Mastery Objective

“Mastery of the Concepts of Base 10 and Place Value – to 100 by the end of the 1st Grade and to 1000 by the end of the 2nd

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?

Paving the ground/ Laying the predicate for Addition/Subtraction

Chapter 1: Place Value in the Decimal System

Beginning with the building blocks before going to10: 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 AoN & 10 e.g. in 6 +7

into either [(5+5) + (1 +2] or (3+3)+7.

Period HTO/Abacus Frame

PERIODS/HTO PLACESBillions, Millions, Thousands, Units

THOUSANDS UNITS

O H T OOnes Hundreds Ten’s One’s

Zero Zero Zero Zero

One Thousand One Hundred Ten OneTwo Thousand Two Hundred Twenty Two

Three Thousand Three Hundred Thirty ThreeFour Thousand Four Hundred Forty FourFive Thousand Five Hundred Fifty FiveSix Thousand Six Hundred Sixty Six

Seven Thousand Seven Hundred Seventy SevenEight Thousand Eight Hundred Eighty EightNine thousand Nine Hundred Ninety 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 ten tens. One thousand is ten hundreds. Ten is ten ones.

One hundred is ten tens. One thousand is ten hundreds. Ten is ten ones. 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.

• 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 AoN and 10 to see them in all their forms and guises. Place Value.

Counting, the Concept of One and Numerosity through the Permanent Counts

Convert into SightCounts

Numerosity in Permanent Counts

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

One Five Nine Ten Eleven (a) I (b) IIII/ (c) IIII/-IIII (d) IIII/-IIII/ (e) IIII/-IIII/-l

H T O H T O H T O H T O H T O— — — — — — — — — — ― — — ― ―― ― ― ― ― ― ― ― ― ― ― ― ― ― ―— — — — — — — — — — — — — — —― ― ― ― ― ― ― ― ― ― ― ― ― ― ―— — — — — — — — — — — — — — —― ― ― ― ― ― ― ― ― ― ― ― ― ― ―— — — — — — — — — — — — — — —— — — — — — — — — — — — — — —— — — — — — — — — — — — — — —

H T O H T O H T O H T O H T O— — — — — — — — — — ― — — ― ―― ― ― ― ― ― ― ― ― ― ― ― ― ― ―— — — — — — — — — — — — — — —― ― ― ― ― ― ― ― ― ― ― ― ― ― ―— — — — — — — — — — — — — — —― ― ― ― ― ― ― ― ― ― ― ― ― ― ―— — — — — — — — — — — — — — —— — — — — — — — — — — — — — —— — — — — — — — — — — — — — —

One Five Nine Ten Eleven (a) I (b) IIII/ (c) IIII/-IIII (d) IIII/-IIII/ (e) IIII/-IIII/-l

1 5 9 10 11 l V lX X Xl

One hundred is ten tens. One thousand is ten hundreds. Ten is ten ones. 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.

(a) Ten Hundreds (b) Ten Ones (c) Ten Tens (d) Addition

1000 10 100 1,110 M X C MCX

M H T O H T O H T O M C X I— — — — — ― ― — ― ― — ―

― ― ― ― ― — ― ― — ―

— — — — — ― — — — —

― ― ― ― ― — ― ― ―

— — — — — ― — — —

― ― ― ― ― — ― ― ―

— — — — — — — — —

— — — — — — — — —

— — — — — — — —

One Thousand Ten One Hundred One thousand one hundred ten

On a Decimal Abacus/HTO Frame

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 NoCount— an empty

column on the abacus

One hundred is ten 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) (b) (c) (d) (e)

H T O H T O H T O H T O H T O 9 4 1 1 0 2− 2 + 6 + _ – _ + 7

= _ = _ _ = 1 0 = 3 = _

(u) (v) (w) (x) (y)

H T O H T O H T O H T O H T O5 7 1 0 1 0 2

– _ + 3 + _ – 2 + _

= 2 = _ _ = 1 0 = _ = 5

5+10 =10 + 5 10+3= 3 + 10 7+10= 10 + 7 10+ 8 = 8 + 10 10+2 =2 + 10H T O

H T OH T O H T O H T O

1 0 3 1 0 8 1 0 + 5 + 1 0 + 7 + 1 0 + 2 = 1 5 = 1 3 = 1 7 = 1 8 = 1 2

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/DrillsMixing Maps of AoN and 10: Footbridges to Regrouping & Multiplication

(a) (b) (c) (d) (e)

HTO Forms C

(u) (v) (w) (x) (y)

6 + 6 = 10 + 2 7 + 7 = 10 + 48 + 8 = 10 + 6

9 + 9 = 10 + 8 6+ 7 = 10 + 3H T O

H T O H T O H T O H T O

2 4 6 8 34 3 2 1 3

+ 6 + 7 + 8 + 9 + 7 = 1 2 = 1 4 = 1 6 = 1 8 = 1 3

5+6 = 10 + 1 6 + 5= 1 + 10 7 + 5 = 2 + 10 8 + 5 = 3 + 10 9+5 = 4 + 10H T O H T O H T O H T O H T O

5 1 2 3 45 5 5 5 5

+ 1 + 5 + 5 + 5 + 5 = 1 1 = 1 1 = 1 2 = 1 3 = 1 4

(f) (g) (h) (i) (j)

• Mixing Maps of 5 and 10 in Regrouping (1) 4 (2) 2 (3) 3 (4) 1

• 5 e1 5 c3 5 d2 5 b4• + 9 = 9 + 7 = 7 + 8 = 8 + 6 = 6• 5110 = 14 5310 = 12 5210 = 13 5410 = 11

• 4+10; 2 + 10, 3 + 10, 1 + 10 Map of 5 0 1 2 3 4 5 6 7 8 9 0 5 4 3 2 1 0 a b c d e f

MAP OF 10T O T O T O T O T O T O T O T O T O T O T O

0 1 2 3 4 5 6 7 8 9 0+1 0 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 +1 0

=1 0 =1 0 =1 0 =1 0 =1 0 =1 0 =1 0 =1 0 =1 0 =1 0 =1 0a b c d e f g h i j k

Map of 9 0 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 0 g h i j k l m n o p

• Mixing Maps of 10 and 9 in Regrouping (5) (5) 8 (6) 4 (7) 1 (8) 5

• 9 o 1 9 k5 9 h8 9 l4• + 9 = 9 + 5= 5 +2 = 2 + 6 = 6• 9110 = 18 9510 = 14 9810 = 11 9410 = 15 • (5) 8+10 (6) 4+10 (7) 1 + 10 (8) 5 + 10

• (9) 7 (10) 3• 9 n2 13 g 10

• + 8 = 8 - 4 = - 4• (9) 9210 = 17 = 7+10 (10) 13 - 4 = 106 &3 = 9

MAP OF 10T O T O T O T O T O T O T O T O T O T O T O

0 1 2 3 4 5 6 7 8 9 0+1 0 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 +1 0

=1 0 =1 0 =1 0 =1 0 =1 0 =1 0 =1 0 =1 0 =1 0 =1 0 =1 0a b c d e f g h i j k

• Pitogo Elementary School in previous years consistently ranked 29th 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, 3x 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 29th is not good enough or nobody cares.