Introduction to the Decimal System ... :: ... : Mathematics I NAMC - North American Montessori Center

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1 Introduction to the Decimal System ~~~ ::... :... 55

2 "Your child can hold a unit bead in one hand and a thousand cube in the other to have a sensory-based impression of the difference between them. A thousand cube is 1,000 times as big as a unit bead, it's 1,000 times as heavy, and it has 1,000 times the value - because it actually contains 1,000 unit beads. You can't get more concrete than that!" - Michael Duffy, Math Works: Montessori Math and the Developing Brain, p B Mathematics I NAMe - North American Montessori Center

3 Introduction to the Decimal System Golden Beads are a materialized abstraction of the decimal system At this point in the math curriculum, the child is ~ comfortable with the names, quantities, and symbols for the numbers 1-10 through her work with the Number Rods, Sandpaper Numbers, Spindle Boxes, and Numbers and Counters activities. She also has a solid understanding that 0 means nothing. Now, she is ready to learn that these numbers can be used to represent all numbers of the decimal system. "The hands are the instruments of man's intelligence." - Maria Montessori, The Absorbent Mind, p. 27. The Decimal System and Its Organization The decimal system, also known as the base-1 0 system, is a number system based on units of ten. In its simplest terms, the decimal system is a mechanism to organize numbers. In the decimal system, the value of each digit is determined by its The child is familiar with the names, quantities, and symbols of numbers 1-10 from prerequisite work in the Numbers to Ten activities 3-6 Mathematics I NAMC. North American Montessori Center 57

4 place value - the value a digit has due to its position within a number. For example, the 2 in 250 represents two hundreds, whereas the 2 in 520 represents two tens. Place value allows the quantity 3, to be represented more simply as 3,925. If place value had not been created, numbers would always be written in long form. The decimal system is a number system based on units of ten The Structure of the Decimal System The structure of the decimal system can be summarized as follows: There are nine in each category: 1-9, 10-90, , and so on. The ratio of any two consecutive categories is always 1:10. There are three categories in each class: units, tens, and hundreds. The ratio of two consecutive classes is always 1:1,000. The classes can be repeated infinitely: simple, thousands, millions, billions, trillions, and so on. The positional values of numbers can also be called categories. In each category, there are nine digits - the integers 1-9 (and 0). Regardless of the category in which a number appears, its absolute value remains the same. For example, in the number 111, each digit has the same absolute value of one. However, the relative value of each digit depends on its position or category. So, the number 1 on the left has a value of one hundred, the number 1 in the middle has a value of one ten, and the number 1 on the right has a value of one unit. The decimal system is made up of three categories that are repeated infinitely within groups called classes or orders of numbers. The three repeating categories are units, tens, and hundreds. The first class of numbers is called the simple class and consists of units, tens, and hundreds. Some Montessori materials in this class are colored white in the Montessori math system. The second class is called the thousands class and consists of units of thousands, tens of thousands, and hundreds of thousands; however, the child works only up to 9,999 in the Montessori early childhood environment. Some Montessori materials in the thousands class are colored gray. The elementary student works with concepts and Montessori materials into the millions class, which is colored black. Montessori Math In the Montessori mathematics program, the decimal system is formally introduced with the Golden Bead Material, which consists of unit beads, 10-bars, 100-squares, and 1,000-cubes. The beads are uniform in size and color so that only their quantities differ. As the child manipulates the Golden Bead Material, he gains a sensorial understanding of place value and the power of ten through sight, touch, and weight. 58

5 The Golden Bead Material represents the decimal system quantities of units, tens, hundreds, and thousands After working with the Golden Bead Material to learn the names and quantities of the decimal categories, the child progresses to working with more abstract material. He is introduced to the written symbols of the categories with the Large and Small Number Cards. The number cards present numbers 1-9, 10-90, , and 1,000-9,000 on a series of wooden cards that incorporate the Montessori place value color-coding: the units cards are green, the tens cards are blue, the hundreds cards are red, and the thousands cards are green. In Montessori mathematics, material that represents place value is always color-coded in the same way, and the same three colors - green, blue, and red - are repeated throughout the classes. As the following table shows, in each class, green is used for units, blue is used for tens, and red is used for hundreds. This consistency helps the child recognize place value more easily, even when the material and concepts are new to him. Montessori Decimal System Color-Coding Class Color Category Color Simple White Units Green Tens Hundreds Blue Red Thousands Gray Units of thousands Green Tens of thousands Hundreds of thousands Blue Red Writing N "l1 When we write numbers, we often record the digits in groups of threes, separating them into their classes. The marker used to indicate the groups can differ from country to country. In North America, classes are often separated with a comma, like this: 45,629; 7,664,125; and 3,187. In Europe, classes are usually indicated with a thin space: and Using this method, four-digit numbers are kept together, like this, Another method is to use a period or center dot: ; ; and And some countries use an apostrophe: 45'629; 7'664'125; and 3'187. To make the numbers easier to read and for formatting reasons, this manual uses commas to separate the classes. On NAMC's Curriculum Support Material, four-digit numbers on Number Slips and Equation Slips do not use commas because the children would not have seen this format in any of the Montessori materials. Millions Black Units of millions Green 59

6 The Small Bead Frame uses Montessori color-coding for place values from units to thousands and indicates the simple class in white and thousands class in gray Using category names makes counting large numbers easier for young children When the child is introduced to quantities with the Golden Bead Material and to numeric symbols with the Large Number Cards, she is also presented with the numbers' names. At this point in the Montessori program, the child is not presented with the conventional names of numbers, which are the names we usually give numbers, such as eleven, twelve, thirteen, and so on. Instead, the focus is on the numbers 1-9 and their repetition within each category. So, for example, to name the quantities 11, 12, and 13, the child uses the category names one ten, one unit (11); one ten, two units (12); and one ten, three units (13). Naming, and subsequently counting, the quantities in this manner has several benefits. First of all, it emphasizes category names, which reinforces the child's understanding of place value and the idea that there is a maximum of nine in each category. Focusing on the category names also furthers the child's understanding of quantity. She can easily connect the name "nine thousands" to the nine 1,OOO-cubes on the mat in front of her. Finally, using category names allows the child to count large quantities with ease. Rather than having to remember a unique name for each number from 1-9,999, she uses the numbers 1-9, which she already knows, and the category names. After working repeatedly with the Golden Bead Material and Large Number Cards separately, the child explores how decimal system quantities and symbols complement each other by working with the two materials in combination. By laying out the full set of Golden Bead Material and the Large Number Cards side by side, the child can see the association of quantities and symbols. In a second activity, the child furthers his understanding of this association by matching number cards to specific quantities of bead material and vice versa. The teacher places a quantity of Golden Bead Material on a tray, such as seven unit beads, and asks the child to retrieve the corresponding Small Number Card, the 7-card. When the child is ready, the teacher introduces more categories, until the child is eventually working with categories to the thousands. This activity also allows the child to practice working with numbers of the decimal system in isolation. This is the first time he works with individual quantities and numbers 60

7 instead of sequencing the material. Forming quantities and numbers in this manner prepares the child for future work in the Four Operations activities in which he will add, subtract, multiply, and divide using the Golden Bead Material and the Large and Small Number Cards. The child is given ample practice working with the decimal system as a static system, exploring the concept that each category contains no more than nine units. He is then invited to discover the dynamic function of the decimal system, discovering what happens when a category exceeds its maximum of nine. Starting with the activity Introducing the Bank, the final two decimal system activities demonstrate how quantities are exchanged once a category has reached its limit of nine (9, 90, 900, or 9,000). For example, the child is invited to count the unit beads and, once he reaches ten, he exchanges ten unit beads for one 10-bar. He then progresses in the same way through the other categories. Working with the decimal system in this manner prepares the child for upcoming work with dynamic operations in the Four Operations activities. The Decimal System and Working from Right to Left up to this point in the Montessori early childhood program, the teacher has consistently demonstrated her movements with the Montessori materials following the Western standard for reading and writing - working from left to right and top to bottom. There is a significant departure from this approach with the introduction of the Decimal System activities and materials. The reason for this deviation concerns the inherent nature of the structure of the decimal system and the usage of the Montessori material associated with it. When working with the decimal system materials, the teacher presents the material starting with the units on the right and progressing to the higher categories, tens, hundreds, and thousands toward the left. This method will be followed throughout the Decimal System activities and the Four Operations activities (except division) when the child will solve arithmetic equations from right to left. Whole to Part Learning The introduction of the decimal system is another example of whole to part learning in Montessori. Once the child has a firm understanding of the structure of the decimal system and its functions (the whole), he then learns about the parts that make up the whole. For example, he learns that 14 is one 10-bar and four unit beads; 78 is seven 1 O-bars and eight unit beads; 156 is one 100-square, five 10-bars, and six unit beads, and so on. To give these parts to the child earlier would distract him from the simplicity and enjoyment of the base-1 0 system. Working with the bank, the child explores what happens when a category exceeds nine and learns about exchanging 3--B Mathematics I NAMC - North American Montessori Center 61

8 /lfofjt/ The Golden Bead Material The Golden Bead Material \:)9::> "The repeated use of 1 a objects ~ (pencils, Metal Inset shapes, and so on) is intentional in Montessori, to reinforce the decimal system." - Angeline Stoll Lillard, Montesson: The Science Behind the Genius, p. 25. The Golden Bead Material, which is also called the Golden Beads, is a set of small gold-colored beads about 1/4 inch (0.64 cm) in diameter that are fashioned into different arrangements. They may be made of glass, ceramic, or plastic. Collectively, the material is referred to as the Golden Bead Material or sometimes the Decimal System Bead Material. The Golden Bead Material represents decimal system categories from units to thousands as follows: One golden bead represents one unit and is commonly referred to as a unit bead. lilll"'" Ten golden beads wired together represent 1 a and are commonly referred to as a 1 a-bar. Ten 1 a-bars wired together form a square that represents 100. They are commonly referred to as a 100-square. Ten 1 OO-squares wired together form a cube that represents 1,000. They are commonly referred to as a 1,OOO-cube. The wooden 1 ~O-squares and 1,ODD-cubes are used interchangeably with the golden bead squares and cubes The golden bead squares and cubes may also be made of wooden slabs and cubes imprinted with a bead design to show that they represent 100 and 1,000 Golden Beads, respectively. 62

9 This material may be used once the child has had ample opportunity to derive the sensorial benefits of working with the glass material and understands that the glass and wood material are equivalent. The Golden Bead Material can be organized in different ways. Two such arrangements that you can put together in your classroom are the Quantity Presentation Tray and the Decimal System Presentation Tray. The Quantity Presentation Tray contains one unit bead, one 1 O-bar, one 1 ~O-square, and one 1,ODD-cube. The Decimal Presentation Tray contains nine unit beads, nine 10-bars, nine 1 ~O-squares, and one 1,000- cube. Many classrooms also create a supply tray or box of Golden Bead Material, sometimes called the bank, which stays in the math area. There is no set quantity for the bank but there should be a sufficient amount of material to carry out exchanges of any type within the first four categories of the decimal system. To add interest, some teachers store the material in a briefcase. However it is stored in the classroom, the Golden Bead Material is always arranged in categories, from the 1,ODD-cubes on the left to the unit beads on the right. The unit beads are usually kept in a small bowl. The original Golden Beads were made of sparkling glass, and the color was chosen specifically to signify the decimal system's high value to society. The first presentations of the Golden Bead Material will be more significant to the child if the material is glass because of the greater weight differences between categories. For example, a 1,ODD-cube is considerably and proportionally heavier (1,000 times) than a single unit bead. The child will already have a solid understanding of size and weight differences from his work with the sensorial dimension materials (i.e., Cylinder Blocks, Pink Tower, Brown Stair, Red Rods, and Knobless Cylinders). The Golden Bead Material presents the first, or simple, class of numbers with the unit beads, 10-bars, and 1 ~O-squares, Quantity Presentation Tray Decimal System Presentation Tray The Bank The supply tray of Golden Bead Material is aptly named the "bank" to emphasize that this is where a large amount of valuable material is kept. Usually, the bank contains at least 55 each of unit beads, 10-bars, and 1 ~O-squares, and a minimum of nine 1,000- cubes. In fact, the bank often has more bead material than that so multiple children can work with the material at the same time. The bank always remains in the math area. When the child requires a quantity of Golden Bead Material, he goes to the bank with a tray and takes the amount he needs. 63

10 Montessori Moment The change from the simple class to the thousands class is represented in the height of the Golden Bead Material. The unit bead, 10-bar, and 1 ~O-square, which make up the simple class, are all the same height. The 1,ODD-cube, however, is ten times greater in height, indicating that it is in a greater class: the thousands class. The presentation tray acts as a guide for the placement of the bead material and part of the second, or thousands, class with the 1,000- cubes. The Golden Bead Material is always carried to and from the work area on a tray. When working with the Quantity Demonstration Tray or the Decimal System Presentation Tray, the tray is used as part of the presentation, acting as an indirect guide for the placement of the material. The teacher removes each material from its place on the tray and places it on the table or mat directly below the tray in that same arrangement. The Golden Bead Material presents the child with his first experience of numbers beyond ten in the Montessori environment, encouraging him to explore multiples of ten in a hands-on way. He already knows that ten units add up to ten; now he will see that ten 1 O-bars add up to 100, and ten 1 ~O-squares add up to 1,000. As the child is familiar with the numbers 1-9, it is a small leap for him to count quantities to the thousands using the Golden Bead Material. This is because of the repetitive structure of the decimal system, in which each category has a limit of nine. Also beneficial is the fact that the child does not use the conventional names of numbers when counting the quantities. For instance, he counts the 1 O-bars as one ten, two tens, three tens, and so on to nine tens. And he counts the 1 ~O-squares as one hundred, two hundreds, and so on to nine hundreds. When the child is first introduced to the Golden Bead Material, he is shown how to lay it out from right to left, starting with the units and progressing to the higher values. The material placed on the mat with the units on the right and progressing to higher values from right to left Initially, each category of the Golden Bead Material is laid out to reference the next higher category. The nine unit beads are placed in a column, working from top to bottom and with the beads touching each other, to look similar to a 10-bar. To the left of the unit beads, the nine 1 O-bars are placed from left to right, vertically and touching each other with their tops aligned, so they look similar to a 1 ~O-square. To the left of the 10-bars, the 1 ~O-squares are placed so 64 3-B Mathematics I NAMC North American Montessori Center

11 their wire loops align with those of the 1 a-bars and they are stacked one on top of the other, building to the size of the 1,000-cube. Finally, the 1,OOO-cube is placed to the left of the 1 OO-squares with the wire loops at the top and bottom. Although a child may learn at home to recite numbers well beyond 10, she may not have a true understanding of number concepts until she gains experience working with real quantities. The Golden Bead Material is an excellent material to introduce the decimal system because the child can handle and manipulate numbers up to 9,999 and all she needs is a solid understanding of the numbers 1-9. In lower elementary, the students learn to name and classify categories into classes: the simple class, the thousands class, and the millions class. Working with real quantities gives the child a true understanding of number concepts Lower elementary students are introduced to geometrical concepts with the Golden Bead Material: a point is represented by a unit bead, a line is represented by a 1 a-bar, a plane is represented by a 100-square, and a solid is represented by a 1,000-cube. Lower elementary students also work with the Golden Bead Material to explore division, working with two- and threedigit dividends and two-digit divisors. Upper elementary students use the Golden Bead Material for more advanced work on the divisibility of numbers. They also use the material to create algebraic expressions and in the exploration of bases, starting with base-10 and progressing to other bases. The Golden Bead Material is used to create algebraic expressions in upper elementary 3-{) Mathematics I NAMC - North American Montessori Center 65

12 I/cfrvrftt Introducing the Golden Bead Material Age years old Direct Aims To become familiar with the names, quantities, and proportions of the categories of units, tens, hundreds, and thousands. To learn the laws that rule the organization of numbers in the decimal system. Indirect Aims To experience the proportional relationships of the decimal system. To visually introduce geometry. Control of Error The teacher. The child's knowledge. Vocabulary One unit, one ten, one hundred,onethousand. Notes This activity is a threeperiod lesson. If working on a table, a piece of felt helps the material stay in place. The unit bead should always be on the righthand side and you work from right to left. Introducing the Golden Bead Material fi?fj5f To help a child become familiar with the names, quantities, and proportions of the first four categories of the decimal system. Quantity Presentation Tray. Invite a child to the math shelf to learn about the decimal system with the Golden Bead Material. 66

13 Ask the child to carry the presentation tray to the table or mat. Ensure that the tray is positioned with the 1,ODD-cube on the left and the unit bead on the right. Leave some space in front of the tray for the material. Point to the bead material in the tray and say, "These are called Golden Beads." Commence with the first period, the naming, "this is" phase of the three-period lesson. Place the unit bead in the palm of your hand and say, "This is one unit. One unit." Place the unit bead in the palm of the child's hand and ask her to repeat the name. Invite her to explore the unit bead: its size, texture, and weight. Ask the child to place the unit bead on the mat below the tray. Place the 1 O-bar in the palm of your hand and say, "This is one ten. One ten." Give the 1 O-bar to the child, and ask her to repeat the name. Invite her to explore the 10-bar: its size, texture, and weight. Ask the child to place the 1 O-bar on the mat. Show her how to position the bar vertically and to the left of the unit bead. Demonstrate that the 1 O-bar is made up of ten unit beads by counting them. Starting at the top, touch each individual bead and count, "One unit, two units, three units... " and so on. Activity Notes The child will already be familiar with the names, quantities, and symbols for one and ten. However, start the three-period lesson with one and ten because the material and category names are new. The specific material names (10-bar, 100- square, etc.) are referenced in the presentation for the teacher's benefit only. It is not intended that the teacher refer to the material in this way when presenting to the child. The presentation tray is positioned to one side in the images to clearly show the bead material. During the presentation, however, center the tray in front of the child and work with the bead material directly below it. Say to the child, "There are ten units in one ten. This is one ten." Place the 1 ~O-square in the palm of your hand and say, "This is one hundred. One hundred." Give the 1 ~O-square to the child, and ask her to repeat the name. Invite her to explore the 1 ~O-square: its size, texture, and weight. The presentation tray is always positioned with the 1,OOO-cube on the left 67

14 Ask the child to place the 1 OO-square to the left of the 1 a-bar with the wire loops at the top and bottom. Demonstrate to the child that the 1 OO-square is made up of 1 a-bars by counting the bars. Starting on the left, trace each bar from top to bottom, and count, "One ten, two tens, three tens..." and so on. Say to the child, "There are ten tens in one hundred. This is one hundred." Teacher showing the child that the 1 a-bar is placed to the left of the unit bead Repeat the previous steps for the 1,OOO-cube, positioning the wire loops at the top and bottom. When counting the squares in the 1,OOO-cube, trace around each side of the 100-square, moving from top to bottom, and count, "One hundred, two hundreds, three hundreds..." and so on. Say to the child, "There are ten hundreds in one thousand. This is one thousand." Continue with the second period, the recognition, "show me" phase, and the third period, the recall, "what is" phase of the three-period lesson. Reinforce the counting and tracing procedures in the second period. Teacher demonstrating how the 1 OO-square is made up of ten 1 a-bars After the child has finished working with the presentation tray, ask her to return the material to the tray and then to the math shelf. Tell the child that she is now free to work with the Quantity Presentation Tray. Performing Stereognostic Exercises a. Invite the child to place one unit, one ten, one hundred, and one thousand in the Mystery Bag, also called the Stereognostic Bag. Ask him to search for one unit and remove it from the bag. Continue with the remaining bead material in the same manner. Child sensorially exploring the 1,OOO-cube b. Invite the child to place one unit, one ten, one hundred, and one thousand in the Mystery Bag. Ask him to feel an item, name it, and then remove it from the bag. Suggest that he 68

15 continue in the same manner with the remaining bead material. c. Invite the child to place two of each bead quantity in the Mystery Bag. Ask him to search for one unit and remove it from the bag. Then, ask him to find its match. Continue with the remaining bead material in the same manner. Removing One Item Invite the child to place the material from the Quantity Presentation Tray in a row on the mat in sequence. Performing stereognostic exercises Ask the child to close her eyes while you remove one item and place it behind your back. Invite the child to open her eyes and to name the missing item. Return the items to the mat. Continue in the same manner with the remaining material. Introducing Montessori Color-Coding Invite the child to learn about the Montessori color-coding system for decimal categories. Ask her to place the material from the Quantity Presentation Tray in a row on the mat in sequence. Position one of the green felt squares on the mat below the unit bead. Invite the child to center the unit bead on the green square. Say to the child, "The color green is for units." Continue in the same manner with the remaining felt squares: blue for tens, red for hundreds, and green for units of thousands. Remove the material from the felt squares and ask the child to repeat the extension in the same manner. Removing one item Introducing Montessori color-coding Cosmic Questions What do you think is the significance of the gold color of the Golden Bead material? How does the color gold symbolize the importance and power of this material? Why doesn't the color change as you move to the next category? 69

16 4fo()~t The Large and Small Number Cards 1000 j A set of Number Cards ~ - "One test of the correctness of educational procedure is the happiness of the child." - Maria Montessori in Paula Polk Lillard's, Montessori: A Modern Approach, p.118. Two sets of number cards are used in the Montessori 3-6 mathematics program - the Large Number Cards and the Small Number Cards. The numbers on these wooden cards are color-coded following the Montessori color scheme for decimal categories, and each set is stored in a wooden compartmentalized box. Each box contains nine units cards (1-9), nine tens cards (10-90), nine hundreds cards ( ), and nine thousands cards (1,000-9,000). The cards vary systematically in length so that the tens cards are twice the length of the units cards, the hundreds cards are three times their length, and the thousands cards are four times their length. 1 tlq The length of each card is proportional to its place value The number cards visually reinforce the important principle of the decimal system that there can only be nine in each category. The proportional length of the number cards means that when the cards are stacked in order, they form a composed number. For example, when number cards 9,000, 900, 90, 70

17 and 9 are superimposed or stacked in order with the units on top and the right edges lined up, all of the zeros are hidden and the child sees the number 9,999. When the cards are separated and arranged into their decimal categories, 9, , the numbers are decomposed. After working with a 4-digit number in its decomposed format, some Montessori teachers demonstrate a "magical" process of revealing this number in its composed format. Align the four cards flush along the left edge of the cards and then slide each card to the right, starting with the hundreds category, highlighting each number as it is revealed. At this point, the child is familiar with the abstract idea that written numbers represent quantities, but his learning has focused on numbers Now that the child has experienced quantities up to 1,000 with the Golden Bead Material, the next step is to marry that understanding to the written numbers representing those quantities. There are enough number cards to create any number from 1 to 9,999, giving the child tremendous freedom to see how the categories interact. And, if there are nine 1,000-cubes available in the Golden Bead Material, the Large Number Cards can be used in parallel with the bead material up to 9,999. This gives the child the resources to move back and forth between the bead material and the cards to explore how they relate. The only difference between the Large and Small Number Cards is the size of the cards. While the Large Number Cards are used with most decimal system activities, the Small Number Cards can be used too. The significance of the cards' size difference is not realized until the child is working on the Four Operations activities. The size of the cards is then used to reinforce the process of the operation. When adding or multiplying, the child uses the Small Number Cards to form the equation and the Large Number Cards for the answer. This is a visual cue that the child is combining two smaller numbers to form a larger The cards are aligned along the right edge as a composed number The cards are separated and arranged into their decimal categories as a decomposed number Did You Know? Roman numerals, which still appear in the front matter of books and on clocks, are a primitive form of the decimal system. The Romans used a system of marks derived from notches on a tally stick. They had symbols for 1, 10, 100, and 1,000 as well as symbols for 5,50, and 500. They then used subtraction or addition for all intermediate values. So, in Roman numerals, the numbers 1-10 are I, II, III, IV, V, VI, VII, VIII, IX, and X. 3-6 Mathematics I NAMC. North American Montessori Center 71

18 Montessori Moment In the elementary classroom, Montessori math materials that relate to place value use the color-coding that is first introduced with the Large and Small Number Cards. The Geometric Hierarchical Material uses color-coding to represent quantities from one unit to one million. The Small and Large Bead Frames use place value color-coding to represent positional value as it relates to addition, subtraction, and multiplication. Colorcoding is also used on the Checkerboard, which elementary students use for multiplication with quantities up to 9,999,999. It is also used in the Long Division Material, a series of color-coded division boards, beads, skittles, and bowls used for division into the millions. The Decimal Checkerboard employs the same color-coding and introduces a similar pattern for the decimal fractions (tenths, hundredths, etc.). The whole number and fraction color-coding are also used in the Decimal Fraction Board and Material, which are used to add, subtract, multiply, and divide decimal fractions and whole numbers. number. Similarly, when subtracting or dividing, the child represents the initial number (the minuend or dividend) with the Large Number Cards. He then works with the Small Number Cards to form the rest of the equation (the subtrahend or divisor) and the answer. Working with the cards in this manner reinforces that the child begins with a large number and ends up with a smaller number. The Large Number Cards are sometimes stored in a wall chart in the math area. Using the wall chart provides a fixed organizational scheme for the material. It also means that the child does not have to organize the cards each time he uses them; a task that can be time consuming. More importantly, using the wall chart incorporates additional movement into the activities as the child must travel to the chart to retrieve the cards. The Bank Game, which is used in the lower and upper elementary Montessori environments, includes number cards that look similar to the Small and Large Number Cards. They employ the same color-coding to indicate place value, continuing to the millions. The Bank Game is used to add and multiply large numbers and to further explore place value and exchanging. OB 9 0 I ~==~n n n, The Bank Game uses Montessori decimal system color-coding into the millions <3 Mathematics I NAMC - North American Montessori Center

19 I/tfrvrfct Introducing the Large Number Cards Age years old Direct Aims To associate the names of the categories of units, tens, hundreds, and thousands with their numeric symbols. To become familiar with the numeric symbols of the decimal system. To become more aware of the laws that rule the organization of numbers in the decimal system. Indirect Aims To experience the proportional relationships of the decimal system. To understand zero's role in the decimal system. Introducing the Large Number Cards Control of Error The teacher. The child's knowledge. To help a child associate the names of the first four categories of the decimal system with their numeric symbols. Large Number Cards. Invite a child to the math shelf to learn about the decimal system with the Large Number Cards. Vocabulary One unit, one ten, one hundred, one thousand. Notes This activity is a threeperiod lesson. The child will already be familiar with the names, quantities, and numeric symbols for one and ten. 73

20 Ask the child to carry the Large Number Cards to the table or mat. Commence with the first period, the naming, "this is" phase of the three-period lesson. The child will be reviewing his knowledge of one and ten and learning the category names for the written numbers. Remove the 1-card from the box and place it on the mat in front of the child. Teacher pointing to the one and asking the child to name it Point to the card and review its name. Say to the child, "What is this?" Confirm that it is one or one unit. Place the 1 a-card to the left of the 1-card. Point to the card and say, "What is this?" Confirm that it is ten or one ten. Demonstrate counting the number of zeros on the 1-card and the 1 a-card. Confirm that there are none and one, respectively. Place the 100-card to the left of the 1 a-card, and point to it while saying, "This is one hundred. One hundred." Ask the child to repeat its name. Demonstrate counting the number of zeros in 100. Confirm that there are two zeros. Child being introduced to the symbol for 100 for the first time Repeat the previous steps for the 1,000-card. Continue with the second period, recognition, "show me" and third period, recall, "what is" phases of the three-period lesson. Reinforce counting the zeros in the second period. After the child has finished working with the Large Number Cards, demonstrate how to return the cards to the compartmentalized box so that the 1, 10, 100, and 1,000 are at the front of the compartments. Remind the child to return the material to the math shelf. Tell the child that he is now free to work with the Large Number Cards for 1, 10, 100, and 1,000. Teacher pointing to 1,000 and saying, "What is this?" 74 3-B Mathematics I NAMC - North American Montessori Center

21 Learning About Montessori Color-Coding Invite the child to learn about Montessori decimal system color-coding with the Large Number Cards. Layout the cards as in the activity. Point to the 1-card and ask the child to name its color. Confirm that it is green and tell him that units are green Learning about Montessori color-coding Repeat with the 10-card and 1 ~O-card in turn, telling the child that tens are blue and hundreds are red. Point to the 1,ODD-card and ask the child to name its color. Confirm that it is green, like the units. Tell the child that units of thousands are green. Working with Montessori Color-Coding Invite the child to learn more about the Montessori colorcoding system for decimal categories. Ask the child to lay out the Golden Beads from the Quantity Presentation Tray on the mat. Ask the child to place the four colored felt squares beneath the corresponding categories of Golden Bead Material. Working with Montessori color-coding Invite the child to place the corresponding Large Number Cards below the bead material, starting with the units. Examining the Size of the Number Cards Invite the child to learn more about the Large Number Cards. Layout the cards in the same way as the activity. Ask the child if he notices any differences in the sizes of the cards. Confirm that each category is a different size. Show the child that the 1-card is the smallest of the cards, the 10-card is twice as long as the 1-card, and so on. Invite the child to place the cards in a column, starting with the 1-card and aligning the cards along the right edge. Ask the child to compare the cards in this arrangement , Examining the size of the number cards 3--B Mathematics I NAMC - North American Montessori Center 75

22 /lcfrf/rftt Counting the Golden Bead Material Age years old Direct Aims To become familiar with the names, quantities, and proportions of the decimal categories. To develop a deeper understanding that a decimal category cannot exceed nine. Indirect Aim To experience the proportional relationships of the decimal system. Control of Error The teacher. The child's knowledge. Counting. Vocabulary Units. Counting the Golden Bead Material Tens. Hundreds. Thousands. Notes The child will already be familiar with the names, quantities, and symbols for one through ten, one hundred, and one thousand. The child is working with large quantities in this activity; however, he only has to count to nine. To help a child become familiar with the names, quantities, and proportions of the first four categories of the decimal system. Decimal System Presentation Tray. Invite a child to the math shelf to learn more about the decimal system with the Golden Bead Material {l Mathematics I NAMC North American Montessori Center

23 Ask the child to carry the presentation tray to the table or mat. Ensure that the tray is positioned on the mat with the 1,OOO-cube on the left and the unit beads on the right. Take one unit bead from the small bowl, and place it below the tray. Ask the child, "How many is this?" The child should say one. Confirm that it is one unit. Place a second unit bead below and touching the first bead. Ask the child how many units there are now. Confirm that there are two units. ~~-.~... 9 ::::: :::::. m~llhh., ':~g~~~gj~... The presentation tray is always positioned with the 1,OOO-cube on the left Continue placing and counting the unit beads in this manner until all nine are on the mat in a column. Ask the child, "What comes after nine units?" Confirm that it is ten units. Remind the child that ten units are the same as one ten. Ask the child to take one ten from the tray and place it vertically and to the left of the column of unit beads. Leave enough space to the right of the 1 O-bar to lay eight more bars. Place a second 1 O-bar to the right of and touching the first 10-bar. Ask the child how many tens there are now and confirm that there are two tens. Teacher asking the child what comes after nine units Ask the child to continue placing and counting the 10-bars in the same manner until all nine 1 O-bars are laid out. Ask the child, "What comes after nine tens?" Confirm that it is ten tens. Remind the child that ten tens are the same as one hundred. Ask the child to take one hundred from the tray and to place it to the left of the 10-bars. Count and stack the 100-squares, placing one on top of the other. After the child has counted nine hundreds, ask him what comes next. Confirm that it is ten hundreds. Child placing the 1 OO-square to the left of the 10-bars 77

24 Remind the child that ten hundreds are the same as one thousand. Ask the child to take one thousand from the tray and place it to the left of the stacked 1 ~O-squares. Point to each category and say the names of each again: "Units, tens, hundreds, thousands." Return the material to the tray. Invite the child to repeat the activity. After the child has finished working with the presentation tray, ask him to return the material to the math shelf. Child placing the 1,OOO-cube to the left of the stack of 100-squares Tell the child that he is now free to work with the Golden Bead Material in the same manner. Perform with relevant vocabulary. Comparing the Golden Bead and Wooden Material Invite the child to compare the Golden Bead Material and wooden material for the 1 ~O-square and the 1,ODD-cube. Give the child a golden and a wooden 1 ~O-square and tell him that both squares are one hundred. Encourage the child to explore with his senses the similarities and differences between the golden and wooden 1 ~O-squares. Comparing the golden bead and wooden material Give the child the golden and wooden 1,ODD-cubes and explain that both cubes are one thousand. Invite him to compare the cubes B Mathematics I NAMC - North American Montessori Center

25 Visualizing the Decimal System Quantities 1 Invite the child to layout the Golden Bead Material on the mat in a different way. Place the first unit bead at the top right side of the mat, counting it as you place it. Place the second unit bead below the first, leaving the length of one 1 a-bar between the beads. Count the bead as you place it. Continue forming a column with the unit beads until you reach nine units, counting each one as you place it. Invite the child to take over when he shows interest. Count the 1 a-bars next. Position them to the left of the unit beads, placing them vertically and aligning the first bead of each 1 a-bar with a unit bead. When he shows interest, ask the child to continue placing and counting the 1 a-bars to nine tens. Count the 1 OO-squares to nine hundreds, forming a column to the left of the 1 a-bars. Finally, place and count one 1,OOO-cube to the left of the 1 OO-square at the top of the mat. Visualizing the Decimal System Quantities 2 Invite the child to repeat the previous extension, this time including nine 1,OOO-cubes in the layout. Show the child how to retrieve eight additional 1,OOO-cubes from the supply tray in the math area and carry them to the mat on a tray.... I!!!!!!!!! I::::::::: I..., I I I I Visualizing the decimal system quantities 1 (partial view) I:::::::::, I I I I Visualizing the decimal system quantities 2 (partial view) 79

26 Playing the "Please Bring Me" Game Invite the child to play the "please bring me" game. In another part of the room, layout the Golden Bead Material on a mat as in the previous extension. Give the child a small tray with a bowl to hold the unit beads and a basket for the 1 O-bars (when needed). Playing the "please bring me" game Ask the child to please bring you a specific quantity of unit beads. Do not ask for more than nine. Ask the child to repeat the number and then ask him to retrieve the quantity. When the child returns, thank him and invite him to count the unit beads, confirming the quantity is correct. If he makes an error, say, for example, "This is four units. Can you please return these beads to the mat and bring me three units?" Once you have verified the quantity, ask the child to return the material to the mat. Repeat with each of the categories separately, asking for nine or less each time. You may find it helpful to record the quantity so you can remember and verify it when the child returns with the material. On another day, ask the child to bring two adjacent categories at once. Start with small quantities of 10-bars and unit beads and progress to larger quantities (and never more than nine in each category). For example, "Please bring me two tens and three units." When the child is ready, invite him to bring three and then four adjacent categories at once. 80

27 4d i {rftr Sequencing the Large Number Cards Age years old Direct Aims To associate the category names of units, tens, hundreds, and thousands with the numeric symbols. To become more familiar with the numeric symbols of the first four categories of the decimal system. To become more familiar with the sequence of numbers up to 9,000. To further understand the laws that rule the organization of numbers in the decimal system. Indirect Aims To experience the proportional relationships of the decimal system. Sequencing the Large Number Cards To help a child associate the category names of units 1-9, tens 10-90, hundreds , and thousands 1,000-9,000 with their numeric symbols. Large Number Cards. To understand zero's role in the decimal system. Control of Error The teacher. The child's knowledge. Note If the child is keenly interested in learning more about numbers after this presentation, introduce her to the conventional names of numbers with the Teen and Tens Boards activities. Invite a child to the math shelf to learn more about the decimal system with the Large Number Cards. 81

28 Ask the child to carry the Large Number Cards to the mat. Remove the units cards from the box and place them, stacked, at the bottom right-hand side of the mat. Place the tens cards in a stack to their left. Repeat with the hundreds and thousands cards. Ensure that each category of cards is sequenced. Position the 1-card in the top right side of the mat and say, "One unit." Teacher positioning the 1-card in the top right corner Invite the child to continue placing the units cards in a column below the 1-card. Name the cards as she places them: "Two units, three units... " and so on, reinforcing the connection between the name and numeric symbol. When the child reaches nine, ask her what comes after nine units. Confirm that it is ten units, and remind the child that ten units are the same as one ten. Ask the child to point to the card for 10. Position the 10-card at the top of the mat, to the left of the units column, and say, "One ten. " Ask the child how many zeros there are in one ten. Confirm that there is one. Invite the child to continue placing the tens cards in a column. Name the cards as she places them: "Two tens, three tens... " and so on as described above. Teacher and child placing the 1 O-card at the top of the column When the child places the last tens card, ask her what comes after nine tens. Confirm that it is ten tens, and remind the child that ten tens are the same as one hundred. Ask the child to point to the card for 100. Position the 100-card at the top of the mat, to the left of the 10-card. Ask the child how many zeros there are in one hundred. Confirm that there are two. 82

29 Continue in the same manner, placing and naming the hundreds cards and then the thousands cards. When the child reaches nine thousands, explain that this is as high as these cards go. Return the cards to the box in order. Invite the child to repeat the activity. After the child has finished working with the Large Number Cards, ask her to return the material to the math shelf. Tell the child that she is now free to work with the Large Number Cards in the same manner. ~b ~8 c:>..c c:> Oh O ' OC o ~ ~o e:. 009 oos OOn oos OOC! Child laying the final number card on the mat Putting the Cards in Order Invite the child to repeat the activity, this time placing the cards in each stack in random order. Ask the child to put the cards in order starting with the 1-card at the top right side of the mat. Invite the child to name each card as she places it in sequence. Invite the child to continue placing all of the cards on the mat in the same fashion as the activity. Working with the Large Number Cards Invite the child to work with the Large Number Cards. Layout the full set of Large Number Cards as in the activity. Ask the child to give you the card for five units, for example. Confirm the number and return the card to the layout. Continue to ask the child for random cards from each of the categories separately. When the child is ready, ask for cards from two and then three and four adjacent categories. ~o L S 00." 008 OOG o 0 1. Working with the Large Number Cards 83

30 Montessori Moment The child has yet to combine the concrete Golden Bead Material with the Large Number Cards, but she will surely notice the parallel between them. Having been introduced to both separately, some children may simply start working with the two materials together to explore the relationship between quantities and numbers. Playing the "Please Bring Me" Game Invite the child to play the "please bring me" game. In another part of the room, layout the Large Number Cards on a mat as in the activity. Give the child a small tray and ask her to please bring you a specific units card. Ask the child to repeat the number and then ask her to retrieve the card. When the child returns, thank her and invite her to say the number aloud, confirming she is correct. If she makes an error, say, for example, "This is six units. Can you please return this number card to the mat and bring me the card for five units?" Once you have verified the number, ask the child to return the card to the mat. Repeat with each of the categories separately. You may find it helpful to record the requested number so you can remember and verify it when the child returns with the card. On another day, ask the child to bring two adjacent categories at once. For example, "Please bring me the cards for two tens and three units." When the child is ready, ask her to bring three and then four adjacent categories at once {) Mathematics I NAMC - North American Montessori Center

31 /fc/;vffrr Associating Quantities and Numbers Age years old Direct Aim To associate quantities and numbers in the decimal system. Indirect Aims To understand zero's role in the decimal system. II To develop a deeper understanding of the decimal system. Control of Error The teacher. Counting. o ~ J oo~ Associating quantities and numbers ~?os"~ To help a child associate quantities and written numbers in the decimal system. Decimal System Presentation Tray. Eight 1,ODD-cubes. Notes This activity allows the child to see that the numbers represent the quantities exactly, both in terms of how many beads there are and in terms of the structure of the decimal categories. The Small Number Cards are used because they take up less space on the mat. If the child is having difficulty with the activity, sequence the Small Number Cards instead of laying them out randomly. Small Number Cards. Tray lined with felt. 85

32 Invite a child to the math shelf to learn more about the decimal system with the Golden Bead Material and the Small Number Cards. Ask the child to carry the Small Number Cards to the mat, while you carry the presentation tray. Child positioning the material on the mat Ask the child to return to the math shelf to retrieve eight 1,ODD-cubes from the supply tray and to carry them on a tray to the mat. The child may need to make several trips. With the child, layout the full set of Golden Bead Material, with the units on the right of the mat and the thousands on the left. Layout the unit beads from top to bottom, leaving a space the length of one 1 O-bar between them. Count each bead as you place it on the mat. Teacher laying out the unit beads Continue to layout and count the other categories of bead material in the same manner: the 10-bars, 1 ~O-squares, and 1,ODD-cubes. Leave enough space to lay the number cards to the right of the bead material. Scatter the units cards on a small mat to the left of the mat of Golden Bead Material. Point to the first unit bead and say, "How many is this?" Confirm that it is one unit. Ask the child to find the card that represents one unit and to place it to the right of the unit bead. Child placing the last 1,ODD-cube Point to the second unit bead and say, "How many units are there now?" Confirm that there are two units. Ask the child to find the card for two and to place it to the right of the second unit bead. Continue in the same manner through to nine unit beads. Ask the child to scatter the tens cards on the small mat. Child placing the 9-card to the right of the ninth unit bead Point to the 1 O-bar at the top of the column and say, "How many is this?" Confirm that it is one ten. 86

33 Ask the child to find the card that represents one ten and to place it to the right of the 10-bar. Continue in the same manner with the remaining tens cards, and then repeat with the hundreds cards and thousands cards. Return the number cards to the box and the Golden Bead Material to the trays. Invite the child to repeat the activity. After the child has finished working with the Golden Bead Material and Small Number Cards, ask her to return the material to the math area. Tell the child that she is now free to work with the Golden Bead Material and Small Number Cards in the same manner. Child continuing to place number cards to the right of the corresponding quantities Associating Numbers and Quantities Invite the child to repeat the activity, this time associating written numbers and quantities in the decimal system. Invite the child to layout the Small Number Cards with the units cards on the right working through to the thousands cards on the left, naming the cards as she places them. Ask the child to count and place the Golden Bead Material to the right of the cards. Associating numbers and quantities 87

34 I/cfrf/{frt Matching Quantities and Numbers Age years old Direct Aims To associate quantities and written numbers in the decimal system. To become familiar with how quantities and numbers correspond in the first four categories of the decimal system. Indirect Aims To further understand the decimal system beyond the categories. To recognize the corresponding categories of quantities and numbers. Control of Error The teacher. Counting. Notes This activity may be an individual or small group presentation. After being introduced to this activity, the children can repeat it in groups of two or three. Although the child is handling large quantities, he is still only counting to nine. Matching quantities and numbers To reinforce a child's understanding of the relationship between quantities and numbers in the decimal system. Decimal System Presentation Tray. Eight 1,OOO-cubes. Small Number Cards. Tray lined with felt and holding a small bowl. 88

35 Invite a child to the math shelf to learn more about the decimal system with the Golden Bead Material and the Small Number Cards. Gather the material together and carry it to the work area. Layout the full set of Golden Bead Material on a mat. On another mat, at a distance away, layout the full set of Small Number Cards. Explain to the child that he will carry his tray between the mats to match quantities and numbers or vice versa. Single Category: Quantities and Numbers Place a quantity of bead material from a single category onto the child's tray, such as five unit beads. Invite the child to count the beads. Child selecting the 5-card to match the quantity of bead material Ask the child to carry the tray to the mat of Small Number Cards and to select the matching number card. Ask him to return to you and to count the quantity and name the number. Confirm that the quantity is correct. Invite the child to return the material to its place on the mats. Repeat with other quantities from a single category (e.g., three 1 a-bars; eight 100-squares; two 1,OOO-cubes; etc.). Invite the child to return the material to the shelf when finished. Single Category: Numbers and Quantities Place a Small Number Card on the child's tray, such as the 8-card. Ask the child to name the number. Child counting the quantity of eight unit beads to match the number card Invite the child to carry the tray to the mat of Golden Bead Material and count the matching quantity of bead material into the small bowl on the tray. 89

36 Ask him to return to you and to name the number and count the quantity. Confirm that the number is correct. Invite the child to return the material to its place on the mats. Repeat with other numbers, each time focusing on a single category (e.g., 500; 7,000; 60; etc.). Invite the child to return the material to the shelf when finished. Multiple Adjacent Categories: Quantities and Numbers Place a quantity of bead material on the child's tray that includes tens and units, such as four 1 O-bars and nine unit beads. Place the unit beads into the bowl on the tray. Invite the child to count the bead material. Child naming the numbers Ask the child to carry the tray to the mat of Small Number Cards and to select the matching number cards, such as 40 and 9. Ask him to return to you and to count the quantities and name the numbers: "Four tens and nine units." Invite the child to return the material to its place on the mats. Repeat with other quantities from multiple adjacent categories, working toward quantities that include all four categories (e.g., three 1,ODD-cubes, nine 1 ~O-squares, four 10-bars, and six unit beads). Invite the child to return the material to the shelf when finished. Multiple Adjacent Categories: Numbers and Quantities Place a tens card and a units card on the child's tray, side by side from left to right, such as 50 and 8. Ask the child to name the numbers. 90

37 Invite the child to carry the tray to the mat of Golden Bead Material and to count the quantities of bead material onto the tray, such as five 1 O-bars and eight unit beads. Ask him to return to you and to name the numbers and count the quantities. Invite the child to return the material to its place on the mats. Repeat with other multiple, adjacent category numbers, including numbers with three categories (e.g., 462) and four categories (e.g., 6,257). Invite the child to return the material to the shelf when finished. Repeat the activity with numbers from multiple categories such as 462, eventually working up to four category numbers Tell the child that he is now free to work with the material in the same manner. Introducing Numbers and Quantities with Zero Invite the child to work with numbers and quantities that contain a zero as a placeholder. Repeat the activity for presenting multiple category numbers and quantities, this time using non-adjacent categories, e.g., seven hundreds, four units. Introducing numbers and quantities with zero Playing the "Please Bring Me" Game Invite the child to play the "please bring me" game. State a number, such as four tens and six units, and ask the child to retrieve the matching number cards (40 and 6) and to then build the quantity on his tray (four 1 O-bars and six unit beads). Repeat with other numbers. Composing Numbers with the Small Number Cards Invite the child to build numbers a different way. Layout the full set of Small Number Cards on the mat. 3-B Mathematics I NAMC - North American Montessori Center 91

38 Invite the child to select one card from the tens column and one from the units column, such as the 20-card and the 3-card. Ask the child to read the two cards in succession. The child may say, "Two tens and three units." Or, he may say, "Twenty-three." Either is correct. Compose the number by sliding the 20-card under the 3- card so that only the numbers 2 and 3 are showing. Composing numbers with the Small Number Cards Ask the child to read the two cards in succession again. If the child does not say "twenty-three," say to him, "This is also called twenty-three." Repeat with other combinations of tens cards and units cards. When the child is ready, compose numbers that include a hundreds card, and later, a thousands card. Using Number Slips Invite the child to retrieve matching quantities and number cards using prepared Number Slips (see NAMC's Curriculum Support Material). Layout the material as in the activity. Invite the child to select a prepared Number Slip and ask him to read the number aloud. 782'-t Invite him to build the matching quantity on his tray and to then retrieve the corresponding number cards. Using Number Slips Repeat with other Number Slips. 92

39 I/cfr(frflt Introducing the Bank Age years old Direct Aims To become familiar with physically exchanging ten of one category for one of the next. To experience the ordering effect of the laws of the decimal system. Indirect Aims To develop a deeper understanding of the decimal system. To prepare to perform dynamic arithmetic operations. Control of Error The teacher. The limited quantity of material on the tray. Introducing the bank!i'(705'f To introduce a child to the process of physically exchanging ten of one category for one of the next. Bank containing a large quantity of Golden Bead Material. Tray lined with felt: Ten unit beads in a small bowl. Nine 10-bars. Nine 1 ~O-squares. Notes This activity prepares the child for dynamic operations in arithmetic, which require exchanging categories. Being able to recognize when a category has been filled and when to exchange for the next category is essential in arithmetic. This introductory activity controls the exchanges to one exchange in each of the categories. 3-6 Mathematics I NAMe - North American Montessori Center 93

40 Invite a child to the math shelf to learn how to make exchanges with the Golden Bead Material. Invite the child to carry the tray of bead material to the table or mat. Ask the child to sort the material into categories on the mat, with the hundreds on the left and the units on the right. Explain to the child that today she will learn how to make exchanges with the bead material. Invite the child to count the unit beads into a single column so they are touching each other. When the child gets to ten, ask her which bead material has ten beads and looks like this column of beads. Confirm that the 1 O-bar has ten beads and explain that the ten unit beads can be exchanged for one 1 O-bar at the bank. Child counting the unit beads into a single column Invite the child to return the ten unit beads to the empty bowl and to place the bowl on the tray. Ask the child to accompany you to the bank, carrying the tray. Show the child how the bank is organized with a substantial supply of unit beads, 10-bars, 100-squares, and 1,OOO-cubes. Demonstrate how to exchange the ten unit beads in the bowl for one 1 O-bar from the bank. Count the ten unit beads individually into the bank. Place one 1 O-bar from the bank on the tray. Ask the child to carry the tray back to the mat. Child returning from the bank with the exchanged 10-bar Invite the child to place the 1 O-bar on the mat with the others. Ask the child to count the 1 O-bars from left to right, pushing them together to form a square. 94 ~ Mathematics I NAMe - North American Montessori Center

41 When the child reaches ten, ask her which bead material has ten 1 O-bars and looks like a square of beads. Confirm that the 1 ~O-square has ten 1 O-bars and explain that the ten 1 O-bars can be exchanged for one 1 ~O-square at the bank. Ask the child to place the 1 O-bars on the tray, and invite her to accompany you to the bank to make the exchange. Invite the child to make the exchange, counting the 10-bars individually into the bank and placing one 1 ~O-square on the tray. Ask her to carry the tray back to the mat and to place the 1 ~O-square on the mat with the others. Child counting the 1 O-bars from left to right Invite the child to count the 1 ~O-squares, stacking them into a cube as she does so. When the child reaches ten, ask her which bead material has ten 1 ~O-squares and looks like a cube of beads. Confirm that the 1,ODD-cube has ten 1 ~O-squares and ask her to exchange the ten 1 ~O-squares for one 1,ODD-cube at the bank. Ask the child to place the 1,ODD-cube on the mat. Discuss with the child how the total amount of bead material on the mat and in the bank has not changed; the material has merely been exchanged for a different form. Child counting and stacking the 1 ~O-squares Return the 1,ODD-cube to the bank and place the original bead material on the tray. Invite the child to repeat the activity. After the child has finished working with the material, ask her to return it to the math area. Tell the child that she is now free to work with the Golden Bead Material in the same manner. Teacher discussing with the child how the total amount of bead material has not changed 95

42 I/tf;vrftt Playing the Change Game Age years old Direct Aims To become more familiar with physically exchanging ten of one category for one of the next. To further experience the ordering effect of the laws of the decimal system. Indirect Aims To develop a deeper understanding of the decimal system. To prepare to perform dynamic arithmetic operations. Control of Error The teacher. Counting. Notes This activity may be an individual or small group presentation. After being introduced to this activity, the children can play it in groups of two or three. Ensure that the child is not working with more than nine 1,OOO-cubes even after exchanges. In this activity, the child makes more than one exchange in a category and sees that material may remain in a category after exchanging. Playing the Change Game To help a child become familiar with the process of physically exchanging ten of one category for one of the next. Bank containing a large quantity of Golden Bead Material. Tray lined with felt and holding a small bowl. 96

43 Invite a child to the math shelf to play the Change Game. With the child, gather a substantial amount of Golden Bead Material from the bank and place it on the tray. Make sure that you have more than ten unit beads, ten 10-bars, and ten 100-squares, and that you will not have more than nine 1,OOO-cubes once exchanged. Carry the tray to the mat. Say to the child, "There is a lot of bead material on this tray. Let's find out how much we have in total. " Ask the child to place the material on the mat, sorting it into categories with the thousands on the left and the units on the right. Invite the child to count the unit beads. When he reaches ten units, ask him, "What can we do with these ten units?" He will say that they can be exchanged for one 10-bar. Confirm that he is correct. The material sorted into categories Ask him to place the ten unit beads into the bowl one at a time. Then, ask him to put the bowl of unit beads on the tray and to take the tray to the bank to make the exchange. When the child returns with one 10-bar, ask him to place it on the mat with the other 10-bars. Ask the child to count the remaining unit beads. If there are ten, ask him to make another exchange. If there are fewer than ten beads, tell the child that no more exchanges can be made with the unit beads. Child counting ten 10-bars Ask the child to count the 10-bars. Remind him to stop counting when he reaches ten 10-bars. Invite the child to exchange the ten 1 O-bars at the bank, and ask him to place the 1 ~O-square with the others on the mat. Ask the child to continue counting and exchanging 10-bars until there are fewer than ten Mathematics I NAMC - North American Montessori Center 97

44 Invite the child to continue in the same manner with the 1 ~O-squares, exchanging for 1,ODD-cubes as necessary. After all the exchanges have been made, count the material on the mat with the child, starting with the unit beads, and say, "Now we know how much bead material we have in total. There are seven thousands, five hundreds, seven tens, and six units." Or, if the child is ready, state the total as a conventional number: "Seven thousand, five hundred, seventy-six. " After the child has finished playing the Change Game, ask him to return the material to the math area. Child counting the 1 DO-squares in preparation for an exchange Tell the child that he is now free to play the Change Game in the same manner. Incorporating the Large Number Cards Invite the child to repeat the activity and represent the final quantity in each category below the bead material with the Large Number Cards. Teacher and child counting the material Playing Another Game Invite a group of three or four children to play another game. Invite one of the children to select a random quantity of beads from each category of the Golden Bead Material. Ask a second child to order the material into categories. Suggest that a third child count the bead material and take it to the banker when he needs to make an exchange. Ask a fourth child, the banker, to exchange the material, which the third child then returns to the mat and adds to the corresponding category. Incorporating the Large Number Cards 98

45 Preparing for Dynamic Subtraction Invite the child to learn to make exchanges in reverse. Place one 1,OOO-cube on the tray and ask the child to carry the tray to the mat. Ask the child if she knows how many 1 OO-squares are equal to the 1,OOO-cube. Confirm that it is ten 100-squares. Invite the child to go to the bank and exchange the 1,OOO-cube for ten 100-squares. When the child returns to the mat, count the 100-squares together by stacking them one at a time on the mat. Pick up one of the 1 OO-squares and ask the child how many 1 a-bars are equal to one 100-square. Confirm that it is ten 1 a-bars. Invite the child to go to the bank and exchange one 1 OO-square for ten 1 a-bars. When the child returns to the mat, count the 1 a-bars together by laying them side by side from left to right. Pick up one of the 1 a-bars and ask the child how many unit beads are equal to one 1 a-bar. Confirm that it is ten unit beads. Preparing for dynamic subtraction Cosmic Question The Change Game is about taking something and repurposing it as something else: units become tens, tens become hundreds, etc. Can you think of examples of other ways we regularly repurpose materials in our daily lives? Invite the child to make the exchange at the bank and then verify the exchange with the child by counting the unit beads in a single column on the mat. Discuss with the child how the total amount of bead material on the mat and in the bank has not changed; the material has merely been exchanged for a different form. 99