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Applying the Spirit of Asian Mathematics VII MAPSA Conference November 2, 2010 Detroit, Michigan by Joan A. Cotter, Ph.D. [email_address] Handout and Presentation: ALabacus.com 7 5 2
Lesson explicitly addresses student motivation, performance, and understanding.
The Flow Principle.
The lesson builds on students’ previous knowledge and supports them in learning the lesson’s big math ideas.
The Unit Principle.
Teacher fits lesson with past and future lessons.
The Adaptive Instruction Principle.
All students do math at their current understanding.
The Preparation Principle.
Coherent lesson plan must be well-thought-out and detailed.
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Adding by Counting From a Child’s Perspective Because we’re so familiar with 1, 2, 3, we’ll use letters. A = 1 B = 2 C = 3 D = 4 E = 5, and so forth
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Adding by Counting From a Child’s Perspective F + E A C D E B A F C D E B
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Adding by Counting From a Child’s Perspective F + E What is the sum? (It must be a letter.) A C D E B A F C D E B
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Adding by Counting From a Child’s Perspective K G I J K H A F C D E B F + E
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Adding by Counting From a Child’s Perspective Now memorize the facts!! E + I G + D H + F C + G D + C
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Subtracting by Counting Back From a Child’s Perspective Try subtracting by ‘taking away’ H – E
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Skip Counting From a Child’s Perspective Try skip counting by B’s to T : B , D , . . . T .
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Place Value From a Child’s Perspective L is written AB because it is A J and B A’s huh?
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Place Value From a Child’s Perspective L is written AB because it is A J and B A’s huh? (12) (one 10) (two 1s). (twelve)
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Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 Sometimes calendars are used for counting.
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Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 Sometimes calendars are used for counting.
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Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 A calendar is NOT like a ruler. On a ruler the numbers are not in the spaces.
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Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 A calendar is NOT like a ruler. On a ruler the numbers are not in the spaces.
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Calendar Math August 8 1 9 2 10 3 4 5 6 7 Always show the whole calendar. A child needs to see the whole before the parts. Children also need to learn to plan ahead.
The field of mathematics is doubling every 7 years.
Math is used in many new ways. The workplace needs analytical thinkers and problem solvers.
State exams require more than arithmetic: including geometry, algebra, probability, and statistics.
Brain research is providing clues on how to better facilitate learning, including math.
Increased emphasis on mathematical reasoning, less emphasis on rules and procedures.
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Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
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Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
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Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
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Memorizing Math Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall
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Visualization “ Think in pictures, because the brain remembers images better than it does anything else.” Ben Pridmore, World Memory Champion, 2009
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5-Month Old Babies Can Add and Subtract Up to 3 Show the baby two teddy bears. Then hide them with a screen. Show the baby a third teddy bear and put it behind the screen.
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5-Month Old Babies Can Add and Subtract Up to 3 Show the baby two teddy bears. Then hide them with a screen. Show the baby a third teddy bear and put it behind the screen.
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5-Month Old Babies Can Add and Subtract Up to 3 Show the baby two teddy bears. Then hide them with a screen. Show the baby a third teddy bear and put it behind the screen.
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5-Month Old Babies Can Add and Subtract Up to 3 Show the baby two teddy bears. Then hide them with a screen. Show the baby a third teddy bear and put it behind the screen.
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5-Month Old Babies Can Add and Subtract Up to 3 Raise screen. Baby seeing 3 won’t look long because it is expected.
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5-Month Old Babies Can Add and Subtract Up to 3 A baby seeing 1 teddy bear will look much longer, because it’s unexpected.
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Yellow is the Sun Yellow is the sun. Six is five and one. Why is the sky so blue? Seven is five and two. Salty is the sea. Eight is five and three. Hear the thunder roar. Nine is five and four. Ducks will swim and dive. Ten is five and five. – Joan A. Cotter Also set to music. Listen and download sheet music from Web site.
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Counting Model How many? Contrast naming quantities with this early counting model.
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Tally Sticks What is 4 apples plus 3 more apples? How would you find the answer without counting?
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Tally Sticks What is 4 apples plus 3 more apples? To remember 4 + 3, the Japanese child is taught to visualize 4 and 3. Then take 1 from the 3 and give it to the 4 to make 5 and 2.
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Materials for Visualizing “ In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic.” Mindy Holte (E I)
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Materials for Visualizing Japanese Council of Mathematics Education Japanese criteria.
Representative of structure of numbers.
Easily manipulated by children.
Imaginable mentally.
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Materials for Visualizing “ The process of connecting symbols to imagery is at the heart of mathematics learning.” Dienes
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Materials for Visualizing “ Mathematics is the activity of creating relationships, many of which are based in visual imagery. ” Wheatley and Cobb
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Materials for Visualizing The role of physical manipulatives was to help the child form those visual images and thus to eliminate the need for the physical manipulatives. Ginsberg and others
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Number Chart 6 1 7 2 8 3 9 4 10 5 To help children learn the symbols.
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AL Abacus Double-sided AL abacus. Side 1 is grouped in 5s. Side 2 allows both addends to be entered before trading. 1000 100 10 1
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4 + 3 = 7 Adding Mentally, think take 1 from 3 and give it to 4, making 5 + 2.
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Sums Adding to Ten 1 and 9; 2 and 8; 3 and 7; and so forth.
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Go to the Dump Game Objective: To to learn the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 Object of the game: To collect the most pairs that equal ten. Children use the abacus while playing this “Go Fish” type game.
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Go to the Dump Game A game viewed from above. Starting
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Go to the Dump Game Each player takes 5 cards. 7 2 7 9 5 7 4 2 6 1 3 8 3 4 9 Starting
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Go to the Dump Game Does YellowCap have any pairs? [no] 7 2 7 9 5 7 2 4 6 1 3 8 3 4 9 Finding pairs
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Go to the Dump Game Does BlueCap have any pairs? [yes, 1] 4 6 7 2 7 9 5 7 2 4 6 1 3 8 3 4 9 Finding pairs
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Go to the Dump Game Does PinkCap have any pairs? [yes, 2] 4 6 7 2 7 9 5 7 2 1 3 8 3 4 9 Finding pairs 7 3
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Go to the Dump Game Does PinkCap have any pairs? [yes, 2] 4 6 7 2 7 9 5 2 1 8 3 4 9 Finding pairs 7 3 2 8
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Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. The player asks the player on his left. 2 4 6 7 3 7 2 7 9 5 1 3 4 9 2 8 Playing
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Go to the Dump Game PinkCap, do you have a 6? Go to the dump. 2 8 5 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1
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Go to the Dump Game YellowCap, do you have a 9? 1 9 2 8 5 4 6 7 3 2 2 7 9 5 4 9 Playing 1
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Go to the Dump Game PinkCap is not out of the game. Her turn ends, but she takes 5 more cards. 1 9 5 4 6 7 3 2 2 7 9 5 4 9 Playing 2 9 1 7 7
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Go to the Dump Game No counting. Combine both stacks. (Shuffling not necessary for next game.) 6 5 1 Winner? 4 5 9 5
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Go to the Dump Game No counting. Combine both stacks. (Shuffling not necessary for next game.) Winner? 4 5 9 6 5 1
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Go to the Dump Game Whose pile is the highest? Winner? 4 6 5 5 9 1
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Part-Whole Circles Whole Part-whole circles help children see relationships and solve problems. Part Part
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Part-Whole Circles 10 4 6 What is the other part?
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Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? A missing addend problem, considered very difficult for first graders. They can do it with a Part-Whole Circles.
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Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 3 a part or whole?
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Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 3 a part or whole? 3
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Part-Whole Circles 3 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 5 a part or whole?
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Part-Whole Circles 3 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 5 a part or whole? 5
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Part-Whole Circles 5 3 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? What is the missing part?
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Part-Whole Circles 5 3 2 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? What is the missing part?
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Part-Whole Circles 5 3 2 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Write the equation. Is this an addition or subtraction problem?
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Part-Whole Circles 5 3 2 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 3 + 2 = 5 5 – 3 = 2 Is this an addition or subtraction problem?
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Part-Whole Circles Part-whole circles help young children solve problems. Writing equations do not.
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“ Math” Way of Counting 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 . . . . . . . . 99 = 9-ten 9 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.
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Language Effect on Counting 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Average Highest Number Counted Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Purple is Chinese. Note jump during school year. Dark green is Korean “math” way. Dotted green is everyday Korean; notice jump during school year. Red is English speakers. They learn same amount between ages 4-5 and 5-6.
Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic.
Just as we first teach the sound of the letters, we first teach the name of the quantity (math way).
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Research Quote “ Rather, the increased gap between Chinese and U.S. students and that of Chinese Americans and Caucasian Americans may be due primarily to the nature of their initial gap prior to formal schooling, such as counting efficiency and base-ten number sense.” Jian Wang and Emily Lin, 2005
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Subtracting 14 From 48 Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones will count 14. Those understanding place value will remove a ten and 4 ones.
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3-ten 3 0 3 0 Place-value card for 3-ten. Point to the 3, saying three and point to 0, saying ten. The 0 makes 3 a ten.
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1 hundred 1 0 0 1 0 0 Of course, we can also read it as one-hun-dred.
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Column Method for Reading Numbers 2 5 8 4 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:
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Column Method for Reading Numbers 2 5 8 4 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:
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Column Method for Reading Numbers 2 5 8 4 2 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:
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Column Method for Reading Numbers 2 5 8 4 2 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: 4
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13 – 9 = Strategy: Going Up Start at 9; go up to 13.
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13 – 9 = Strategy: Going Up Start at 9; go up to 13. To go up, start with 9.
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13 – 9 = Strategy: Going Up Start at 9; go up to 13. Then complete the 10 and 3 more.
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13 – 9 = Strategy: Going Up Start at 9; go up to 13. Then complete the 10 and 3 more.
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13 – 9 = Strategy: Going Up Start at 9; go up to 13. 1 + 3 =
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13 – 9 = Strategy: Going Up Start at 9; go up to 13. 1 + 3 = 4
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Traditional Names 4-ten = forty 4-ten has another name: “forty.” The “ty” means ten.
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Traditional Names 6-ten = sixty The same is true for 60, 70, 80, and 90.
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Traditional Names 3-ten = thirty The “thir” is more common than “three,” 3rd in line, 1/3, 13, and 30.
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Traditional Names 5-ten = fifty The same is true for “fif.”
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Traditional Names 2-ten = twenty Twenty is twice ten or twin ten. Note “two” is spelled with a “w.”
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Traditional Names A word game fireplace place-fire paper-news box-mail mailbox newspaper Say the syllables backward. This is how we say the teen numbers.
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Traditional Names ten 4 Ten 4 becomes teen 4 (teen = ten) and then fourteen. Similar for other teens. teen 4 fourteen
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Traditional Names a one left 1000 yrs ago, people thought a good name for this number would be “a one left.” They said it backward: a left-one, which became: eleven. a left-one eleven
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Mental Addition You need to find twenty-four plus thirty-eight. How do you do it? You are sitting at your desk with a calculator, paper and pencil, and a box of teddy bears. Research shows a majority of people do it mentally. “How would you do it mentally?” Discuss methods.
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Mental Addition 24 + 38 = + 30 24 + 8 = A very efficient way, taught to Dutch children, especially oral.
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Evens To experience “evens”, touch each row with two fingers, (e-ven).
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Odds To experience “odd”, touch each row with two fingers. Last row will feel odd.
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36 58 + 2 7 38 96 1 Addition Notice the number of yellow beads. [3] Notice the number of purple beads left. [3] Coincidence? No, because 13 – 10 = 3. 1000 100 10 1
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Skip Counting Patterns Fours Notice the ones repeat in the second row. 4 4 8 8 2 2 6 6 0 0
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Skip Counting Patterns Sixes and Eights Also with the 6s and 8s, the ones repeat in the second row. Also, the ones in the eights are counting by 2s backward, 8, 6, 4, 2, 0. 6 6 2 2 8 8 4 4 0 0 8 8 6 6 4 4 2 2 0 0
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Skip Counting Patterns 6x4 Sixes and Eights 8x7 6 x 4 is the fourth number (multiple).
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Skip Counting Patterns Nines Second row done backward to see digits reversing. Also the digits in each number add to 9. 9 18 27 36 45 90 81 72 63 54
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Skip Counting Patterns 15 5 12 18 21 24 27 3 6 9 30 Threes 2 8 1 4 7 3 6 9 0 Threes have several patterns. First see 0, 1, 2, 3, . . . 9.
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Skip Counting Patterns 1 2 1 5 1 8 2 1 2 4 2 7 3 6 9 30 Threes The tens in each column are 0, 1, 2.
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Skip Counting Patterns 6 15 24 6 12 21 3 30 Threes The second column. [6] And the third column – the 9s. Now add the digits in each number in the first column. [3] 18 27 9 18 27 9 12 21 3 30 15 24 6
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Skip Counting Patterns Sevens 28 35 42 49 56 63 7 14 21 70 Start in the upper right to see the 1, 2, 3 pattern. 8 9 7 0 5 6 4 2 3 1
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6 4 (6 taken 4 times) Multiplying on the Abacus
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5 7 (30 + 5) Multiplying on the Abacus Groups of 5s to make 10s.
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7 7 = Multiplying on the Abacus 25 + 10 + 10 + 4
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“ Pie” Model Try to compare 4/5 and 5/6 with this model.
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“ Pie” Model Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com Specialists also suggest refraining from using more than one pie chart for comparison. www.statcan.ca
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Simplifying Fractions The fraction 4/8 can be reduced on the multiplication table as 1/2. The fraction 21/28 can be reduced on the multiplication table as 3/4. 21 28 45 72
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Simplifying Fractions 6/8 needs further simplifying. 12/16 could have put here originally. 12 16
Cotter, Joan. “Using Language and Visualization to Teach Place Value.” Teaching Children Mathematics 7 (October, 2000): 108-114.
Also reprinted in NCTM (National Council of Teachers of Mathematics) On-Math Journal and in Growing Professionally: Readings from NCTM Publications for Grades K-8 , in 2008.
Manipulatives: representative of math concept, for
children’s use, and imaginable mentally.
Less time spent reviewing.
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Applying the Spirit of Asian Mathematics VII MAPSA Conference November 2, 2010 Detroit, Michigan by Joan A. Cotter, Ph.D. [email_address] Handout and Presentation: ALabacus.com 7 5 2
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