2 Computation Algorithms in Everyday Mathematics Instead of learning a prescribed (and limited) set of algorithms, Everyday Mathematics encourages students to be flexible in their thinking about numbers and arithmetic. Students begin to realize that problems can be solved in more than one way. They also improve their understanding of place value and sharpen their estimation and mental-computation skills.The following slides are offered as an extension to the parent communication from your child’s teacher. We encourage you to value the thinking that is evident when children use such algorithms—there really is more than one way to solve a problem!
3 Before selecting an algorithm, consider how you would solve the following problem. 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 and then= = = 847What was your thinking?
4 Important Qualities of Algorithms An algorithm consists of a precisely specified sequence of steps that will lead to a complete solution for a certain class of problems.Important Qualities of AlgorithmsAccuracyDoes it always lead to a right answer if you do it right?GeneralityFor what kinds of numbers does this work? (The larger the set of numbers the better.)EfficiencyHow quick is it? Do students persist?Ease of correct useDoes it minimize errors?Transparency (versus opacity)Can you SEE the mathematical ideas behind the algorithm?Hyman Bass. “Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician’s Perspective.” Teaching Children Mathematics. February, 2003.
5 Click on the algorithm you’d like to see! Table of ContentsPartial SumsPartial ProductsPartial DifferencesTrade FirstPartial QuotientsLattice MultiplicationClick on the algorithm you’d like to see!
6 Click to proceed at your own speed! Partial Sums735+ 246900Add the hundreds ( )70Add the tens ( )+11Add the ones (5 + 6)981Add the partial sums( )
7 + 247 500 +13 603 90 Try another one! 356 Add the hundreds (300 + 200) Add the tens ( )+13Add the ones (6 + 7)603Add the partial sums( )
8 Click here to go back to the menu. Try one on your own!429+ 989Nicework!1300100+ 181418Click here to go back to the menu.
9 Click to proceed at your own speed! Partial ProductsClick to proceed at your own speed!56×824,00050 X 8010050 X 24806 X 8012+6 X 2Add the partial products4,592
10 How flexible is your thinking How flexible is your thinking? Did you notice that we chose to multiply in a different order this time?52×76Try another one!3,50070 X 5014070 X 23006 X 5012+6 X 2Add the partial products3,952
11 A Geometrical Representation of Partial Products (Area Model) 52×465022,0003004062000808012300122,392Click here to go back to the menu.
12 Trade-FirstStudents complete all regrouping before doing the subtraction. This can be done from left to right. 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.11136127 2 34 5 9Next, we need to regroup a 10 into 10 ones. The 12 tens is now 11 tens and the 3 ones is now 13 ones.264Now, we complete the subtraction. We have 6 hundreds minus 4 hundreds, 11 tens minus 5 tens, and 13 ones minus 9 ones.
13 Click here to go back to the menu. Try a couple more!91213167108148 0 29 4 62 7 45 6 8528378Click here to go back to the menu.
14 – 2 4 5 5 0 0 4 9 1 Partial Differences Subtract the hundreds 7 3 6– 2 4 55 0 0Subtract the hundreds(700 – 200)1 0Subtract the tens(30 – 40)1Subtract the ones(6 – 5)4 9 1Add the partial differences(500 + (-10) + 1)
15 Click here to go back to the menu. Try another one!4 1 2– 3 3 51 0 0Subtract the hundreds(400 – 300)2 0Subtract the tens(10 – 30)3Subtract the ones(2 – 5)7 7Add the partial differences(100 + (-20) + (-3))Click here to go back to the menu.
16 Students begin by choosing partial quotients that they recognize! I know x 12 will work…PartialQuotientsClick to proceed at your own speed!2 3 11 21 2 01 01 1 1Add the partial quotients, and record the quotient along with the remainder.6 05Students begin by choosing partial quotients that they recognize!5 14 8431 9
17 8 5 R6another one!Try3 25 0Compare the partial quotients used here to the ones that you chose!8 0 02 53 2 63 2 01 068 5Click here to go back to the menu.
18 Compare to partial products! Click to proceed at your own speed!Lattice Multiplication5353×7221355 × 73 × 772350010021063816+3Compare to partial products!165 × 23 × 28Add the numbers on the diagonals.16
19 Click here to go back to the menu. Try Another One!1616×23122232003012018368+318368Click here to go back to the menu.