Assessment): Visual - the stair, color divisions The smallest rod Vocabulary: one, two, three, four, five, six, seven, eight, nine, ten, red, blue, rods, long, longer, longest, short, shorter, shortest Extensions : Place numerals with rods in stair.

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Place numerals on rods which are in random array. Adaptations: Use smaller rods for a child who is not able to use the large rods on the floor mat. Standards: HCPS III -Standard 4: Measurement: FLUENCY WITH MEASUREMENT: Understand attributes, units, and systems of units in measurement; and develop and use techniques, tools, and formulas for measuring -Topic Measurement Attributes and Units -Benchmark MA.K.4.1 -Rubric Advanced

Use and describe an appropriate method to compare and order objects according to length, weight, capacity, area, and volume, with accuracy 

Proficient

Compare and order objects according to length, weight, capacity, area, and volume, with no significant errors 

Partially Proficient

Compare and order objects according to length, weight, capacity, area, and volume, with a few significant errors 

Novice

Use an inappropriate method to compare and order objects according to length, weight, capacity, area, and volume 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 2: Recognize and create patterns and become aware of relationships.

©2004 Dr. Louise Bogart. All rights reserved.

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NUMERATION TO 10

Age: 3 1/2 - 6

Symbol - Sandpaper Numerals

Materials: Table/desk Sandpaper numerals 0 to 9 Presentation (Lesson) : 1. Invite the child(ren). 2. Place numerals on table. 3. Choose two numerals that look different. Provide name and show how to trace using the index and middle finger of the appropriate hand. State name of numeral at end of tracing motion. 4. Using a three period lesson, teach the names of the two numerals. 5. Continue the process described in step 3 until all numerals can be named and traced correctly. 6. Return numerals to the shelf. Direct Aim (Objective): To teach the names for the numerals zero through nine and how to write them. Indirect Aims (Scaffolding): Preparation for higher math Preparation for science Preparation for language Preparation for art Develop visual memory, and physical internalization of the base ten Develop order, concentration, coordination, and independence Points of Interest (Motivation): Sandpaper Rough and smooth surfaces Numerals shapes and names Color on backing Three period lesson Control of Error (Self-Assessment): Visual - shape of numeral Kinesthetic - hand moving off the sandpaper Vocabulary: zero, one, two, three, four, five, six, seven, eight, nine, sandpaper, rough, smooth, color of background Extensions: Associate the sandpaper numerals to the numeral cards that will be used to label the number rods.

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Adaptations: Help the child to trace the numbers. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Number and Number Systems -Benchmark MA.K.1.1 -Rubric Advanced

Count and compare groups of objects up to 30 according to the number of objects in each group, with accuracy 

Proficient

Partially Proficient

Novice

Count and compare groups of objects up to 30 according to the number of objects in each group, with no significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with a few significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

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NUMERATION TO 10 Association of Quantities and Symbols Age: 4 - 6 Materials: Mat Ten number rods (1-10) Ten symbols (1-10)

Presentation 1 (Lesson) 1. Invite the child(ren) and have them bring the rods and numerals to the floor mat. Place rods and symbols in mixed array. 2. Teacher holds up a numeral. A child identifies it and finds the rod that matches. If a group activity, children take turns placing numerals with corresponding rods. 3. Return rods and numerals to shelf. 4. Return mat. Presentation 2 (Lesson) 1. Invite the child(ren) to bring the rods and numeral cards to the floor mat placing rods in mixed array. 2. Have the child(ren) arrange rods in sequence and then place numerals with corresponding rod. 3. Return rods and numerals to the shelf. 4. Return mat. Presentation 3 (Lesson) 1. With rods on mat, set the ten rod above and apart from others. 2. Teacher says, “Let’s see how many ways we can make 10 using only two rods.� Teacher places 9 rod below the ten and asks a child which rod is needed to make ten. The child who answers gets the rod and places it next to the nine rod. Continue until all combinations are done. 3. The same procedure can be used for combinations that make 9, 8, etc. Note: After the introduction to addition with Golden Beads/Base Ten Blocks, child(ren) can be shown how to illustrate an equation for addition using signs for plus and equal with numeral cards. Direct Aim (Objective): Lessons 1 & 2: To associate quantity and symbol. Lesson 3: To find combinations that make 10 Indirect Aims (Scaffolding): Preparation for higher math, science, language, art Develop visual memory Develop order, concentration, coordination, and independence Points of Interest (Motivation):

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Hunt to match Working with friends Colors Rods Control of Error (Self-Assessment): Teacher or other children Control chart Vocabulary : number, numeral, quantity, symbol, names 1-10, combinations Adaptations: Place the symbols in order and have the child find the matching quantity or vice versa. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Number and Number Systems -Benchmark MA.K.1.1 -Rubric Advanced

Count and compare groups of objects up to 30 according to the number of objects in each group, with accuracy 

Proficient

Partially Proficient

Novice

Count and compare groups of objects up to 30 according to the number of objects in each group, with no significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with a few significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

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NUMERATION TO 10 Age: 4 - 6

Spindle Boxes

Materials: Mat Two wooden boxes each with five divisions 0 - 4 and 5 - 9 Box of 45 wooden spindles

Presentation (Lesson): 1. Invite the child(ren). 2. On table or floor mat, place spindle boxes and box of spindles in front of you. 3. Teacher explains that the purpose of this activity is to place the correct number of spindles in each division. 4. Point to the numeral one and say, “one.” Using your dominant hand to count and the other to hold the spindles, count one spindle and place it into the hand. 5. Using two hands, slide the spindle(s) into the corresponding division. 6. Follow the procedure in steps four and five until divisions 1 - 9 are filled. 7. Return to the division marked “0” and indicate that there are no spindles left for this division because zero is the number property for the empty set. 8. Return spindles to box and boxes to shelf. 9. Return Mat. Direct Aim (Objective): To introduce the concept of making quantities (sets) from discrete units/objects. To introduce the concept of zero as the empty set. Indirect Aims (Scaffolding): Preparation for higher math Preparation for science Preparation for language Preparation for art Develop visual memory Develop order, concentration, coordination, and independence Points of Interest (Motivation): Spindles zero number sequence divisions for sets Control of Error (Self-Assessment): Just enough spindles (45) Vocabulary: zero, empty set, sets, divisions/compartments, spindles Variation: Use objects to follow class theme in place of spindles. 17


Adaptations: Use larger or smaller objects. Use larger or smaller symbols with containers that match the size of the objects. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Number and Number Systems -Benchmark MA.K.1.1 -Rubric Advanced

Count and compare groups of objects up to 30 according to the number of objects in each group, with accuracy 

Proficient

Partially Proficient

Novice

Count and compare groups of objects up to 30 according to the number of objects in each group, with no significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with a few significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

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NUMERATION TO 10 Cards and Counters

Age: 4 to 6 Materials: Mat Numeral cards labeled 1 - 10 55 counters Presentation (Lesson): 1. Invite the child(ren). 2. Place container in upper left corner of mat. 3. Place numeral cards in mixed array on mat. Indicate that the numerals will be placed in order along top of mat and that they must be spaced to accommodate objects to be counted. 4. Place the corresponding number of counters below each numeral card and demonstrate how to arrange them in pairs or sets of two. 5. Count the counters as you place them below the numerals as you proceed 1 - 10. 6. Return materials to container and return container to shelf. 7. Return mat. Direct Aim (Objective): To provide practice in association of symbol and quantity. To introduce the concept of odd and even. Indirect Aims (Scaffolding): Preparation for higher math Preparation for science Preparation for language Preparation for art Develop visual memory Develop order, concentration, coordination, and independence Points of Interest (Motivation): Counters Pattern made with counters Control of Error (Self-Assessment): Just enough counters (55) Pattern Vocabulary: pattern, counters, odd and even Extensions: Use objects to follow class theme in place of disks. Ask children to point out pattern and to explain concept of odd and even 19


Adaptations: Have children write or draw object that match the corresponding number. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Number and Number Systems -Benchmark MA.K.1.1 -Rubric Advanced

Count and compare groups of objects up to 30 according to the number of objects in each group, with accuracy 

Proficient

Partially Proficient

Novice

Count and compare groups of objects up to 30 according to the number of objects in each group, with no significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with a few significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

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NUMERATION TO 10 Mystery Game

Age: 4 - 6 Materials: Mat Folded numeral cards labeled 1 - 10 55 counters Presentation (Lesson): 1. Invite the child(ren). 2. Place container in upper left corner of mat. 3. Choose any numeral, unfold the card, then count out the correct number of counters. 4. Continue with the procedure described in step 3 until all cards are unfolded and each one has the correct number of counters with it. 5. Check work. 6. Return materials to shelf. 7. Return mat. Direct Aim (Objective): To test for mastery of numeration to 10. Indirect Aims (Scaffolding): Preparation for higher math Preparation for science Preparation for language Preparation for art Develop visual memory Develop order, concentration, coordination, and independence Points of Interest (Motivation): 1. Counters 2. Folded numerals 3. The mystery Control of Error (Self-Assessment): Just enough counters (55) Vocabulary: mystery, folded, names of numerals and quantities zero to ten Variation: Use object to follow class theme.

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Adaptations: Pair children one can read the number the other can get the quantity of the symbol. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Number and Number Systems -Benchmark MA.K.1.1 -Rubric Advanced

Count and compare groups of objects up to 30 according to the number of objects in each group, with accuracy 

Proficient

Partially Proficient

Novice

Count and compare groups of objects up to 30 according to the number of objects in each group, with no significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with a few significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

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NUMERATION TO 10

Introduction to the Short Bead Stair Age: 4 - 6 Materials: Mat Bead bars of lengths for quantities 1 - 10 (each quantity a different color) Presentation (Lesson): 1. Invite the child(ren). 2. Place container in upper left corner of mat. 3. Place bead bars in mixed array on the mat. 4. Select the red “1” and place it at the left side of the mat. 5. Select the green “2” and place it below the red bar. 6. Continue on through the ten bar. 7. Using the three period lesson, teach the names of all the bars. 8. Replace the beads and return container to shelf. 9. Return mat. Direct Aim (Objective): To introduce the colored bead stair Indirect Aims (Scaffolding): Preparation for further study of math Develop visual memory Develop order, concentration, coordination, and independence Points of Interest (Motivation): Colors Stair formation Control of Error (Self-Assessment): Stair formation Vocabulary: colors, names of quantities Extensions: Match each bead bar to its corresponding symbol. Adaptations: Develop larger materials for easier handling. Use cuisenaire rods to develop a stair.

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Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Number and Number Systems -Benchmark MA.K.1.1 -Rubric Advanced

Count and compare groups of objects up to 30 according to the number of objects in each group, with accuracy 

Proficient

Partially Proficient

Novice

Count and compare groups of objects up to 30 according to the number of objects in each group, with no significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with a few significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

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NUMERATION TO 10 Search For Ten

Age: 4 - 6 Materials: Mat Bead bars of lengths for quantities 1 - 9 (each quantity a different color) Five tens Set of black and white place savers

Presentation (Lesson): 1. Invite the child(ren). 2. Place container in upper left corner of mat. 3. Place bead bars in a stair on the mat. 4. Place the black and white beads in a stair on the mat. 5. Place the tens on the mat. 6. Arrange the colored bead bars in a snake configuration across the bottom of the mat mixing them up. 7. Start counting the colored beads until you get to ten. If there are any uncounted beads on the bead bar that is part of the ten, use a black and white place saver in place of the uncounted beads. Then place a ten in the snake and remove the colored beads. 8. Continue the procedure described in step 7 until the colored beads have all been counted. 9. Replace beads on shelf. 10. Return mat. Direct Aim (Objective): To practice numeration to ten. Indirect Aims (Scaffolding): Preparation for further study of math Develop visual memory Develop order, concentration, coordination, and independence Points of Interest (Motivation): Colors Stair Formation Changing colored beads to gold. Control of Error (Self-Assessment): Correct number for exchange Teacher or other children Vocabulary: colors, counting 1 - 10, place savers, change, exchange

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Extensions: Use more bars and have exchange not come out even. Adaptations: Use cuisenaire rods. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent whole numbers up to 30 in a variety of ways, using different forms of representation 

Represent whole numbers up to 30 in a variety of ways, with no significant errors 

Represent whole numbers up to 30 in one way, with no significant errors 

Represent whole numbers up to 30 in one way, with significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

26


BASE TEN PLACE VALUE INTRODUCTION TO GOLDEN BEAD MATERIAL Age: 4 to 7

Introduction to Quantity

Materials: 1 one 1 ten 1 hundred 1 thousand Mat Presentation (Lesson): 1. Invite child(ren). 2. Carry introduction tray for quantity to mat prepared on table or floor and place it on the left side of the mat. (If a child has not experienced the “golden beads” allow him the chance to just hold and feel the beads before you begin). 3. As child(ren) already know quantities 1 - 10, use third period of 3 period lesson and show the “one” and ask “What is this?” Wait for response. Show the “ten” and ask “What is this?” Wait for response. 4. Using three period lesson, teach vocabulary for 100 and 1000. 5. Return golden beads to tray. 6. Return tray to shelf. 7. Return mat. Direct Aim (Objective): To introduce the child(ren) to the language for base ten place value quantities Indirect Aims (Scaffolding): Preparation for higher math Preparation for science Preparation for language Preparation for art Internalize place value kinesthetically Independence Develop visual memory Concentration Points of Interest (Motivation): Golden beads Size and shape of beads Control of Error (Self Assessment): Teacher Vocabulary: One, ten, hundred, thousand, place value, base ten, golden beads Extension: Montessori Wooden Bead Materials: Before using the wooden material in an exercise, the teacher should bring out a hundred bead square and a hundred wooden square for the child to compare. The same is done with the bead cube and wooden cube. Teacher can explain the reason for using wood is that beads are very expensive so we cannot afford a lot of them.

27


Adaptations: Use larger materials or cuisenaire rods. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent whole numbers up to 30 in a variety of ways, using different forms of representation 

Represent whole numbers up to 30 in a variety of ways, with no significant errors 

Represent whole numbers up to 30 in one way, with no significant errors 

Represent whole numbers up to 30 in one way, with significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

28


BASE TEN PLACE VALUE INTRODUCTION TO GOLDEN BEAD MATERIAL Introduction to Symbol

Age: 4 to 7 Materials: Large numeral cards for 1 (green), 10 (blue), 100 (red), 1000 (green) Mat Presentation (Lesson): 1. Invite the child(ren). 2. Carry introduction tray for symbols to mat prepared on table or on the floor. 3. As child(ren) already know quantities 1 - 10, use third period of 3 period lesson and show the “one” and ask “What is this?” Wait for response. Show the “ten” and ask “What is this?” Wait for response. 4. Using three period lesson, teach vocabulary for 100 and 1000. 5. Return symbols to tray. 6. Return tray to shelf. 7. Return mat. Direct Aim (Objective): To introduce numerals for base ten place value Indirect Aims (Scaffolding): Preparation for higher math Preparation for science Preparation for language Internalize place value kinesthetically Preparation for art Develop visual memory Independence Concentration Points of Interest (Motivation): Three period lesson Colors on numerals and numerals themselves Control of Error (Self Assessment): Teacher Vocabulary: Symbol, names for 1, 10, 100, 1000, base ten, place value Extension: Stack cards showing all zeroes, then slide to right and expose all ones composing the numeral. Read numeral.

29


Adaptations: Use larger cards or smaller cards. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent whole numbers up to 30 in a variety of ways, using different forms of representation 

Represent whole numbers up to 30 in a variety of ways, with no significant errors 

Represent whole numbers up to 30 in one way, with no significant errors 

Represent whole numbers up to 30 in one way, with significant errors 

HPCS -Domain IV Cognitive Development Mathematics -Standard 2: Recognize and create patterns and become aware of relationships.

©2004 Dr. Louise Bogart. All rights reserved.

30


BASE TEN PLACE VALUE INTRODUCTION TO GOLDEN BEAD MATERIAL Association of Quantity and Symbols

Age: 4 to 7 Materials: Large numeral cards for 1, 10, 100, 1000 One each of golden bead quantities 1, 10, 100, 1000 Mat Presentation (Lesson): 1. Invite the child(ren). 2. Take materials to mat. 3. Place both the quantity and the symbols on the mat, in mixed array. 4. Teacher shows the one bead and asks, “What is this?” When the child responds, “One”, teacher asks her to find the symbol that matches the quantity, that is, the symbol that shows how to write “one.” The “one” is matched with the symbol “1 .” 5. Teacher repeats this procedure for 10, 100, 1000. 6. Return materials to respective trays. 7. Return trays to shelf. 8. Return mat. Direct Aim (Objective): Associate base ten place value symbols with quantities. Indirect Aims (Scaffolding): Preparation for higher math Preparation for science Preparation for language Internalize place value kinesthetically Independence Concentration Preparation for art Develop visual memory Points of Interest (Motivation): beads numerals matching Control of Error (Self Assessment): Teacher Vocabulary: numeral, symbol, quantity, base ten place value 31


Adaptations: Place the symbols in order and have the child find the matching quantity or vice versa. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.1 -Rubric Advanced

Count and compare groups of objects up to 30 according to the number of objects in each group, with accuracy 

Proficient

Partially Proficient

Novice

Count and compare groups of objects up to 30 according to the number of objects in each group, with no significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with a few significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

32


BASE TEN PLACE VALUE INTRODUCTION TO GOLDEN BEAD MATERIAL Age: 4 to 7

Crisis of Nine

Materials: Mat 9 Ones 9 Tens 9 Hundreds 1 Thousand Presentation (Lesson): 1. Invite child(ren) to join activity. 2. Lay out a mat. 3. Place tray with materials on the left side of the mat. 4. Beginning at the right side of the mat, count the 9 ones as you place them in a line vertically. Place the ones so they touch each other. 5. When you get to nine, say, “And one more would be...?” Wait for student to say “10.” 6. Teacher says, “Yes. And 10 has its own place.” As you say, “own place,” put one 10 to the left of the ones so that children can see it is exactly one more than the 9 ones. Pause. Remove the ones. 7. Teacher counts,” One ten, two tens, three tens, etc. and after counting 9 tens teacher says, “And one more ten would be...?” Wait for student to answer “10 tens.” 8. Teacher then responds, “And 10 tens is the same as...?’ (Show 100 as you ask that question.) Wait for a response. 9. “Yes, 100 and 100 has its own place.” Put 100 to the left of the tens so that children can see that it is exactly one more than 9 tens. Pause. Remove the tens. 10. Continue with the lesson in the same manner counting 9 hundreds to 1000. 11. Return beads to the tray. 12. Return the beads and mat to shelf. Direct Aim (Objective): To introduce the concept that in the base ten, the largest digit that can be written in any place is 9. Therefore, after counting 9 you must change place. Indirect Aim (Scaffolding): Preparation for higher math, science, art, language Develop visual memory, and physical internalization of place value and base ten. Develop order, concentration, and independence. Points of Interest (Motivation) Materials exchanging and moving to next place counting Control of Error (Self Assessment): Teacher

33


Vocabulary: names of quantities 1 - 9, ones, tens, hundreds, thousands, place value, base ten Extensions: Exchanging game Do the crisis for other bases using the material in the bead cabinet Adaptations: Use cuisenaire rods. Standards: HCPS III -Standard 10: Patterns, Functions, and Algebra: SYMBOLIC REPRESENTATION: Use symbolic forms to represent, model, and analyze mathematical situations -Topic Numeric and Algebraic Representations -Benchmark MA.K.10.1 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent simple numerical situations with objects and number sentences, with accuracy 

Represent simple numerical situations with objects and number sentences, with no significant errors 

Represent simple numerical situations with either objects or number sentences, but not both 

Have difficulty representing simple numerical situations with objects or number sentences 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 2: Recognize and create patterns and become aware of relationships.

©2004 Dr. Louise Bogart. All rights reserved.

34


BASE TEN PLACE VALUE INTRODUCTION TO GOLDEN BEAD MATERIAL Quantity Layout

Age: 4 1/2 to 8 Materials: Mat 9 ones 9 tens 9 hundreds 9 thousands

Presentation (Lesson): 1. Invite child(ren) to sit next to you at mat on floor. 2. Spread out mat and place quantity layout tray to left. 3. Teacher counts out 9 ones starting in top right corner of mat and works top to bottom leaving about 5 inches between each bead (hands width or use a ten). When she gets to 9 teacher asks, “And one more would be...?” Wait for response. 4. “Yes, and ten has its own place.” 5. Teacher places ten to left of one bead at top of mat leaving space between rows. She counts out 9 tens working top to bottom as with ones, saying “one ten, two tens, etc. When she gets to nine tens, she asks “And one more would be...?” Wait for response. 6. “Yes, ten tens which is the same as...?” (Show the 100 as you ask.) Wait for response. After students respond, teacher says, “Yes, and 100 has its own place.” 7. Teacher places 100 to left of first ten leaving space between rows and continues the procedures in steps 5 - 7 through 9000. 8. When layout is complete, have child count it all by himself. Later child can lay it out by himself, counting. 9. Return materials to tray. 10. Return tray to shelf. 11. Return mat. Direct Aim (Objective): To practice place value with quantity Indirect Aims (Scaffolding): Develop visual memory of place value. Internalize place value kinesthetically Preparation for higher math, science, language, art Point of Interest Motivation): Beads Pattern Layout Using hand or ten to measure space Control of Error (Self Assessment): Teacher Material on tray

35


Vocabulary: Layout, pattern, ones, tens, hundreds, thousands, place value, base ten Extension: Fetching Games Adaptations: Help the child count and lay out the materials. Use smaller materials. Standards: HCPS III -Standard 10: Patterns, Functions, and Algebra: SYMBOLIC REPRESENTATION: Use symbolic forms to represent, model, and analyze mathematical situations -Topic Numeric and Algebraic Representations -Benchmark MA.K.10.1 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent simple numerical situations with objects and number sentences, with accuracy 

Represent simple numerical situations with objects and number sentences, with no significant errors 

Represent simple numerical situations with either objects or number sentences, but not both 

Have difficulty representing simple numerical situations with objects or number sentences 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 2: Recognize and create patterns and become aware of relationships.

©2004 Dr. Louise Bogart. All rights reserved.

36


BASE TEN PLACE VALUE INTRODUCTION TO GOLDEN BEAD MATERIAL Fetching Quantities

Age: 4 1/2 to 8 Materials: Tray Mat 9 ones 9 tens 9 hundreds 9 thousands

Presentation (Lesson): 1. Child(ren) lay out quantities. 2. Teacher says “We’re going to make some large numbers. Let’s make 1 thousand, 8 hundreds, 4 tens, 2 ones. Can you get me the quantity 1842?” 3. Child gets quantity on tray and takes to teacher. 4. Teacher asks child, “What did you bring?” Child reads quantity. 5. Teacher verifies. 6. Child returns quantity to layout and returns to teacher with empty tray. 7. Teacher gives child another quantity to fetch. 8. Procedure continues as long as child is interested. 9. Child(ren) return materials to tray. 10. Child(ren) return trays to shelf. 11. Return mat. Direct Aim (Objective): Practice with base ten place value Indirect Aims (Scaffolding): Preparation for higher math, science, language, art Develop auditory memory Points of Interest (Motivation): bead layout shapes of quantities pattern Control of Error (Self Assessment): Teacher material Vocabulary: Names of base ten place values 37


Variations: If child cannot remember four places, try three, then 2, if necessary Adaptations: Depending on the child’s ability ask for quantities you know they can fetch. Use different materials. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent whole numbers up to 30 in a variety of ways, using different forms of representation 

Represent whole numbers up to 30 in a variety of ways, with no significant errors 

Represent whole numbers up to 30 in one way, with no significant errors 

Represent whole numbers up to 30 in one way, with significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

38


BASE TEN PLACE VALUE INTRODUCTION TO GOLDEN BEAD MATERIAL Symbols Layout

Age: 4 ½ to 8 Materials: Mat Large Numeral Cards 1 - 9000 Presentation ( Lesson): 1. Invite child(ren) to sit next to you at mat on floor. 2. Spread out mat and place quantity layout tray to left. 3. Teacher counts 1- 9, starting at top right corner of mat and works top to bottom, while placing cards for 1 - 9 in a row vertically. When she gets to “9” teacher asks, “And one more would be...?” Wait for response. 4. “Yes, and ten has its own place.” 5. Teacher places 10 to left of 1 at top of mat leaving space between rows. She counts out 9 tens working top to bottom as with 1 - 9, saying “one ten, two tens” etc. When she gets to nine tens, she asks “And one more would be...?” Wait for response. 6. “Yes, ten tens which is the same as...?” (Show the 100 as you ask.) Wait for a response. After students respond, teacher says, “Yes, and 100 has its own place.” 7. Teacher places 100 to left of first ten leaving space between rows and continues the procedures in steps 5-7 through 9000. 8. When layout is complete, have child count it all by himself. Later child can lay it out by himself, counting. 9. Return materials to tray. 10. Return tray to shelf and return mat. Direct Aim (Objective): To practice place value with symbols Indirect Aims (Scaffolding): Develop visual memory of place value Internalize place value kinesthetically Preparation for higher math, science, language, art Point of Interest (Motivation): Symbols and colors Pattern Layout Seeing large numerals Control of Error (Self Assessment): Teacher Material on tray Vocabulary: Layout, pattern, ones, tens, hundreds, thousands, place value, base ten

39


Extension: Fetching Games Adaptations: Use larger or smaller cards with numbers. Standards: HCPS III -Standard 10: Patterns, Functions, and Algebra: SYMBOLIC REPRESENTATION: Use symbolic forms to represent, model, and analyze mathematical situations -Topic Numeric and Algebraic Representations -Benchmark MA.K.10.1 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent simple numerical situations with objects and number sentences, with accuracy 

Represent simple numerical situations with objects and number sentences, with no significant errors 

Represent simple numerical situations with either objects or number sentences, but not both 

Have difficulty representing simple numerical situations with objects or number sentences 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 2: Recognize and create patterns and become aware of relationships.

©2004 Dr. Louise Bogart. All rights reserved.

40


BASE TEN PLACE VALUE INTRODUCTION TO GOLDEN BEAD MATERIAL Fetching Symbols

Age: 4 1/2 to 8 Materials: Tray Mat Large numeral cards 1 - 9000 Presentation (Lesson): 1. Child(ren) lay out symbols. 2. Teacher says “We re going to make some large numerals. Let s make 1 thousand, 8 hundreds, 4 tens, 2 ones. Can you get me the numeral 1842?” 3. Child gets symbols on tray and takes to teacher. 4. Teacher asks child, “What did you bring?” Child reads numeral. 5. Teacher verifies. 6. Child returns symbols to layout and returns to teacher with empty tray. 7. Teacher gives child another numeral to fetch. 8. Procedure continues as long as child is interested. 9. Child(ren) return materials to tray. 10. Child(ren) return trays to shelf. 11. Return mat. Direct Aim (Objective): Practice with base ten place value using symbols Indirect Aims (Scaffolding): Preparation for higher math, science, language, art Develop auditory memory Points of Interest (Motivation): symbols layout colors pattern Control of Error (Self Assessment): Teacher material Vocabulary: Names of base ten place values Variations: If child cannot remember four places, try three, then 2, if necessary.

41


Adaptations: Depending on the child’s ability ask for quantities you know they can fetch. Use different materials. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent whole numbers up to 30 in a variety of ways, using different forms of representation 

Represent whole numbers up to 30 in a variety of ways, with no significant errors 

Represent whole numbers up to 30 in one way, with no significant errors 

Represent whole numbers up to 30 in one way, with significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

42


BASE TEN PLACE VALUE INTRODUCTION TO GOLDEN BEAD MATERIAL Birdseye View (45 Layout)

Age: 4 1/2 to 8 Materials: Large Mat Set of Large Numeral Cards 45 ones, 45 tens, 45 hundreds, 45 thousands Presentation (Lesson): 1. Invite children to sit next to you on the floor. 2. Spread out mat and place materials on the left side of the mat. 3. Teacher asks students to layout the large numeral cards being careful to show children how to space the places value to allow space for the quantities. 4. Teacher demonstrates placing a one bead beside the numeral one, two ones beside the numeral two, and three ones beside the numeral three. Children are then asked to continue the process to the numeral nine. 5. Teacher begins placing the tens beside the numerals for one ten and two ten and then asks children to continue to nine tens. 6. teacher begins placing hundreds beside the numerals for one hundred and two hundred, being careful to place the quantities side by side moving from right to left. Children are asked to continue the process to nine hundred. 7. Follow step 6 for thousands place. 8. When the layout is complete, children marvel at their creation and help to collect the numerals and quantity. 9. Return materials to shelf. 10. Fold or roll mat and return it. Direct Aim (Objective): To build place value with symbols and matching quantities Indirect Aims (Scaffolding): Develop visual memory of place value symbols with matching quantities The step pattern that is created by this layout Preparation for higher math, science, language and art Points of Interest (Motivation): Beads and symbols pattern layout building on large mat Control of Error (Self Assessment): Teacher or other children 45 of each quantity symbols 43


Vocabulary: layout, pattern, one, tens, hundreds, thousands, base ten, place value Adaptations: Have children work together. Use smaller materials to work at a table. Standards: HCPS III -Standard 10: Patterns, Functions, and Algebra: SYMBOLIC REPRESENTATION: Use symbolic forms to represent, model, and analyze mathematical situations -Topic Numeric and Algebraic Representations -Benchmark MA.K.10.1 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent simple numerical situations with objects and number sentences, with accuracy 

Represent simple numerical situations with objects and number sentences, with no significant errors 

Represent simple numerical situations with either objects or number sentences, but not both 

Have difficulty representing simple numerical situations with objects or number sentences 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

44


LINEAR COUNTING

TEENS: 11 - 19 Quantity Age: 4 to 7 Material: Short bead stair 1 - 9 9 ten bars Counter Mat Presentation (Lesson): 1. Invite the child(ren). 2. Build short bead stair in upper left corner of mat with ten bars below 3. Take one ten and place the red 1 bead next to it, “One ten one has another name, 11” 4. To the right of 11, place another 1 ten with the green 2 next to it. “One ten two has another name, 12.” 5. Continue this procedure through nineteen. 6. Using 3 Period Lesson teach names of bead combinations 11 - 19. 7. When child(ren) know all combinations, have him put them in sequence. 8. Return beads to box. 9. Return box to shelf. 10. Return mat. Direct Aim (Objective): To learn quantities 11 – 19, their sequence, and their names Indirect Aim (Scaffolding): Preparation for higher math Points of Interest (Motivation): Bead stair Combinations of bead bars. Control of Error (Self Assessment): Teacher Materials Vocabulary: eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen Extension: Using other sets of bead stairs, make other combinations of 11 - 19 besides 10 + one to nine example: 9+2=11 8+3=11 7+4=11

45


Adaptations: Use cuiseniare rods. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent whole numbers up to 30 in a variety of ways, using different forms of representation 

Represent whole numbers up to 30 in a variety of ways, with no significant errors 

Represent whole numbers up to 30 in one way, with no significant errors 

Represent whole numbers up to 30 in one way, with significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

46


LINEAR COUNTING TEENS: Symbols

Age: 4 to 7 Materials: Teen Boards - Two wooden boards with vertical rows of nine 10’s. A slot is provided to cover the zero with a numeral to compose the numerals from 11 to 19 Mat Presentation (Lesson): 1. Invite the child(ren). 2. Place the boards in front of the child(ren) with symbols stacked to right of board. 3. Pointing to top “10” teacher names “one ten one” sliding in symbols for “1,” “is eleven.” “This is the way we write 11.” 4. Teacher presents symbols for 12 through 19 in the same manner. 5. Using a three period lesson, teacher teaches names for the numerals. 6. Return boards to box. 7. Return box to shelf. 8. Return mat. Direct Aim (Objective): To learn symbols, sequence and names for numerals 11 - 19 Indirect Aims (Scaffolding): Preparation for high math Points of Interest (Motivation): New numerals Sliding in numbers Control of Error (Self Assessment): Teacher Materials Vocabulary: Names for symbols 11 - 19 Variation: Mix up numerals and have children place quantity Adaptations: Place the numbers for the ones place in order from 1-9.

47


Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent whole numbers up to 30 in a variety of ways, using different forms of representation 

Represent whole numbers up to 30 in a variety of ways, with no significant errors 

Represent whole numbers up to 30 in one way, with no significant errors 

Represent whole numbers up to 30 in one way, with significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

48


LINEAR COUNTING

TEENS: Quantity and Symbols Age: 4 to 7 Materials : Mat Teen boards Short bead stair 9 ten bars Presentation (Lesson): 1. Invite the chiId(ren) 2. Place boards on the mat in front of the child (ren). 3. Place beads for ten and one beside the board and say “one ten one is eleven.” 4. Slip symbol into board over the first zero and say “this is how we write eleven.” 5. Follow same procedure for remaining quantities and symbols. 6. Return boxes to shelf. 7. Return mat. Direct Aim (Objective): Associate quantity and symbols 11 - 19 Indirect Aim (Scaffolding): Preparation for high math Points of Interest (Motivation): Short bead stair Slots Sequence Control of Error (Self Assessment): Teacher Materials Vocabulary: Names for quantities and symbols for 11 - 19 Extension: Use other combinations of bead stair 1 - 9 to make teen quantities Mix numerals and have child(ren) put the matching bead combinations alongside each numeral Mix quantities next to board and have child(ren) put matching numerals on the board

49


Adaptations: Place the symbols in order and have the child find the matching quantity or vice versa. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.1 -Rubric Advanced

Count and compare groups of objects up to 30 according to the number of objects in each group, with accuracy 

Proficient

Partially Proficient

Novice

Count and compare groups of objects up to 30 according to the number of objects in each group, with no significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with a few significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

50


LINEAR COUNTING TENS: Quantity

Age: 4 1/2 to 7 Materials: 45 tens Mat Presentation (Lesson): 1. Invite the child(ren). 2. Place the materials in front of the child(ren). 3. Place one ten at the left side of the mat and ask child (ren) to name it. Place two tens nearby and to the right of ten. “Two tens has another name, twenty.” 4. Continue with this procedure through ninety. 5. Using a Three period lesson, teach the names of twenty to ninety. 6. When child(ren) know all quantities, have them put the beads in sequence. 7. Return materials to box. 8. Return box to shelf. 9. Return mat. Direct Aim (Objective): Learn names for quantities 20 – 90 and their sequence Indirect Aim (Scaffolding): Preparation for higher math Points of Interest (Motivation): Beads New vocabulary Control of Error (Self Assessment): Teacher Number of Beads Vocabulary: Ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety Adaptations: Use larger materials or cuisenaire rods. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems

51


-Topic Numbers and Number Systems -Benchmark MA.K.1.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent whole numbers up to 30 in a variety of ways, using different forms of representation 

Represent whole numbers up to 30 in a variety of ways, with no significant errors 

Represent whole numbers up to 30 in one way, with no significant errors 

Represent whole numbers up to 30 in one way, with significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

52


LINEAR COUNTING TENS: Symbols

Age: 4 1/2 to 7 Materials: Ten boards Mat Presentation (Lesson): 1. Invite the child(ren). 2. Place materials at left side of mat. 3. Remove boards and place in front of the child(ren). 4. Ask child(ren) to name the ten. Then, “Two tens has another name, twenty, and this is how we write twenty.� 5. Continue with procedure thirty through ninety. 6. Using a 3 period lesson, teach the names for the symbols 20, 30, 40, 50, 60, 70, 80, 90. 7. Return materials box. 8. Return box to shelf. 9. Return mat. Direct Aim (Objective): Learn names of symbols 20-90 and their sequence Indirect Aim (Scaffolding): Preparation for higher math Points of Interest (Motivation): Wooden boxes Boards Numerals and their names Control of Error (Self Assessment): Teacher Materials Vocabulary: Names 10 - 90 Adaptations: Place the numbers for the ones place in order from 1-9. Standards: HCPS III

53


-Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Represent whole numbers up to 30 in a variety of ways, using different forms of representation 

Represent whole numbers up to 30 in a variety of ways, with no significant errors 

Represent whole numbers up to 30 in one way, with no significant errors 

Represent whole numbers up to 30 in one way, with significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

54


LINEAR COUNTING

TENS: Quantity and Symbol Age: 4 1/2 to 7 Materials: ten boards Mat 9 tens and 9 ones

Presentation (Lesson): 1. Lay ten boards out in front of child(ren). 2. Teacher places a ten alongside of the numeral 10, saying “Ten,” and placing one bead with the ten, says “One ten one” is eleven.” 3. Pointing to “10” and sliding “1” into slot, teacher says “This is the way we write 11.” 4. Teacher removes a symbol (symbols are always kept face down in beginning) and continues with 12 - 19, adding one bead and changing the symbol to match. 5. “9 and one more is…?” Wait for response. Exchange for ten “now we have two tens which is the same as…?” Wait for response. Move the two tens down to the numeral 20 and say “this is how we write twenty.” 6. Continue procedure 21 through 99. 7. Let child(ren) do it as soon as he realizes what to do. 8. Return materials to boxes. 9. Return boxes to shelf. 10. Return mat. NOTE: Children should go at least to number 30 during their first lesson. Always bring the beads down so the child can see the progression. Direct Aims (Objective): To associate quantity, name and symbols for 11 - 99 Build numbering system one bead at a time Indirect Aim (Scaffolding): Preparation for higher math Point of Interest (Motivation): Beads increasing and sliding down with each 10 Control of Error (Self Assessment): Teacher Materials Vocabulary: Names for numerals and quantities 11 - 99 55


Extension: Provide materials so that children can make a number roll. Adaptations: Place the symbols in order and have the child find the matching quantity or vice versa. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.1 -Rubric Advanced

Count and compare groups of objects up to 30 according to the number of objects in each group, with accuracy 

Proficient

Partially Proficient

Novice

Count and compare groups of objects up to 30 according to the number of objects in each group, with no significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with a few significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

56


LINEAR COUNTING Hundred Board

Age: 5 - 8 Materials: Board with 10 rows of 10 Tiles 1 - 100 Mat Presentation (Lesson): 1. Invite the child(ren). 2. Sort tiles by tens. 3. Place first group of tiles (1 - 10) in mixed array. 4. Starting in upper left hand corner and working left to right child places tiles in correct numerical sequence. 5. Child(ren) work(s) chart row by row with appropriate group of tiles until board is filled 1 - 100. OR Some children may choose to work from mixed array to fill the board with tiles 1 - 100 as a matter of learning style. Allow them to do so. Note: The tiles should not be presorted according to tens. Doing so does not meet the objective of this lesson nor does it allow for development of concentration. Points of Interest (Motivation): Matrix Tiles Filling in all the blanks Sorting

Direct Aim (Objective): Test for ten boards (test knowledge of 1 - 100) Indirect Aims (Scaffolding): Practice in counting 1 - 100 Preparation for higher math Control of Error (Self Assessment): Visual - pattern of numbers Exact amount of tiles Control chart Vocabulary: Hundred, board & tiles, tens, sort, numerals 1-100 Extensions: Skip count by 2’s, 5’s, and 10’s 57


Adaptations: Put the hundred board on a piece of paper. Fill in some spaces to help children fill out the hundred board. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems -Topic Numbers and Number Systems -Benchmark MA.K.1.1 -Rubric Advanced

Count and compare groups of objects up to 30 according to the number of objects in each group, with accuracy 

Proficient

Partially Proficient

Novice

Count and compare groups of objects up to 30 according to the number of objects in each group, with no significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with a few significant errors 

Count and compare groups of objects up to 30 according to the number of objects in each group, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 2: Recognize and create patterns and become aware of relationships.

©2004 Dr. Louise Bogart. All rights reserved.

58


LINEAR COUNTING HUNDRED CHAIN Age: 5 – 7 Materials: 100 chain 100 square Arrows 10 - 100 Mat Presentation (Lesson): 1. Teacher shows the child(ren) how to carry chain to mat and lay it out. 2. Teacher shows 100 square saying “This is 100.” 3. Teacher folds 100 chain into square and compares to square - “This is 100 also.” 4. Pull chain out to make line. 5. Place arrows in mixed array on mat. 6. Start counting chain from the left “1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10” Select arrow that says “10” and place it on 10 bead. 7. Continue counting individual beads until 100 and mark each ten with an arrow. When you get to 100, place 100 square at end of chain. 8. Read arrows. Child(ren) may record numerals on tags in journals. 9. Replace chain and arrows on shelf. 10. Return mat. Direct Aims (Objective): To give practice in linear counting To introduce skip counting (by 10’s) Indirect Aim (Scaffolding): Preparation for higher math Points of Interest (Motivation): Folding and unfolding square Arrows Counting by 10 Vocabulary: Numerals 1 - 100, square, chain, skip counting, arrows Variation: After this linear counting is very familiar, the child gets skip counting aspect and does it that way.

59


Extension: After the 100 chain comes the 5 chain in skip counting, then 2, 3, 4, 6, 7, 8, 9. Adaptations: Use cuisenaire rods with large counters and symbols. Have a paper with the hundreds chain for the child to match the quantities to. Standards: HCPS III -Standard 9: Patterns, Functions, and Algebra: PATTERNS AND FUNCTIONAL RELATIONSHIPS: Understand various types of patterns and functional relationships -Topic Patterns -Benchmark MA.K.9.1 -Rubric Advanced

Proficient

Partially Proficient

Novice

Demonstrate repeating patterns involving shapes, objects, sounds, and movements, with accuracy 

Demonstrate repeating patterns involving shapes, objects, sounds, and movements, with no significant errors 

Demonstrate repeating patterns involving shapes, objects, sounds, and movements, with a few significant errors 

Demonstrate repeating patterns involving shapes, objects, sounds, and movements, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 2: Recognize and create patterns and become aware of relationships.

©2004 Dr. Louise Bogart. All rights reserved.

60


LINEAR COUNTING 1000 Chain

Age: 5+ Materials: 1000 chain 1000 cube 100 chain 9 - 100 squares Arrows 10 - 1000 Large mat Presentation (Lesson): 1. Teacher shows child(ren) how to carry chain to mat and lay it out. 2. Teacher shows 1000 cube saying “This is 1000.” 3. Teacher folds 1000 chain into ten 100 squares then stacks ten 100 squares to compare to cube - “This is 1000 also.” 4. Start counting chain from the left by 10’s placing arrows, measuring with 100 chains and placing 100 squares at proper places. 5. Stack 100 squares to make 1000 cube at end. 6. Replace chains and arrows. 7. Return mat. Direct Aim(Objective): To give practice in linear and skip counting Indirect Aim (Scaffolding): Preparation for higher math Points of Interest (Motivation): Length of chain Folding and unfolding Arrows Control of Error (Self Assessment): Teacher Beads Vocabulary: Names of numerals 1 - 1000, chain, square, cube, arrow, skip counting Adaptations: Use cuisenaire rods with large counters and symbols. Have a paper with the thousand chain for the child to match the quantities to.

61


Standards: HCPS III -Standard 9: Patterns, Functions, and Algebra: PATTERNS AND FUNCTIONAL RELATIONSHIPS: Understand various types of patterns and functional relationships -Topic Patterns -Benchmark MA.K.9.1 -Rubric Advanced

Proficient

Partially Proficient

Novice

Demonstrate repeating patterns involving shapes, objects, sounds, and movements, with accuracy 

Demonstrate repeating patterns involving shapes, objects, sounds, and movements, with no significant errors 

Demonstrate repeating patterns involving shapes, objects, sounds, and movements, with a few significant errors 

Demonstrate repeating patterns involving shapes, objects, sounds, and movements, with many significant errors 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 2: Recognize and create patterns and become aware of relationships.

©2004 Dr. Louise Bogart. All rights reserved.

62


WHOLE NUMBER OPERATIONS GOLDEN BEADS Addition - Static - No Exchanging

Age: 41/2 to 7 Materials: Trays Mat Yarn or ribbon Addition sign Base ten blocks or Golden Beads One large and two small sets of numeral cards Presentation (Lesson): 1. Invite children to the lesson. 2. Ask children to lay out the three sets of numeral cards. 3. Teacher asks two students individually to get a tray and to bring a numeral from one of the two sets of small numeral cards. Teacher is careful to provide two numerals that do not require exchanging. 4. As each student returns, (s)he reads the numeral, the teacher verifies and asks the student to get the quantity that matches the numeral and put it in his/her tray. 5. When students return with quantity, the students read their quantity, the teacher verifies that it is correct. 6. Teacher takes numeral from one of the trays and places it on the left side of the mat. To the right and parallel to the numeral, the teacher places the quantity. 7. The teacher then asks the other student to place his/her numerals below the first student’s numeral and to place the quantity below the quantity already on the mat. 8. The teacher recaps “name brought amount and name brought amount and asks, “I wonder how much we’d have if we put them all together?” 9. Teacher places yarn, to simulate a line, just below the two numerals and quantities. 10. The teacher then asks one of the students to please bring the ones below the line and count them. The student does so and tells the group how many there are. 11. The teacher then asks a student to get the numeral, from the large set of numerals, that matches the quantity just counted. 12. This procedure continues for tens, hundreds and thousands until all places are counted and the numeral card that matches has been retrieved. 13. The teacher then asks one of the students to compose the numeral and another to read it. 14. The teacher recaps the entire process: “name brought amount and name brought amount. When we put them together we counted total. This operation is called addition and addition has a special sign.” Place sign. 15. Students return materials to learning center. Direct Aim (Objective): To provide a concrete experience with the concept of addition without exchanging.

63


Indirect Aim (Scaffolding): Preparation for higher math Cooperative learning Points of Interest (Motivation): Materials Yarn or ribbon Addition sign Working with friends Operating with “big numbers” Control of Error (Self Assessment): Teacher or another student Control chart Vocabulary: add, addition, plus, addend, total, sum Adaptations: Use cuisenaire rods. Create papers with matching pictures of the quantity of the objects for the child to add. Standards: HCPS III -Standard 2: Numbers and Operations: OPERATION SENSE: Understand the meaning of operations and how they relate to each other -Topic Operations -Benchmark MA.K.2.1 -Rubric Advanced

Proficient

Partially Proficient

Novice

Demonstrate addition as "putting together" or "combining sets" and determine the total, with accuracy 

Demonstrate addition as "putting together" or "combining sets" and determine the total, with no significant errors 

Demonstrate addition as "putting together" or "combining sets" and determine the total, with a few significant errors 

Have difficulty demonstrating addition as "putting together" or "combining sets" and are unable to determine the total 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

64


WHOLE NUMBER OPERATIONS GOLDEN BEADS Addition - Dynamic - With Exchanging

Age: 41/2 to 7 Materials: Trays Mat Yarn or ribbon Addition sign Base ten blocks or Golden Beads One large and two small sets of numeral cards Presentation (Lesson): 1. Invite children to the lesson. 2. Ask children to lay out the three sets of numeral cards. 3. Teacher asks two students individually get a tray and to bring a numeral from one of the two sets of small numeral cards. Teacher is careful to provide two numerals that do not require exchanging in the thousands place. 4. As each student returns, (s)he reads the numeral, the teacher verifies and asks the student to get the quantity that matches the numeral and put it in his/her tray. 5. When students return with quantity, the students read their quantity, the teacher verifies that it is correct. 6. Teacher takes numeral from one of the trays and places it on the left side of the mat. To the right and parallel to the numeral, the teacher places the quantity. 7. The teacher then asks the other student to place his/her numerals below the first student’s numeral and to place the quantity below the quantity already on the mat. 8. The teacher recaps “name brought amount and name brought amount and asks, “I wonder how much we’d have if we put them all together?” 9. Teacher places yarn, to simulate a line, just below the two numerals and quantities. 10. The teacher then asks one of the students to please bring the ones below the line and count them. The student does so and tells the group how many there are and then makes the appropriate exchange. 11. The teacher then asks a student to get the numeral, from the large set of numerals, that matches the quantity just counted. 12. This procedure continues for tens, hundreds and thousands until all places are counted and the numeral card that matches has been retrieved. 13. The teacher then asks one of the students to compose the numeral and another to read it. 14. The teacher recaps the entire process: “name brought amount and name brought amount. When we put them together we counted total. This operation is called addition and addition has a special sign.” Place sign. 15. Students return materials to learning center.\ Direct Aim (Objective): To provide a concrete experience with the concept of addition with exchanging.

65


Indirect Aim (Scaffolding): Preparation for higher math Cooperative learning Points of Interest (Motivation): Materials Yarn or ribbon Addition sign Working with friends Operating with “big numbers” Control of Error (Self Assessment): Teacher or another student Control chart Vocabulary: add, addition, plus, addend, total, sum, exchange, change, regroup, rename, trade Adaptations: Use cuisenaire rods. Create papers with matching pictures of the quantity of the objects for the child to add. Standards: HCPS III -Standard 2: Numbers and Operations: OPERATION SENSE: Understand the meaning of operations and how they relate to each other -Topic Operations -Benchmark MA.K.2.1 -Rubric Advanced

Proficient

Partially Proficient

Novice

Demonstrate addition as "putting together" or "combining sets" and determine the total, with accuracy 

Demonstrate addition as "putting together" or "combining sets" and determine the total, with no significant errors 

Demonstrate addition as "putting together" or "combining sets" and determine the total, with a few significant errors 

Have difficulty demonstrating addition as "putting together" or "combining sets" and are unable to determine the total 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations. ©2004 Dr. Louise Bogart. All rights reserved.

66


WHOLE NUMBER OPERATIONS GOLDEN BEADS

Multiplication - Static Without Exchanging Age: 41/2 to 7 Materials: Trays Mat Yarn or ribbon Multiplication sign Base ten material or Golden bead material One large and three small sets of numeral cards Presentation (Lesson): 1. Invite children. 2. Have children lay out all sets of numeral cards. 3. Teacher asks three children individually to bring a four digit numeral from one of the three sets of small numeral cards. Teacher is careful to give a numeral that will not require exchanging. 4. As each student returns, (s)he reads the numeral, teacher verifies then asks the student to get the quantity that matches the numeral. Upon return, student reads quantity and teacher verifies. 5. Each student places numeral and quantity on the mat using the same format that was used for addition. 6. Teacher recaps problem: “name brought amount, name brought amount, name brought amount. The three of you took amount 1 time, 2 times, three times. I wonder how many we have all together?” Place ribbon across bottom of materials. 7. Individual students are invited to bring quantities below the line and count. The numeral that matches is retrieved from the large set of numeral cards. 8. When all places are counted and labeled, one student is asked to compose the numeral and another is asked to read it. 9. Teacher recaps the entire process: “name brought amount, name brought amount, name brought amount. So you took amount 1 time, 2 times, three times. When we put them all together we counted product. This operation is called multiplication and multiplication has a special sign.” 10. Children return materials and mat to the shelf. Direct Aim (Objective): To introduce the concept of multiplication without exchanging through a concrete experience. Indirect Aim (Scaffolding): Preparation for higher math Cooperative Learning Points of Interest (Motivation): Materials 67


Multiplication sign Yarn or ribbon Working with friends Operating with “big numbers” Control of Error (Self Assessment): Teacher or another student Control cards or charts Vocabulary: multiply, multiplication, product, take, multiplier, multiplicand, times, factor Extension: Teacher explains: “There is a special way to record multiplication. Because we took the same amount 3 times, we only need to write the amount once.” Remove second 2 numerals and replace them with a 3 and the multiplication sign saying, “You took amount 1 time, 2 times, 3 times.” Place the 3 as you say “three” and the X as you say “times”. Adaptations: Use cuisenaire rods. Create papers with matching pictures of the quantity of the objects for the child to multiply. Standards: HCPS III -Standard 2: Numbers and Operations: OPERATION SENSE: Understand the meaning of operations and how they relate to each other -Topic Operations -Benchmark MA.K.2.1 -Rubric Advanced

Proficient

Partially Proficient

Novice

Demonstrate addition as "putting together" or "combining sets" and determine the total, with accuracy 

Demonstrate addition as "putting together" or "combining sets" and determine the total, with no significant errors 

Demonstrate addition as "putting together" or "combining sets" and determine the total, with a few significant errors 

Have difficulty demonstrating addition as "putting together" or "combining sets" and are unable to determine the total 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations. ©2004 Dr. Louise Bogart. All rights reserved.

68


WHOLE NUMBER OPERATIONS GOLDEN BEADS

Multiplication - Dynamic - With Exchanging Age: 5 to 7 Materials: Yarn or ribbon Mat Multiplication sign Tray Base ten material or Golden bead material One large and three small sets of numeral cards Presentation (Lesson): 1. Invite children. 2. Have children lay out all sets of numeral cards. 3. Teacher asks three children individually to bring a four digit numeral from one of the three sets of small numeral cards. Teacher is careful to give a numeral that will not require exchanging in the thousands place. 4. As each student returns, (s)he reads the numeral, teacher verifies then asks the student to get the quantity that matches the numeral. Upon return, student reads quantity and teacher verifies. 5. Each student places numeral and quantity on the mat using the same format that was used for addition. 6. Teacher recaps problem: “name brought amount, name brought amount, name brought amount. The three of you took amount 1 time, 2 times, three times. I wonder how many we have all together?” Place ribbon across bottom of materials. 7. Individual students are invited to bring quantities below time line and count, exchanging as appropriate. The numeral that matches is retrieved from the large set of numeral cards. 8. When all places are counted and labeled, one student is asked to compose the numeral and another is asked to read it. 9. Teacher recaps the entire process: “name brought amount, name brought amount, name brought amount. So you took amount 1 time, 2 times, three times. When we put them all together we counted product. This operation is called multiplication and multiplication has a special sign.” 10. Children return materials and mat to the shelf. Direct Aim (Objective): To introduce the concept of multiplication with exchanging through a concrete experience. Indirect Aim (Scaffolding): Preparation for higher math

Cooperative Learning

Points of Interest (Motivation): 69


Materials Multiplication sign Operating with “big numbers”

Yarn or ribbon Working with friends

Control of Error (Self Assessment): Teacher or another student Control cards or charts Vocabulary: Multiply, multiplication, product, take, multiplier, multiplicand, times, factor, change, exchange, regroup, rename, trade. Extension: Teacher explains: “There is a special way to record multiplication. Because we took the same amount 3 times, we only need to write the amount once.” Remove second 2 numerals and replace them with a 3 and the multiplication sign saying, “You took amount 1 time, 2 times, 3 times.” Place the 3 as you say “three” and the X as you Adaptations: Use cuisenaire rods. Create papers with matching pictures of the quantity of the objects for the child to multiply. Standards: HCPS III -Standard 2: Numbers and Operations: OPERATION SENSE: Understand the meaning of operations and how they relate to each other -Topic Operations -Benchmark MA.K.2.1 -Rubric Advanced

Proficient

Partially Proficient

Novice

Demonstrate addition as "putting together" or "combining sets" and determine the total, with accuracy 

Demonstrate addition as "putting together" or "combining sets" and determine the total, with no significant errors 

Demonstrate addition as "putting together" or "combining sets" and determine the total, with a few significant errors 

Have difficulty demonstrating addition as "putting together" or "combining sets" and are unable to determine the total 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

70


WHOLE NUMBER OPERATIONS GOLDEN BEADS

Subtraction - Static - Without Exchanging Age: 5 to 8 Materials: Mat Trays Subtraction sign Yarn or ribbon Base Ten Materials 1 large and 2 small sets of numeral cards Presentation (Lesson): 1. Invite children. 2. Have children lay out all sets of numeral cards. 3. Teacher asks two children individually to bring a four digit numeral, the first from the large set of numeral cards and the second from one of the small sets of numerals. Teacher is careful to give a numeral that will not require exchanging. 4. Each student returns, (s)he reads the numeral, teacher verifies. Only the student with the large numeral cards is asked to get the quantity that matches the numeral. Student reads quantity and teacher verifies. 5. The student with the large numeral cards places his/her numeral and quantity on the mat using the same format as was used in addition. 6. The student with the small numeral cards takes the quantity that matches his/her numeral from the quantity that has been placed on the mat by the first student and places the quantity on his/her tray. 7. Teacher recaps problem: “name brought amount. name took his/her quantity from name and put it on his/her tray. I wonder how much name has left?� Place ribbon across bottom of materials. 8. Individual students are invited to bring quantities below the line and count. The numeral that matches is retrieved from the unused small set of numeral cards. 9. When all places are counted and labeled- one student is asked to compose the numeral and another is asked to read it. 10. Teacher recaps the entire process: name brought amount. name took his/her quantity from name and put it on a tray. When we counted what name had left, we got difference. This operation is called subtraction and subtraction has a special sign. 11. Children return materials and mat to the shelf. Direct Aim (Objective): To introduce the concept of subtraction without exchanging through a concrete experience. Indirect Aims (Scaffolding): Preparation for higher math 71


Cooperative Learning Points of Interest (Motivation): Materials Yarn or ribbon Subtraction sign Working with friends Operating with “big numbers” Control of Error (Self Assessment): Teacher or another student Control cards or charts Vocabulary: subtraction, subtract, take away, minus, minuend, subtrahend, difference. Adaptations: Use cuisenaire rods. Create papers with matching pictures of the quantity of the objects for the child to subtract. Standards: HCPS III -Standard 2: Numbers and Operations: OPERATION SENSE: Understand the meaning of operations and how they relate to each other -Topic Operations -Benchmark MA.K.2.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with accuracy 

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with no significant errors 

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with a few significant errors 

Have difficulty demonstrating subtraction as "taking away," "separating sets," or "counting back" and are unable to determine the difference 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

72


Š2004 Dr. Louise Bogart. All rights reserved.

WHOLE NUMBER OPERATIONS GOLDEN BEADS

Subtraction - Dynamic - With Exchanging Age: 5 to 8 Materials: Trays Mat Subtraction sign Yarn or ribbon Base Ten Materials 1 large and 2 small sets of numeral cards Presentation (Lesson): 1. Invite children. 2. Have children lay out all sets of numeral cards. 3. Teacher asks two children individually to bring a four digit numeral, the first from the large set of numeral cards. and the second from one of the small sets of numerals. 4. Each student returns, (s)he reads the numeral, teacher verifies. Only the student with the large numeral cards is asked to get the quantity that matches the numeral. Student reads quantity and teacher verifies. 5. The student with the large numeral cards places his/her numeral and quantity on the mat using the same format as was used in addition. 6. The student with the small numeral cards takes the quantity that matches his/her numeral from the quantity that has been placed on the mat by the first student and places the quantity on his/her tray. 7. Teacher recaps problem: “name brought amount. name took his/her quantity from name and put it on his/her tray. I wonder how much name has left?� Place ribbon across bottom of materials. 8. Individual students are invited to bring quantities below the line and count, exchanging as necessary. The numeral that matches is retrieved from the unused small set of numeral cards. 9. When all places are counted and labeled, one student is asked to compose the numeral and another is asked to read it 10. Teacher recaps the entire process: name brought amount. name took his/her quantity from name and put it on a tray. When we counted what name had left, we got difference. This operation is called subtraction and subtraction has a special sign. 11. Children return materials and mat to the shelf. Direct Aim (Objective): To introduce the concept of subtraction without exchanging through a concrete experience. Indirect Aims (Scaffolding): 73


Preparation for higher math Cooperative Learning Points of Interest (Motivation): Materials Yarn or ribbon Subtraction sign Working with friends Operating with “big numbers” Control of Error (Self Assessment): Teacher or another student Control cards or charts Vocabulary: subtraction, subtract, take away, minus, minuend, subtrahend, difference, exchange, change, rename, regroup, trade Adaptations: Use cuisenaire rods. Create papers with matching pictures of the quantity of the objects for the child to subtract. Standards: HCPS III -Standard 2: Numbers and Operations: OPERATION SENSE: Understand the meaning of operations and how they relate to each other -Topic Operations -Benchmark MA.K.2.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with accuracy 

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with no significant errors 

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with a few significant errors 

Have difficulty demonstrating subtraction as "taking away," "separating sets," or "counting back" and are unable to determine the difference 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved

74


WHOLE NUMBER OPERATIONS GOLDEN BEADS Division - Static - Without Exchanging

Age: 5 to 8 Materials: Trays Mat Yarn or ribbon Division sign Base Ten Materials 1 large and 3 small sets of numeral cards Presentation (Lesson): 1. Invite children. 2. Have children lay out all sets of numeral cards. 3. Teacher asks one student to bring a four digit numeral from the large set of numeral cards. Teacher is careful to give a numeral that will not require exchanging. 4. The student returns, (s)he reads the numeral, teacher verifies. Student with the large numeral cards is asked to get the quantity that matches the numeral. Student reads quantity and teacher verifies. 5. The student with the large numeral cards places his/her numeral and quantity on the mat using the same format as was used in addition. 6. Each student involved in the lesson gets a tray (no more than 9) and sits around the mat containing the quantity. 7. The teacher tells the students that they are going to share equally the quantity on the mat. The teacher shares the quantity equally among trays starting with thousands. Teacher stops after each round and has children check to make sure they all have the same amount (an equal share). Continue until all quantities that can be are shared equally. 8. Teacher recaps problem: “name brought amount. We then took that quantity and shared it equally among the three of you. I wonder how much each of you has.” 9. Individual students are invited to count the quantity on their trays. Each student retrieves the numeral that matches from one of the small set of numeral cards. 10. Each student is asked to compose the numeral on his/her tray and read it.. 11. Teacher recaps process: name brought amount. We shared it equally among the three of you. When each of you counted what was on your tray, each of you had quotient. So, when we take quantity and divide it into three equal shares, each share has quotient.” Place quantity ÷ 3 = quotient while recapping. “This operation is called division and division has a special sign.” 12. Children return materials and mat to the shelf. Direct Aim (Objective): To introduce the concept of division without exchanging through a concrete experience. 75


Indirect Aim (Scaffolding): Preparation for higher math Cooperative learning Points of Interest (Motivation): Materials Yarn or ribbon Division sign Working with friends Operating with “big numbers” Control of Error (Assessment): Teacher or another student Control cards or charts Vocabulary: division, divide, divisor, dividend, quotient, equals Adaptations: Use cuisenaire rods. Create papers with matching pictures of the quantity of the objects for the child to divide. Standards: HCPS III -Standard 2: Numbers and Operations: OPERATION SENSE: Understand the meaning of operations and how they relate to each other -Topic Operations -Benchmark MA.K.2.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with accuracy 

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with no significant errors 

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with a few significant errors 

Have difficulty demonstrating subtraction as "taking away," "separating sets," or "counting back" and are unable to determine the difference 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations. ©2004 Dr. Louise Bogart. All rights reserved.

76


WHOLE NUMBER OPERATIONS GOLDEN BEADS Division - Dynamic - With Exchanging

Age: 5 - 8 Materials: Trays Mat Yarn or ribbon Division sign Base Ten Materials 1 large and 3 small sets of numeral cards Presentation (Lesson): 1. Invite children. 2. Have children lay art all sets of numeral cards. 3. Teacher asks one student to bring a four digit numeral from the large set of numeral cards. 4. The student returns, (s)he reads the numeral, teacher verifies. The student with the large numeral cards is asked to get the quantity that matches the numeral. Student reads quantity and teacher verifies. 5. The student with the large numeral cards places his/her numeral and quantity on the mat using the same format as was used in addition. 6. Each student involved in the lesson gets a tray (no more than 9) and sits around the mat containing the quantity. 7. The teacher tells the students that they are going to share equally the quantity on the mat. The teacher shares the quantity equally among trays starting with thousands. Teacher stops after each round and has children check to make sure they all have the same amount (an equal share). Continue until all quantities that can be are shared equally. 8. Teacher recaps problem: “name brought amount. We then took that quantity and shared it equally among the three of you. I wonder how much each of you has.” 9. Individual students are invited to count the quantity on their trays. Each student retrieves the numeral that matches from one of the small set of numeral cards. 10. Each student is asked to compose the numeral on his/her tray and read it.. 11. Teacher recaps process: name brought amount. We shared it equally among the three of you. When each of you counted what was on your tray, each of you had quotient. So, when we take quantity and divide it into three equal shares, each share has quotient.” Place quantity ÷ 3 = quotient while recapping. “This operation is called division and division has a special sign.” 12. Children return materials and mat to the shelf. Direct Aim (Objective): To introduce the concept of division with exchanging through a concrete experience. Indirect Aim (Scaffolding): 77


Preparation for higher math

Cooperative learning

Control of Error (Self Assessment): Teacher or another student

Control cards or charts

Points of Interest (Motivation): Materials Yarn or ribbon Division sign Working with friends Operating with “big numbers” Vocabulary: division, divide, divisor, dividend, quotient, equal, exchange, change, regroup, rename, trade Note: The quotient is defined as one share or “what one skittle gets.” Adaptations: Use cuisenaire rods. Create papers with matching pictures of the quantity of the objects for the child to divide. Standards: HCPS III -Standard 2: Numbers and Operations: OPERATION SENSE: Understand the meaning of operations and how they relate to each other -Topic Operations -Benchmark MA.K.2.2 -Rubric Advanced

Proficient

Partially Proficient

Novice

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with accuracy 

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with no significant errors 

Demonstrate subtraction as "taking away," "separating sets," or "counting back" and determine the difference, with a few significant errors 

Have difficulty demonstrating subtraction as "taking away," "separating sets," or "counting back" and are unable to determine the difference 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations. ©2004 Dr. Louise Bogart. All rights reserved.

78


FRACTIONS

Concept Development - Regional Model Age: 4 to 8 Materials: Set of ten circle insets, from one whole to circles divided into equal parts from 2 through 10; circles are red and frames are green. Apple, knife, cutting board, and plastic gloves One golden bead A sphere of playdough or clay Slips of paper and marker to write on them Mat Presentation 1 (Lesson 1): 1. Invite the child(ren). 2. Ask if they have ever shared food with someone. Ask them to talk about how they shared. 3. Show the child(ren) the apple. We can share this apple equally by cutting it. Cut the apple in half. “Now the apple is no longer whole because we have divided it into two equal parts. How many children could have a share.” (Wait for response.) “Each child would get one-half of the apple.” 4. Cut each half in half. “Now how many children can have a share?” (Wait for response.) “Each child receives one-fourth of the apple.” 5. If you have more children, divide the fourths into eighths so there are enough parts for each child to have a piece. If there are more children, have enough apples to repeat the process. 6. Pass out the pieces of apple. Presentation 2 (Lesson 2): 1. Invite the child(ren). 2. Prepare a long floor mat and bring two trays of circle insets to the mat. 3. Show child(ren) the golden one bead and explain that in order to find numbers smaller than one we would have to divide this “one” into equal pieces. An easy way to do that is to flatten it into a circle. 4. Take the sphere of playdough or clay and flatten it into a circle. Now I can divide this circle into equal parts. Relate this circle to the whole circle inset. 5. Trace the whole circle and cut it into two pieces that are not equal explaining that although there are two pieces because they are not equal, they are not halves. Explain that when we divide the whole (one) into equal parts we call each part a fraction. 6. Relate fractions to everyday life, for example, time, recipes, food buying. 7. Return materials to the shelf. 8. Return mat.

79


Presentation 3 (Lesson 3): 1. Invite the child(ren). 2. Prepare a long floor mat and bring the two trays of fractional insets to the mat. 3. Show the whole circle and ask the children how many circles you have. 4. Write “1” on a slip of paper. 5. Reminding child(ren) that the whole can be divided into equal parts, move on to the circle divided into two equal parts. Pick up the two parts and put them back to back to show they are equal. 6. “How many parts is this circle divided into?” (Write the line for division as you say the word “divided” and wait for response.) 7. Write the “2” below the line indicating that “2” is the family name, i.e., it is the part of their name that they all have in common. 8. Remove one-half from the inset and place it on the mat. Ask, “How many members of the family of “2” did I place on the mat?” Write “1” above the line and place the label on the fractional part on the mat. 9. Ask, “How many members of the family of “2” remain in the inset?” Write “1” above the line and place the label on the fractional part still in the inset. 10. Continue this procedure until all fractional parts are labeled. 11. Show child(ren) the prepared slips that will be with the material for them to work with. 12. Return materials to the shelf. 13. Return mat. Presentation 4 (Lesson 4): 1. Invite the child(ren). 2. Prepare a long floor mat and bring the two trays of fractional insets to the mat. 3. Take the fractional insets for halves out of the inset and ask the child(ren)< “How many parts is this circle divided into?” (Write the line for division as you say the word “divided” and wait for a response.) 4. Write “2” below the line and ask, “How many family members am I pointing to?” Write the “1” above the line. 5. Provide the language for numerator and denominator indicating the number below the line is the name of the family and tells us how many parts the circle is divided into. This number is called the denominator. The number above the line tells us how many of the family members we have to work with and that is called the numerator. 6. Do a three period lesson to teach vocabulary. 7. Return materials to the shelf. 8. Return mat. Direct Aim (Objective): To help children develop the concept of fractional parts and the numbering system that goes with them. Indirect Aims (Scaffolding): Preparation for higher math

80


Preparation for science Preparation for art Preparation for language Develop visual memory and internalization of sizes of fractional parts Points of Interest (Motivation): Colors Circles, Insets and Frames Different sizes of fractional parts New numbering system New vocabulary Control of Error (Self-Assessment): Inset frames and insets Teacher Vocabulary: Circles, insets, frames, fractions, fractional parts, numerator, denominator, family, family members, whole, half, halves, thirds, fourths, fifths, sixths, sevenths, eighths, ninths, tenths Extensions: Use other geometric shapes Have children trace fractional parts into their math journals, color and label them Have children trace fractional parts onto construction paper, cut them out, glue them on large poster board, and label them for display Have students trace fractional parts onto construction paper, cut them out , glue them onto pages to make a booklet and label them Add derivations of words: Fraction is from the Latin “frangere” which means to break or to divide Numerator has the same root as number Denominator has the same root as denomination which is used for money and for delineation of groups like religions Add history: Romans wrote division as 8 fractus 4 and later modified it to 8 F 4 A further modification was 8/4 and today we write 8 ÷ 4 Because of this notation for division, we write fractions the way we do Note: Lesson on concept development should also be done with linear & set models. Adaptations: Use different materials or objects. Use dye cut circles to create different fractions. Standards: HCPS III -Standard 1: Numbers and Operations: NUMBER SENSE: Understand numbers, ways of representing numbers, relationships among numbers, and number systems

81


-Topic Numbers and Number Systems -Benchmark MA.1.1.2 -Rubric Advanced

Proficient

Consistently Usually identify identify representations of representations of simple fractions  simple fractions and provide representations of common fractions 

Partially Proficient

Novice

Sometimes Rarely identify identify representations of representations of simple fractions  simple fractions 

HPCS -Domain IV: Cognitive Development Mathematics -Standard 1: Learn about number, numerical representation, and simple numerical operations.

©2004 Dr. Louise Bogart. All rights reserved.

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MEASUREMENT ACTIVITY AGES • 4 to 6 years MATERIALS • String • Paper • Scissors • Small Red Rod • Tape • Chart PRESENTATION 1. Invite child 2. Child stands back against the wall and teacher marks the paper to the top of their head 3. Child holds one end of string on floor and teacher cuts the string at the mark on the wall 4. Child uses small Red Rod to measure length of string 5. Chart the length 6. Repeat lesson with as many students who show interest 7. Have children explain what they see on the chart DIRECT AIMS • Introduction to measurement • Introduction to charts INDIRECT AIMS • Making comparisons POINTS OF INTEREST • Height on wall • Length of string • Using Red Rods in new ways • Chart CONTROL OF ERROR • Teacher overview

VOCABULARY • Height • Length • Chart VARIATIONS • Use different lengths of Red Rods ADAPTATION • For elementary age children, trace whole body on large paper and find height, width, perimeter, and area

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HAWAII PRESCHOOL CONTENT STANDARDS Domain IV: Cognitive Development Mathematics â&#x20AC;˘ Standard 4: Develop and use measurement concepts HAWAII CONTENT AND PERFORMANCE STANDARDS Standard 4: Measurement: FLUENCY WITH MEASUREMENT: Understand attributes, units, and systems of units in measurement; and develop and use techniques, tools, and formulas for measuring

Topic Benchmark MA.K.4.1 Sample Performance Assessment (SPA)

Rubric Advanced Proficient Use and describe an Compare and order appropriate method to objects according to compare and order length, weight, objects according to capacity, area, and length, weight, volume, with no capacity, area, and significant errors volume, with accuracy

Measurement Attributes and Units Compare and order objects according to length, weight, capacity, area, and volume The student: Compares objects, physically or virtually, to each other (e.g., places two objects side-by-side to determine which is longer; picks up two objects to determine which is heavier; fills a liquid from one container to another to determine which one has more capacity). Partially Proficient Compare and order objects according to length, weight, capacity, area, and volume, with a few significant errors

Novice Use an inappropriate method to compare and order objects according to length, weight, capacity, area, and volume

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Lesson Plan Numeration to Ten Cards and Counters Age: 3 to 4 years Materials: Mat Cards with number and printed objects 55 counters Presentation: (Lesson) 1. Invite the child(ren). 2. Place container in upper left corner of mat. 3. Place the card in any order on the mat and match them with the number in the story. 4. Read the numeric name and trace the symbol with two fingers and match it with the counters. 5. Count each counters as you place them beside the card, from 1 to 10. 6. Return materials to container and return container to shelf. 7. Return the mat.

Direct Aim (Objective):

To provide practice in association of symbol and quantity.

To introduce the concept of odd and even.

Indirect Aims (Scaffolding): Preparation for higher Math

Develop visual memory Preparation for language

Preparation for science

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Points of Interest (Motivation): Colorful counters Colorful story page

Control of Error (Self-Assessment): Exact amount of counters

Number of pictures

Vocabulary: Pounce, crawl, petals, slither, blossoms, nibble, odd and even

Extension: Let the child(ren) group the objects and check the ones without a partner.

Adaptation:

Let the child explore and read the story as he wishes.

Put the counters and the cards beside the story book.

Standards: HPCS

Domain III- Communication, Language development and Literacy

Standard 3- Acquire increasingly rich vocabulary and sentence structure.

Domain IV- Cognitive development

Standard 1- Learn about number, numerical representation and simple numerical operations.

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Topic

Numbers and Number Systems Count and compare groups of Benchmark MA.K.1.1 objects up to 30 according to the number of objects in each group Sample Performance Assessment The student: Counts the number of (SPA) objects in one group and the number of objects in a second group and indicates which group has more.  Rubric Partially Advanced Proficient Novice Proficient Count and Count and Count and Count and compare groups compare groups compare groups compare groups of objects up to of objects up to of objects up to of objects up to 30 according to 30 according to 30 according to 30 according to the number of the number of the number of the number of objects in each objects in each objects in each objects in each group, with group, with no group, with a few group, with many accuracy  significant errors  significant errors  significant errors 

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How many do you have? Ages: 4-6 years Materials: Mat Folded numeral cards labeled 1-10 55 counters Container Presentation: 1. Invite the child 2. Lay out the mat 3. Place the container between the child and yourself 4. Teacher takes a folded numeral card, looks at it and hides it without showing anyone 5. Teacher gets the appropriate number of counters her folded numeral card says 6. Ask the child if he/she would like to choose a number 7. Child picks a folded number card, looks at the number and gets the number of counters needed. 8. Teacher counts the number of counters she has, modeling one to one correspondence, touching each clothespin as she counts. 9. Teacher states how many counters she had and shows the folded number card she has 10. The child completes steps 8 and 9 11. If child is still interested teacher and child grab another number and repeat the same steps Direct Aim: Test for mastery of numeration to 10 One to one correspondence Indirect Aim: Preparation for higher math Develop visual memory Order Concentration Coordination Independence Points of Interest:

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The mystery Folded numeral cards Counters Control of Error: Just enough counters (55) Vocabulary: mystery, folded, names of numerals, clothespin Variations: Different types of counters Adaptation: Take two or more cards and add the numbers together if the child is beyond 10. For children who need a little more help with counting or number recognition do a three period lesson. Standards: HCPS III ! -Standard 1: Numbers and Operations: NUMBER SENSE: Understand ! numbers, ways of representing numbers, relationships among ! numbers, and number systems. ! -Topic: Numbers and Number Systems ! -Benchmark MA.K.1.2 ! -Rubric: Advanced Proficient Partially Proficient Novice Represent whole Represent whole Represent whole numbers up to 30 numbers up to 30 numbers up to 30 in a variety of ways, in a variety of ways, in one way, with no using different with no significant significant errors  forms of errors  representation 

Represent whole numbers up to 30 in one way, with significant errors

HPCS ! -Domain IV: COGNITIVE DEVELOPMENT MATHEMATICS ! -Standard 1: Learn about number, numerical representation, and simple ! numerical operations.

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Math Album-Jodie Unten