Page 1 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/9/15) Milestone A1: Instructions The purpose of this document is to learn the Numbering System. A supervisor shall be available in the classroom to help you. 1. Answer the following question. Check your answers with the answers provided right after the questions. 2. Continue with this milestone only if some understanding is missing. There are 10 lessons in this document. 3. Answer the questions for each lesson. Check your answers with the answers provided right after the questions. 4. Review a lesson only if some understanding is missing. 5. Make sure that you understand the main point given after each lesson. 6. Learn to see things as they are. Do not assume anything. By the time you complete all lessons you will know how to read and write numbers in millions, billions, and trillions. Main Point: Know the numbers and their place values. Milestone A1: Answer the following questions. 1. What are the ten numbers that follow 1,095? 2. Read the following numbers. (a) 25,807 (b) 357,000 (c) 3,007,002 (d) 3,000,009,133 1. (e) 325,601,213,000,102 3. Write the following numbers. (a) Seventy-seven thousand, three hundred nine (b) Forty-two thousand, seven (c) Six million, sixty-six thousand, sixty (d) Four hundred forty-four thousand, four (e) One hundred eight billion, four hundred fifty-six million, eighty-seven
Page 2 of 22 4. What are the place values of the underlined digits in the following numbers. (a) 10 (b) 145 (c) 3,257 (d) 5,000,400 (e) 31,952,833 Check the answers below. Continue with this milestone only if some understanding is missing. Milestone A1: Answer 1. 1096, 1097, 1098, 1099, 1100, 1101, 1102, 1103, 1104, 1105. 2. (a) Twenty-five thousand, eight hundred seven (b) Three hundred fifty-seven thousand (c) Three million, seven thousand, two (d) Three billion, nine thousand, one hundred thirty-three (e) Three hundred twenty-five trillion, six hundred one billion, two hundred thirteen million, one hundred two 3. (a) 77,309 (b) 42,007 (c) 6,066,060 (d) 444,004 (e) 108,456,000,087. 4. (a) Ten (b) One Hundred (c) One Hundred (d) Ten Thousand (e) Ten Million
Page 3 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/9/15) Lesson 1: Counting on fingers 1. You may count from ONE to FIVE with fingers of one hand as follows.. 2. You may count from SIX to TEN with fingers on two hands as follows.
Page 4 of 22 3. You count the following chairs as ONE TWO THREE FOUR FIVE Five chairs. 4. Counting starts from ONE. These counting numbers are called NATURAL NUMBERS. Main Point: Mathematics starts with counting. Lesson 1: Check your Understanding 1. Count the fingers on your two hands. How many fingers are there? 2. How will you show the number SEVEN using your fingers? 3. Count ten items around you. 4. What are Natural Numbers? What is the smallest natural number? Check the answers below. Review the lesson only if some understanding is missing. Lesson 1: Answer 1) Normally there are ten fingers. 2) See Lesson, or get a supervisor to check your answer. 3) See Lesson, or get a supervisor to check your answer. 4) Natural numbers are the counting numbers. The smallest natural number is ONE.
Page 5 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/9/15) Lesson 2: Counting on Abacus 1. An ABACUS has ten wires. Each wire has ten beads. NOTE: You may get a simple wooden abacus from a Toy Shop. 2. You count on abacus by moving beads from left to right. The beads on the right show the count. The beads on the left are in storage. 3. When all the beads are on the left, and no beads on the right, we may say that the count is ZERO. 4. ZERO means there is no count. For example, think of How many elephants are in the room? Then look around.
Page 6 of 22 5. Beads are counted to the right as follows. NOTE: All these counts are on the first wire. Main Point: The count is beads on the right on a wire of abacus. Lesson 2: Check your Understanding 1. On an Abacus, point to the beads counted; point to the beads in storage. 2. What is the maximum count on a wire of abacus? 3. What is zero? Show zero on abacus. Check the answers below. Review the lesson only if some understanding is missing. Lesson 2: Answer 1) See Lesson, or get a supervisor to check your answer. 2) You can only count up to ten on a wire (unit may differ depending on the wire). 3) Zero is no count. See the lesson.
Page 7 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/9/15) Lesson 3: The Rule of Abacus 1. We may replace ten pennies with a dime because both have the same value. 2. On abacus, we may replace a count of TEN beads on the first wire by a count of ONE bead on the second wire. They both have the same value. Therefore, ten ONES (like ten pennies ) on the first wire become one TEN (like one dime ) on the second wire. 3. This is called THE RULE OF ABACUS: WHENEVER ALL THE BEADS ARE TO THE RIGHT ON A WIRE, THEY ARE RETURNED TO THE LEFT AND REPLACED BY ONE BEAD TO THE RIGHT ON THE NEXT WIRE. 4. We count by ONES on the first wire. At the count of ten we apply the Rule of Abacus. This gives us ZERO on first wire and a TEN on the second wire. 5. The count does not increase with the application of the rule of abacus. The count is still ten. We then start counting on the first wire again.
Page 8 of 22 6. The following is the count from nine to twelve on abacus. 7. After ten we count: eleven (11), twelve (12), thirteen (13), fourteen (14), fifteen (15), sixteen (16), seventeen (17), eighteen (18), and nineteen (19), and so on. Main Point: On abacus, ten ONES on the first wire become one TEN on the second wire per the Rule of Abacus. Lesson 3: Check your Understanding 1. What do you do when you run out of beads to count on a wire? 2. How do ten ONES differ from one TEN in counting? 3. Does the count increase when you apply the Rule of Abacus? 4. Count up to thirteen on the first wire. Check the answers below. Review the lesson only if some Lesson 3: Answer 1) You apply the Rule of Abacus when you run out of beads to count. 2) Both represent the same count. 2) The count does not increase when you apply the Rule of Abacus. 3) See Lesson, or get a supervisor to check your answer.
Page 9 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/9/15) Lesson 4: From Abacus to Digits 1. Each count shown on abacus is written as a digit on paper. There is one digit for ZERO, and nine more digits for counts ONE to NINE. At TEN no new digit is used because there is regrouping. Therefore, there are ten digits. 2. After regrouping the count of TEN is written with two existing digits 1 and 0 as 10. 3. Each wire has its own digit showing how many beads are on that wire. The counts after ten are: eleven (11), twelve (12), thirteen (13), fourteen (14), fifteen (15), sixteen (16), seventeen (17), eighteen (18), and nineteen (19), as follows.
Page 10 of 22 4. At count TWENTY; regrouping takes place once again. Therefore TWENTY is written as 20. 5. Each bead on the second wire is a TEN. Therefore, we count the beads on the second wire as Ten (10), twenty (20), thirty (30), forty (40), fifty (50), sixty (60), seventy (70), eighty (80), and ninety (90). Regrouping takes place with the next bead. 6. We show larger numbers as a combination of TENS and ONES. The a) The number THIRTY-FOUR is made up of 3 TENS and 4 ONES. b) The number FIFTY-EIGHT is made up of 5 TENS and 8 ONES. c) The number NINETY-NINE is made up of 9 TENS and 9 ONES.
Page 11 of 22 Main Point: The numbers of beads on a wire are written as DIGITS. Lesson 4: Check your Understanding 1. Why do we need DIGITS? 2. How many different digits are there? 3. Why is there no new digit for TEN? 4. Start counting on abacus from 1, until you have applied the Rule of Abacus enough times to feel confident about it. 5. The number sixteen is made up of TEN and ONES. 6. The number seventy-five is made up of TENS and ONES. 7. Show the following counts on abacus. Write them down using digits. (a) Thirty-seven (b) Seventy-three (c) Ninety-seven 8. Read the following numbers: (a) 55 (b) 83 (c) 94 Check the answers below. Review the lesson only if some understanding is missing. Lesson 4: Answer 1) We need digits to write the count on paper.. 2) There are ten different digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. 3) Because after regrouping TEN is written with two existing digits 1 and 0 as 10. 4) Get a supervisor to check your regroupings (this is important). 5) 1 TEN and 6 ONES 6) 7 TENS and 5 ONES 7) (a) 37 (b) 73 (c) 97 8) (a) Fifty-five (b) Eighty-three (c) Ninety-four
Page 12 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/9/15) Lesson 5: HUNDREDS, TENS and ONES 1. One more count from NINETY-NINE (9 TENS and 9 ONES) makes it 9 TENS and 10 ONES. This regroups as 10 TENS and no ONES. This regroups once again as 1 HUNDRED (no TEN, no ONE). 2. Each bead on the third wire is a HUNDRED. We count on the third wire by HUNDREDS as: one hundred (100), two hundreds (200), three hundreds (300), four hundreds (400), five hundreds (500), and so on 3. We show larger numbers as a combination of HUNDREDS, TENS and ONES.
Page 13 of 22 Main Point: We regroup twice at the count of One Hundred. Lesson 5: Check your Understanding 1. How do ten TENS differ from one HUNDRED in counting? 2. Show NINETY-SEVEN on abacus. Then count up to ONE HUNDRED THREE on abacus showing the regroupings. 3. Three hundred seven is made up of HUNDREDS, TENS and ONES. 4. Show the following counts on abacus, and then write them down using digits: (a) Three hundred twelve (b) Three hundred twenty-one (c) Five hundred eight 5. Read the following numbers: (a) 111 (b) 277 (c) 658 Check the answers below. Review the lesson only if some understanding is missing. Lesson 5: Answer 1) Both represent the same count. 2) See Lesson, or get a supervisor to check your answer. 3) 3 HUNDREDS, 0 TENS and 7 ONES 4) (a) 312 (b) 321 (c) 580 5) (a) One hundred eleven (b) Two hundred seventy-seven (c) Six hundred fifty-eight
Page 14 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/11/15) Lesson 6: ONE-TEN-HUNDRED & THOUSAND 1. The number three hundred ninety-five is made up of 3 HUNDREDS. 9 TENS, 5 ONES. The place values from right to left are ONE, TEN and HUNDRED. 2. The values depend on the place of the digit in a number. A place value to the right is ten times the place value to its left. 3. The place values of ONE, TEN and HUNDRED form a group. 4. 10 HUNDREDS become 1 THOUSAND because of regrouping. Therefore the group of ONE-TEN-HUNDRED repeats to the left as THOUSAND. A comma is used to separate the thousands. 5. We read the above number as: Eight hundred twenty-six (826) thousand, five hundred ninety-three (593). 6. When no count exists for a place value, a zero is placed there. For example, in the following number the count for HUNDRED is missing. We read this number as 683 thousand, 53
Page 15 of 22 Main Point: The place values ONE-TEN-HUNDRED form a group that repeats for THOUSAND. Lesson 6: Check your Understanding 1. What are place values? What is the relationship of one place value to the next? 2. Place a comma to separate the thousands group from the basic group. (a) 3829 (b) 56942 (c) 419736 (d) 100001 (e) 350093 3. Read the following numbers. (a) 1,111 (c) 532,658 (e) 300,005 (b) 23,277 (d) 500,074 (f) 101,010 4. Write the following numbers in digits. (a) Six thousand, three hundred sixty-five (b) Ninety Eight thousand, eight hundred one (c) Two hundred sixty thousand, four hundred twenty seven (d) Nine hundred thousand, ninety-nine (e) Three hundred twenty-nine thousand, five hundred forty-two (f) Seventy-seven thousand, six hundred Check the answers below. Review the lesson only if some understanding is missing. Lesson 6: Answer 1) Place values are the values, such as ONE, TEN, HUNDRED, etc., assigned to the digits based on their place in the number. Each place value to the right is ten times the place value to its left. 2) (a) 3,829 (b) 56,942 (c) 419,736 (d) 100,001 (e) 350,093 3) (a) One thousand, one hundred eleven (b) Twenty-three thousand, two hundred seventy-seven (c) Five hundred thirty-two thousand, six hundred fifty-eight (d) Five hundred thousand, seventy-four (e) Three hundred thousand, five (f) One hundred one thousand, ten 4) (a) 6,365 (b) 98,801 (c) 260,427 (d) 900, 099 (e) 329,542 (f) 77,600
Page 16 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/11/15) Lesson 7: MILLION, BILLION & TRILLION 1. Each place value is ten times the previous place value. 2. 10 HUNDRED THOUSAND become 1 MILLION. 10 HUNDRED MILLION become 1 BILLION. 10 HUNDRED BILLION become 1 TRILLION. Therefore, the group of ONE-TEN-HUNDRED repeats again to the left of THOUSAND as MILLION, BILLION & TRILLION. A comma is used to separate the groups. We read the above number as: 714 million, 826 thousand, 593. 3. We put zero at the places of no count. We read the above number as 600 million, 83 thousand, and 3. 4. The group of billion is added to the left of the group of million. We read the above number as 132 billion, 603 million, 41 thousand, 299
Page 17 of 22 5. The group of trillion is added to the left of the group of billion. Note that there are no thousands in the following number. So the thousand s places are filled with zeroes. We read the above number as 302 trillion, 4 billion, 865 million, and 7. 6. You do not need to know the groups beyond trillion, but here they are anyway if you are curious: Thousand, million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, and decillion. Main Point: The group of ONE-TEN-HUNDRED repeats for Millions, Billions, Trillions, etc. Lesson 7: Check your Understanding 1. Place commas at the correct place in the following numbers (a) 8268268 (c) 45806032650 (e) 76098305009023 (b) 82682682 (d) 700005973037 (f) 801000006000759 2. Read the following numbers. (a) 52,762,869 (c) 75,765,532,658 (e) 409,008,007,006,834 (b) 273,045,008 (d) 30,006,000,074 (f) 590,000,060,000,001 3. Write the following numbers. (a) 2 million, 304 thousand, and 516 (b) 45 million, 464 thousand, and 801 (c) 1 billion, 5 million, and 6 (d) 25 billion, 43 million, 60 thousand, and fifty (e) 43 trillion, 6 thousand, and 35 (f) 608 trillion, 45 million, and 529 Check the answers below. Review the lesson only if some understanding is missing. Lesson 7: Answer 1) (a) 8,268,268 (b) 82,682,682 (c) 826,826,826 (d) 100,000,000 (e) 305,009,023 2) (a) 52 million, 762 thousand, and 869 (b) 273 million, 45 thousand, and 8 (c) 75 billion, 765 million, 532 thousand, and 658 (d) 30 billion, 6 million, and 74 (e) 409 trillion, 8 billion, 7 million, 6 thousand, and 834 (f) 590 trillion, 60 million, and 1 3) (a) 2,304,516 (b) 45,464,801 (c) 1,005,000,006 (d) 25,043,060,050 (e) 43,000,000,006,035 (f) 608,000,045,000,529
Page 18 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/9/15) Lesson 8: Numbers as Patterns 1. Numbers are patterns. We can see the pattern of three in 3 clocks, 3 balls and 3 chairs. The digit 3 is merely a symbol. 2. Look around the room and find some patterns of three. Main Point: Numbers are patterns that show how many things are there. Lesson 8: Check your Understanding 1. What is a number? 2. Show the pattern of 7 using (a) Pennies (b) Fingers (c) Beads on abacus Check the answers below. Review the lesson only if some understanding is missing. Lesson 8: Answer 1) Numbers are patterns that show how many. 2) Get a supervisor to check your answer.
Page 19 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/9/15) Lesson 9: Numbers and Digits 1. Digits are like letters because they are used to write numbers, just like letters are used to write words. The number three hundred ninety-five is written with digits: 3, 9, and 5. 2. Single digits numbers are written with one digit only. 5 is a number written with one digit, just like I is a word written with one letter. From 1 to 9 we have NINE single-digit numbers. 3. Double-digit numbers are numbers written with two digits. 35 is a number written with two digits, just like ME is a word written with two letters. From 10 to 99 there are NINETY double-digit numbers. 4. Three-digit numbers are numbers written with three digits. 164 is a number written with three digits, just like YOU is a word written with three letters. From 100 to 999 there are NINE HUNDRED three-digit numbers. Main Point: Numbers are written with digits. Lesson 9: Check your Understanding 1. What are digits used for? 2. What is a single-digit number? Give an example. 3. Beside zero, how many single-digit numbers are there? 4. How many double-digit numbers are there? 5. What are the smallest and largest three-digit numbers? Check the answers below. Review the lesson only if some understanding is missing.
Page 20 of 22 Lesson 9: Answer 1) Digits are used to write numbers. 2) It is a number written with one digit only; for example, 7. 3) There are nine single-digit numbers beside zero.. 4) Ninety 5) 100 and 999
Page 21 of 22 MATH MILESTONE # A1 NUMBERS & PLACE VALUES Researched and written by Vinay Agarwala (Revised 4/9/15) Lesson 10: Mathematics 1. When we count, we learn how many things are there. Thus, counting may be looked upon as a tool for learning. 2. Mathematics is made up tools, such as, counting, adding, subtracting, multiplying, dividing, etc. Mathematics is a set of tools for learning. 3. One counts in the sequence: ONE, TWO, THREE, FOUR, FIVE, etc., forever. When one knows the system of counting one can easily tell the next number. 4. Mathematics helps you think systematically to get answers. Main Point: Mathematics gives you tools to get answers. Lesson 10: Check your Understanding 1. How many people are there in the room where you are? 2. How did your learn that? 3. What number comes after 1099? 4. Why is there a need for mathematics? Check the answers below. Review the lesson only if some understanding is missing. Lesson 10: Answer 1) As counted. You may get a supervisor to check your answer. 2) You learned that by counting. 3) 1100 4) Math helps you get answers.
Page 22 of 22 SUMMARY The first action in Mathematics is counting. From counting one learns to compute. The basis of computations is the Numbering System. This system uses place-values to read and write numbers. The place values are based on the following Rule of Abacus. WHEN ALL THE BEADS ARE TO THE RIGHT ON A WIRE, THEY ARE REGROUPED AS ONE BEAD ON THE NEXT WIRE. Each place-value is ten times the previous place-value. The place-value system makes it possible to write large numbers very simply. The best sequence to learn mathematics is, a) Count on fingers b) Count on abacus with beads c) Compute mentally with the help of digits on paper. d) Compute on calculator while checking the results to be in the ball park mentally.