!!!!! Abacus Resource Guide Al Mendle

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1 !!!!! Abacus Resource Guide Al Mendle

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7 5 Parts of an Abacus Upper Deck { Reckoning Bar or Beam { Lower Deck Rod } Earth Beads The abacus above is showing a value of zero. This is called a cleared abacus. All of the beads are moved away from the reckoning bar.

8 6 A Few Pictures to Ponder The Japanese Soroban A life-size abacus A Schoty, a Russian abacus still widely used A model of an ancient Roman abacus Quipu, an Incan abacus

9 Popsicle Stick Abacus - 8 Rods 7 For each abacus in this version, you will need: 9 craft sticks 4 4-inch pieces 1 8! of inch diameter doweling. 28 pony beads white glue Use the grid on the right as a template. The dotted lines are not part of the abacus, but they show where the dowels will be placed. glue diagram diagram 1 1. Glue two popsicle sticks together. Repeat this two more times. Place glued pairs!!!!on top of the template, diagram Put a line of glue on top of the! three popsicle stick pairs as!! shown in diagram 1. diagram 2 3. Put 7 pony beads on a piece of!!!!doweling and place on top of the!!!!template, directly above the glue,!!!!as shown in diagram Repeat, gluing the beaded dowels! and the template. 5. Glue the remaining popsicle sticks! above the popsicle sticks on the!!! base.

10 8 Place dowels and craft sticks on top of this template 4 inch template for the 4 dowel pieces Place dowels and craft sticks on top of this template 4 inch template for the 4 dowel pieces

11 Popsicle Stick Abacus 9 For each abacus in this version, you will need: 12 craft sticks 8 4-inch pieces 1 of 8! inch diameter doweling. 56 pony beads white glue Use the grid on the right as a template. The dotted lines are not part of the abacus, but they show where the dowels will be placed. glue diagram 1 1. Glue two popsicle sticks together. Repeat this five more times. Place glued!!!!!!!pairs!on top of the template, diagram Put a line of glue on top of the three popsicle sticks as shown in diagram Put 7 pony beads on a piece of!!!!doweling and place on top of the!!!!template, directly above the glue,!!!!as shown in diagram 2. diagram 2

12 10 4. Continue to glue the dowels!!!!with the beads as shown in!!!!diagram 3. Be sure to use the!!!!template (diagram 1) as a guide. diagram 3 5. Glue the last three popsicle!!!sticks on the top of the!!!assembly. diagram 4 The idea for this project came from a webpage entitled, The Popsicle Stick Abacus by Mr. Barinque. The url is:

13 Place dowels and craft sticks on top of this template diagram 1 4 inch template for the 8 dowel pieces 11 Place dowels and craft sticks on top of this template diagram 1 4 inch template for the 8 dowel pieces

14 12 The abacus is a natural device to amplify important place value concepts. Use it in conjunction with place value charts.

15 A B 13 C D E F

16 G H 14 I J K L

17 15 millions The abacus is a natural device to amplify important place value concepts. Use it in conjunction with place value charts tenths hundredths 10 thousandths "1 "2 "3 Use different colors of beads to represent different whole number periods. Expressing powers of 10 with exponents is another nice fit. Using colors can also represent decimal fractions.

18 millions 1 2 millions 16 millions 3 4 millions Name:

19 millions 5 6 millions 17 millions 7 8 millions Name:

20 millions 9 10 millions 18 millions millions Name:

21 millions millions 19 millions millions Name:

22 The following pages contain some scenarios meant to provide some direction for using the abacus. They are not meant to be a how-to guide. Let the students derive their own strategies to enhance their understanding of base-ten concepts. Let the students use the abacus to invent strategies. Using a Craft There is some good research that suggests the following: Students who invent strategies perform at least as well on standardized tests. When given the chance to invent strategies, student errors change. Wrong answers are within the realm of possibility. Students who are taught only procedurally tend to be more prone to answers that don t fit the problem. Traditional algorithms should not be taught until after students have explored and invented their own strategies. 20 Teachers in upper grades, 5-8, are not in a position to help students acquire understanding before the algorithm is taught. The students have already received algorithmic instruction, and it is quite possible that students are immersed in the procedural realm. In this sense, the abacus might be used to help a student acquire the conceptual foundations. Stick Abacus Why? The abacus is viewed as a toy that children seem to enjoy making and using. Other than making the abacus in class, the students can be encouraged to explore on their own. Thinking out of the box is almost required because it s not easy to use the Chinese abacus to mimic the traditional algorithms. Class can then be used for students to briefly share their understanding and strategies.

23 21 Begin with a clear abacus. Build 297 Add 700. Need to add 80. I don t have 80. So, I ll add 100. Oops, too Remove 20. Phew, now we really added 80. Need to add 4. I don t have 4. So, I ll add 5. Oops, too Two Possible Remove 1. Phew, now we really added Make fair trades and evaluate your answer. Is there another way to use the abacus to preform this Begin with a clear abacus. Build 297 Add 800. Oops, that s too big! Remove 16. Phew, now we really added 800. Make fair trades and evaluate your answer.

24 Begin with a clear abacus. Build Need to subtract 700. I don t have 700. So, I ll subtract Add 300. Phew, now I really subtracted 700. Need to subtract 60. Need to subtract 7. I don t have 7. I don t even have ten. So, I ll subtract 100. Oops, too big! I need to add 93. I don t have 93. So, I ll add 95. Oops, that s too much! I ll subtract 2. Phew, now I really added 93. Can you show another representation of the same calculation?

25 3 8 = 23 Builds on the idea that multiplication is repeated addition. Begin with a clear abacus. I need to add 3 groups of 8. I ll put 1 group in. I ll use the left to keep a tally of each group. I can t add another group of 8. So, I ll add 10. Oops, that s too I ll subtract 2. Phew, now I only added 8 this time. I m still keeping a tally on the I ll add my third group of 8. I m still keeping a tally on the I ll make a fair trade and evaluate my

26 75 96 = Builds on the distributive property. Use to keep track of one factor! 24 Begin with a clear abacus. 5 6= =450. I moved the five bead on the second rod from the left because I am done multiplying by =420. I don t have 420, so I ll add 500. Oops! I ll subtract 80. Phew. Now I added exactly =6300. I don t have 6300, so I ll add Oops! I ll subtract 700. Phew. Now I added exactly I also moved the 70 from the far left rod because am done. Evaluate the final answer.

27 47 6 = Builds on the concept of division as repeated subtraction. Begin with a clear abacus. Build 47. Remove one group of 6. Use the left column to tally the groups Need to remove another group of 6. I don t have 6. I ll remove 10. Oops!...Keeping a tally on the left. Need to remove another group of 6. I don t have 6. I ll remove 10. Oops!...Keeping a tally on the left. Need to remove another group of 6. Keeping a tally on the left. Need to remove another group of 6. I don t have 6. I ll remove 10. Oops!...Keeping a tally on the left Make a fair trade. Need to remove another group of 6. Keeping a tally on the left. Need to remove another group of 6. I don t have 6. I ll remove 10. Oops!...Keeping a tally on the left. Make a fair trade. Evaluate the answer. Where s the quotient? Where s the remainder?

28 = Use these rods to tally the groups you subtract.! 26 Builds on the concept of division as repeated subtraction. Begin with a clear abacus. Build 713. Hmm. I know is 460, so I should remove 460. I don t have 460, so I ll remove 500. Oops! I ll put 40 back. Phew, now I removed exactly 460. I ll use the far left to tally the 20 groups of 23. Thinking ahead, I going to skip making a fair trade, Instead, I ll remove & tally another or 230. Theres exactly 31 to remove, I ll move a bead on the 2nd rod from the left to tally Now, I ll evaluate the answer. Where s the quotient? Where s the remainder?

29 1! = Builds on the concept of division as repeated subtraction ! Use these rods to tally the groups you subtract. Begin with a clear abacus. Build Hmm. I know =9,200, so I should remove I don t have 9200, so I ll remove 10,000.Oops! I ll put 800 back. Phew, now I removed exactly 9,200. I ll use the far left to tally the 100. I ll make a fair trade. What trade was made? 9 I ll make a another fair trade. What trade was made? I want to remove another 100 groups of 92 or 9200, but I ll settle for Oops! I ll put 300 back. Phew, now I removed exactly 9,200. I ll tally this 100 on the far left. Hmm is 184. I can t remove that, so I ll remove 200. Oops! I ll put 16 back. Phew, I removed exactly 184. Use the 3rd rod from left to tally the 2 groups. Thinking ahead, I going to skip making a fair trade, Instead, I ll remove & tally another Make a fair trade. Evaluate the answer. Where s the quotient? Where s the remainder?

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

31 !"#$%&()"*)&+%,$)&!"#$%&(&%)*+#,#-.(%&/01)# =+/2#1/&5)2#/+26),L1,2%#4#"#$%&26#2$#/#00#/0&%"2,#$2#&B)44#& #6+B2+D%10); V+K+WV#J1/DV#26V)#/1/D7%4 V+K+1&#&2%0)/2#/02)#$6),7,1)/045&12)0)01$#2)02+C#J1/D 7%/0#C)/2#4C#26)C#21$&)/D#D1/D#/0$+C3,)6)/&1"4);E61&3#D)6#& 1/2),)&21/D)Q34+,#21+/&#/01/7+,C#21+/B ),)/2#"#$1;.20+)& &2#2)26#24)#,/1/D2+%&)#/#"#$%&1&)#&5F#/0.261/J261&1&/+22,%)7+, C#/5+7%&; X1Y+%,/#4 =+/2#1/&#41224)$+CC)/2#,5+/26) "#$%&#/0#L1,2%#4#"#$%&26#2+/) $#/%&)2+#00#/0#00#/0&%"2,#$2; E61&>9E%")L10)+&6+B&26#2$6140,)/H&)Q2,)C)0)L+21+/2+26)#"#$%& $#/C#J)26)C$#4$%4#2)1/#41D62)/1/D7#&2C#//),;E61&0+)&/+2C)#/ 26#2#44%&),&&6+%40#$Z%1,)261&&3))0; E61&1&#L10)+1/2,+0%$21+/2+%&1/D"+2626)Y#3#/)&)#/026)=61/)&) #"#$1; -./01$#2)&#,)&+%,$)3#,21$%4#,456)437%47+,89:%&);

32 30 Bibliography* Bernazzani, David (2005) The soroban/abacus handbook, retrieved from Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E. & Empson, S. B. (1998) A longitudinal study of invention and understanding in childrens multidigit addition and subtraction. Journal for research in mathematics education, 29, Carpenter, T. P., Franke, M. L., & Levi, L. (2003) Thinking mathematically: integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann Dilson, J. (1968) The abacus, New York: St. Martins Press Kammii, C., and Dominic, A. (1997) To teach or not to teach algorithms. Journal of mathematical behavior, 16, Knuth, E. J., Alibali, M. W., Hattikudur, S., McNeil, A & Stephens. The importance of equal sign understanding in the middle grades. Mathematics Teaching in the Middle School. 13, Ma, L. (1999) Knowing and teaching elementary mathematics. Mahwa, NJ: Lawrence Erlbaum Associates, Inc. Smith, D. (1925) History of mathematics, New York: Dover Van De Walle, J. (2004) Elementary and middle school mathematics, teaching developmentally, Boston: Pearson. *See also list of Web Resources, pp

Count By Tens and Hundreds

Count By Tens and Hundreds Student Probe Sarah had 70 stickers. Then she got 30 more stickers. How many stickers does Lesson Description This lesson helps students develop an understanding of counting