CA2950116A1 - Visual arithmetic teaching device - Google Patents

Abstract

An educational device is provided which can be used to teach arithmetic. The educational device comprises a set of transparent dotted tiles, a set of opaque frames matched to the size of tiles, a manual of directions for parents/teachers, and a storage container with sunken pits matched to the size of each tile. Tiles are patterned in odd and even arrays of dots. No numerals appear on these tiles. Each number is assigned a distinct colour. Each tile is outlined with a fine black line. Three distinct stages are recommended for optimal use of QC tiles in "number sense" development. These are Free Play, Guided Play, and Formal Instruction. Problem-solving becomes embedded from the Guided Play stage onward.

Description

VISUAL ARITHMETIC TEACHING DEVICE
FIELD OF THE INVENTION
This invention relates to an educational device which can be used to teach arithmetic.
BACKGROUND OF THE INVENTION
Currently students, and in particular, elementary school age children, learn arithmetic by memorizing the "facts" of addition, subtraction, multiplication and division.
The students then practice their arithmetic to help "drill" in the "facts". However, not all children learn well this way. Rather, many children are visual learners. It would thus be beneficial to provide a device which can be used by children to visualize arithmetic problems, and, which will hopefully be fun for the children to use.
In the early 1950s, Caleb Gattegno popularised a set of coloured number rods created by the Belgian primary school teacher Georges Cuisenaire (1891-1975), who called the rods reglettes. Cuisenaire rods are mathematics learning aids for students that provide a hands-on elementary school way to explore mathematics and learn mathematical concepts, such as the four basic arithmetical operations, working with fractions and finding divisors.
Examples of prior art include U.S. Patent No. 7,695,283 which used blocks and tracks so that the student could add and remove blocks to the track such that addition and subtraction "facts" could be visually taught.
SUMMARY OF THE INVETNION
According to one embodiment of the invention, an educational device is provided which can be used to teach arithmetic. The educational device comprises a set of transparent dotted tiles, a set of opaque frames matched to the size of tiles, a manual of directions for parents/teachers, and a storage container with sunken pits matched to the size of each tile.
Tiles are patterned in odd and even arrays of dots. No numerals appear on these tiles.
Each number is assigned a distinct colour. Each tile is outlined with a fine black line.
An advantage of this educational device is that it is easier for a child to understand how numbers relate to each other, to recognize patterns, and to use numbers to solve problems when size, shape, and colour are held constant, DETAILED DESCRIPTION
Rationale for Tiles (QC Tiles):
QC Tiles are founded on two principles essential to creating a solid foundation for understanding the concept of "number". The first of these is QUANTITY. Each numeral has a value related to single units. The second is CONSTANCY. Each quantity is deliberately held constant in size, shape, and colour. (Colours coincide with those used by Cusienaire Rods, to facilitate easy transition to those manipulative tools as well.) When size, shape, and colour are held constant, it is easier for a child to understand how numbers relate to each other, to recognize patterns, and to use numbers to solve problems.
This QC Guide recognizes and respects the need for clear and accurate use of vocabulary required for the development of numeracy skills. Essential words and phrases are presented in bold script.
Educators have long recognized the need for children to use as many modalities as possible while learning. Being able to manipulate QC tiles ensures that each child is visually and kinaesthetically engaged.

2 Because of their ability to imprint number concepts visually, QC Tiles are also an ideal remedial tool for reviewing basic concepts.
Three distinct stages are recommended for optimal use of QC tiles in "number sense"
development. These are Free Play, Guided Play, and Formal Instruction. Problem-solving becomes embedded from the Guided Play stage onward.
QC Tiles - Physical Description:
The QC Number Tile Set has four components: a set of transparent dotted tiles, a set of opaque frames matched to the size of tiles, a manual of directions for parents/teachers, and a storage container with sunken pits matched to the size of each tile Tiles: As shown in Image No. 1, QC Number Tiles are a set of transparent plastic tiles (preferably soft enough to avoid too much noisy clatter) patterned in odd and even arrays of dots as shown in the photographs. No numerals appear on these tiles. Each number is assigned a distinct colour. Each tile is outlined with a fine black line.
Colour Codes:
1-white 2-red

3- light green

4- purple (light shade like "plum")

5- yellow

6- dark (bright) green

7- black

8- brown

9- blue

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Manual: The parent/teacher is provided with a manual of how best to develop "number sense" in the child. Students progress from free, creative play, to guided play.
Frames: QC As shown in Image No. 1, tiles are supplemented with opaque white frames in dual-track odd-even pairings. These are in increments of 10, interlocking to form "trains" of various lengths up to 120. The base of each frame is marked with a red line to denote one full set often. Numerals appear in the bottom right-hand corners of these frames and are clearly visible when covered with a transparent dotted tile.
Four blank frames are also included in the set.
Storage Box: As shown in Image No. 2, the tiles come in a storage box that has a sunken pit to accommodate the tiles as shown in the figure below, stressing combinations often.
There are six tiles of each colour. Each tile area bears the numeral corresponding to the tile that belongs there. Grooves are provided for finger access to each tile (so that the child does not need to dump out the entire lot). The child will begin to associate the number represented on the tile with the numeral imprinted at the bottom of its pit.

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Image No.: 2 STAGE ONE ¨ FREE PLAY
Children are introduced to QC tiles through free play, as shown in Image No.
3. They are encouraged to design, create, and build with the tiles by manipulating them in any way they see fit.
Tiles are identified by colour at this stage; identification by number is not necessary, but is acceptable if the child can do so very comfortably.

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r---i, itiL 1111 1 IIP . to to -...........=====01401 Image No.: 3 At this stage the parent's role is to admire the child's efforts, to provide encouragement, and to develop the vocabulary that will enhance concept development.
Essential Vocabulary words and phrases appear in bold print.
Join the child in creating designs. Encourage the child to "turn" or "rotate"
them, to "flip them over", to "overlap" them (put one on top of the other).
Note tiles that:
- are the same size - have more dots than another - have less dots than another Note that if the child says his design is a horse, it's definitely a horse.
Enjoy this period of becoming familiar with the tiles.
Note that some tiles have one extra dot sticking out. Do not be afraid to introduce the words "odd" and "even" at this early stage. "Evens make pairs" like socks and boots and mittens; "odds have one left over" or "one extra". Note how we can fit two "odds"
together to make an "even", by "flipping" one tile over and fitting the two left-over single dots together to make a pair.
Challenge the child to make a design using only even tiles/only odd tiles.
If more than one child is playing, encourage paired play. (Paired activity ¨
unlike larger groupings - ensures the engagement of EVERY child.) ROLE OF THE STORAGE TRAY
The storage tray plays an important part in early learning. By storing the tiles in order as indicated by lines in the tray, as shown in Image No. 4, children become familiar with colours that add to make ten, and begin to associate each quantity with the printed numeral.

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Image No.: 4 When does Stage Two begin? When the child can:
- identify each tile by colour - can explain that each tiles with the same colour is the same size.
This will vary from child to child. For that reason, these directions are not intended to be followed slavishly.
STAGE TWO ¨ GUIDED PLAY
Note: Every session of activity should be preceded or terminated with 10-15 minutes of free play with the tiles. Make use of this time to practice the essential vocabulary provided Introducing "the Language of Math" within the context of actual meaningful play and manipulation, is essential for concept development. Critical words and phrases appear in bold print.

Note: All work at this level is done orally, even when words for each number are used.
(Written work begins at Step 3, Formal Instruction.) Once children understand how they can rotate, flip, and overlap these tiles, other activities can be introduced.
COUNTING
- the use of number words along with colour is now introduced. Most children have been exposed to numbers on their media devices.
- Not all children will count accurately. Many may lack 'One-to-one correspondence' (i.e. will say numbers in sequence, but speak too quickly.) The game of "Touchdown" may help: Tell the child he has to very quiet and only say a number when his 'pointing finger' touches a dot.
- Arrange the tiles in counting order from I to 10 (Some children will have a tendency to start on the right. Glue a star or marker of some kind to the child's work space to remind him that when we do school work, this is the 'starting side'.
As shown in Image No. 5, any arrangement ¨ from left to right, is acceptable.

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Play a quick guessing games to consolidate the association of QC tile colours with numbers:
- "Hold up the tile that has 3 (etc) dots? What colour is it?"
- "I am hiding a red tile. How many dots does it have?"
As each number is identified, relate each to objects familiar to the child.
Ask child for ideas; list and post these.
1 ¨ only 1 of each of us, one head, one mouth, etc. Also only one sun, one moon, for earth. 1 horn on a unicorn.
2 ¨ eyes, ears, hands, two feet, socks, mittens, wheels on a bicycle, horns on a rhinoceros, wings on a bird or plane.
3 ¨ wheels on a tricycle, sides on a triangle, horns on a triceratops, toes on a sloth.

4 - corners in a room, sides on a square/rectangle, tires on a car, wheels on a skateboard.
¨ fingers, toes, days of school every week 6 ¨ highest number when you roll dice, sides on a box, legs on an insect, colours of the rainbow (ROYGBV).
7 ¨ days in a week, 8 ¨ reindeer on Santa's sleigh, 8 legs on a spider; 8 arms on an octopus. 8 fingers if thumbs aren't counted 9 ¨ planets in our solar system, 9 lives for a cat (?) ¨ fingers, toes, months of school every year.
Create a poster with pictures of favourite items with numbers to provide a visual reference. Do not underestimate the value of visual imprint in learning. Be creative!
Always link to subjects of interest to the child.
Play various "hunting" and guessing games:
- "Can you hold up the number 6?" "What colour is it?"
- "I am hiding a brown tile. How many dots it have?"
Develop speed and accuracy through frequent practice.
PRESERVATION OF QUANTITY
Take any tile. Count its dots. Turn that tile sideways:
- "How many dots now?" The child should, without counting, immediately say the same number. If not, re-count the dots, then put the tile in yet another position and ask the same question. If there is any doubt, come back to this concept daily until it is clear.
COMPARING
See Image No. 6

11 - "Which tiles have more dots than yellow?" and "Which is bigger/smaller than yellow?"
- "I am hiding a tile greater than 5. What colour could it be?" (Yes, it could be 6 or 7 or 8 or 10, but I am holding 9.) - "I am hiding a tile less than 8. What colour could it be?"
- "Can you think of an odd (or even) number less than 6?"
- "Can you think of an odd (or even) number greater than 4?"
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See Image No. 7 - "Take one tile of each colour. Put them in a neat row. Put the smallest one first.
Put the greatest one last."

12 - Now say the order of the numbers in sequence, having the child touch each one as you progress. (This develops both one-to-one correspondence and a visual association of the quantity of each number.) - Repeat the above activity starting with the largest number to the smallest one.
Children love to count backwards from 10, always finishing with "Blast Off!"
- Separate odd and even tiles. Arrange these in ascending and descending order.
Repeat each pattern orally as it is established.
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Only the quantities have to be in proper sequence. (An additional activity might be to have the child copy each of the above sequences in isolation, providing a baseline -like a ruler - to provide a "starting line".)

13 INTRODUCING ADDITION
Ask the child to take one tile of each colour and arrange them in order of size starting with 1.
- "Now take another white 1 and put it beside the first one."
Explain that when we put more dots with some we already have we say we add.
¨ "Now, we had a one and we added another one. Can you find a different colour of tile that will cover the one and one added together'?" (Red) "Yes, red is the same as a one and one added together, so one and one added together makes.. .("Two") - "Now take a red tile. Add a white one to it. What colour has just as many dots as a red and a white tile added together?" (Light green). "So two and one more makes ... THREE!"
- Continue until you find that nine and one more makes ten.
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14 - Finish by arranging the tiles as shown above. Chant the following while putting a finger on each combination:
"One and one more makes two; two and one more makes three, three and one more makes four, etc."
-"What happens to an even number when we add one?"
-"What happens to an odd number when we add one??
(Odds become evens; even become odds.) Note: for purposes of illustration, Image No. 8 shows combinations side-by-side.
In reality, the tile on the right can be overlapped with the +1 combination to prove they have exactly the same number of dots.
SUBTRACTION AS "TAKING AWAY"
Start with the orange (10) tile:
- "What happens when we hide one dot? Let's pretend to take one away by covering one corner dot with our pointing finger." Demonstrate and make sure the child can do this.
- "When we hide one, how many can we still see? (9)"
- "What colour can cover those nine dots?" (blue) - "Now let's take another nine tile. Let's hide the extra one that sticks out. Let's take it away by hiding it." Demonstrate and provide help if necessary.
- "How many are left?" (8) What colour will cover them?"
Continue until you have following pattern. (The 1 can be removed from any corner of even numbered tiles, but must be the 'extra one' on odd numbered tiles.) m, Ilir ' - r ¨
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-Image No.: 9 (In Image No. 9 above, tiles are deliberately off-set only to clarify their placement. Note that the child will have the same answer regardless of how he places his tiles.
LINKING ADDITION AND SUBTRACTION (Taking Awayl See Image No. 10 Use the above arrays to review adding one.
- "If we add one more to the 9 tile, what will we get?"
- "If we have 10 and we take 1 away, how many will we have?"
Play with each number in this way, not necessarily in any order.
Using smaller numbers, complete these concepts by asking:
- "I had 3, but now I have 4. What did I do?" (added one) - "I had three, but now I have 2. What did I do?" (took one away). Etc.
Consolidate the concept by stacking the tiles one on top of the other starting with the ten:

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4 041111K -17=p-Image No.: 10 (Whether the child stacks upward or downward, with odd number "extra dots" on left or right, doesn't matter, but insist that all those "extra dots" appear on the same side.) TWO-NUMBER COMBINATIONS
Understanding the concept of adding numbers, opens up the possibility of finding numerous two-tile combinations to create a larger number.
- "Find two tiles that you can add to make a dark green one." Praise all correct responses (yellow and white, red and purple, two light green).
- "Can you tell me which NUMBERS we add to make a six?"
Put a dark green tile on display and ask the child to identify it by the number it represents.
Now ask the child to put the two tiles that he can use to make that number beside the 6 tile to make the same rectangle shape as 6.

- "We used a purple and a red. We put the purple one up here and the red one down here. Let's count the dots." (6) - "Now let's take another red one and put it first. Now lets' count the dots." (6) - "Does it matter which tile we put first?" (No) Repeat this process using the 5 and 1 tiles.
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- Now pick up any 2-number tile combinations and use them to create a different shape; invite the child to do the same:

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, ' oir Image No.: 12 Referring to Image No. 12 - "When we added red and purple together on top of the dark green tile, we found out that they made six. Now look at the new picture we made with a red and a purple. Count the dots. Ilow many are there?" (six) - "If we put a 4 and a 2 together, will the picture we make change the big number we make?"
Have child try to copy the design shown and create new ones.
(If the child happens to cover up a dot while creating a design, remind him that covering up is the same as "taking away". Right now we are adding, so we can't hide any dots.) In every case, count the number of dots that result.
- "Now, let's make a rectangle for six using a 4 and a 2."
Put this combination beside the dark green 6 tile.
- "If we take the two away (demonstrate) what will we have left?" (4) - "Put the 2 and 4 back together. This time let's take the 4 away."
(Demonstrate) "What is left?"
Repeat the above process with any other number. Find two numbers that can be added together to make that tile. Then review "taking away" by removing one of the two tiles. (Note: this concept is more easily demonstrated using even numbers.
If an odd number is used, the child may need help placing the even number first (since he already knows that, when adding, order does not matter.) (Refer to Image No.
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i tr.,,,!:,,,.44 ish 1 ______________________ , , , -õ j I , 7 Image No.: 13 Begin a systematic discovery of addition combinations for each number.
Review "taking away" as the opposite of adding.

Staring with 2, find two tiles that can add to that number.
(Combinations fir 2, 3 and 4 could be covered in one session; beyond that, one number per session is recommended.) Remember to review "taking away" as well as adding.
Before going on to combinations for 5 and beyond, review the concept that order does not matter when we add:
(Use even numbers and Is only for this purpose; odds cannot be reversed ¨ but can be counted to prove equality.) (Refer to Image No. 14) = = .130 Jim `411 =
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11 L. a ip IIIIIM t, 0,- ri PlLio 111,.4,, 1 µ'---111, = i i.. %IP' :*:,4,r Image No.: 14 Referring to Image No. 14, these can be arranged in any direction/order the child chooses.
Combinations for 8 and 9:
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(Extended to review counting forward and backward.) (Refer to Image No. 16) - . .
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Image No.: 17 "THE DIFFERENCE BETWEEN" (subtraction) (preparing for problem-solving in the future) Subtraction is not always "taking away"; it can always distinguish the "difference between" two quantities. The child is not expected to know the word 'subtraction', but does need to explore the following relationships between quantities:
Introduce this concept by comparing the number of legs on an ant and the number of legs on a spider (or "arachnid") - "Which has more legs? How many more? (2) - "WE can say the difference in the number of legs is 2."
Now take a 6 tile and an 8 tile and place them side by side.
- "Which has more dots?" (the 8) - "How many more?" (2) - "Let's cover up the six legs an insect has. If we want the six to have just as many dots as the 8, how many dots will we have to add?" (2) - "So we can say the difference between 6 and 8 is....2.
Use other examples such as:
- wheels on their school bus and wheels on a car - legs on a spider and legs on an ant.
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Image No.: 18 After each example, use tiles to clarify the difference. See Image No. 18.
SPATIAL AWARENESS
(Templates provided in lid of storage box) (See Image No. 19) Fill in grids using any arrangement of tiles. (Multiple possibilities.) These can be supplemented with any additional design.
"Try to fill in using only even (or odd) tiles."

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______________ m-10101 101 r"0"400õ, 141 11)i H 11414.-Image No.: 19 FOUNDATIONS FOR MULTIPLICATION AND DIVISION
"Same Number" activities:
Instruct child to take two 6 tiles.
= -"I low many 3s make a 6?" Cover one dark green with two light green tiles.
= -"How many 2s make a six?" Cover the other dark green tiles.
= "What does this show us?"
- "Two 3s make 6 and three 2s make six."
- "There are different ways to make the same number."
- "There are many ways to make any number."
"When two patterns have the same number of dots we can say the number of dots are equal." (See Image No. 20) FACT ' = o al .µ = 4, _ .
Image No.: 20 Referring to Image No. 21, repeat the above pattern using 4s and 2s to make 8, then using 5s and 2s to make 10. Stress use of the word equal as meaning the same number of dots.
(Note that the colours are not equal, only the number of dots are...) Review the concept by asking:
"How many 2's do I need to cover a 10?"
"How many 4s cover an 8?"
Challenge: "1 Low many 3s cover a 9?" (More complicated manipulation.) UJöLJ
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4, Image No.: 21 Encourage the child to use the term equal in describing the above relationships:
"Five 2s are equal to 10."
"Two 5s are equal to 10."
"Two fives are equal to five 2s".
"IDENTICAL" versus "EQUAL"
Referring to Image No. 22, review all the different ways we can build a 9 tile using any combination of tiles available, including those making use of more than two tiles. (The photo below shows only some possibilities.) õ 1 r-c-i r--1 . 1 ' f , Pr 0 411 , ____ t ' - I ' r .. 4, _ , 4--1 161* 40 *...J
Image No.: 22 Now hold up a blue tile and ask the child to find another tile that is identical to your tile:
"Listen carefully, when I say identical I mean it has the same shape, the same colour, AND the same number of dots. Everything about it has to be the same as my tile."
The child should identify another blue tile.

Repeat the same using other tiles.
Then review the meaning of "equal", by asking the child to find two colours that can equal another by being added together.
ENRICHMENT (Problem solving) The following questions are designed to review and to challenge. Not all concepts need be fully mastered before moving on to the Formal Stage; every concept will be reviewed once numerals and paperwork are introduced.
Allow use of the tiles to find solutions to the following types of questions:
- "How many purple tiles cover a brown?"
- "Find two identical tiles that can cover the orange one."
- "Use three different-coloured tiles to cover the blue one."
- "Find three identical tiles that are equal to the blue one."
- "What three identical tiles are equal to a dark green?"
- "Show me how to make an 8 with two even numbers."
- "Can you make a ten with two even numbers?"
- "Can you make a ten with two identical even numbers?"
- "What number do you have to add to 6 make 10?"
- "If you have 9, but you want to have only 6, how many will you have to take away?"
- "If you have 8 dots and you take away three, how many dots will be left?"
- "Brett is 5 years old and his little brother is 3. What is the difference in their ages?"
- "I had some marbles in my hand. I dropped two, but I have 6 left. How many marbles did I start with?"

Have the child make up problems to solve.
Stage 3: FORMAL INSTRUCTION
Note: Formal instruction is usually done at school, but can be done effectively at home by anyone who recognizes the essential role arithmetic skills play in real life, and in all future mathematical learning.
NUMBERS, NUMERALS, DIGITS
Technically, "numbers" are abstract concepts of quantity, while "numerals" are the printed symbols that represent those numbers. For the purposes of using QC
Tiles with young children, the word "number" will be used exclusively to include both concepts ¨ as it is in common practice.
"Digit", however, will need to be used as soon as the number 10 is introduced.
A digit is any one of the 10 symbols we use when representing numbers: 0, 1, 2,3, 4, 5, 6, 7, 8, 9.
The word digit should be introduced when teaching the child to form the number 10.
"Some numbers, like 1, 2, 3, 4, 5, 6, 7, 8, 9 have only ONE digit, but 10 uses two digits ¨ a 1 and a round symbol ¨ like this - that we call 'zero'. Zero means nothing, or no more.
It will be useful to point out that it's like using up all 10 fingers:
"When we have 10 fingers we have 1 person and no more fingers."
Game: "War" with a deck of playing cards and one orange QC tile each (See Image No.
23). Markers to track wins. Eliminate face cards and use Ace as 1. Each person gets all red cards, shuffled; the other all the black ones, shuffled. Turn over cards simultaneously;
person with higher number gets a marker to place on his ten tile. This is a fast game.
"How many markers do you have? How many more do you need to win?" Make sure you explain this is a game of chance only.

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Image No.: 23 A NOTE ABOUT PAPERWORK
By now your child will have a substantial understanding of the quantities associated with numbers to 10, of how these quantities relate to each other, and how they can be manipulated and compared.
Most children will also already be familiar ¨ given exposure to TV and technological devices - with the numerals associated with these quantities. The QC storage tray will also have enhanced this association of quantity with printed number.
It is now time to learn how to print these numbers and to begin to represent arithmetical concepts in symbolic terms.
The use of squared grid paper for all printed work is highly recommended. It allows for easy replication of QC Tile patterns and helps keep work organized. 1-cm grid paper is suitable for young children. (Each square should hold one digit only.) A necessary note about REVERSALS: Children who reverse numerals need a reference set of numerals taped to their work area. Reversals can be a sign of visual anomalies or processing differences; they are NOT indicative of a lack of intelligence.
(The same can be said of all paperwork copying difficulties.) Darkening lines or providing squared paper can help guide the child who has difficulty with spacing It is essential that children be formally taught efficient ways to form numerals on paper.
Careless encoding slows progress. The D'Nealian style (illustrated below) is recommended, but any carefully designed program is acceptable. Many of these formats are available on-line without cost. (See Image No. 24) i23 Li 5 7 = 0 Numbers Set IT ALL STel RTS WITH ADDITION
= Image No.: 24 INTRODUCING PLUS AND EQUAL SIGNS and "EQUATIONS"
Begin with a review of basic addition concepts:

- "When we add two tiles together - when we put them together - we say 4 AND 2. But when we do work with numbers we have a special sign with a special name to mean that we put two numbers together."
Write 4 + 2.
- "Do you see the sign between 4 and 2? That + is called a PLUS sign. The plus sign tells us we put two numbers together. When we see 4+2 it means we have put a 4 and a 2 together."
Have the child duplicate this combination.
- "Which single tile can cover 4 PLUS 2 ?" ("dark green" or "6") - 4 Plus 2 has the same number of dots as 6; 4+2 makes six, and the dark green tile has 6. When both sides have the same number, we can say they are EQUAL.
- "Are the colours the same?" (No) - "What is the same?" (The number of dots.) - "That's right. The colours are different, but the number is the same."
- "When the number for both sides is the same, we say the value is the same."
- "Things that look different can have the same value."
Note: This concept will avoid confusions when working with money, with equivalent fractions, and, eventually, with algebraic equations.
Write 4+2 = 6 Have the child read this as a sentence: "4 plus 2 equals 6".
"This number story has an equal sign in it. We have a special name for a sentence that tells us two things are equal. That word is `equation'."
"Let's read this equation again." Repeat the reading.
To consolidate the concepts of "same value" and use of the equal sign, it is useful to use a simple balance scale (a ruler over a half-circle block with an equal sign on it, will do) and place six blocks of one colour on one side, four and two of different colours on the other side to make it balance.) (See Image No. 25) -, - -Image No.: 25 To prove they have the same value, remove one block from one side, or add one to the other. If the child wants to play with this apparatus to test other equations, give him time.
Note: Thinking of the equal sign as the fulcrum of a balance scale helps in later years when working with algebra. The equal sign makes values on each side perfectly balanced.
"WHEN WE ADD, ORDER DOESN' T MATTER." (Commutative Law of Addition) "What if we change the order and put the 2 first? Will that change the value?"
(No) Have the child do this to demonstrate that the value will still be 6.
Now have the child construct the combination and write the appropriate math sentence as shown below in Image No. 26:

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Image No.: 26 Have the child find all the possible two-number combinations that add up to six.
Have him record the number sentence that explains what he has discovered.
"Can you find two other tiles that can make six?"
These combinations can be arranged vertically or horizontally, or even covering the dark green tile. The numerical concepts have being reviewed; the critical element at this stage is to make sure the child can print the number sentences ("equations") that express each combination.

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Image No.: 27 Use the above method to review combinations for all numbers from 2 to 10. It is increasingly important for the child to arrange his work in an organized sequence. Refer back to pages 13-16 for examples. Note that combinations for 10 are most critical. See Image No. 27.
"When we add, it doesn't matter which number comes first."
MENTAL ARITHMETIC
By the time all combinations have been discovered and recorded in print, the parent can begin to challenge the child to work without the tiles, though the tiles should always be available if needed.
Sample mental arithmetic questions:
"I am thinking of two numbers that are the same size and add up to 6 (or any even number)."

"I want to cover a ten tile, but I only have a brown one. What colour will I
need to add so that I can make 10?"
"TAKING AWAY": The MINUS SIGN
Referring to Image No. 28, using a 7and a 3, have the child make an array of ten and write 7 + 3 = 10 "Can you remember how to take away? If we make a 10 with a 7 and a 3, what happens if we take the 7 away?" (We will have 3 left.) Demonstrate and have child do the same.
"What happens to the value of a number when we take something away?" ("The value /number gets smaller") "That's right. When we start with number and take a smaller number away, our value gets smaller."
"We have another sign we can use to tell us that we took something away. Look.
I
started with my 10, and I took away 3. 1 can write it this way: 10 ¨3 "This is called the MINUS sign "When I took 3 away from 10, what was left?" (7) "So I can write 10 minus 3 equals 7 like this: 10 ¨3 = 7 Have the child write the equation and read it back using words.
_AIM 7+3=10 -- ,--- \
i _oft, Image No.: 28 Repeat this activity using other combinations for 10.
MINUS AS "THE DIFFERENCE BETWEEN"
Referring to Image No. 29, we use the minus sign when we take away and when we want to find the difference between two numbers.
"What is the difference between 10 and 4?" (Allow time for the child to think about this and reply.) "When we want to compare two values, we always put the larger number first and we can cover up part of it with the smaller number, just as if we were going to take it away."
Do this with the 10 and 4 tiles. Note that, no matter where you place the 4 on the ten, 6 dots will remain uncovered.
"We can use the minus sign to show what we did. WE put down the large number and covered it with the smaller number, just as if we were going to take away some dots.
How many were left?" (6) "So we can write: 10 ¨4 = 6. The difference between 10 and 4 is 6. Or ten minus 4 is 6."
When we find the difference we use the minus sign.

St Image No.: 29 When we use the minus sign, whether we take away or find the difference, we say we are subtracting.

Note:
When we ADD, the value gets larger/bigger.
When we SUBTRACT, the value gets smaller.
Draw attention to the fact that, when we subtract, order DOES matter.
The right language at the right time, is critical.
PROBLEM SOLVING
Refer to Image No. 30 "Here's a question. Can you use your tiles to find the answer?"
"Brad's is 6 years old. His big sister is 9 years old. What is the difference in their ages?"
Children may need help to put the 9 tile down first, then cover it with the 6 to see how many dots remain.
When we want to find the difference between two numbers, we put the larger one first and "take away" the smaller one.
"How will we write what we did? What is the equation that tells this story?
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- Legs on spider and legs on ant - Wheels on car compared to number of wheels on school bus.
- etc.
WRITING EQUATIONS
"Now we have had lots of practice adding tiles and comparing them. Here is a combination:" (See Image No. 31) Fol OP 0, 4110, laroi'saw Image No.: 31 "What does this picture tell us?"
Prompt the following number sentences:
4 + 3 = 7 3 + 4 = 7 7 ¨ 3 = 4 7 ¨ 4 = 3 Have the child produce the array with his tiles, write the equations, and read them orally.
Language Practice:
When I add 4 and 3 what will I get?

If I add 3 and 4 what will I get?
If I start with 7 and take 4 away, what will I have left?
If I start with 7 and take 3 away, what will I have left?
What is the difference between 7 and 4?
What is the difference between 3 and 7?
Which is bigger, 4 or 7? By how much?
Which is bigger, 7 or 3? How much bigger?
What equation tells me that if I add 3 and 5 I get 8?
(Add a review of any terms or expressions that have confused the child.) ADDITION AND SUBTRACTION AS "NUMBER FAMILIES"
"Every time we put two DIFFERENT numbers together, we can write 4 different equations."
"What is 5+4 equal to?" (9) "Well, if I know 5 + 4 = 9, what else do I know?"
4 + 5 = 9 9 ¨ 4 = 5 9 ¨ 5 = 4 Show how these facts can be written vertically as well:

+5 +4 -5 -4 Every time we add two numbers we can make up a "family" of equations.
When we write these "up and down" we start at the top and read downward.

Refer back to the two-number combinations you completed (page 13). Have the child take each number in turn and list the possible combinations using the plus and minus signs.
Provide grid paper for child to design the first equation in each set, as shown in Image No. 31. These should be posted somewhere prominent for reference. (The intent of posting things on walls is to make use of the brain's ability to visually imprint information, helping with memory. Do not clutter walls with too much material;
keep postings separate and tidy.) ,:e- i .9 - i i 31 11 , I
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Image No.: 32 .... Etc.
This work can also be done vertically, making sure that numbers appear appropriately one under the other. The 10 with its two digits will use an extra square to the left of the zero. (This would signify the beginning of understanding "place value".) Note: Completing all these number "Families" will take about two weeks.
Intersperse lessons with a language drills, problem solving, and mental arithmetic.
IT CANNOT BE, STRESSED ENOUGH THAT THESE BASIC FACTS ARE THE
FOUNDATION FOR A THOROUGH UNDERSTANDING OF NUMBER. Do NOT
rush. Allow /hr use of tiles at any time to consolidate understanding.
INTRODUCING FRAMES
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Review "Zero":
"I am holding an odd (or even) tile behind my back. What could it be?"
Repeat similar questions three or four times. (Be creative ¨ give different types of clues, like "the number we get when we add 3 and 4", or "the number that is one less than nine", etc.
Now hold a closed, empty, fist out and have the child guess how many dots you have in your hand. Eventually open the fist to show that you are holding none.
"How many dots do you see?" ("None." "It's empty." "Nothing.") "Do you know a symbol that means nothing?"
If child doesn't know "zero", give him the word.
Draw a zero and ask what the zero looks like. (Donut, wheel, tire, circle.) Talk about the donut analogy to stress that the middle of the donut has NOTHING in it, and that's what "zero" means: "nothing, empty, none".
"NOW look at the number 10. We used 2 symbols to make the ten. What are they?"

("ONE and ZERO") "Today we are going to find out WHY we use a 1 and a zero."
Hand the child a blank 10-frame and ask him to count the number of squares in it.
Identify this as a "frame".
"A frame is like a railroad car where the dots sit side-by-side. Sometimes the dot has a partner to make an even number; sometimes one sits alone to make an odd number.
Have them place the orange 10 tile over the frame to reinforce quantity of 10.
Cover up the five dots on one side of the frame.

"How many dots do you see on this side?" (5) Now cover up the five on the other side.
"How many dots do you see on this side?" (5) "What did we get when we put 5 and 5 together?" (10) Now hold up another empty 10-frame.
How many dots will fit on this frame? (10) So we will call each of these frames a "10-frame".
"Listen very carefully to my next question:
How many orange tiles can I put on each frame?" (ONE) ("I did not ask how many dots; I asked how many frames.") Hold up two empty frames and ask how many orange tiles you can use to cover the two frames. ("Two") Spend as much time as necessary to clarify the concept of one frame for each orange tile.
"So, how many DOTS does it take to fill up ONE frame?" ("Ten") "When I cover one frame with one orange tile, do I have any dots left over?"
("No") "So when I fill up one 10-frame I have NO extra dots."
"What symbol tells me there are NONE left over?" (ZERO) "Now you know why we use a 1 and a 0 to make a ten. It means we have one group of ten ¨ like an orange tile ¨ and no more dots left over!"
Finish this introduction to the use of frames by having the children write the numerals in odd/even order down a replica of a QC frame. Have the child identify the numbers on one side and on the other (odds and evens).

Using Frames for Addition:
Referring to Image No. 34, have the child practice putting each individual tile on the frame, Note that the even side of odd tiles will have to sit on spaces 1 and 2, and that the odd one of 1,3,5,7,9 will always end up on the left side of the frame. This will require some flipping and reversing of tiles. (All good practice for "transformations"
in the future.) Practice placing all tiles in one pile on the frame in reverse order starting with 10...9...8...7, etc.
BE PATIENT. HASTE MAKES WASTE... AND FRUSTRATION!
Note: The term "left" will create a problem for even some very capable children. Have her draw a tracing of her left hand, cut that out, print the word LEFT on it in large letters and place it to the left of a doorway so that it is always visible for reference.
All addition facts up to 10 can now be reviewed by placing tiles on the frame.
The child will quickly realize that there is a problem when he tries to add 5+4, or any other combination that starts with an odd number.
Until now, it has been possible to place the tiles in any position and simply count to establish the total. Frames establish a left-right, sequential pattern with which the child is already familiar. This brings us to the two fundamental rules for using frames:
Rules for using Frames:
- all spaces must be used in sequence - no spaces between tiles can be left empty So, what to do with 5+4?
Review what we know about adding: "When we add, order doesn't matter."
When we are using the frame, we can change the order and put the odd one first;
now we can see that we have the right answer 4+5=9, and so does 5+4.

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Image No.: 34 (Pick the combination up and spin it around to review the concept if necessary.) Review: Our number system uses only ten digits: 0-1-2-3-4-5-6-7-8-9. When we have "TEN" of something, we use a 1 and a 0 to indicate that we have "One Group of Ten and zero more". When we have 10 dots we have 1 full frame and no more; no more =
ZERO more.
Counting to twenty:
Place two orange tiles side-by-side and ask the children to count the dots.
Practice this a few times. (Children typically have difficulty with the change in rhythm between "twelve" and "thirteen". Counting fluently to 20 does not have to be mastered at this stage, since each of the following steps will review and eventually consolidate that skill meaningfully.) About 10 minutes should be plenty of time for counting to 20 as an introduction to higher numbers.

Introduce 11:
First ask the child to use one orange tile to add 6 + 4.
"What number do we get when we ADD 6 AND 4?" (10) "Now I want you to start with the 6 again but this time I want you to add 5."
The orange tile will be full, and one extra dot will stick out.
"Count the dots. How many do we have?" ("Eleven") "Oops. We have one extra dot. So we need a new number that is bigger/greater than 10."
"TEN was 1 group of ten and 0 more; but now we have 1 group of ten and 1 more, so we write 11 as a 1 and another 1 beside it. The first 1 means 1 group often; the other 1 means we have one more than ten."
"We can show it this way:" Beside the 6 and 5 tiles, place one orange and one white tile.
"Here is our answer. 6 + 5 is the same as 10 + 1 because eleven is ten and one more."
Have the child build these two patterns for 11 and place them side by side.
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Complete the written equation: 6 + 5 = 11.
"If 6 + 5 = 11, what will 5+6 equal?" ("11") To consolidate this fact, hold the two tiles horizontally first, then spin them 180 degrees, to show that nothing is lost or gained.
"So I know I have 11 when I add a six and a five or a five and a six. What will I have left if I take the 6 away?" Demonstrate. ("You will have 5 left.") "What will happen if I start with 11 and take the 5 away?" Demonstrate. ("You will have 6 left.") You may want to try the following to review subtraction as "the difference between":
Hold up the 10+1 combination for 11 in one hand and just a 6 in the other.
"Can you tell me the difference between 11 and 6?" (5) "Can you tell me the difference between 11 and 5?" (6) (If the above leads to confusion, leave it for a later time. It always pays to expose children to more challenging concepts even if they do not totally master them.) "Now we can write four number sentences about 11? What are they?"
6 + 5 = 11 5 + 6 = 11 11 ¨ 6 = 5 11 ¨ 5 = 6 "What other two numbers can we use to make 11?"
Use one orange tile and one white tile as the model for eleven.
Allow the child to experiment. At this stage, the child can place the tiles in any position as long as they total 11 when they are counted. (She will discover that, if you are trying to place your tiles on top of the 10+1 model, and try to put an odd number first (like 7+4) there will be a gap in the pattern. If this happens, remind her to put the even number first because we know that when we add, order doesn't matter.) Eventually you should have a list of all the combinations for 11. Have the child write the four number sentences for each combination. Organize these in ascending or descending order.
+ 1 = 11 1 + 10 = 11 11 ¨ 1 = 10 11 ¨ 10 = 1 9 + 2 = 11 2 + 9 = 11 11 - 2 = 9 11- 9 = 2 8 + 3 = 11 3 + 8 = 11 11 ¨ 3 = 8 11- 8 = 3 7 + 4 = 11 4 + 7 = 11 11 ¨ 4 = 7 11- 7 = 4 6 + 5 = 11 5 + 6 = 11 11 ¨ 5 = 6 11 - 6 = 5 Just for fun:
- Can two even numbers make 11? Why not?
- Can two odd numbers make 11?
Numbers to 20:
Introduce all numbers to twenty by counting the dots in the arrangement shown below in Image No. 36. Count one array at a time, drawing attention to the fact that it uses one group of 10 and then some more:
12 is one group of 10 and 2 more; that is why we use the digits 1 and 2 to write it.
13 is one group of ten and 3 more.
14 is one group of ten and 4 more.
etc.

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II 12 13 lit 15 16 17 IS 141 20 Image No.: 36 "When we say 20, how many tens do we have?" (2) "Do we have any more?" (No!) "So that's why we use two digits - a 2 and a 0. We have 2 tens and nothing more."
Now begin developing all the combinations of two numbers that make each of the above 2-digit numbers.
The number 12 lends itself to the introduction of the egg carton, the concept of a dozen, and games using dice.
"Lucky" Game: Odds and Evens Label and number an egg carton as shown in Image No. 37. Each player chooses either odd or even number side of carton to fill with one marker for each winning combination on dice.

(This simple game provides lots of practice with addition facts to twelve.
However, ALWAYS stress that this is only a game of pure chance. "Who's lucky today?"
(Everyone's a winner when they practice adding quickly!) out fr =
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t Image No.: 37 This game a can be extended to adding any number of dice, according to the ability level of the child. See Image No. 38.
THE CHILD WHO CANNOT CALCULA FE, CANNOT ESTIMATE.
Continue with all two-number combinations to 20. Include writing some questions vertically to stress the placement of the tens digit. Without the introduction of enrichment activities this may become a tedious process, Note: While QC tiles provide constancy and are intended to be the backbone of arithmetic learning, they are not intended to be the only material to which a child is exposed. Lessons can be broken up by making use of many of the items pictured below:
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µta, di 1114 476 711, -Image No.: 38 Also: Cuisenaire Rods, Unifix Cubes, Tic-Tac-Toe, Snakes and Ladders, Score 4 Once combinations to 20 have been introduced (not necessarily mastered), introduce a second blank frame and have the child fill in the numbers from one to 20.
Have the child show you and repeat all the even numbers and all the odd numbers. Note that all the evens end in 2,4,6,8,0 which appear on the right and all the odds end in 1,3,5,7,9 on the left.
Introduce the numbered second frame and hook it up like a train car, to the first one. Use this extended "train" to practice all the addition combinations you have studied.
Remember, all calculations are based in a thorough understanding of addition facts.
important.) Stress doubles 6+6, 7+7, 8+8, 9+9 and show how these help us remember 6+7 (1 more than 6+6 or 1 less than 7+7) Devise problems and mental math activities to measure how much has been learned.
Review of all number facts should be on-going. This is not "rote" drill, since tiles give meaning to quantity and can be used at any time to model answers.
Making use of 10s:
Referring to Image No. 39, when adding numbers that total more then 10, it is useful to draw the child's attention to the fact that, when he adds 6 + 7, for example, he uses 4 of the 7 dots to complete the 10. Because he knows that 7 is made up of 4 + 3, he should know that he will now have a 10 and 3 more, i.e. "13".
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,41. 40 _ -6+1 10+3 Image No.: 39 Introduce numbers to 100 using the ten frames that hook into each other.
Review the concept of place value by noting that 30 is 3 groups of ten and no more, 40 is four groups of ten and no more, etc Count to 100 by ones, pointing to each number in turn (left, right, left, right).
Count to 100 by 10's.
Along with the 100 odd-even train of QC Frames, the child should have a 100 chart in traditional 10-by-10 mode for additional reference.
Practice printing numbers to 100.
BRIDGING FACTS (Beyond 201 Referring to Image No. 40, because the child has been exposed to adding beyond 10, he is now able to add numbers well beyond 10, relating this to the earlier learning, using his QC Tiles. These can be referred to as "bridging facts". If 6 + 4 = 10, then 16 + 4 = 20, 26 + 4 = 30, etc.
At this stage, work should be written in vertical fashion to allow for easy transition to the concept of adding "with carrying".
Using 7 + 5 as the example, develop the following:

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Draw attention to the repetition of the 7 + 5 "bridging" the red tens line in each case.
Remember to review subtraction as "the undoing of addition":
- Since 7 + 5 = 12, then 12 ¨ 7 = 5, and 12 ¨ 5 = 7 Repeat the above process adding a 1-digit number to any 2-digit number.
- Since 7 + 5 =12, then 17 + 5 = 22, 27 + 5 = 32, 37 + 5 = 42, etc.
- Link to related subtraction facts: 22-5 =17, 32 ¨5 = 27, 42¨ 5 = 37 (The degree to which the child needs master these bridging facts, will depend on the method chosen for teaching "subtraction with borrowing" in future.
Understanding the above relationships may be sufficient if you select Method 2 for subtraction.) ADDITION OF 2-DIGIT NUMBERS "WITHOUT CARRYING"
Using only QC tiles (no frames), ask the child to construct a 24 and a 53.
Have them place these two models one below the other as shown below in Image No. 41:
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\-----1 ¨, 71 _,..., Image No.: 41 "How much will we have if we add these two numbers together?"
"How many tens will we have?" (7) "How many single dots will we have when we add the 4 and the 3? (when we add the 1 and the 3? (7) "What number means six tens and 4 more?" (64) Write the question vertically: 2 4 +53 Draw two lines below each column of numbers as shown.
Important Rules:
1. "When we add 2-digit numbers, we always add the smaller numbers first. We will call these ones because they don't make a group of 10. After we have done that, then we will add the 10s." ("First we'll practice doing some questions like this one and then I'll show you some different questions that will explain why we have this rule.) 2. "When we have numbers one above the other, there is always only ONE DIGIT
IN EACH SPACE."
Point to the little lines below the question.
"Each of these is a space. This one ¨ on the right (point) is the one we will use first to tell us how many single ones will be in our answer. Then we will use the one on the left (point) to tell us how many groups of 10 we will have."
Point out the 4 and 3 one above the other.
"We can write only ONE digit under here when we add. What did we say that digit will be? 4 + 3 = ? (7) Write the 7 in the ones place.
"Now we can add the groups often. How many groups of 10 do we have?" (7) Write the 7 in the tens space.
"What number has 7 tens and 7 ones?" (77) "Let's check our work on the 100 train."
Build 77 on frames by placing all the tens in sequence, then adding the 4 and Do a few more examples with the child, checking to make sure she is adding the ones side first.
ADDITION WITH CARRYING
Refer to Image No. 42.
"Now we are ready to look at why we add the ones side first."
Using tiles only, construct the model for the following question:

+2 7 "Which do we add first, tens or ones?" (ones) "What happens when we add 6+7?" (We get 13) "How many digits in 13?" (Two ¨a 1 and a 3) "But how many digits can I write in one place?" (Only one!) "All right. Let's see how we can solve this problem."
Take the 6+7 ones and move them to the side.
"What did we say 6+7 makes?" (13) Write the number 13 off to the side.
"Let's look at 13. Why do we use the digits 1 and 3 to write 13? (Because we have 1 group of 10 and 3 more.) "So let's change the 6+7 for a 10+3. Is that fair?" ("Yes") "Why?" (Because they are equal. They have the same value.) Have the child duplicate this action with his own model.
"Let's look at what we have now..."
"How many single ones do we have?" (3) Write the 3 in the ones space of the answer.
"Now look at our tens. We had 3 here and 2 more and that made 5. But look, we made another one!"
"To show that we have 1 more ten, I am going to write an extra 1 on the tens side - up here - above the 3. Now I have to add 3+2+1 more. What will my answer be on the tens side?" (6) What number has 6 tens and 3 ones? (63) 1õ., ilz Arco = 4 ,,40 = 4Z-,-F 410 ly --. __________________________________________ 3, === 41) . = 414) =
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el A; .,t i. t .::_.2. 1 ....i Image No.: 42 SUBTRACTION "WITHOUT BORROWING"
To tackle subtraction of two-digit numbers review the concept of covering up the amount you are "taking away". Refer to Image No. 43.

t , , Image No.: 43 Have the child construct 48 using 4 tens and an 8 "If I am going to take away 23, how many tens will I cover up?" (2) "How many ones will I cover up?" (3) "What's left?" (2 tens and 5 ones, or 25) Review comparing with Problems I have 36 Pokemon cards. My friend has 24. Who has more? How many more?
Brad weighs 78 pounds. His little sister weighs 52 pounds. How much more does Brad weigh?
A skateboard costs $96 dollars. I have $ 45. How much more money do I need?
SUBTRACTION "WITH BORROWING" - Method I (Traditional' Refer to Image No. 44 below. To solve 44-26 (for example) Tr) , , , 7' . -4 ' ,µ,.,...
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Alt- --1. Nr. = i Image No.: 44 In the traditional model, we have the child "borrow" a ten from the tens column and add that 10 to the 4 on the right by writing a little 1 beside the 4, making it 14. The child then subtracts 6 from the 14 using his knowledge of "bridging" facts.
SUBTRACTION "WITH BORROWING" - Method 2 (Korean) Much easier is this method explained to me by an 8-year-old Korean boy who always came up with the right solution to subtraction questions in half the time, regardless of the number of digits - including zeros in the top number. (See Image No. 45) 11 ft 11 Ng, " **f" =
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t tc' 12; r t 1 = .
Image No.: 45 The Korean model uses only skill with facts to 10. Again the child borrows, and indicates in the tens column that he has done so, but he subtracts the lower number from that 10 and calculates how many ones he will have left. He adds those ones to the original ones and has his answer. He does not need to have mastered the more challenging "bridging" facts (above 10).
Translating the work to paper, the two models would appear as shown below in Image No. 46:
Tr a ci flan o 1.4 /
2_ Image No.: 46 The Korean method proved to be significantly faster; only the borrowing from the lOs column was indicated on the child's work. (I have included crossing out the 6 and adding the 4, only to clarify the process.) PREPARING FOR MULTIPLICATION
Introducing new language:
"Let's look at our tiles again. If I pick up a tile with yellow dots, how many dots will it have?" (5) "Yes, there are 5 yellow dots on this one tile, so we can call it one group of 5. The yellow tile is one group of 5."
"Which tile shows one group of 8?" (the brown one) "Which is one group of 10?" (orange) Now reverse the language:
"What colour is one group of 4?" (purple) "What colour is one group of 7?"
Add a mental calculation:
If I add two groups of 5, how many dots will I have? (10) - show this as 5 + 5 = 10 (write this vertically as well) If I add three groups of 2 together, how many dots will I have? (6) - show this as 2 + 2 + 2 = 6 (also write vertically) If I add three groups of 3 together, how many dots will I have? (9) - show this as 3 + 3 + 3 = 9 (also write vertically) What is the value of four groups of 2? (Always allow time for manipulation if necessary.) (8) - show this as 2 + 2 + 2 + 2 = 8 (also write vertically) Once the child indicates that he is comfortable with the above language and calculations (using tiles when necessary), proceed to the next step.

Introduce the X sign as "GROUPS OF":
"Can you show me what six groups of 2 makes?" (Child makes the array with tiles and discovers it makes 12.) "How would we write this?"
2 + 2 + 2 + 2 + 2 + 2 = 12 "Yes, we have taken six red tiles and added them all together. But that's a long equation.
There is a much shorter way of writing this.
When we add the same number over and over again, we can use this sign: X
It is the GROUPS OF sign."
"Look. Instead of writing all those 2s, I can just write 6 x 2, because that means 6 groups of 2. The X tells me how many times I added a number together. That is why it can also be called the TIMES sign.
I can say:
"6 TIMES 2"
or "6 GROUPS OF 2."

2.

g+ 2+1+ 1+2+1 =12 11 I 2 ecris 12' 1,45 2 p 2 p Ls 2. p1,4.3,.. p 44.5 P
_ Ea, j fo wr fe % =12 , E as i er +0 y " 6 altia; 2eortuLt 12' le t.s Image No.: 47 Referring to Image No. 47, practice by doing repeated addition using tiles and frames, then writing the long addition equation, and finally re-writing the shorter equation using the X sign. READ each equation orally, using "GROUPS OF" for the X sign.
e.g. 5 + 5 + 5 + 5 = 20, becomes 4 x 5 = 20, READ as "Four groups of 5 equals 20."
"When you use the X sign you are doing multiplication.
MULTIPLICATION is just a quick way to add the same number over and over again."
Perform additional examples, recording both the longer addition equation, and the corresponding multiplication equation.
"WHEN WE MULTIPLY, ORDER DOESN'T MATTER" (The Commutative Property of Multiplication) Review the meaning of one dozen. (12) "How can you make one dozen?" (Accept all possible combinations, including 3-and 4-tile additions.) Record all of these as equations, including those given that use the X
sign.
"But now I'm going to change the rules. Listen carefully. Use your first two frames and see if you can you make one dozen if you can only use one colour of tile?"
Give the child time to experiment and find a solution.
Prompt if necessary to discover that we can use red ones, dark green ones, purple ones, and light green ones. (Background for "factors" later on...) "Use the groups of 2 ¨ the red ones only ¨ to make a dozen."
"How many red tiles did you need?" (six) "Right, you needed 6 groups of 2 to make 12."
"How will we write that?" (6 x 2 = 12) "Right. Let's read that: 6 groups of 2 equals 12."
"Now, use only the dark greens ¨the groups of 6 to make 12."
"How many dark green tiles did you need?" (two) "How will we write that?" (2 x 6 = 12) "Right. Let's read that: 2 groups of 6 equals 12."
Draw attention to the fact that, since both 2x6 and 6x2 equal six, it looks like order doesn't matter when we multiply. (This shouldn't surprise us since "multiplying is just a quick way of adding the same number over and over again." (See Image No.
48) "Let's try another question to see if this is true."
"How many frames do we need to make 20?" (two) "Take two frames and show me how many groups of four you need to make 20." (5) , t ---- a -1 -i ; x3 =
1 _ I t ' , J
, .
¨,, ' ...,.....0 --J
6 le 2r- 12 2%6 la ,../.A&
5263, L ,e50 5%4. 1-,,, .5.-Image No.: 48 Try a few more examples, using frames and tiles to prove that:
"WHEN WE MULTIPLY, THE ORDER DOESN'T MATTER."
MANTRA: "SHOW ME WITH YOUR TILES!-or "PROVE IT WITH YOUR TILES!"
FACTORS and MULTIPLES
"We made 20 using 4s and Ss. Can you make a 20 using sixes?" (No. You Can make and 24, but not 20.) "Can you make 20 using 2s?" (Yes) "How many?" (10) Record in print: 10 x 2 =

"Can you make a 20 using 3s?" (No. You can make 18 and 21, but not 20) "Can you make 20 using lOs?" (Yes) "How many?" (2) Record in print: 2 x 10 =

"Let's list the numbers we can use to make 20: 4, 5, 2, 10."
We have a special name for small numbers that can be multiplied to make a big number.
They are called factors.
2, 4, 5, and 10 are FACTORS of 20.
What are the numbers we multiplied to get 12?
2, 3, 4, 6.
Try other numbers: 5, 8, 9. They are NOT factors of 12.
The larger number that factors make when we multiply them is called a "multiple".
Post the following for visual imprint of terms:
x 6 = 30 3 x 4 = 12 Factors Multiples (When working with young children, the introduction of the terms "factor" and "multiple" can be considered enrichment only, but it is essential to introduce to older student who are using QC Tiles as a remedial tool.) THOSE MULTIPLICATION TABLES
Begin by using 2s on frames to 20 Make use of Commutative Property 1 group of 2 = 2 has the same value as 2 groups of 1:
1 x 2 = 2 1 x 2 = 2 2 groups of 2 = 4 (identical in either direction) 2 x 2 = 4 3 groups of 2 = 6 has the same value as 2 groups of 3 3 x 2 = 6 2 x 3 = 6 . Each set of multiples should be recorded on three different number charts, one depicting the QC frames, one a grid to 100, and finally, a "Times Table" grid, working in both vertical and horizontal directions to indicate that the commutative property is understood.
The use of colour can help with memorization.
Only ONE set of multiples should be represented on a grid during these early stages. (See Image No. 49) . p.
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Image No.: 49 Note that when we repeat a two, the multiples are always even numbers.
"Two always makes a pair, so our values will always be even numbers."
Threes "Can we make pairs when we have 3?" (No, we have 1 left over ¨ or ¨ "one extra".) "Let's see what happens when we repeat the 3 to make multiples." (Refer to Image No.
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1 ay MI 10 1 i t 30r ¨ L-1¨ 1 ... 1 Image No.: 50 Note that when we repeat an odd number, the pattern goes from odd to even, to odd to even, etc. Review how the "odd ones sticking out" fit together to form an even every time we have two of them, accounting for the odd-even changes.
Continue with all numbers to 10.
The recommended pattern is; 10s, 5s, s, 6s, 7s, 8s, 9s.
Always draw attention to odd even patterns and final digit patterns.
Ask the student to identify any other interesting patterns he can see:

= CA 02950116 2016-11-30 The 100 grid patterns should be posted for constant reference. See Image No.
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Image No.: 51 Odds and Evens in multiplication patterns (helping memory) See Image No. 52:
- an even group of even numbers always results in an even: 8 x 4 = 32 - an odd group of even numbers will still result in an even: 3 x 4 = 12 (no matter how many even numbers you have, they will remain even).

- An even group of odd numbers will always result in an even number since every pair of odd numbers create an even. 4 x 3 =12 (AND it's just the reverse of 3 x 4) - only an odd number of odds result in an odd number, since, after all the odds are paired, you will still have one "extra one" sticking out. 3 x 5 = 15 8 )( 4=342 3v3/4.- 12 3 x 5 = i5 e ven x evcm = even Odd Y even . cven acid s cock odd - 1 7....._ I zip , r , 1 '':...jil litit I
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rinB, 31' _3p 3 , Image No.: 52 DIVISION
Introducing Division as "Un-doing" multiplication When we were adding, we discovered that every two-number addition resulted in FOUR
equations. For instance, if I added a 7 and a five what were the four equations I could write?
7+5 = 12, 5 + 7 =12, 12-5 = 7, 12 ¨ 7 =5 Now we know that multiplying is just a quick way of adding a number over and over again. Well, just like subtraction is "the opposite of addition," WE CAN DO
SOMETHING THAT IS THE OPPOSITE OF MULTIPLYING, IT IS CALLED
'DIVIDING".
Here is how it works. I can give you a multiple like 8 and ask you:
How many 2's does it take to make an 8? Or how many 2's are in 8 How many 2s make 8? Show me with tiles.
Now start with an 8 and see how many tiles it takes to cover it.
OR:
Have the child copy what you are doing:
"It's like seeing how many times I can take 2 away from 8, before I have nothing left.
Start at 8 on your 10-frame, cover up the first two squares with a two. Now you only have 6, cover up another 2, now you have 4, cover up 2, now you have 2, cover up that 2 and you have finished because there are no more spaces left. How many times did we cover up a two?" (4) Multiplication is like adding the same number over and over again; division is like subtracting the same number over and over again and counting how many times you can do it before you have nothing left to cover.
Dividing is easy to write, but difficult to read.
The division sign looks like this:
=
It means "divided by"
We can put that sign between 8 and 2 and read the question as:
"8 divided by 2."
But if we want to understand what that means, we have to read from the middle and go backwards, like this:
"How many 2s are there in 8?"
NOW it makes sense. And we know the answer. There are four 2s in eight. We know that because 4 x 2 =8.
Go on to other examples like:
(My computer does not allow for a dividing sign. It needs to be used below:
divided by 5. (Have child change this to "How many 5s make 10?") divided by 4 ("How many 4s make 20?") 18 divided by 6 (How many 6es make 18?) Now review the multiplication facts by creating number families as we did for addition and subtraction. (See Image No. 53) e.g.
We know 6 x 5 = 30 and that, therefore, 5 x 6= 30 Now we also know that there arc six 5s in 30 (30 divided by 5 = 6) And that there are five sixes in 30 (30 divided by 6 = 5) 411111 , _ .

... 1 141 ' Division.
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t't becOme5:
_ , Haw many 5 in 30?
i How many is in 30?
The si5n eon division is -4-.
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Image No.: 53 = We know that 6 x 8 = 48 and 8 x 6 = 48 So we know that 48 divided by 6 = 8 and that 48 divided by 8 = 6 (Use tiles as necessary when not sure.) What does 7 x 9 equal? (63) Then what else do you know?
9 x 7 = 63 63 divided by 7 = 9 63 divided by 9 7 RE-TIIINKING with QC TILES
Through the use of QC Tiles, the child can re-think groupings:
"I can't remember 6 x 8, but when I picture 6 brown tiles, I can separate those into two groups of 3 brown tiles. 3 x 8 = 24 and I have 2 groups that size. 2 x 24 =
48."
"I can't remember 9 x 7, but I know ten black tiles (7s) make 70. If I take away one black tile, that makes 63."
Providing a fixed representation of each number makes it easier to visualize and manipulate quantities when working mentally.
PRIME NUMBERS
"Can you show me all the tiles that can repeat to make 12?" (2, 3, 4, 6) "We called these factors of 12. The numbers we can multiply to get 12."
"Now... can you show me the factors of 15?" (3 and 5) "Now, can you show me the factors for 13?"
Students will discover that the only number that repeats to make a 13 is a 1.
"You are right. Some numbers have no factors except 1 and itself. I can write 1 x 13 = 13 And that's right, isn't it?" (yes) "So the factors of 13 are 1 and itself- 13."
"Can you find another number that has no factors other than 1 and itself?"
If students have difficulty, guide them toward 11, 17, 19 "We have a special name for numbers like this. They are called PRIME NUMBERS"
Build a Factor Chart to 23 (or higher Note: Every activity of this type reinforces the understanding of the way our number system works, and review those critical multiplication tables.
Let the child develop this list himself using his tiles and frames:
then Develop a FACTOR CHART:
/ - / 14 - 1, 2, 7, 14 2 - 1 2 15 -1,2,3,5, 15 3 - 1 3 16 - 1, 2, 4, 16 4 -1,2, 4 17 - 1 17 - 1 5 18 - 1, 2, 3, 6, 9, 18 6-1,2,3,6 19 - 1 19 7 - 1 7 20 - 1, 2, 4, 5, 10, 20 8-1,2,4,8 21 - 1, 3, 7, 21 9 - 1, 3, 9 22 - 1, 2, 11, 22 - 1, 2, 5, 10 23- 1,23 11 - 1 11 24 - 1, 2, 3, 4, 6, 8, 12,24 12 - 1, 2, 3, 4, 6, 12 25 - 1, 5, 25 26 - 1, 2, 13, 26 "Which number has the most factors?" (24) "Which number has the least number of factors?" (1) "That makes 1 a very special number. It belongs in a group all by itself. Some mathematicians call it "Unity". All we need to remember is that it is in a group that is different from all other numbers."
Circle the 1 and its single factor (itself) in the list, to indicate that it is separate from other groups.
"Now let's find all those PRIME numbers ¨ the ones that have only two factors ¨ 1 and itself" Have the child underline these: 2,3,7,11,13,17,19,23 "What is the smallest prime number?" (2) "What kind of number is 2 ¨ odd or even?" (Even) "Are there any other even numbers that are PRIME?" (No) "Why do you think this is the case?" (Because every other even number can be divided by 2 ¨ in addition to 1 and itself.) Prompt for this concept. Check through the even numbers and find the 2 in each case.
"So 2 is the only prime number because, like other even numbers, it can be divided by 2, but that 2 happens to be itself."
"Look at your list again. Which numbers have more factors, odds or evens?"
(evens) "Why do you think that happens?" (This is a stretch and may need prompting....
"Even numbers can be made by multiplying an even by an even, or an odd by an even, but odd numbers can only be made by multiplying an odd by another odd") OTHER PROPERTIES OF NUMBER
The Associative Property of Addition:
When adding three or more tiles, you can add in any order (See Image No. 54):
- red plus white plus yellow = yellow plus white plus red = white plus yellow plus red - 2+1+5 = 5+1+2 = 1+5+2 = 5 + 2 +1, etc , r (i __ _ 4 , ____________________________________ 1 ri-1 3 Li i I rfol , ' _ ¨
t tr -g 1 -- ......=1 _i H
H
Image No.: 54 This make it possible to add the "easiest" combinations when adding a series of numbers:
e.g. 7 + 8 + 2 + 3. Add the 7 + 3 to make ten, then the 8 + 2 to make another ten and you have 20.
Note: when we add, even if we are adding many numbers, we only perform one operation at a time, adding only TWO numbers at each step. Emphasis is always, therefore, on adding TWO numbers. Adding a series of numbers prepares children for re-grouping numbers for efficiency with larger numbers.
STRESS COMBINATIONS FOR 10- these are critical for more advanced calculations.
The Associative Property of Multiplication:
When multiplying any number of numbers, you can multiply in any order:
(order still doesn't matter) (See Image No. 55) 2 x 4 x 3=3 x2 x4=4 x3 x2 =24,etc ,2,_ x li x 3 . 3 x 2 x If ____,......._ _ 20. s s.5(3 =24 3x2, g Cyilz2it 4)6:- tz 12s2:21-1 1 , II ¨
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1 1- i , p , , Image No.: 55 As in addition, this allows the student to multiply = in any or der convenient, and to check his answer by multiplying in a different order.
The Distributive Law of Multiplication:
e.g. 2(6+5) = (2x6) + (2x5) Use frames to show that 5+6 =11. Two times 11=22 = Then show two sixes added to two fives.
(For purposes of clarity, Image No. 56 below show these tiles side-by-side. In reality they would be placed on the frame and overlapped with each other.) , , -ill-. ,,,,, 0 0 al iii 41;;t1 ---., , . 1 .
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Image No.: 56 The same principle can be applied to division:
18 is 10 + 8, so 18 divided by 2 is the same as 10 divided by 2 and 8 divided by 2:
18/2= 10/2 + 8/2 = 5 + 4 (A very useful concept when working with fractions and dividing larger numbers.
OC TILES for REMEDIATION
All of the preceding basic concepts are critical for progress through the higher grades;
they can be presented at any grade level to fill in conceptual gaps.
The scope of the claims should not be limited by the preferred embodiments set forth in the examples, but should be given the broadest interpretation consistent with the description as a whole.

Claims (5)

The embodiments of the invention in which an exclusive property or privilege is claimed are defined as follows:

1. An educational device that teaches arithmetic, comprising: a frame in dual-track odd-even pairings, said pairings being in increments of 10, interlocking to form groups of various lengths up to 120.

2. The educational device of claim 1 wherein the base of each frame is marked with a line to denote one full set of ten.

3. The educational device of claims 1 or 2 wherein numerals appear in the bottom right-hand corners of the frames and are clearly visible when covered with a transparent dotted line.

4. The educational device of any one of claims 1 to 3 further including a storage box which has a sunken pit, said pit having a floor and opposed sides defining a channel, the blocks having a width such that the blocks can be removably received in the channel.

5. The educational device of claim 4 wherein the track includes a stop at one end thereof.