Math Algorithms: Enhancing Computation Skills

1. Algorithm Workshop

2. A Quote from a California Legislator • All you have to do in mathematics is add, subtract, multiply and divide. Do you agree or disagree with this statement? Please discuss this at your tables.

3. Math Message: • List 5 ways you use computation in your daily life (adding, subtracting, multiplying and dividing). Do not include ways you do this with children, but rather how computation is part of your daily activities.

4. The essence of mathematics is not to make simple things complicated, but to make complicated things simple.S. Grudder

5. Famous Quote • One can flatly state that if students do not feel comfortable with the mathematical reasoning used to justify the standard algorithm for whole numbers, then their chances for success in Algebra are exceedingly small. Hung-Hsi-Wu

6. Algorithm Development • What is an algorithm? Think-Pair-Share

7. Algorithm Development 45 + 21 = More or Less than 100?

8. Algorithm Development • Discuss how you solved the problem…

9. Before selecting an algorithm, consider how you would solve the following problem. 48 + 799 We are trying to develop flexible thinkers who recognize that this problem can be readily computed in their heads! One way to approach it is to notice that 48 can be renamed as 1 + 47 and then 48 + 799 = 47 + 1 + 799 = 47 + 800 = 847 What was your thinking?

10. Algorithm Development 43 * 26 More or Less than 1000?

11. Algorithm Development • Discuss how you solved the problem…

12. Algorithm Development • Find the EXACT answer to these problems using mental math strategies (NOT PENCIL and PAPER algorithms). Then record how you solved the problem. You may use words, numbers, or symbols to describe your solution strategy.

13. Algorithm Development 1004 – 97 365 + 399 + 148 68 * 5 196 / 7

14. Focus Algorithms • Why are there focus algorithms? • These algorithms were selected for their mathematical “payoff” – they are powerful, relatively efficient, and they are easy to understand and learn.

15. Table of Contents Partial Sums 2nd Partial Products 3rd Partial Differences 3rd Trade First 2nd Partial Quotients 4th Lattice Multiplication 3rd Click on the algorithm you’d like to see!

16. Add the hundreds(700 + 200) Add the partial sums (900 + 70 + 11) +11 Add the ones (5 + 6) Click to proceed at your own speed! Partial Sums 735 + 246 900 70 Add the tens (30 + 40) 981

17. Add the hundreds(300 + 200) Add the tens (50 + 40) Add the ones (6 + 7) Add the partial sums (500 + 90 + 13) +13 356 Try another one! + 247 500 90 603

18. + 18 Try one on your own! 429 + 989 Nice work! 1300 100 1418 Click here to go back to the menu.

19. 50 X 80 50 X 2 6 X 80 + 6 X 2 Add the partial products Click to proceed at your own speed! Partial Products 5 6 × 8 2 4,000 100 480 12 4,592

20. 5 2 × 7 6 70 X 50 70 X 2 6 X 50 + 6 X 2 Add the partial products How flexible is your thinking? Did you notice that we chose to multiply in a different order this time? Try another one! 3,500 140 300 12 3,952

21. 5 2 × 4 6 40 6 A Geometrical Representation of Partial Products (Area Model) 50 2 2,000 300 2000 80 80 12 12 300 2,392 Click here to go back to the menu.

22. 4 5 × 4 6 40 6 A Geometrical Representation of Partial Products (Area Model) 40 5 1,600 240 1600 200 200 30 240 30 2,070 Click here to go back to the menu.

23. Trade-First Students complete all regrouping before doing the subtraction. This can be done from left to right or right to left. In this case, we need to regroup a 100 into 10 tens. The 7 hundreds is now 6 hundreds and the 2 tens is now 12 tens. 11 13 6 12 723 459 6 2 4 Next, we need to regroup a 10 into 10 ones. The 12 tens is now 11 tens and the 3 ones is now 13 ones. Now, we complete the subtraction. We have 6 hundreds minus 4 hundreds, 11 tens minus 5 tens, and 13 ones minus 9 ones.

24. Try a couple more! 13 9 16 12 14 10 8 7 946 802 568 274 7 2 3 8 5 8 Click here to go back to the menu.

25. 10 Partial Differences 736 –245 500 Subtract the hundreds (700 – 200) Subtract the tens (30 – 40) 1 • Subtract the ones • (6 – 5) 491 Add the partial differences (500 + (-10) + 1)

26. 20 3 Try another one! 412 –335 100 Subtract the hundreds (400 – 300) Subtract the tens (10 – 30) • Subtract the ones • (2 – 5) 77 Add the partial differences (100 + (-20) + (-3)) Click here to go back to the menu.

27. 19R3 12 231 120 I know 10 x 12 will work… Partial Quotients Click to proceed at your own speed! 10 111 Add the partial quotients, and record the quotient along with the remainder. 60 5 Students begin by choosing partial quotients that they recognize! 51 48 4 19 3

28. 85R6 Try another one! 32 2726 50 1600 1126 Compare the partial quotients used here to the ones that you chose! 800 25 326 10 320 6 85 Click here to go back to the menu.

29. 5 3 3 2 7 3500 5 1 100 1 0 210 2 0 6 6 + 3816 Click to proceed at your own speed! Lattice Multiplication 5 3 7 2 × 5× 7 3× 7 3 Compare to partial products! 3× 2 5× 2 8 Add the numbers on the diagonals. 6 1

30. 1 6 0 1 2 200 2 2 30 0 1 120 3 3 8 18 + 368 Try Another One! 1 6 2 3 × 3 8 6 Click here to go back to the menu.

31. Famous Quotes • Do not worry about your problems with Mathematics, I assure you mine are greater. • It’s not that I am so smart, it’s just that I stay with the problems longer. Albert Einstein

32. What we know is not much. What we do not know is immense. Pierre Simon La Place Remember: Thanks for coming!