Algorithms In Everyday Mathematics

Algorithms in Everyday Mathematics What is an algorithm?  An algorithm is a well-defined procedure or set of rules used to solve a problem.  A good algorithm is efficient, unambiguous, and reliable. Why do we teach students algorithms?  To give them important problem-solving skills  To build computational confidence  To develop sound number sense, including a good understanding of place value What are some examples of algorithms that have been taught in the past?  Long division  Subtraction with regrouping  Multi-digit multiplication Our new math curriculum, Everyday Mathematics, teaches a number of different ways to do grade-level computations. Some of them are hundreds of years old, but simply aren’t the way we have traditionally taught in U.S. schools. A few of these algorithms are emphasized more than others, and they are called “focus algorithms”. What are the new “focus algorithms” in Everyday Mathematics? Mathematical Operation Addition Subtraction Multiplication Division Now, let’s do some math…

Name of Algorithm Partial-Sums Addition Trade-First Subtraction Partial-Products Multiplication Partial-Quotients Division

Partial-Sums Addition First, add these numbers any way you’d like: 6,907 + 485 =

In Everyday Math, students are taught the Partial Sums algorithm, which relies on deep understanding of place value, and of what each digit in a number truly means:

Add the thousands: 6,000 + 0 Add the hundreds: 900 + 400 Add the tens: 0 + 80 Add the ones: 7+5 Add the partial sums: 6,000 + 1,300 + 80 + 12 =

6,907 + 485 6,000 1,300 80 + 12 7,392

Discussion Questions: What do you notice that’s different about this method? Do you think it will help students better understand the addition of large numbers? Why or why not?

To firm up our understanding, we’ll do one more sum using the Partial Sums algorithm: 15,384 + 3,602 =

Trade-First Subtraction First, subtract these numbers any way you’d like: 9,062 – 4,738 =

Traditionally, students are taught to regroup, or “borrow” as needed in order to solve the problem. This can lead to confusion, and to unnecessary regrouping when students don’t understand when regrouping is needed. Everyday Math teaches students the Trade-First Algorithm, in which all the trading (regrouping) is done before all the subtraction: There are three steps: 1. Examine all columns and trade as necessary so that the top number in each place is as large or larger than the bottom number. (The trades can be done from left to right OR right to left!) 2. Check that the top number in each place is at least as large as the bottom number. 3. Subtract column by column. 9, 0 6 2 - 4, 7 3 8

Discussion Questions: How might this be less confusing for students? Where might they most need support when learning this method?

Let’s try one more: 4,826 – 3,934 =

Partial-Products Multiplication First, multiply these numbers any way you’d like: 67 x 53 =

In Everyday Math, students are taught to think of numbers as sums of their parts, e.g. 67 = 60 + 7 and 53 = 50 + 3. When multiplying, then, each part of one factor can be multiplied by each part of the other factor. To get the solution, all of those partial products are added together:

50 x 60 50 x 7 3 x 60 3x7

   

67 x 53 3,000 350 180 + 21 3,551

Discussion Questions: How is this different from the traditional algorithm? Where might students most need support when learning this method?

Let’s try two more for practice: 45 x 82 =

134 x 65 =

Partial-Quotients Division First, divide 157 by 12 using any method you’d like:

The traditional method for long division has often been frustrating for students. There are many steps, and many skills required, to get consistently correct calculations. Partial-Quotients Division relies on known facts, estimation, and number sense to help students make sense of division. It may look more complicated, but follow the logic… 157 ÷ 12 = ? Question: How many groups of 12 are in 157? Estimate: There are at least 10 groups of 12 in 157, because 10 x 12 = 120, and 120 is still less than 157. If we take 120 from 157, we are left with 37. Next question: How many groups of 12 are in 37? Estimate: There are 3 groups of 12 in 37, because 3 x 12 = 36. If we take 36 from 37, we have only one number remaining. Solution: The final result is the sum of the partial quotients of 10 + 3 = 13 with a remainder of 1.

12 157 -120 10 37 - 36 3 1 13

Discussion Questions: What’s challenging about mastering this method? How is it different from the traditional algorithm for long division? Let’s try one more: 758 ÷ 28 =