Computational Strategies

 

Development of Number Sense and Mental Computational Strategies

 

Preferred Mental Strategies

Addition              Partial Sums

Subtraction        Counting Up

Multiplication   Area Model  Partial Products

Division               Partial Quotients

Fractions             Area Model

 

Problem Solving          Strategies for Problem Solving

                                        Writing mathematically

                                        Mathematical displays

                                        Setting out mathematically

 

Students need to know mentally:


1)     A variety of Addition, Subtraction, Multiplication and Division strategies.


2)     Doubles and near doubles (eg. 18+19=37)


3)     Compliments of 10 (eg. 3&7), 100 (eg. 64&36) & 1000 (eg. 234 + 766)


4)      Skip counting by all numbers 2-10 plus 15 and 25. (And starting at a variety of numbers other than zero)



5)     Times Tables to 9 x 9 instantly.

 

 

 

 

 

Addition


In Summary:

Students begin counting by 1's using individual Number Lines (1-10 then 0-20) then move onto individual Hundreds Charts.

Progress students to count by 10's and 1's on their Hundreds Charts.

Next is 100's, 10's and 1's using their Hundreds Charts for support

For each of these encourage students writing the Open Number Line summary, Number Sentence, Fact Families and a picture or sentence describing problem solved for each.

Students are then ready to progress through understanding the Partial Sums approach to addition. This emphasises use of Place Value and Mental Computational strategies. Begin with two digit numbers then move onto three digit.


Partial Sums (1).

Pulling both numbers apart into Place Value components:

     123 + 678 =

     100 + 600 = 700

        20  +  70 = 90

             3 +  8  = 11

                               700 + 90 + 11 = 801

 

Partial Sums (2).

Pulling one number into Place Value components:

   123 + 678 =

                123 + 600 = 723

                    723 + 70 = 793

                        793 + 8 = 801



This counting on method is excellent for adding time. E.g., 7.45am + 2 hours and 35 minutes.




OR




                             123    +    678

    (  100 + 20 + 3        +      600 + 70 + 8 )

Combine the hundreds, then tens, then ones. 

                   700   +      90    +        11      = 801

 

 

 

 

These lead into a formal algorithm for addition. Note working left to right as useful for approximation of solution.

   123

+678

   700

  + 90

  + 11

   801

 

 

 

Subtraction

Use the relationship between addition and Subtraction. Addition is more easily solved mentally than subtraction using counting up strategies.

E.g., Change from $5.00 for a $3.75 coffee, mentally most people count up.

3.75 + .25 = 4.00              4.00 + 1.00 = 5.00           Change = 1.25

No renaming, no borrowing, no crossing out the 5 making it 4, changing the 0 to 10, etc.

Uses Part/Part/Whole thinking.

Part

Part

Whole

 

100(Whole) – 37(Part) = ? (Part)      

Rearrange as      37(Part) + ? (Part)= 100 (Whole)

37 +3 = 40

40 +60  = 100

Therefore:    37 + 63 = 100

 

 

Solve 813- 586 = ?

Rearrange to 586 + ? = 813

586 + 4      >   590 + 10        >  600 + 200       >    800 + 13        >   813

             4                      +10                       +200                        +13   = 227.

 

Counting up is excellent for Elapsed Time.

E.g., 3.12pm to 5.05 am

3.12pm. to 3.12am. is 12 hours.

3.12am to 4.00am. is 48 minutes.

4.00am to 5.05am is 1 hour and 5 minutes.

Total elapsed time 12+1 hours and 48+5 minutes = 13 hours & 53 minutes.

 

 

 

 

Multiplication

Partial Products using Area Model.

123 x 5

x

100

20

3

5

500

100

15

 

500 + 100 + 15 = 615

 

123 x 567

x

100

20

3

500

50000

10000

1500

60

6000

1200

180

7

700

140

21

 

Find sum of individual products. Encourage mental addition.

This strategy utilises multiplication of one digit by one digit only.

More simple products can be found mentally.

123 X 6  

100 x 6 = 600        20 x 6 = 120     3 x 6 = 18          600 + 120 + 18 = 738

 

This is the strategy we also encourage for algebraic multiplication.

(x+2)(x-3)

 

x

+2

x

X^2

+2x

-3

-3x

-6

 

X^2    +2x      -3x      -6 

=      x^2 - x - 6



This strategy will also work when multiplying different units.

4 lots of 3 hours, 27 minutes and 8 seconds

x

3 Hours

20 Minutes

7 Minutes

8 Seconds

4

12 Hours

80 Minutes

28 Minutes

32 Seconds

 

Convert and add. 12 Hours + 1 Hour and 20 Min. + 28 Min. + 32 Sec.

13 Hours, 48 Minutes and 32 Seconds.

Multiplication using Skip Counting Patterns.

Run through the Counting Patterns with the students completing and look for patterns (in the ones place).

1

2

3

4

5

6

7

8

9

2

4

6

8

0

2

4

6

8

3

6

9

2

 

8

1

4

7

4

8

2

6

 

4

8

2

6

5

0

5

0

 

0

5

0

5

6

 

8

 

 

 

2

 

4

7

 

1

 

 

 

9

 

3

8

 

4

 

 

 

6

 

2

9

 

7

 

 

 

3

 

1

0

 

0

 

 

 

0

 

0

 

Note we know the 1’s, 5’s and 9’s patterns already.

Note 8’s is the reverse of 2’s, 7’s the reverse of 3’s and 6’s reverse of 4’s.

So if we write down the 2’s, 3’s and 4’s we have the 8’s, 7’s, and 6’s. So bold above are the only ones we need to remember.

This compliments the Area Partial Products method as using this strategy we only ever need 1 digit by 1 digit multiplication.

 


 

Division

Similar to how we use addition strategies to find the solution to subtraction problems, we here use multiplication to find the solution to division problems.

This also utilises Part/Part/Whole

Part

Part

Whole

 

124(Whole) ÷ 4 (Part) = ? (Part)

Rearrange to          4 (Part) x ?(Part) = 124(Whole)

It is much easier to think of multiplication than division.

 

Partial Quotients Method

1220 ÷ 16 = ?      

Prepare first a table of some easy multiples of the divisor; say twice (2 x 16 = 32) and five times the divisor (5 x 16 = 80).

Then we work towards the answer by finding the difference left after each multiplication.

In the example at right, 1220 divided by 16, we may have made a note first that 2 x 16 = 32 and 5 x 16 = 80.

Then we work up towards 1220.

50 x 16 = 800 subtract from 1220, leaves 420;

20 x 16 = 320; etc..

     ------  |

  16 ) 1220  |

      - 800  |  50

       ----  |

        420  |

      - 320  |  20

        ---  |

        100  |

      -  80  |   5

        ---  |

         20  |

      -  16  |   1

         --  |  --

          4  |  76

 

  ans: 76 R4    

 

http://www.khanacademy.org/video/partial-quotient-division?playlist=Arithmetic


Fractions

 

 

 

 

 

 

 

 

 

 

 

 

Use the area model. This reinforces the links between multiplication & division and fractions.  It also gives a powerful visual for making sense of fractions.

1/3 + 1/4  = ?                                          

 

 

 

 

1/3 of this block is 4 pieces                                         

1/4 of this block is 3 pieces.

That’s a total of 7 pieces out of the 12,  or  7/12.

 

Note that this strategy can be also be used for subtraction of fractions

2/3  –  1/4 =

with uncommon denominators.

2/3 (8 pieces) less 1/4 (3 pieces) is 5 pieces out of 12 or 5/12.

 

Multiplication of fractions utilises this strategy traditionally.

The area coloured twice is 2 out of 12   or   2/12   or   1/6.

 

 

 

 

 

 

 

 

 

 

 

 

 

From these visual and manipulative strategies find shortcuts for finding the solution to fractional problems.

E.g., 

1/2 + 1/3 = 5/6

 


Multiply denominators to find Denominator

(because it makes an array)

Add denominators to find numerator when 1 is numerator in question (because that is the number of pieces the other fraction breaks into)

 

1/2 + 2/3 = 7/6

 

Multiply denominators to find Denominator.

Cross multiply and add denominators to find numerator when numerator in question is not 1.

Comments