Place Value Strategies

What are place value strategies and how do they relate to standard algorithms? The Common Core State Standards describe the requirements for fluent use of the standard algorithms for addition, subtraction, multiplication and division as follows:

4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.
5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.
6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.
6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

Prior to developing fluency with standard algorithms the CCSS emphasize place value strategies. This strong focus on place value strategies plays a critical role in the development of mental and written computation strategies, while providing students with the opportunity to develop a deep understanding of how the standard algorithms work. Common Core Standards explicitly referencing strategies based on place value include 1.NBT4, 2.NBT.5, 2.NBT.6, 2.NBT.7, 3.NBT.2, 3.NBT.3, 4.NBT.5, 4.NBT.6, 5.NBT.6, and 5.NBT.7.

So, what do place value strategies look like? Many different strategies based on place value exist for both written and mental computations. Some examples of written methods for Grades 1-5 are shown below.

Possible 1st Grade Place Value Strategies:

Draw Base 10 blocks: 33 + 20 = 53

45 + 30 =75

Draw jumps on an empty number line: 2 digit + multiple of 10

Possible 2nd Grade Place Value Strategies:

Draw jumps on an empty number line:

Partial Sums (Expanded form layout): Each addend is represented using expanded notation. Like place values are added or subtracted.

       123 + 234 =                                                           238 + 473 =

       100 + 20 + 3                                                         200 + 30 + 8
    + 200 + 30 + 4                                                      + 400 + 70 + 3
        300 + 50 + 7 = 357                                              600 + 100 + 11= 711

        548 - 325                                                       614 - 459

        500 + 40 + 8                                                 600 + 10 + 4    becomes      500 + 100 + 14
     - 300 + 20 + 5                                               - 400 + 50 + 9 - 400 +   50 + 9
        200 + 20 + 3 = 223                                                                                    100 + 50 + 5= 155

Possible 3rd Grade Place Value Strategies:

Partial Sums: Expanded Form layout as above or vertical format.
632 + 325=

                               632
                            + 325
                               900
50
                                  7
                               957

Partial Differences: Each number is represented using expanded notation. Like place values are grouped and subtracted. Negative place values may result.

752 - 436 523 - 259=

   700 + 50 + 2 500 + 20 + 3
- 400 + 30 + 6    - 200+ 50 + 9
   300+ 20 - 4 = 316 300 - 30 - 6 = 264

Use multiplication facts and place value to multiply by multiples of ten:
9 x 80 =

9 x80 = 9 x 8 tens
= 72 tens = 720
9 x80 = 720

Use the distributive property to multiply within 100:
15 x 5 =

15 x 5 = (10 x 5) + (5x5)
             = 50 + 25
             = 75

Possible 4th Grade Place Value Strategies:

Partial Products: (2 digit x 2 digit)

                                 32
                            x   34
                               900   (30 x 30)
                               120   (30 x 4)
                                 60   (2 x 30)
                                   8   (2 x 4)
                              1,088

Area Model: (1 digit x 3 digit)

Area Model: (2 digit x 2 digit)

Partial Quotients:   7725/6

                    1204 r 1
                6) 7225
                 - 6000 ( 1000 x 6)
                     1225
                   - 1200 (200 x 6)
                          25
                       - 24 (4 x 6)
1

Partition the Dividend: Partition the dividend into
multiples of the divisor.

                 292/4

                70 + 3 = 73
           4) 280 + 12

Possible 5th Grade Place Value Strategies:

Add decimals on an empty number line:

35.8 + 8.3 =

Subtract decimals on an empty number line: (Count up to find the difference)

                         126.4 - 58.7 =

Draw Base-Ten Blocks: Division with decimals

Subtracting decimals on an empty number line

Start at 58.7 and jump up 1.3 to 60, then jump 40 to 100, then jump 26.4 to 126.4. Add the jumps:
40 + 26.4 + 1.3 = 67.7

Area Model: Multiplying decimals

Regardless of which place value strategies are taught it is important that there is consistency across each grade level, and that a clear progression is maintained from one grade level to the next within a school. Time needs to be allocated to school wide discussions to ensure that place value strategies are being used or adopted. The following questions can be used to promote discussion and the selection of 1-2 focus strategies per grade for each operation:

Which written methods for addition, subtraction, multiplication and division do we currently teach as a school?
Do we have enough emphasis on place value strategies throughout the school?

Are there written methods we don’t use at the moment? Do we need to adopt them?

What mental calculation skills are needed in order for students to use written methods based on place value? Do our students have the necessary mental calculations skills needed?
How can we develop whole school agreement on the written methods that we will teach for addition, subtraction, multiplication and division? How will consistency and progression be maintained?