Mastery Objective Mastery of the Concepts of Base

Mastery Objective “Mastery of the Concepts of Base 10 and Place Value – to 100 by the end of the 1 st Grade and to 1000 by the end of the 2 nd Grade. ” (National Council of Teachers of Mathematics of America or NCTM)

MASTERY OF THE CONCEPT OF BASE 10: EXPONENTIAL NOTATION? 100 = 10²; 1000 = 10³; 10 = 10¹; 1 = 10 ?

Chapter 1: Place Value in the Decimal System Paving the ground/ Laying the predicate for Addition/Subtraction Beginning with the building blocks before going to 10: 10 as tip of iceberg What comes with knowing 10? If we are constructing in the minds of children the Concept of Place Value in the Decimal System, Where do we begin and what are the building blocks that we need? [Kindergarten: (a) Understand/ can explain Meaning of Place Value in 10 and 11 in the Decimal System: ; (b) Map & Explain Maps of Decimal Alphabet & 10; (c) Add/Subtract without regrouping to/from 19; (d) and (enrichment, optional) Simple Regrouping using Maps of Ao. N & 10 e. g. in 6 +7 into either [(5+5) + (1 +2] or (3+3)+7.

Period HTO/Abacus Frame

PERIODS/HTO PLACES Billions, Millions, Thousands, Units THOUSANDS UNITS O H T O Ones Hundreds Ten’s One’s Zero One Thousand Two Thousand Three Thousand Four Thousand Five Thousand Six Thousand Seven Thousand Eight Thousand Nine thousand One Hundred Two Hundred Three Hundred Four Hundred Five Hundred Six Hundred Seven Hundred Eight Hundred Nine Hundred Ten Twenty Thirty Forty Fifty Sixty Seventy Eighty Ninety One Two Three Four Five Six Seven Eight Nine Ten and one, one and ten is eleven; ten and two, two and ten is twelve; ten and four, three and ten, ten and three is thirteen; ten and four, four and ten is fourteen; ten and five, five and ten is fifteen; ten and six, six and ten is sixteen; ten and seven, seven and ten is seventeen; ten and eight, eight and ten is eighteen; ten and nine, nine and ten is nineteen. Ten is ten ones; twenty is two tens; thirty is three tens; forty is four tens; fifty is five tens; sixty is six tens; seventy is seven tens; eight is eight tens; ninety is nine tens. One hundred is tens. One thousand is ten hundreds. Ten is ten ones.

One hundred is tens. One thousand is ten hundreds. Ten is ten ones; twenty is two tens; thirty is three tens; forty is four tens; fifty is five tens; sixty is six tens; seventy is seven tens; eight is eight tens; ninety is nine tens.

Counting, the Concept of One and Numerosity through the Permanent Counts • Building the meaning of 10 in children’s minds using complementary building blocks/ teaching aids (manipulative & visual aids) • The HTO Frame, the Number words under the HTO Places, the decimal alphabet of 1 to 9 and 0, not 1 to 10, counting as action word, not as naming word, rote counting to ten, counting the digits of the hands to ten, the concept of one, the concept of numerosity in the permanent counts, regrouping, the concepts of addition/subtraction, also the abacus frame, maps of Ao. N and 10 to see them in all their forms and guises. Place Value.

Convert into Sight. Counts

Numerosity in Permanent Counts • (a) One, • (a) l (b) Five, (c) Nine, (d) Ten, (e) Eleven (b) llll/ (c) llll/-llll (d) llll/-llll/ (e) llll/- l

One hundred is tens. One thousand is ten hundreds. Ten is ten ones; twenty is two tens; thirty is three tens; forty is four tens; fifty is five tens; sixty is six tens; seventy is seven tens; eighty is eight tens; ninety is nine tens.

On a Decimal Abacus/HTO Frame (a) Ten Hundreds (b) Ten Ones (c) Tens M H T O H T — — — ― ― ― ― — — — ― ― ― — ― ― ― — — — ― ― ― — — — — — — — One Thousand 1000 M Ten 10 X O (d) Addition ― M C X I ― — — — One Hundred 100 C One thousand one hundred ten 1, 110 MCX

In the decimal system, in 1, 110 for example, any 1 is ten times the 1 to its right, or 1/10 of the 1 to its left; while zero, 0, is a Placeholder for No. Count— an empty column on the abacus One hundred is tens. One thousand is ten hundreds. Ten is ten ones.

Addition/Subtraction to 9 and 10 in all their Forms and Guises (End of Kindergarten: Addition/Subtraction Speed/Diagnostic Test) (a) H T − (b) O 9 2 H _ = _ _ = = O 4 6 + (u) H T (c) H O 5 – _ 2 H T O 7 + 3 = _ _ O H (e) H _ T O 1 0 – _ = 1 0 = 3 = 1 + (v) T T (d) (w) H O 0 _ H = 1 0 = + T 1 (x) T O 1 0 – 2 _ T O 2 + 7 _ (y) H T O 2 + _ = 5

Thinking 10: Introduction to Simple Regrouping: to 18 (a) 10+2 = (b) 7 = 5+2 (c) 6+5 = (1+ 5) + 5 = (d) 7+6 = (2+5) + (1+5) (e) 7+6 = (3+4)+6 Regular, Open Maps Class-Wide Games/Drills Mixing Maps of Ao. N and 10: Footbridges to Regrouping & Multiplication (a) (b) 5+10 =10 + 5 H T O 1 + = 0 5 1 10+ 8 = 8 + 10 10+2 =2 + 10 H H T = = O 1 3 0 1 3 T = (g) + 5 5 1 1 1 (e) 7+10= 10 + 7 (f) 5+6 = 10 + 1 H T O (d) 10+3= 3 + 10 + 5 (c) = (u) 1 1 + 0 7 1 7 T + = 8 0 1 8 1 = 1 (v) = 2 = (w) 1 1 + 0 2 1 2 9+5 = 4 + 10 H T O 3 5 5 + O (j) 8 + 5 = 3 + 10 H T O 2 5 5 + T (i) 7 + 5 = 2 + 10 H T O 1 5 5 O 1 (h) 6 + 5= 1 + 10 H T O + O 4 5 5 + 3 = ( x) 1 4 (y) 6 + 6 = 10 + 2 7 + 7 = 10 + 4 8 + 8 = 10 + 6 9 + 9 = 10 + 8 6+ 7 = 10 + 3 H H H = T O + 2 4 6 1 2 = T O + 4 3 7 1 4 = T O + 6 2 8 1 6 HTO Forms C = T O + 8 1 9 1 8 = T O + 3 3 7 1 3

Mixing Maps of 5 and 10 in Regrouping • 2 5 c 3 +7= 7 3 5 d 2 +8=8 5 1 10 = 14 5 3 10 = 12 5 2 10 = 13 (1) • • • 4 5 e 1 +9=9 (2 ) 4+10; • 1 5 b 4 +6= 6 (3 ) 2 + 10, (4) 5 4 10 = 11 3 + 10, 1 + 10 Map of 5 0 5 a 1 2 3 4 5 4 3 2 1 0 b c d e f 67890 MAP OF 10 T O +1 =1 0 0 0 a T O 1 9 0 + =1 b T O 2 8 0 + =1 c T O 3 7 0 + =1 d T O 4 6 0 + =1 e T O 5 5 0 + =1 f T O 6 4 0 + =1 g T O 7 3 0 + =1 h T O 8 2 0 + =1 i T O T 9 1 +1 0 =1 + =1 j O 0 0 0 k

Mixing Maps of 10 and 9 in Regrouping • (5 ) 8 o 1 (5) 9 • • • 9 1 10 = 18 • (5) 8+10 4 9 k 5 (6) + 9 = 9 • • • 9 9 + 5= 5 +2 = 2 +6= 6 9 5 10 = 14 9 8 10 = 11 9 4 10 = 15 (7) 1 7 + 10 (8) 5 + 10 3 (10) n 2 13 g 10 +8= 8 - 4 = -4 (9) 9 210 = 17 = 7+10 • 5 l 4 (8) 9 h 8 (6) 4+10 (9) 1 (7) (10) 13 - 4 = 10 6 &3 = 9 Map of 9 0 1 2 3 4 5 6 9 8 7 6 5 4 3 g h i j k l m 7 2 n 8 1 o 9 0 p MAP OF 10 T O +1 =1 0 0 0 a T O 1 9 0 + =1 b T O 2 8 0 + =1 c T O 3 7 0 + =1 d T O 4 6 0 + =1 e T O 5 5 0 + =1 f T O 6 4 0 + =1 g T O 7 3 0 + =1 h T O 8 2 0 + =1 i T O T 9 1 +1 0 =1 + =1 j O 0 0 0 k

• Pitogo Elementary School in previous years consistently ranked 29 th out of 29 elementary schools in Makati. A month before the city-wide test in March 2001, with the permission of the Math Coordinator, a Ms Griarte, we reviewed, pro bono, the three sections in Grade 1 for four weeks, 3 x a week - total of 12 hours. Months later, after I inquired, I was told of the achievement test result: For Grade 1 - a jump to the top 3 level. • • But nobody noticed the change. Either a jump to the top 3 level from 29 th is not good enough or nobody cares.