math album-jodie unten

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Math Lessons

Edited by: Jodie Unten

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By: Dr. Louise BogartRevised 2009

Published by:The Christina Trudeau Education and Research Foundation

5862 Fountainhead St. SE Salem, OR 97306-9020 (503) 364-4362http://www.cterf.org

All right reserved.Copyright ©2004 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 [email protected]

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, itʼs 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 childʼs 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 childʼs 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 10Quantity - Number Rods

Age: 3 1/2 - 6

Materials:Matten wooden rods measuring from one decimeter to one meter in length with alternatedivisions 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 mathPreparation for sciencePreparation for languagePreparation for artDevelop visual memory, and physical internalization of the base tenDevelop order, concentration, coordination, and independence

Points of Interest (Motivation):Color pattern Making stairHandling the rods 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 Proficient Partially Proficient Novice

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

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

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

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 10Symbol - Sandpaper Numerals

Age: 3 1/2 - 6

Materials:Table/deskSandpaper 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 mathPreparation for sciencePreparation for languagePreparation for artDevelop visual memory, and physical internalization of the base tenDevelop order, concentration, coordination, and independence

Points of Interest (Motivation):Sandpaper Rough and smooth surfacesNumerals shapes and names Color on backingThree period lesson

Control of Error (Self-Assessment):Visual - shape of numeralKinesthetic - 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 thenumber 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


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

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.


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NUMERATION TO 10Association of Quantities and Symbols

Age: 4 - 6

Materials:MatTen 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 usingsigns 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, artDevelop visual memoryDevelop order, concentration, coordination, and independencePoints of Interest (Motivation):

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Hunt to match Working with friendsColors Rods

Control of Error (Self-Assessment):Teacher or other childrenControl 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.









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NUMERATION TO 10Spindle Boxes

Age: 4 - 6

Materials:MatTwo wooden boxes each with five divisions 0 - 4 and 5 - 9Box 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 sciencePreparation for language Preparation for artDevelop visual memoryDevelop order, concentration, coordination, and independence

Points of Interest (Motivation):Spindles zeronumber 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.

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Adaptations:Use larger or smaller objects. Use larger or smaller symbols with containers that match the size of the objects.









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NUMERATION TO 10Cards and Counters

Age: 4 to 6

Materials:MatNumeral cards labeled 1 - 1055 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 sciencePreparation for language Preparation for artDevelop visual memoryDevelop order, concentration, coordination, and independence

Points of Interest (Motivation):CountersPattern 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

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Adaptations:Have children write or draw object that match the corresponding number.









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NUMERATION TO 10Mystery Game

Age: 4 - 6

Materials:MatFolded numeral cards labeled 1 - 1055 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 mathPreparation for sciencePreparation for languagePreparation for artDevelop visual memoryDevelop order, concentration, coordination, and independence

Points of Interest (Motivation):1. Counters2. Folded numerals3. 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.









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NUMERATION TO 10Introduction to the Short Bead Stair

Age: 4 - 6

Materials:MatBead 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 mathDevelop visual memoryDevelop order, concentration, coordination, and independence

Points of Interest (Motivation):ColorsStair 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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NUMERATION TO 10Search For Ten

Age: 4 - 6

Materials:MatBead bars of lengths for quantities 1 - 9 (each quantity a different color)Five tensSet 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 memoryDevelop order, concentration, coordination, and independence

Points of Interest (Motivation):Colors Stair FormationChanging 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


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



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BASE TEN PLACE VALUEINTRODUCTION TO GOLDEN BEAD MATERIAL

Introduction to QuantityAge: 4 to 7

Materials:1 one 1 ten 1 hundred 1 thousandMat

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 sciencePreparation for language Preparation for artInternalize place value kinesthetically IndependenceDevelop 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 ahundred bead square and a hundred wooden square for the child to compare. Thesame is done with the bead cube and wooden cube. Teacher can explain the reasonfor using wood is that beads are very expensive so we cannot afford a lot of them.

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Adaptations:Use larger materials or cuisenaire rods.









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Introduction to SymbolAge: 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 sciencePreparation for language Internalize place value kinestheticallyPreparation for art Develop visual memoryIndependence Concentration

Points of Interest (Motivation):Three period lessonColors on numerals and numerals themselves


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 thenumeral. Read numeral.

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Adaptations:Use larger cards or smaller cards.







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


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Association of Quantity and SymbolsAge: 4 to 7

Materials:Large numeral cards for 1, 10, 100, 1000 One each of golden bead quantities 1, 10, 100, 1000Mat

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 mathPreparation for sciencePreparation for languageInternalize place value kinestheticallyIndependenceConcentrationPreparation for artDevelop visual memory

Points of Interest (Motivation):beadsnumeralsmatching


Vocabulary:numeral, symbol, quantity, base ten place value

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Crisis of NineAge: 4 to 7

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 inany place is 9. Therefore, after counting 9 you must change place.

Indirect Aim (Scaffolding):Preparation for higher math, science, art, languageDevelop visual memory, and physical internalization of place value and base ten.Develop order, concentration, and independence.

Points of Interest (Motivation)Materialsexchanging and moving to next place counting


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Vocabulary:names of quantities 1 - 9, ones, tens, hundreds, thousands, place value, base ten

Extensions:Exchanging gameDo 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


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



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Quantity LayoutAge: 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 kinestheticallyPreparation for higher math, science, language, art

Point of Interest Motivation):Beads PatternLayout Using hand or ten to measure space

Control of Error (Self Assessment):Teacher Material on tray

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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.









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Fetching QuantitiesAge: 4 1/2 to 8

Materials:Tray Mat9 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, artDevelop auditory memory

Points of Interest (Motivation):bead layout shapes of quantitiespattern

Control of Error (Self Assessment):Teachermaterial

Vocabulary:Names of base ten place values

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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.









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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 kinestheticallyPreparation for higher math, science, language, art

Point of Interest (Motivation):Symbols and colors PatternLayout Seeing large numerals

Control of Error (Self Assessment):Teacher Material on tray

Vocabulary:Layout, pattern, ones, tens, hundreds, thousands, place value, base ten

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Extension:Fetching Games

Adaptations:Use larger or smaller cards with numbers.









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Fetching SymbolsAge: 4 1/2 to 8

Materials:TrayMatLarge 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, artDevelop auditory memory

Points of Interest (Motivation):symbols layoutcolors pattern

Control of Error (Self Assessment):Teachermaterial

Vocabulary:Names of base ten place values

Variations:If child cannot remember four places, try three, then 2, if necessary.

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Adaptations:Depending on the child’s ability ask for quantities you know they can fetch. Use different materials.









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Birdseye View (45 Layout)Age: 4 1/2 to 8

Materials: Large Mat Set of Large Numeral Cards45 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 layoutPreparation for higher math, science, language and art

Points of Interest (Motivation):Beads and symbols patternlayout building on large mat

Control of Error (Self Assessment):Teacher or other children 45 of each quantitysymbols

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Vocabulary:layout, pattern, one, tens, hundreds, thousands, base ten, place value

Adaptations:Have children work together. Use smaller materials to work at a table.









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LINEAR COUNTINGTEENS: 11 - 19 Quantity

Age: 4 to 7

Material:Short bead stair 1 - 99 ten barsCounterMat

Presentation (Lesson):1. Invite the child(ren).2. Build short bead stair in upper left corner of mat with ten bars below3. 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 stairCombinations of bead bars.

Control of Error (Self Assessment):TeacherMaterials

Vocabulary:eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen

Extension:Using other sets of bead stairs, make other combinations of 11 - 19 besides10 + one to nine example: 9+2=11 8+3=11 7+4=11

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Adaptations:Use cuiseniare rods.









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LINEAR COUNTINGTEENS: Symbols

Age: 4 to 7

Materials:Teen Boards - Two wooden boards with vertical rows of nine 10’s. A slot is providedto cover the zero with a numeral to compose the numerals from 11 to 19Mat

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 numeralsSliding in numbers


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.

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LINEAR COUNTINGTEENS: Quantity and Symbols

Age: 4 to 7

Materials :MatTeen boardsShort bead stair9 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 stairSlotsSequence


Vocabulary:Names for quantities and symbols for 11 - 19

Extension:Use other combinations of bead stair 1 - 9 to make teen quantitiesMix numerals and have child(ren) put the matching bead combinations alongsideeach numeralMix quantities next to board and have child(ren) put matching numerals on theboard

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LINEAR COUNTINGTENS: Quantity

Age: 4 1/2 to 7

Materials:45 tensMat

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


Points of Interest (Motivation):BeadsNew vocabulary

Control of Error (Self Assessment):TeacherNumber 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

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








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LINEAR COUNTINGTENS: Symbols

Age: 4 1/2 to 7

Materials:Ten boardsMat

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


Points of Interest (Motivation):Wooden boxesBoardsNumerals and their names


Vocabulary:Names 10 - 90

Adaptations:Place the numbers for the ones place in order from 1-9.

Standards:HCPS III

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-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








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LINEAR COUNTINGTENS: Quantity and Symbol

Age: 4 1/2 to 7

Materials:ten boards Mat9 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. Alwaysbring the beads down so the child can see the progression.

Direct Aims (Objective):To associate quantity, name and symbols for 11 - 99Build numbering system one bead at a time


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

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Extension:Provide materials so that children can make a number roll.










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LINEAR COUNTINGHundred Board

Age: 5 - 8

Materials:Board with 10 rows of 10Tiles 1 - 100Mat

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 TilesFilling 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 tilesControl chart

Vocabulary:Hundred, board & tiles, tens, sort, numerals 1-100

Extensions:Skip count by 2’s, 5’s, and 10’s

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Adaptations:Put the hundred board on a piece of paper. Fill in some spaces to help children fill out the hundred board.









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LINEAR COUNTINGHUNDRED CHAIN

Age: 5 – 7

Materials:100 chain100 squareArrows 10 - 100Mat

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 countingTo introduce skip counting (by 10’s)


Points of Interest (Motivation):Folding and unfolding squareArrowsCounting 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 anddoes it that way.

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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


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



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LINEAR COUNTING1000 Chain

Age: 5+

Materials:1000 chain1000 cube100 chain9 - 100 squaresArrows 10 - 1000Large 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


Points of Interest (Motivation):Length of chainFolding and unfoldingArrows

Control of Error (Self Assessment):TeacherBeads

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.

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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


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



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WHOLE NUMBER OPERATIONSGOLDEN BEADS

Addition - Static - No ExchangingAge: 41/2 to 7

Materials:Trays MatYarn or ribbon Addition signBase ten blocks or Golden BeadsOne 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.

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Indirect Aim (Scaffolding):Preparation for higher mathCooperative learning

Points of Interest (Motivation):MaterialsYarn or ribbonAddition signWorking with friendsOperating with “big numbers”

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


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



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Addition - Dynamic - With ExchangingAge: 41/2 to 7

Materials:Trays MatYarn or ribbon Addition signBase ten blocks or Golden BeadsOne 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.

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Points of Interest (Motivation):MaterialsYarn or ribbonAddition signWorking with friendsOperating with “big numbers”

Control of Error (Self Assessment):Teacher or another studentControl 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.









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Multiplication - Static Without Exchanging

Age: 41/2 to 7

Materials:Trays MatYarn or ribbon Multiplication signBase ten material or Golden bead materialOne 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 concreteexperience.

Indirect Aim (Scaffolding):Preparation for higher mathCooperative Learning

Points of Interest (Motivation):Materials

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Multiplication signYarn or ribbon Working with friendsOperating with “big numbers”

Control of Error (Self Assessment):Teacher or another studentControl 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 tookthe same amount 3 times, we only need to write the amount once.” Remove second2 numerals and replace them with a 3 and the multiplication sign saying, “You tookamount 1 time, 2 times, 3 times.” Place the 3 as you say “three” and the X as yousay “times”.

Adaptations:Use cuisenaire rods. Create papers with matching pictures of the quantity of the objects for the child to multiply.









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Multiplication - Dynamic - With Exchanging

Age: 5 to 7

Materials: Yarn or ribbon MatMultiplication sign TrayBase 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 concreteexperience.

Indirect Aim (Scaffolding):Preparation for higher math Cooperative Learning

Points of Interest (Motivation):

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Materials Yarn or ribbonMultiplication sign Working with friendsOperating with “big numbers”


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 tookthe same amount 3 times, we only need to write the amount once.” Remove second2 numerals and replace them with a 3 and the multiplication sign saying, “You tookamount 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.









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Subtraction - Static - Without Exchanging

Age: 5 to 8

Materials:Mat TraysSubtraction sign Yarn or ribbonBase Ten Materials1 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 concreteexperience.

Indirect Aims (Scaffolding):Preparation for higher math

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Cooperative Learning

Points of Interest (Motivation):Materials Yarn or ribbonSubtraction sign Working with friendsOperating with “big numbers”


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.



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


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Subtraction - Dynamic - With Exchanging

Age: 5 to 8

Materials:Trays MatSubtraction sign Yarn or ribbonBase Ten Materials1 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 it10. 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 concreteexperience.

Indirect Aims (Scaffolding):

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Preparation for higher mathCooperative Learning

Points of Interest (Motivation):Materials Yarn or ribbonSubtraction sign Working with friendsOperating with “big numbers”


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.








©2004 Dr. Louise Bogart. All rights reserved

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Division - Static - Without Exchanging

Age: 5 to 8

Materials:Trays MatYarn or ribbon Division signBase Ten Materials1 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 hasquotient.” 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 concreteexperience.

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Points of Interest (Motivation):MaterialsYarn or ribbonDivision signWorking with friendsOperating with “big numbers”

Control of Error (Assessment):Teacher or another studentControl 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.









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Division - Dynamic - With ExchangingAge: 5 - 8

Materials:Trays MatYarn or ribbon Division signBase Ten Materials1 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):

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Preparation for higher math Cooperative learning

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

Points of Interest (Motivation):MaterialsYarn or ribbonDivision signWorking with friendsOperating 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.









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FRACTIONSConcept Development - Regional Model

Age: 4 to 8

Materials:Set of ten circle insets, from one whole to circles divided into equal parts from2 through 10; circles are red and frames are green.Apple, knife, cutting board, and plastic glovesOne golden beadA sphere of playdough or claySlips of paper and marker to write on themMat

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.

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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 systemthat goes with them.

Indirect Aims (Scaffolding):Preparation for higher math

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Preparation for sciencePreparation for art Preparation for languageDevelop visual memory and internalization of sizes of fractional parts

Points of Interest (Motivation):Colors Circles, Insets and FramesDifferent sizes of fractional parts New numbering systemNew vocabulary

Control of Error (Self-Assessment):Inset frames and insetsTeacher

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 shapesHave children trace fractional parts into their math journals, color and label themHave children trace fractional parts onto construction paper, cut them out, glue themon large poster board, and label them for displayHave students trace fractional parts onto construction paper, cut them out , gluethem onto pages to make a booklet and label themAdd derivations of words:Fraction is from the Latin “frangere” which means to break or to divideNumerator has the same root as numberDenominator has the same root as denomination which is used for money and fordelineation of groups like religionsAdd history:Romans wrote division as 8 fractus 4 and later modified it to 8 F 4A further modification was 8/4 and today we write 8 ÷ 4Because 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

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


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

Usually identify representations of simple fractions

Sometimes identify representations of simple fractions

Rarely identify representations of simple fractions



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MEASUREMENT ACTIVITY

AGES• 4 to 6 years

MATERIALS• String• Paper• Scissors• Small Red Rod• Tape• Chart

PRESENTATION1. Invite child2. Child stands back against the wall and teacher marks the paper to

the top of their head3. Child holds one end of string on floor and teacher cuts the string at

the mark on the wall4. Child uses small Red Rod to measure length of string5. Chart the length6. Repeat lesson with as many students who show interest7. 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 STANDARDSDomain IV: Cognitive Development Mathematics

• Standard 4: Develop and use measurement concepts

HAWAII CONTENT AND PERFORMANCE STANDARDSStandard 4: Measurement: FLUENCY WITH MEASUREMENT: Understand attributes, units, and systems of units in measurement; and develop and use techniques, tools, and formulas for measuring

TopicTopic Measurement Attributes and UnitsMeasurement Attributes and Units

Benchmark MA.K.4.1Benchmark MA.K.4.1Compare and order objects according to length, weight, capacity, area, and volumeCompare and order objects according to length, weight, capacity, area, and volume

Sample Performance Assessment (SPA)Sample Performance Assessment (SPA) 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).

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).

RubricRubricRubricRubricAdvanced Proficient Partially Proficient NoviceUse and describe an appropriate method to compare and order objects according to length, weight, capacity, area, and volume, with accuracy

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

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

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

http://165.248.30.40/hcpsv3/imr/report_by_code.jsp?code=MA.K.4.1


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Lesson PlanNumeration to TenCards and Counters

Age: 3 to 4 years

Materials: MatCards with number and printed objects55 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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TopicTopic Numbers and Number SystemsNumbers and Number Systems

Benchmark MA.K.1.1Benchmark MA.K.1.1Count and compare groups of objects up to 30 according to the number of objects in each group

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

Sample Performance Assessment (SPA)Sample Performance Assessment (SPA)

The student: Counts the number of objects in one group and the number of objects in a second group and indicates which group has more.

The student: Counts the number of objects in one group and the number of objects in a second group and indicates which group has more.

RubricRubricRubricRubric








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How many do you have?

Ages: 4-6 years

Materials:MatFolded numeral cards labeled 1-1055 countersContainer

Presentation:1. Invite the child2. Lay out the mat3. Place the container between the child and

yourself4. Teacher takes a folded numeral card, looks at

it and hides it without showing anyone5. Teacher gets the appropriate number of

counters her folded numeral card says6. Ask the child if he/she would like to choose a

number7. 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 911. If child is still interested teacher and child grab another number and repeat

the same steps

Direct Aim:Test for mastery of numeration to 10One to one correspondence

Indirect Aim:Preparation for higher mathDevelop visual memoryOrderConcentrationCoordinationIndependence

Points of Interest:

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The mysteryFolded numeral cardsCountersControl 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:






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

math album-jodie unten

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