Introduction to the numeration of two-place numbersAuthor(s): HITOSHI IKEDA and MASUE ANDOSource: The Arithmetic Teacher, Vol. 16, No. 4 (APRIL 1969), pp. 249-251Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41187501 .

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Introduction to the numeration of two-place numbers HITOSHI IKEDA and MASUE ANDO

Hiioshi lkeda is a professor at the Imperiai Valley Campus, San Diego State College, and has taught at the Hilo Campus of the University of Hawaii the past summers.

Masue Ando is a Resource Teacher in Mathematics, Hawaii District Schools, Department of Education, Hilo, Hawaii. She is also a lecturer at the University of Hawaii, Hilo Campus.

Ahe primary teacher has one of the most important tasks in the teaching of arithme- tic - that of introducing and instilling in the minds of children a firm foundation of the concept of place value. Much of the success children will have in working with the operations of addition, subtraction, multiplication, and division depends on their concept of place value. With ad- mirable patience and stamina, the primary grade teacher has circulated amid the bundles of counters, popsicle sticks, tongue depressors, and what-have-you to help chil- dren gain this concept. This is well and good.

An equally important aspect of teaching place value is that of symbolizing the idea. Children should be led to see that numerals are records of a kind and that there needs to be agreement as to what kind of form this record is to assume.

The following series of lessons is pre- sented to outline a way of stressing the record-keeping aspect in the teaching of place value to children. Teachers may wish to expand and extend these lessons as necessary.

Lesson 1 : The use of records

Discuss the word "record" (noun form). Children usually associate this word with phonograph records. This is a good start- ing point for a discussion of the wider ap- plication of the word. Discuss how a record "remembers" something, that is, it plays

back for you what was recorded when you want to hear it. Go on to other records that help to remember things, such as shopping lists, calendars, attendance rec- ords, etc.

Introduce the following situation to carry on the discussion. Suppose we want to remember the number of boys and girls in the class who come to school each day of the week. We can keep a record of this by writing down how many boys and how many girls are in class each day. For example :

Monday 1 2 boys and 1 5 girls Tuesday 1 1 boys and 1 6 girls Wednesday 12 boys and 14 girls Thursday 10 boys and 15 girls Friday 11 boys and 16 girls

This is a kind of record, but it is a very long record. How many times did we have to write the words "boys," "and," and "girls"? How can we shorten this record or do less work in making it? Children may suggest using В for boys and G for girls. There may be other suggestions, some good and some not appropriate in terms of doing less work in keeping a record. Follow through on good sugges- tions offered, but lead them gradually to the following form:

Boys Girls

Monday 12 15 Tuesday 11 16 Wednesday 12 14 Thursday 10 15 Friday 11 16

April 1969 249

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that, in order not to need to remember many names for different-sized groups, we should just use the same grouping and remember the name for that group.

Now ask children to group by tens. The name of the group is agreed upon as "Tens." Start a record on the board:

Tens Extras

Elaine 2 7 Craig 3 5 John 2 5 Maxine 3 0

Call on children whose names are not on the board to tell the class how many "Tens" and "Extras" they have. Point to columns and ask what they would enter in each column.

Specific objectives. - (1) The child groups objects into piles of tens and relates the numbers of tens and extras orally. (2) The child writes the number of "Tens" and "Extras" in appropriately labeled col- umns.

Lesson 3: To develop understanding of two-place numerals

Pass out counters as in Lesson 2. Have the children group by tens and start a record similar to that obtained at the end of Lesson 2. Suppose the record begins:

Tens Extras

Lynne 2 9 Allen 3 2 Mark 2 7

After a few more entries on the board call on students to tell the class what they would record beside their names.

Discuss with children how we can agree on what each column is to be. In this way we do not need to label the columns. Emphasize that the people who use our symbols have agreed to use the first column for the number of tens and the second column for the number of extras. At this point the word "ones" can be introduced as another name for "extras."

Referring to Lynne above, ask children

250 The Arithmetic Teacher

Now ask children if they can be "lazier" yet. What else can be left out of the record and still make sense? Lead children to the agreement of using the first column for the number of boys and the second column for the number of girls. With this agreement, the columns need not be labeled "Boys" and "Girls." Check to see whether children understand the agree- ment. They should respond by filling the columns like this:

Monday 12 15 Tuesday 11 16

Specific objective. - The child uses col- umn notation as a shortcut to record the number of things.

Lesson 2: Making a record of groups of things

Pass out twenty to forty counters per child. Give each child a different number of counters. Direct children to make piles of four counters. They should make as many piles of four counters as they are able to. Have children agree on a made-up name for a pile of four counters. Suppose they agree on "Blims" as the name for the pile of four counters. Now start a record on the board:

Blims Extras

Joyce 5 3 (Joyce has 5 groups of John 6 1 fours (20) and 3 more.)

(Continue this record for about ten children.)

Call on children not listed on the board to tell the class how many "Blims" and "Extras" they have.

Change the grouping amount and give a new name for this grouping. Start a new record on the board and compare the entries of the same child in the first and second record. Suppose the grouping is by fives. Joyce (see above) now records 4 and 3 (4 groups of five and 3 more) and John records 5 and 0(5 groups of five and no more). This should be done to impress children that the record will be different when the grouping changes and

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what to record if Lynne had one more counter. Should we record 2 | 10? Discuss this situation. Note that with the columns separated and agreement as to what each column stands for, 2 | 10 gives the same information as 3 | 0.

In order to be lazier yet in our record- keeping, an agreement must be made to use only one symbol in each column. With this agreement, the line separating the columns is not necessary. Impress this point on the children.

Have children read the following as to the number of tens and ones.

Steven 2 5 Jill 3 2 David 3 8 Jan 2 8

In agreeing to use only one symbol per column, there may be youngsters who might ask what to do if they have 10 tens or more. Work with these youngsters sep- arately, showing how a new group is formed by using another column to des- ignate the name of this new group (Hun- dreds). Others may not be ready to deal with larger amounts yet.

Specific objectives. - ( 1 ) When columns are not labeled, the child agrees to and does use the first column for the number of tens and the second column for the number of ones. (2) When columns are not separated by a line, the child can in- dicate the number of tens and ones given a two-place numeral.

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April 1969 251

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