Adding Sequences

Achievement Objectives
NA4-1: Use a range of multiplicative strategies when operating on whole numbers.
Specific Learning Outcomes

Use multiplication to solve addition and subtraction problems.

Description of Mathematics

Number Framework Stage 7

Required Resource Materials
Two sets of cards numbered 1 to 20.
Activity

The famous mathematician Carl Friedrich Gauss was asked as a five year old to work  out 1 + 2 + 3 + ... + 99 + 100. He almost instantly replied the answer was 5050. How  could he have got the answer so fast?

Using Materials

This is the method Gauss used to add up the numbers 1 to 10.
Place cards from 1 to 10 in order

 sequences1.

Place 10 under 1, 2 under 9, and so on.

 sequences2.

Discuss the total of each pair. (Answer: it is 11.)
Discuss why the total is 5 x 11 = 55. (Answer: There are 5 pairs each adding to 11.)

Repeat for 1 + 2 + 3 + 4 + ... + 8, 1 + 2 + 3 + 4 + ... + 12, 1 + 2 + 3 + 4 + ... + 14

Using Number Properties

Solve Gauss’ problem: 1 + 2 + 3 + 4 + ... + 98 + 99 + 100.
(Answer: 1 and 100 = 101, 2 and 99 = 101 ... So the series adds up to 50 _ 101 = 5050.)

Repeat for 1 + 2 + 3 + 4 + ... + 999 + 1000,
1 + 2 + 3 + 4 + ... + 46,
1 + 2 + 3 + 4 + ... + 88.

Understanding Number Properties:

Find a formula for 1 + 2 + 3 + 4 + ... + 2 n. (Answer: n(2n + 1).)

 

 


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