Table of Contents - NUMBERS TO TEN
- DECIMAL SYSTEM
- TEENS AND TENS
- Teens, Formation of Quantities 11-19 with Golden Beads
- Formation of 11-19 with Teen Board
- Combination of Teen Board and Beads to Form 11 to 19
- Formation of 10-90 with the Golden Beads
- Formation of 10-90 with the Ten Board
- Combination of Quantities 10-90 with Golden Beads and Ten Boards
- Formation of 11-99 with Golden Beads and Ten Board
- Changing Exercise
- Addition
- Multiplication
- Subtraction
- Division
- The Stamp Game
- Linear Counting: Chains of 100 and 1000 Golden Beads
- Skip Counting
- EXPLORATION AND MEMORIZATION OF TABLES
- PASSAGE TO ABSTRACTION
[top] Number Rods
[top] Sandpaper Numbers
[top] NUMBER RODS & CARDS
[top] SPINDLE BOXES
[top] MEMORY GAME OF NUMBERS
[top] CARDS AND COUNTERS
[top] PRESENTATION WITH THE GOLDEN BEAD MATERIAL
Presentation 1:
Presentation 2:
Count through each hierarchy in order from units to thousands. Each time nine is reached, state that if we had one more we would have ten. Instead of having ten loose beads (or tens, or hundreds) we can have a ten bead bar (or a hundred square, or a thousand cube).
Presentation 3: A small group exercise. Show the tray of Golden Beads to the children. Each child gets an empty tray with a castor. Ask each child, individually, to fetch a quantity of beads from the tray - one quantity per child. Upon their return ask each child, "What amount did you bring me?" The children will replace the beads in the tray. Continue as above according to the interest of the children. On another day the quantity may include two hierarchies until all the hierarchies have been included. Establish that when counting the category the name is included ie. five tens, six hundred, etc.
[top] PRESENTATION WITH THE LARGE NUMBER CARDS
Presentation 1: Individual exercise. Bring a box of Large Number Cards to a table/mat. Remove the number card for 1, 10, 100, 1000. Familiarise the child with the color coding and review 1 and 10 which he has learned previously by referring to the Golden Beads. Place them as headers at the top of the mat. Give a Three Period Lesson with the new numerals, i.e., 100 and 1000. At the end of the lesson put the number cards in correct sequence with the thousand on the left and the one on the right.
Presentation 2: Individual exercise. Remove all the Large Number Cards from the box. Stack the cards for each category in sequential order with the thousands on the left and the units on the right at the bottom of the mat. Place the '1' at the top right corner and continue placing and counting the unit cards in a vertical column until nine is reached. Then place the 10 - 90 to the left of the units, the 100 - 900 to the left of the tens and the 1000 - 9000 to the extreme left of the hundreds. As you place the number cards count them using their category name, i.e., one unit, two units, three units, etc. Show the child how to replace the number cards by stacking them in their categories with all the 'ones' showing on top. Bind them with a well fitting elastic band and replace them in the box.
Presentation 3: Small group exercise. Ask the child to help you lay out the number cards as in exercise 2. Give each child an empty tray. Ask each of them to bring you certain number cards beginning with the unit category and building up as in Presentation 3 with the Golden Beads.
[top] FORMATION OF LARGE NUMBER CARDS WITH BEADS
Indirect: Preparation for working with: - The hierarchy of numbers, i.e. that while the numerals are always from 1-9, it is the place they occupy in the large number that gives them their value. - In a number '0' holds a place for a specific category. - Only 1-9 cards of each category is necessary to form any number.
Layout: Set out two mats on the floor. Bring a tray of beads to one mat. Lay out the beads, similar to the layout for the number cards. Beginning with the units, lay out the beads for each category, individually, in a straight column working from top to bottom. Count each bead using the category name as you lay it out. On the other mat the children lay out the number cards as in the previous exercise. Note: only 1000 card corresponding to only 1 thousand cube is necessary at this stage. Each child gets a tray with a castor for holding the unit beads. 1. Single Category - Cards to Beads e.g. Units Place one number card of a single category on each child's tray. Ask the child to identify it and to obtain that quantity of beads from the other mat. When the child returns, he reads the card and counts the beads. The child returns the card and the beads to their mats. 2. Single Category - Beads to Cards Place quantity of beads from one category on the child's tray. The child counts the beads and obtains the corresponding card from the other mat. Check. 3. Two Adjacent Categories - Cards to Beads As in presentation 1a except place two cards which represent two adjacent categories on to the tray. Show the child how to superimpose the number cards and how to read the combined number, e.g., two tens and eight units. 4. Two Adjacent Categories - Beads to Cards As in presentation 1b except place a quantity of beads from two adjacent categories on to the tray. Continue in this manner until all four categories have been included and proceed to non-adjacent categories as the child progresses.
[top] COMBINATION OF GOLDEN BEAD AND LARGE NUMBER CARDS
The teacher shows the child how to place the number card '1' on the top right-hand corner of the mat and how to place the quantity, one golden unit, to the right of the card. The child continues in this way until he has used all the golden bead material for the units. The teacher shows the child how to place the number card for '10' and the ten bead bar parallel to the number card and the child continues until he reaches '90'. He then continues with the hundreds and the thousand.
[top] TEENS, FORMATION OF QUANTITIES 11-19 WITH GOLDEN BEADS
The lessons on Teens and Tens are usually given after the introduction to the decimal system exercises, but may be presented when the need for the terminology arises. At a table place the colored bead stair in disarray on the felt mat. Ask the child to count the beads on each bar and place them in stair formation. Give the child time to acquaint himself with the material. 1. Take three of the ten bead bars from the box. Place one of the ten bead bars in a vertical position in front of the child. Place the one red unit bead touching the 'one' on the ten bead bar and say, "This is eleven." Repeat the name a few times. Bring the 'eleven' to the left side and place another ten bead bar as before with the green, two bar touching the first two units of the ten bead bar and say, "This is twelve - twelve ... twelve." Do the same for thirteen. 2. Place the three quantities in front of the child and ask him to point to each quantity in non-sequential order. 3. Place one quantity in front of the child and ask him to name it. Always end with your material in order. Proceed with 14, 15, 16 and then 17, 18, 19 always reviewing the previous step.
[top] FORMATION OF 11-19 WITH TEEN BOARD
At a mat on the floor place the boards in a vertical column. Stack the cards from 1 - 9 at the top right corner of the boards. The child identifies the 10's. Using the THREE PERIOD LESSON teach the names of the numerals by sliding the unit cards over the zeros to create the numerals 11-19. At first teach 11, 12, and 13 and continue if the child shows interest until the child knows 11-19. Playfully, ask the child to name the numerals in non-sequential order. End the lesson by asking the child to read the numerals in sequential order. Show how to replace the number cards and the boards and to return them to the shelf.
[top] COMBINATION OF TEEN BOARD AND BEADS TO FORM 11 TO 19
Invite the child to bring the Teen Boards to a mat on the floor. Show the child how to lay out the boards as before and then to arrange the beads to the left of the board. Slide in the first card and ask the child to read the numeral and to make the quantity. He makes eleven with the beads directly to the left of the numeral. Ask him what comes after eleven and if he says twelve he may proceed to build the quantity for twelve.
[top] FORMATION OF 10 - 90 WITH THE GOLDEN BEADS
[top] FORMATION OF 10 - 90 WITH THE TEN BOARD
The child has already learned the names in association with the quantities and therefore he can be taught the names of of all the numerals in one lesson through the Three Period Lesson.
[top] COMBINATION OF 10 - 90 WITH GOLDEN BEADS AND TEN BOARDS
Invite the child to place the boards in order on the floor and to place the box with the bead bars to the left of the board. Ask the child to read the numerals on the board beginning with '10'. Show him how to place one ten bead bar immediately to the left of the numeral. Ask him to read the next numeral. He will probably say 'two tens'. Say, "Yes, you are quite right but remember it has a special name - it is 'twenty'." "Can you say 'twenty'?" "Now we are going to make 'twenty' with the ten bead bars." "How many will we need?" "Yes, two." "Two tens make 'twenty'." Proceed in this way saying, "This is how we write thirty." "Can you make thirty for me?"
[top] FORMATION OF 11 - 99 WITH GOLDEN BEADS AND TEN BOARD
Two children can work together at this stage with the teacher's guidance. Ask the child to place the Ten Boards in position and to place the nine ten bead bars together with the nine golden units to the far left of the '10' section leaving space for the quantities 11 - 99 immediately to the left of the Board. Ask the other child to stack the number cards in order 1 to 9. Ask one child to make 'eleven' with one ten bead bar and one unit - the unit touching the 'one' bead in the ten bead bar. Ask the other child to make the numeral 11 by slipping the '1' unit card over the zero in 10. Say, "Ten and one make eleven. This is how we write 11." Continue in this way until the children have reached '19' and say if we had one more golden unit we would have ten. Instead of having ten loose unit beads we can replace the units to the left and exchange them for a ten bead bar. Each time a number card is removed, place it face downward in a pile. Bring down the ten bar from the 10 section and with the 'new' ten place them to the left of the 20 section and say this is 'twenty'. Now let us make 'twenty one' by placing a golden unit to the right of the ten bead bars touching the 'one' bead in the ten bead bar nearest the Board. Say, "Now we have made 'twenty-one'." Turn the number cards to show the '1' at the top and proceed to slip the unit cards over the '0' in '20' as the other child makes twenty-one to twenty-nine with the Golden Bead material. Continue in this manner until '99' is reached when the nine ten bead bars and the nine Golden Units show at the left of the Board and the numeral 99 appears on the last section of the Board.
[top] CHANGING EXERCISE
- Units to Tens A good supply of units and tens on a tray and a castor for each child. Invite the children to help you count the units. Begin counting the units into a castor - stop when you reach ten. Remind the children that there are ten units in one ten. Explain that when you count to 10 you must exchange the 10 units for 1 ten. When all the exchanges have been made, read the quantity of beads, e.g., seven tens and four units/seventy-four. - Tens to Hundreds Using the same procedure exchange 10 tens for 1 hundred. - Hundreds to Thousands Using the same procedure exchange 10 hundreds for 1 thousand. EXERCISE 1: The child works with the material as shown.
- Thousands to Hundreds A good supply of thousands at a mat. Remind the children that 10 hundreds are exchanged for a thousand. Explain that you can reverse this procedure: exchanging 1 thousand for 10 hundred. Children exchange thousands in a systematic manner. - Hundreds to Tens Using the same procedure exchanging 1 hundred for 10 tens. - Tens to Units Using the same procedure exchanging 1 ten for 10 units. EXERCISE 2: The child works with the material as shown.
Place a quantity of units, tens, hundreds and thousands all together in a pile on a mat. Playfully, say to the children, "I wonder how much we have here?" "How can we find out?" Hopefully the children will suggest sorting the quantity into their hierarchies and then count them beginning with the units and changing when necessary until they have all been counted. Get the child to use the Large Number Cards to show the quantity in written form. EXERCISE 3: The child works with the material as shown.
[top] ADDITION
This work can be done at a large table or at a mat on the floor. Invite one child to lay out the large number cards, another sets out three sets of small number cards in the same formation as the large number cards. Ask another child to take care of the tray of Golden Bead Material acting as a 'banker'. Give a tray to each of three children with a castor to hold the unit beads. - Static Addition (Without changing) Place a low four digit number on each of the three trays using the small number cards - each tray should have a different number. The children read the number on their tray. Show them how to approach the 'banker' and to ask politely for that amount of golden bead. The children return to the work area where the teacher removes the golden beads in hierarchical order from each tray placing them on the mat in three separate quantities (addends). Place the small number cards in problem formation at the left corner of the mat (representing addends). Add the units by bringing the three amounts together in the center of the mat. Ask the child in charge of the large number cards to give you the numeral which represents the answer. Place it below the units. Proceed to work in the same way with the tens, hundreds and thousands. Superimpose the large number cards and place them below the small number cards (sum). Review the process and say, "Now we have done 'addition'." ______ brought me ______ ______ brought me ______ and ______ brought me ______ and when we added them together we found that we had _________________. Point to each of the small number cards and say, "This is an addend, this is an addend and this is an addend." Point to the Large Number Cards and say, "This is the sum." The children will pick up the terminology quite naturally. - Dynamic Addition (With changing) Follow the same procedure as for the static addition but now you select small number cards whose sum will require exchanging. - Individual Work When the children have understood the process of addition they may work on their own taking a tray of golden bead to their table or taking what they need from a 'pool' of golden beads. Later, they may progress from using the number cards to having the teacher write problems in their work book or taking a sheet of prepared problems and working with them. However, this needs some preparation. The children need to be shown how to write problems in their work books and how to space the problems and use a ruler to draw straight lines. Neat work habits are fostered in this way.
Note: The process is the focus of the decimal system. When working with the golden beads do not correct the answer. [top] MULTIPLICATION
EXERCISE 2: Introduce multiplier card: After the children have completed several examples, review the completed equation. Explain that it is not necessary to lay the multiplicand out so many times - a multiplier card may be used (use a small unit card which is placed under the multiplicand as in a written problem).
[top] SUBTRACTION
Static Subtraction: Select a high number in the large number cards and the corresponding amount of golden bead material (minuend). Place the golden bead in hierarchical order in the center of the work area. Place the large number cards at the top right corner of the mat. Select a low number in the small number cards and place them on a child's tray (subtrahend). Tell him he may take that quantity from the golden bead on the table, beginning taking from the units. He proceeds in this way until he has taken the quantity specified in the small number cards on his tray. The teacher then states the quantity that is left on the table and asks a child to find the small number cards and place them below the beads (difference). Review the process and place the small number cards from the child's tray (subtrahend) below the large number cards (minuend) and place the small number cards representing the difference, below the subtrahend. State, now we have done 'subtraction'. Dynamic Subtraction: This can follow when the child understands the process of subtraction. Individual Work: The child can take the tray of golden bead and work at his table or he may take what he needs from a 'pool' of golden bead material. EXERCISE 1: As in the presentation. Presentation & EXERCISE 2: Minuend of 9000 (When the child has mastered Exercise 1.) Select a minuend of 9000. Proceed as for dynamic subtraction showing the child how to exchange a thousand for hundreds; a hundred for tens and a ten for units before he can begin subtracting.
[top] DIVISION
a) Static Division: Select a high number in the large number cards and the corresponding amount of golden bead material (quotient). The amount selected must be divisible evenly by the number of children who represent the divisor. Place the golden bead in hierarchical order in the center of the work area. Place the large number cards at the top right corner of the mat. Explain that you are going to share the quantity of golden beads equally between the number of children. Begin with the thousands by placing one thousand in each tray and proceeding in this way until all the thousands have been used. The child in charge of the small number cards gives each child the small card which represents the amount of thousands on their tray. Proceed in this way sharing hundreds, tens and units until all the golden bead is used. Review the process and state that in division our answer is what one person gets and therefore we need only take a set of small number cards from one tray. Place these small number cards representing the dividend above the large number cards. State that now we have done 'division'. b) Dynamic Division (no remainder) c) Dynamic Division (with remainder) EXERCISE 1: As in the presentation Presentation & EXERCISE 2: Individual Work: Introduce the 'Divisor' card and green skittles to represent the number of children. Share the golden bead between the skittles and the answer will be what one unit skittle gets.
[top] THE STAMP GAME Note: The process is still important, however if answers are frequently incorrect re-present. This material is used by the children for individual work with the Decimal System, following the group exercises done with the golden bead material.
Note: Use correct terminology with each operation. Addition - addends, sum. Multiplication - multiplicand, multiplier, product. Subtraction - minuend, subtrahend, difference. Division - dividend, divisor, quotient. Introduce the signs used to symbolise, e.g., + for addition; - for subtraction; x for multiplication and ¸ for division. EXERCISE 1: Static Addition With the child's input, write two addends which will not require carrying. Draw a line under the addends and include a plus sign. Point out the use of a plus sign denotes this is addition. Read the problem with the child. The child lays out the appropriate stamps for the first addend. Encourage the child to check by reading the quantity made with the stamps. Place a ruler under the first addend and have the child lay out the second addend. Check. Remove the ruler. Remind the child of the necessary process to find the answer - combine categories and count beginning with the units. To combine the categories push the stamps up towards the top of the table until they form a double column per category. Count stamps using the category name. As each stamp is counted move it slightly toward you. The child records the answer in the units place - below the equal line. Have the child repeat the process for the other categories: tens, hundreds, thousands respectively. Review the problem with the child. Dynamic Addition Follows the same procedure except when counting, exchange the categories as necessary by removing one stamp of the next higher category from the box and replacing the ten stamps, which have been counted, into their appropriate place in the box. Presentation & EXERCISE 2: Multiplication With the child's input, write a multiplicand. The child chooses a multiplier of 2 or 3, which is written in the units column below the multiplicand. Introduce the multiplication sign. Read the equation with the child. Proceed as in dynamic addition. The child lays out the multiplicand the appropriate number of times combines the categories and counts exchanging as necessary and records the answer for each category as he counts. After some experience use '0' in the multiplicand. Presentation & EXERCISE 3: Static Subtraction With the child's input, write a minuend. Write a subtrahend which does not necessitate exchanging. Introduce the subtraction sign. Read the problem with the child. The child lays out the appropriate stamps for the minuend. Beginning with the units the child takes away the necessary number of stamps and replaces them into the box. The child counts the remaining number of units and records the answer. He repeats this process for the remaining categories in their respective order. Review the problem with the child. Dynamic Subtraction Follows the same procedure except when subtracting exchange categories as necessary by replacing one stamp of the next higher category into the box and removing ten stamps of the needed category (using a ruler for spacing purposes). Presentation & EXERCISE 4: Static Short Division With the child's input write a dividend as a statement and as in a process. Introduce the division sign and read the problem with the child. Remind the child that the skittles represent the divisor. Set out the appropriate number of skittles horizontally. Stack the appropriate stamps for the dividend to the left of the skittles in hierarchical order. Review the procedure for division: we start with the highest category and we give an equal number of stamps to each skittle. Share out the stamps underneath the skittles. Remind the child that the answer is what one unit received. The child counts the stamps under one skittle and records the quotient above the dividend. Read the problem with the child. Dynamic Short Division (No remainder) Follows the same procedure except to exchange categories as necessary. Dynamic Short Division (With remainder) Follows the same procedure as dynamic short division, except to introduce 'remainder'. Write the remainder to the right of the quotient with a small case 'r' before it. Explain that the 'r' is an abbreviation of remainder.
[top] LINEAR COUNTING: CHAINS OF 100 AND 1000 GOLDEN BEADS
EXERCISE 2: Introduce the special mat and set it out. Introduce the 1000 chain. Show how to carry the chain. Note the chain's rings. Hold the left hand perpendicular to the floor, close to the cabinet. Lift the chain from the left hook and over on to your hand, so that the ring lies on top of the index finger. Proceed as above for each ring, then return the chain to the cabinet one ring at a time. Invite the child to remove the chain from the cabinet and carry it to the mat. Lay out the chain horizontally on the mat. With the child, fold the chain as above. The ring denotes the end of each square which leaves a space. After each square is folded get a hundred square from the cabinet and superimpose it on to the folded chain. When the chain is folded count the hundreds. Get the 1000 cube from the cabinet - compare by stacking the hundreds. Pull out the chain - place a square above the last bar of each section at the right, place a cube at the end of the chain. Lay out the arrows, sorting by color and by hundreds. Count as above, establishing the repeating process. When the chain has been counted, note the number of beads in the chain. Read the arrows in sequence. Show the child how to pick up the chain by every other ring.
[top] SKIP COUNTING
EXERCISE 2: Begin with the cube of five chain (long chains). Proceed in the same manner as the 1000 chain but 'skip-count' at the end of the exercise. Encourage the child to skip-count the other cube chains in the same manner.
[top] EXPLORATION AND MEMORIZATION OF TABLES ADDITION SNAKE GAME
EXERCISE 2: The child uses any combination of colored bead bars to make the snake. Use a black and white bar for any remaining colored beads less than 10. When checking the child may need to change to make the appropriate combination - use the supply of colored bead bars in the box. Note: use only two colored bead bars to make a ten. Match one colored bead bar to the black and white bead bar in remainder. EXERCISE 2: Isolating combinations After child has worked well with the snake game, proceed as before. To count, pull down two bead bars isolating them at the bottom of the mat. Count to 10 as above using the black and white bar for remainder. Compare two sets of beads visually. Place colored bars into box. Move ten bar up into snake and pull down next colored bead bar. If two bead bars are less than 10 exchange them for a black and white bar. Proceed until snake is counted. Check as above. EXERCISE 3: Showing multiples - second check Proceed as above, this time make a snake using a number of the same colored bead bars (choose three of your favourite numbers or colors). To check - Arrange tens and remainder as before. Arrange the colored bead bars horizontally, grouping like beads bars parallel to one another from left to right, in diminishing size. Beginning with the largest set of colored bead bars, note how many there are in the set. Count into tens - laying tens and remainder vertically below (use tens and colored bead supply). Repeat this process for the other sets of colored beads. Note: that there should be the same number of beads in each of the three areas. Check by placing the tens (from colored beads) below the other tens (from the golden snake). Change the remaining colored bars into tens. Child counts top 10's and the bottom 10's to see if they match.
[top] ADDITION STRIP BOARD
EXERCISE 2A: Ask the child to choose a number, ie. 6 or 7. Write the number centerd on the page. State the goal: to find all the pairs of numbers which make up that amount. Begin with the 'one' blue strip - laying it on the grid as before. Ask the child what is needed to add to '1' to make the chosen amount. (The child may count the squares to find the answer.) The child places the appropriate strip to the right of the blue '1'. Read the equation, ie. 1 + 5 = 6. The child writes the equation. Repeat the procedure until all the possible combinations are on the board. Check the work with the Addition Control Chart #1. The child continues through all sums 2 - 18. EXERCISE 2B: When the child has completed the above exercise, have him make a chart of all the numbers and their combinations on graph paper. ie. 2 = 1 + 1 3 = 1 + 2, 2 + 1 4 = 1 + 3, 2 + 2, 3 + 1 Check chart with Addition Control Chart #1. EXERCISE 3A: Commutative Law Write an equation, ie. 2 + 4 =. The child uses the board to find the answer (the blue strip for the first addend, the red for the second addend) and records it. Write another equation reversing the above addends, ie. 4 + 2 =. The child uses the board as above to find and record the answer. Ask if the same numbers were used to make the sum. Check by comparing the strips (place equivalent strips underneath each other). Note that it does not matter which side of the plus sign the addends are on the answer is the same. Repeat with a few more examples. EXERCISE 3B: Using the chart the child made in Exercise 2b make a second chart which does not include duplicates. ie. 2 = 1 + 1 3 = 1 + 2 4 = 1 + 3, 2 + 2 5 = 1 + 4, 2 + 3 Check chart with Addition Control Chart #2.
[top] ADDITION CHARTS 3, 4, 5 AND 6 (BLANK)
CHART 3
CHART 4
CHART 5
CHART 6
[top] NEGATIVE SNAKE GAME
EXERCISE 2: Isolating subtraction - Proceed as above. When the child has counted back and (at a later stage) found the remainder, pull down all the bead bars involved in the counting back process to the bottom of the mat. Arrange the bead bars horizontally, so that the bead bars counted back on, match the grey bar and the black and white remainder. Note the quantity of the golden bead bar and/or the black and white bead bar, ie. 10 and 4 make 14. Note the quantity of the grey bead bar and the remainder, ie. 7 and 7 make 14. "So 14 (pick up golden bead bar and/or black and white bead bar) take away 7 (pick up grey bead bar) is 7 (leave remaining black and white bar on the table)." Place the bead bars in your hand in their respective places as before. Place the remaining black and white bead bar back into the snake and rejoin the snake. Continue in the same manner. Check as in the presentation.
[top] NEGATIVE STRIP BOARD
EXERCISE 2: Set out the materials as before. The child sets out the red strips in stair formation to the left of the board. The child chooses a number, ie. 6 or 7. Using the wooden strip cover the extra numerals to the right of the minuend. The child writes the minuend centerd at the top of the page and builds a table. Beginning with the red strips build a table as in the Addition Strip Board Exercise 2a beginning with the largest addend. The last row will be the minuend represented by one blue strip. This is only possible for minuends less than 10. "Now we're going to take away the blue strips." Using the board, note the minuend, ie. 6. The child writes '6'. Take away the first blue strip, ie. 6, sliding it to the bottom right of the board. 'What's left? - Nothing.' The child finishes the equation, ie. 6 - 6 = 0. Note: the difference is indicated by the red strip or its absence. Proceed in the same manner, working up through the rows. Check the work with Subtraction Control Chart #1.
[top] SUBTRACTION CHART 2 AND 3 (BLANK)
CHART 2
CHART 3
[top] MULTIPLICATION BEAD BAR LAYOUT
EXERCISE 2: Multiplying by 10 At a table, the child chooses a multiplicand, ie. 4. The child writes the multiplicand centerd on a page and takes out 10 four bars from the answer box and arranges them horizontally. The child counts as in the presentation, and lays out the product vertically. Say, "When we have 10 fours it's the same as 4 tens or forty." "When you multiply by 10 all you have to do is add a zero to the multiplicand." Repeat with a few more examples. Then child may continue on his own. EXERCISE 3: Divisibility of product Select a number, ie. 12. Find the number of ways to make 12. Begin with 1 and see if you can make 12 by counting, adding each bar on one at a time. The use of 1 works however at this point note that the multiplier should be less than 10. Continue in the same manner for 2 through 9. Leave out the combinations which make 12. Child may write down combinations. EXERCISE 4: Commutative Law At a table, using the answer box layout place 7 five bars horizontally as before. Then 5 seven bars. The child counts the beads placing the answer vertically below. Compare the answers. They are the same. Turn the beads to compare. Review and write the equations. 7 x 5 = 5 x 7 Try a few more examples. Conclude that it does not matter which side of the multiplication sign the multiplicand and the multiplier are on - the answer is the same. EXERCISE 5: Making the decanomial Build the decanomial square (see the sensorial album). Set out two mats. On a mat set out the supply of bead bars. Use the other mat to build the decanomial square, in the same manner as above, using the bead bars. When the decanomial is complete visually explore the layout. Note the squares on the diagonal and have the child exchange them with the squares from the bead cabinet. Note: the number of bars in the band is the number squared; the number of beads in the band is the number cubed.
[top] MULTIPLICATION BOARD
[top] MULTIPLICATION CHARTS 3, 4 AND 5 (BLANK)
CHART 3
CHART 4
CHART 5
[top] UNIT DIVISION BOARD
EXERCISE 2: Bring both the unit division and the multiplication bead board to the table. Write an equation, ie. 7 x 3 =. The child uses the multiplication board to find the answer. Write the corresponding division equation, ie. 21 ¸ 7 =. The child uses the unit division board to find the answer. Compare the results with the child to show what you build in multiplication, you take apart in division (division is the inverse process of multiplication). Explore with a few more examples.
[top] DIVISION CHARTS 1 AND 2
CHART 1
CHART 2
EXTENSION: The teacher explains prime numbers and factoring. When the child inquires about the numbers in white explain they are 'prime numbers'. "That means they can be only divided by 1 and themselves." The child gets the chart he compiled and paper and pencil. Select a quotient, ie. 56, the child writes it in the center of the page. Look it up on the chart and record the two numbers which make it up, ie. 7 and 8, below and a bit apart with arrows pointing down from 56. Look on the chart again to find out the numbers which make up 7 and then 8. Continue on (showing factoring). Since 7 and 2 are prime numbers they do not have any factors. Repeat according to the interest of the child. 56 7 8 2 4 2 2
[top] THE DOT GAME
DESCRIPTION OF MATERIAL: Paper which is squared and has columns headed 1, 10, 100, 1000 and 10,000. The columns are divided into small squares so that there are ten in each horizontal row. At the foot of each column are two spaces, the upper one for indicating the changing process with dots, and lower one for the result. There is a blank column on the right side in which the problem to be done is written. A lead pencil, an orange colored pencil and a ruler.
EXERCISE 2: Begin as before. This time put out the dots according to the category starting with the units (ie. put out the dots for all the units in all the addends - count as above, only now the orange dot(s) will appear at the beginning of the top row - repeat the process for tens, hundreds, thousands).
[top] SMALL BEAD FRAME Purpose Direct: The exercises of numeration recapitulate the function of the decimal system by making the child realize once again the following: 1) That ten of one category make one of the next higher category and then in each category there can be no more than 9. 2) The value of the numerals is determined by the place they hold. 3) The function of zero is that of a place holder. The whole set of exercises brings the child to the realization that when he writes down addition and subtraction problems, he must place all the figures belonging to the same category in the same vertical column. The exercises also provide the children with opportunities to apply all they have learned before and thus prepare them further for abstraction. Indirect: Both the exercises on numeration and the multiplication exercises prepare the child for the distributive law of multiplication where it becomes necessary to analyze the numbers into their hierarchical values.
EXERCISE 2: Dynamic Addition - Proceed as above except this time add all the units first, then the tens, hundreds and thousands. The child reads the numbers in the units column - sets out the first, then adds on the second, exchanging as necessary and records the sum. Repeat for the remaining categories. Read the problem. EXERCISE 3: Static Subtraction - Write a problem on the notation paper that does not require exchanging. The child reads and sets out the minuend. Remind the child to begin with the units. The child reads the units in the subtrahend and takes them away by counting the appropriate number of beads from right to left. The child reads the number of beads remaining at the right of the frame, which is the difference and records. Repeat for the remaining categories, tens to thousands respectively. Read the problem. Dynamic Subtraction - Write an equation that necessitates exchanging. The child reads and sets out the minuend. Beginning at the units, the child notes the number of beads which must be taken away. The child counts the corresponding number beads from right to left, however the child will run out of beads. Remind the child of how to get more units. Slide one ten bead to the left and slide the 10 unit beads to the right. The child continues to count the appropriate number of beads to the left. The child reads the number of beads remaining at the right of the frame which is the difference and records it. Repeat for the remaining categories. Read the problem.
Note: Multiplication is done as for Addition. The child can be shown how to take the multiplicand a certain number of times (multiplier). [top] |