Units, Tens, and Beyond: Understanding Place Value

Lesson 5: Units, Tens, and Beyond

Introduction

• Many children struggle with understanding hundreds, tens, and ones, despite having been taught these concepts.

• Observed issues with class 6 and 8 students unable to read 5-digit numbers, and lacking understanding of place value.

• Example: In the number 20013, students incorrectly identified the place value of '2' as hundreds, thousands, lakhs, or crores.

• Lack of understanding is also evident when solving addition and subtraction problems.

• Children often add column-wise without understanding the underlying principles because they are told to by the teacher.

• They may incorrectly align digits in addition problems, leading to unusual answers.

• Instead of getting frustrated, educators should try to understand why children struggle with place value.

• Possible reasons include insufficient activities for concept understanding or the educators themselves lacking a proper understanding.

• This lesson aims to provide an in-depth understanding of these problems and explore various teaching approaches.

• The goal is to instill confidence and interest in educators to explore new ways of teaching place value in the decimal system.

Objective

• Create different types of activities for learners to develop understanding of units, tens, and hundreds.

• Test the learner's understanding of units, tens and hundreds.

• Create activities to understand and expand the concept of place value.

Increase Understanding

• A 7-year-old niece who had just learned to write large numbers at school read the title of a book called '203 Cats' as '23 Cats'.

• However, when asked to read 213, she read it correctly.

• This raises the question of whether she understood the significance of the position of a digit in a number or if the problem was related to her understanding of zero.

Traditional Teaching Method (Example 1)

• Teacher writes numbers 0-19 on the board & asks children to repeat them and write them in their homework book.

• Memorizing single-digit numbers first.

• Children copy what is written on the board (e.g., 10, 11, 12,…19).

• Teacher says names of numbers while pointing to each number.

• Children write each number five times.

• Example: Write 0 with 1, ten; write 1 with 1, eleven; … write 9 with 1, nineteen.

• Next year, teaching numbers from 101 to 1000 in the same way.

• If asked to write 152, first write 100, 10 & 1.

• Explained 100 and 1100 means 100, 100 and 100.

• Since 152 has one hundred, 50 and 200, write 1 under 100, 5 under 10 and 2 under 10.

• Lots of practice is given in writing two and three digit numbers.

• The teacher explained the meaning of the position of a digit and its place value, giving examples like the place values of 4, 2, and 7 in 427 being 4 \times 100, 2 \times 10, and 7 \times 1 respectively.

• Example homework question: How many tens are there in 251?

• Common answer: 5. Correct Answer: 25

• The traditional method is very mechanical.

• Children may not understand that one hundred is equal to 10 tens.

Alternative Method Using Concrete Activities (Example 2)

• Using beads to make groups of 10, calling a row of 10 beads a 'string'.

• 10 strings make a necklace (garland).

• Making a connection between the garland and the string in her mind.

• Asked how many strings she would give in exchange for one garland.

• Asked how many strings can be made with 107 beads. After thinking for a while she said, "10 strings, and 7 beads will be left."

• Creating a system to write numbers: Ma/L/Mo (garlands/strings/beads).

• Writing the number of beads under Mo, the number of strings under L, and the number of garlands under Ma.

• Introducing 0 as a placeholder: writing 0 under L when there are no strings.

• Then S.D.I. was written above M.L.M..

• Asking, "How much is 325?" and getting the correct response: "3 beads 2 strings 5 beads".

• Playing a game to make as many numbers as possible with a given set of points, then arranging them in descending order to utilize the concept.

• Realization: Children should be given enough time and opportunity to learn and practice concepts without pressure.

Key Points

• Numerals are written symbols for representing a number (e.g., 25 or ggt represent the same number).

• It is important to repeat concepts periodically to reinforce understanding.

Problems Related to Implementing Operations

• Testing Class 4 students on addition and subtraction problems horizontally and vertically, with and without 'gain' or borrowing.

• Children who got all questions correct or incorrect provided limited feedback on understanding.

• Insights were gained from analyzing the mistakes of children who got some questions wrong.

Observations

• Higher percentage of correct answers for vertically written questions.

• More success with questions not requiring 'gain' (carrying) or borrowing.

• Some children changed the places of digits while writing or copying, resulting in drastically different answers, yet they did not recognize these answers as strange.

• Didn't rethink their way of solving the question when they got the wrong answer.

Little Mukesh’s Problem (Example 3)

• Mukesh was doing vertical addition homework.

• He understood the process that teacher had told him, but made a mistake in the values.

• For example, in 68 + 45, he wrote 13 below 5, then 10 below 4, getting 1013 as the answer.

• The teacher's method involved carrying over the '1' to the next column.

• Many classmates also made mistakes.

• The teacher moved on to new concepts, leaving some children behind.

• Children did not ask questions, indicating a lack of understanding of place value.

Radha's Problem (Example 4)

• Radha needed help with addition problems.

• She added 15, 10 and 25 by adding the units digits first (5 + 0 + 5 = 10), then the tens digits (1 + 2 + 1 = 4), and wrote the answer as 410.

• Using matchsticks to create bundles of 10 to represent tens and individual sticks to represent ones.

• Asked to give 5 matches, then 10 matches, understanding that 1 bundle is 10 matches.

• Bundles and sticks were used till the child understands which bundles represent tens and single sticks represent units.

• Asked to give back 35 sticks; the child gave 3 bundles and 5 sticks.

• Asked for 60 sticks and gave 6 bundles.

• Game with stones, dice, cards, and colored beads. The dice amount shown corresponds to the number of stones to pick up.

• Stones could be traded with the dealer for cards, if enough stones are accumulated.

• Collecting 10 stones is equal to one card, and collecting 10 cards is equal to one pearl.

• The person with five cards, or three pearls, wins the game.

• The game was adapted to collect 10 cards for each pearl, and played with two dice to make it quicker. Two dice resulted in a more complex game.

Principle of Exchange

• The principle of exchange is demonstrated when we start moving from one number to the next moving right to left. 10 ones is equal to 1 ten, and so on.

Mental Maths Activities

• Mental maths can assess how easily a child operates within hundreds, tens, and ones.

• (i) Mentally adding 1, 10, or 100 (or multiples) to a number (especially with the digit 9, e.g., 93).

• (ii) Mentally subtracting 1, 10, 100 (or multiples) from a number (especially with the digit 0, e.g., 804).

• Blackboard activity: Write headings like 'subtract 1', 'add 2', 'subtract 100', 'add 20'. Write numbers under each and have groups pass the column forward.

• Another activity is to give each child an answer sheet with headings and ask them to complete 8 steps in a given time, performing operations mentally. If the 'Subtract 100' column starts with 801, the child will write 801, 701, 601, 501, …. 1.

• An activity example for 11 years and above:

  • subtract 1 penny

  • add 2 paise

  • subtract 1 rupee

  • add 20 paise

  • Rs. 404

  • Rs. 184

  • Rs. 9.01

  • Rs. 1.50

Variations for 9-10 year olds

• Use a grid with directions for moving horizontally and vertically (add 1, subtract 1, add 10, add 100, subtract 5, etc.).

• Write a starting number in the upper left corner and the final number in the lower right corner.

• Children can fill these out themselves, following the instructions from left to right along a row and from top to bottom along a column.

Number Chart Activities

• Number Chart activities (10 x 10) to enhance concrete and abstract understanding of S.E.

• Ask the child to choose any number on the chart and compare it to the numbers above and below.

• Find the difference between the two numbers by subtracting the smaller number from the larger number.

• Find out how much the number chosen is less or more than its neighbors in each direction.

• Explore number patterns, such as directions of decreasing/increasing, and directions of tens differences.

Example 5: Rani's Example using a Number Chart

• 7 year old Rani was asked to add 37 and 26 using a number chart.

• She divided 26 into 13 and 13.

• Then 13 into 10 and 3.

• Realized that 26 is 20 + 6

• So the teacher assisted with how to add 37 and 26 by first adding 20 and then 6.

• She moved in the number chart two squares down, and then six squares to the right to reach the edge finally counting to 63.

• Rani was learning how to divide and add a number into units and tens to understand the meaning of the result.

• Children should understand division with concerete things.

Adding Two-Digit Numbers

• Gradually, you can start adding two-digit numbers by writing them in units-tens form. For example,

  \begin{aligned}
  &1.  34 = 3 \text{ tens and } 4 \text{ units} \\
  &+ 28 = 2 \text{ tens and } 8 \text{ units} \\
  &\hline
  &5 \text{ tens and } 12 \text{ units} \\
  &= 5 \text{ tens and } 1 \text{ten and } 2 \text{ units} \\
  &= 6 \text{ tens and } 2 \text{ units} \\
  &= 62
  \end{aligned}
  

• Solving these problems helps see why numbers are being added the column.

What is Place Value?

• In decimal system all numbers are written using only 10 symbols or digits.

• Each depends on place in the number; therefore, even the largest numbers can be written in smaller form.

• Ten, hundred, and thousand are represented by the same two digits, 1 and 0, with the only difference being in the places of the digits.

• Primary school children might not understand the broad concept of place value.

• We assign a different value to each place

• Values are called local values.

• In the decimal system, the column values from right to left are 1, 10, 100, etc.
  10^0, 10^1, 10^2

• The value of 4 in 1420 will be 4 \times 100 = 400

Place value with base 7 system:

The place values going from right to left will be 1,7,49,343, etc.

• Values from right to left 1, 7^1, 7^2, 7^3

• In this system, the value of 4 in 1420 will be 4 \times 7^2 = 196

Binary System

How will we write 10 in binary?

• Local letters are written in units, twos, fours, eights, etc

• Values will be. Since 10 = (8 \times 1) + 2 = (8 \times 1) + (4 \times 0) + (2 \times 1) + (1\times 0), so in binary system ten will be written as 1010.

Other numeral systems

• Binary System: Used in computers.

• Five-digit system: Used in abacus.

• Twelve-digit system: Used to count things by dozen.

• Sixty-digit system: Used to represent time and angles.

• Roman system: Representation of the symbols is from left to right. The symbol of the largest number is written on the extreme left. As the numbers get bigger, their symbols become more complex. For example, 337 is written as CCCXXXVII.

• Hindu-Arabic numerals: Numbers written on the base ten or in the decimal system are called Hindu-Arabic numerals.

Summary

• Some of the problems children face because they do not understand the units/tens concept properly.

• Discussed group activities that can help them understand these concepts.

• Observed some activities in which children dealt with numbers in which at least one digit was zero.