Unit Ill/ Math and the Mind's Eye Activities Modeling Whole Numbers Grouping and Numeration ath and the Mind's Eye materials Base number pieces arc used ro examine rhc role of grouping and place values in recording numbers The results are extended to other bases are intended for use in grades 4-9 Linear Measure and Dimension Base number pieces are used to imroduce linear measure The relationship between rhe dimensions and rhe area of rectangular regions is discussed They are written so teachers can adapt them to fit student backgrounds and grade levels A single activity can be extended over several days or used in parr Arithmetic with Number Pieces Base number pieces are used to perform arirhmcrical operations Emphasis is placed on modeling arirhmedcal operations rather than developing paper-andpencil processes A catalog of Math and the Mind's Eye Base 10 Numeration PO Box 3226, Salem, OR 97302, I 800 Base 10 number pieces are used to examine the roles of grouping and place value in a base 10 numeration system 575-8130 or (503) 370-8130 Fax: (503) Base 10 Addition and Subtraction Base 10 number pieces are used multidigit numbers to portray methods fOr adding and subtracting materials and teaching supplies is available from The Math Learning Center, 370-7961 Learn more about The Math Learning Center at: ww\v.mlc.pdx.edu Number Piece Rectangles ｂＺＮｌｾ･ 10 number pieces are used to find the area and dimensions of rectangles as a preliminary to developing models for multiplication and division Base 10 Multiplication Base 10 number pieces and base 10 grid paper are used to porrray methods of multiplying whole numbers Base 10 Division Base 10 number pieces and base 10 grid paper are used to portray methods of dividing whole numbers Math and the Mind's Eye Copyright© l 'JHS The 1\tnh Le:1rning Center The !vLnh Lc:1rning Cemcr granrs permission to classroom te:JChers to n:produce the student Ktiviry page.> in appropriate quanritie; fi1r their classroom me ｔ ｨ ･ ｾ ･ nJ;lteriah were prepared with the suppon or National Science Foundation Grant l'vlDR-840371 ISBN 1-886131-l'i-5 Unit Ill• Activity Grouping and Numeration Prerequisite Activity None Materials Base number pieces (see Comment 1), grid paper Comments Actions Distribute base mats, strips and units to each student Tell the students the names of the pieces Ask them to determine the number of unjts in each piece and ask them to describe the relationships among the pieces Base number pieces can be made by copying page of this activity on tagboard and cutting along the indicated lines You may wish to have your students the cutting Each student should have a supply of about mats, 10 strips and 15 units The large oblong pieces will be used later in this activity and should be distributed then Mat Strip Unit Base Pieces A small square is a unit square or simply a unit A group of units arranged in a row is a strip and a group of strips arranged in a square is a mat Each mat contains 25 units Ask each student to form a collection of mats, strips and units Have the students determine (a) the number of pieces and (b) the total number of units in this collection Each mat, strip and unit is counted as a single piece Hence this collection contains + + or 14 pieces The total number of units is 4x25 + 3x5 + or 122 Have the students find different collections of number pieces which total 79 units Make a chart on the chalkboard or overhead listing the various collections the students find Include a column for the number of pieces in the collection Some possible collections are listed below The asterisk marks the collection with the fewest number of pieces M s u 2 14 79 19 Unit Ill• Activity 0 No of Pieces 19 11 19 79 7* 23 ©Copyright 1986, Math Learning Center Comments Actions Discuss the base representation of79 The collection which totals 79 units and contains the least number of pieces is the collection which contains mats, strips and units It can be described by the notation 3045 This is called the base representation of 79 Thus 3045 = 79 Write the following chart on the chalkboard or overhead: The completed chart is: Total Units M s u 113 95 21 50 4 Tell the students that, from now on, all collections are to contain the fewest number of pieces Work with the students to complete the first two lines of the chart Write numerical statements for these two lines Then ask each student to complete the chart and write numerical statements for the remaining lines Unit Ill• Activity Total Units M s u 113 59 95 21 28 50 124 4 4 0 4 The corresponding numerical statements are: 113 59 95 21 28 50 124 =423 =2145 =3405 = 41 = 1035 =2oo5 =4445 Students may write 0415 instead of 415 This is correct, however zeroes on the left are usually not recorded since no information is lost if they are omitted Math and the Mind's Eye Comments Actions Provide each student with one of the large oblong number pieces Discuss with the students what larger base number pieces might look like Provide names for the new pieces introduced Base Mat-Mat Each large oblong piece is a group of mats arranged ·in a row Thus, it is a strip of mats or strip-mat It contains a total of 125 units The next larger base piece is a group of strip-mats These are arranged to form a square of 25 mats Hence, this piece is a mat of mats or matmat It contains 625 units Base Strip-Mat This process of forming base pieces can be continued indefinitely Thus, matmats are grouped to form a strip of mat-mats or strip-mat-mat (3125 units) Five strip-mat-mats are grouped to form a mat-mat-mat (15,625 units), etc Note that base number pieces are successive groups of five Ask students to find the base representations of 200 and 2000 The collection of 200 units which contains the fewest number of base pieces consists of strip-mat, mats, strips and units Thus, 200 = 13005 The collection for 2000 with the fewest pieces contains mat-mats, strip-mat, mats, strips and units Hence, 2000 = 310005 Discuss why 0, 1, 2, and are the only digits which occur in base representations The collection which contains the fewest number of pieces will never contain of the same kind of piece If it did, the pieces could be exchanged for the next larger piece and the number of pieces in the collection reduced Unit Ill • Activity Math and the Mind's Eye Ask the students to imagine base number pieces Ask for volunteers to describe what individual pieces look like and the number of units they contain The first three base pieces are illustrated A base strip-mat contains 512 units and a mat-mat contains 4096 units Mat Strip Unit (64 Units) (8 Units) (1 Units) Base Pieces 10 Ask the students to find the base representations of 180 and 1000 10 The collection of 180 units which contains the fewest number of base pieces consists of mats, strips and units Thus, 180 = 2648 The collection with the fewest pieces for 1000 consists of stripmat, mats, strips and units Thus, 1000 = 17508 11 Have the students imagine base 10 number pieces Ask them to describe the collection with the fewest pieces that contains 1275 units 11 A base 10 strip is a group of 10 units, a mat groups 10 strips and totals 100 units, a strip-mat groups 10 mats and totals 1000 units The collection of 1275 units which contains the fewest number of base 10 pieces consists of strip-mat, mats, strips and units This collection can be denoted as 127510 However, if the base is 10, it is customary to omit the subscript indicating the base Conversely, if no base is indicated, it is assumed to be 10 Unit Ill• Activity Math and the Mind's Eye 12 (Optional.) Have the students cut out base number pieces 12 The ftrSt base pieces may be cut from a 16x16 grid Page of this activity is a master for centimeter grid paper Strip-Mat-Mat-Mat Mat-Mat-Mat Strip-Mat-Mat Mat-Mat (128 units) (64 units) (32 units) (16 units) Strip-Mat (8 units) Mat Strip Unit (4 units) (2 units) (1 unit) Base Pieces 13 (Optional.) Ask the students to find the base 2, or binary, representations of9, 23, and 100 Point out the "onoff' nature of binary representations 13 The collection of units which contains the fewest number of base pieces consists of strip-mat, mats, strips and unit Thus = 10012 Also, 23 = 101112 and 100 = Ｑ Ｑ Ｐ Ｐ Ｑ Ｐ ｾ Ｎ The two digits, and 1, that occur in base representations can be interpreted as the two positions of an electrical switch, say, is "on" and is "off' Using the binary representation of a whole number allows it to be represented as a sequence of switches in on or off positions This is analagous to the way computers store numerical information Unit Ill • Activity Math and the Mind's Eye Actions Comments 14 (Optional.) Have the students visualize base 16 pieces Ask them to find the collection with the fewest number of pieces that contains (a) 100 units, (b) 500 units Discuss the base 16 representations of 100 and 500 Mat Strip (256 Units) (16 Units) Base 16 Pieces 14 A base 16 mat contains 162 or 256 units, a strip-mat contains 163 or 4096 units The collection of 100 units which contains the fewest number of.base 16 pieces consists of strips and units; that for 500 units consists of mat, 15 strips and units Unit (1 ｕ ｮ ｾ Ｉ Base 16 representations require 16 digits New digits representing 10, 11, 12, 13, 14 and 15 must be added to the standard collection of digits, 0, 1, 2, 3, 4, 5, 6, 7, and Students may wish to invent their own symbols for these additional digits In machine language compter programming, which uses base 16, or hexadecimal, representation, it is customary to use the symbola A, B, C, D, E and F to represent 10 through 15, respectively Thus D6B16 represents a collection of base 16 pieces consisting of 13 mats, strips and 11 units for a total of 13x256 + 6x16 + 11 or 3435 units The base 16 representations of 100 and 500 are 6416 and 1E416• respectively Unit Ill • Activity Math and the Mind's Eye Base Number Pieces Cut on heavy lines Unit Ill• Activity Math and the Mind's Eye Centimeter Grid Paper Unit Ill • Activity Math and the Mind's Eye Actions Comments Continued (c) A collection of number pieces equivalent to mats and strips cannot be arranged in a rectangle which has 13 as one of its dimension However, if strips is exchanged for 10 units, the resulting collection can be arranged in a 13 x 22 recatangle with units left over Hence 290 + 13 = 22 with a remainder of [J Q Q Q 13 Remainder ｾ ｾ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｒ Ｒ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ Ｍ ｾ If the students are familiar with fractions, you may want to point out that the remainder may be written in fractional form Note that if each of the left over units were sliced into 13 equal parts and these were distributed among the rows of the rectangle, each row would get of these parts and the result would be a 13 x 22 4j13 rectangle Thus 290 + 13 = 22 4/13 Remainder: J i 13 -+ +++HI - - - 224/13 ｾ Continued next page Unit Ill• Activity Math and the Mind's Eye Actions Comments (c) Continued Alternatively, instead of dividing each of the remaining units into 13 parts, they could be placed in a row This row could then be divided into 13 equal parts and these parts distributed, one to each row of the rectangle · Remainder: \ 'J y i 13 [>I -224/13 The remainder can also be considered in the context of the grouping method of division In this method, 220 + 13 is the number of groups of 13 units that can be formed from 220 units In this case, there are 22 such groups plus a partial group of units The partial group contains parts of the 13 needed for another group, i e it is 4/13 of a group So the number of groups of 13 is 22 4ft3 - r- r- r- r- r- - - - - - - - r- - r- r- r- r- r- - - - - - - - - r- 1- r- r- f- - - - t - - 22 4/13 groups of 13 Unit Ill• Activity r- fff- - - ｾ Math and the Mind's Eye Actions Comments Distribute base 10 grid paper to each student Ask the students to draw a sketch of a rectangle that will enable them to find the quotient 322 + 14, without using any arithmetical procedures other than counting Discuss the methods the students use A master for base 10 grid paper is attached to Unit ill, Activity 7, Base 10 Multiplication Each student will need at least sheets Pencil sketches show up better on dittoed copies than on blackline masters The quotient may be found by sketching a rectangle which has an area of 322 and a dimension of 14 The other dimension will be the quotient Some students may have difficulty sketching an appropriate rectangle It may help to have them enclose a region on their grid paper whose area is mats, strips and units: Then discuss with them ways to construct a rectangle which encloses the same area and has a side of length 14 There are a number of ways to construct a rectangle of area 322 with one dimension of 14 The following method is used in Comment to describe the long division algorithm First, determine how many bands of width 10 and edge 14 can be incorporated into the Band of Width 10 ｾＺｕＺｧＡＺＢ･｡ｾ ｾ ｾ ｾ ［ Ｚ ｾ ､ ｳ , - ,.A - strips '\ ( Continued next page Unit Ill • Activity Math and the Mind's Eye Continued Two bands of width 10 provide an area of mats and strips Adding another band of width 10 creates too large an area, so bands of width and edge 14 are added until an area equivalent to mats, strips and units is obtained Bands of Width 10 Bands of Width A _ r ｾ Y\ 14 "'I 23 Unit Ill• Activity Math and the Mind's Eye Ask the students to fmd the following quotients by drawing sketches on base 10 paper (a) 182+ 13 (b) 315 +21 (c) Attempting to construct a rectangle of 181 square units so that one dimension of the rectangle is 11, results in a 11 x 16 rectangle with unit squares remaining Hence 181 + 11 = 16 with a remainder of (c) 181 + 11 _.o1lll 11 181 ｾ 16 The remainder can be written in fractional form Similar to the example in Comment 3c, each of the units in the remainder could be divided into 11 parts and distributed among the rows of the rectangle The result would be an 11 x 16 5fll rectangle Thus 181 + 11 = 16 5/11 (Optional) Discuss the paper and pencil procedure for finding quotients that is commonly called "long division" i 23 The long division procedure can be related to the method of sketching rectangles discussed in Comment For example, finding 584 + 23 by the long division method can be related to sketching a rectangle with area 584 and 23 as one dimension You may want to point out the similarity between a sketch of a rectangle with a missing dimension and the notation for a long division with a missing quotient ＿ Ｍ ｟ ｟ ｟ Ｌ ｾ 584 ++ ? 23f5a4 Ｑ ｾ ｟ ｟ ｟ Ｌ Continued next page Unit Ill• Activity Math and the Mind's Eye Unit Ill I Math and the Mind's Eye The Math Learning Center PO Box3226 Salem, Oregon 97302 Catalog #MET3 Centimeter Grid Paper Math and the Mind's Eye Unit 111 • Activity ©Copyright 1989, The Math Learning Center Base Number Pieces Math and the Mind's Eye Unit Ill • Activities 1, 2, © Copyright 1989, The Math Learning Center Complete the table Finish drawing segments D, E and F so their length are those given in the table A B c D E F = -D 16 10 E - Activity Sheet 111-2-A Math and the Mind's Eye Unit Ill • Activity • Actions 3, e Copyright 1989, The Math Learning Center ＮｾＮＮＮ ｾ · w " "• Record the length of this line segment using base notation c: 15 c: LO Q) m C) c: ·u; ::l (/) c: 15> c: Q) C) c: ·u; 5Q E Q) c: c: '' ' ' 40 ' ', ' Complete the line segments so they have the indicated length Activity Sheet 111-2-B Math and the Mind's Eye Unit Ill ÃActivity ã Action âCopyright 1989, The Math Learning Center Write the areas and dimensions in base notation ｉ ｾ I Area: Area: ｉ ｾ Area: Area: I Activity Sheet 111-2-C Math and the Mind's Eye Unit Ill• Activity ·Action 10 @ Copyright 1989, The Math Learning Center Base Grid Paper Math and the Mind's Eye Unit Ill· Activity ·Actions 5-7 e Copyright 1989, The Math Learning Center Base 10 Nurn ber p·1eces Cut on heavy lines I Math and the Mind's Eye Unit Ill • Activities 4-8 and Unit IV ·Activities 6-9 © Copyright 1989, The Math learning Center Activity Sheet 111-6 Math and the Mind's Eye Unit Ill • Activity ã Actions 1, âCopyright 1989, The Math Learning Center Base 10 Grid Paper Math and the Mind's Eye Unit Ill· Activities 7, e Copyright 1989, The Math Learning Center ... mats and each light-shaded region can be converted into strips 233 + 31 25 32 1 -1 43 325 x 245 404 + 135 (a) 233 5 + 31 25 (b) 32 1 - 1 435 = = 11005 1 235 (c) 32 5 x 24 = 14 235 (d) 4045 + 135 = 23s A... from the following: (a) 1021 +32 45 (b) 131 45 -421 (c) 33 x 225 (d) 20 x 41s (e) 1 232 5 +225 (f) 4245 + 145 Answers: (a) 14005 (b) 34 35 (c) 133 1 (d) 132 05 (e) 31 (f) 225 w/ rem 11s Notice the remainder... addition", i.e they will fmd x 34 by forming collections for 34 , combining them, and then making exchanges: (a) x 34 (b) 13 X 24 Groups of 34 Observe the processes the students use and discuss their methods