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O: Operations and calculations
Here you will find additional support, guidance, activities and ideas for every sub-section in O Operations and calculations. If you’re accessing these pages via the QR codes in the book, the cloud symbols will help you locate where ideas and activities expand discussion in the main text of the book.
O1: Number lines
page 64
Singapore Bar Model
John Bessant and Phil Bagge “Number Lines and Vertical Methods” Maths Video Help Files
page 67
Henry Liebling“Using an Urdhu Number Grid” [View T14]
O2: The story of 24
Rudy Rucker, author of Mind Tools
Factors of 24 and templates for number sentences available in the Resources section
O3: Arithmetic operations
More ideas page 77
Possible resources to choose from:
- number line or track
- time line
- blank number line
- number grid
- fingers
- number cards
- Cuisenaire rods
- multilink or cubes
- teddy counters
- base 10 equipment
- money
- bundles of straws, etc.
More ideas page 79
More ideas page 81
The Treviso Arithmetic contained the first printed version of the Gelosia (Lattice) method of multiplication
“In the footsteps of mathematics” blog explains the history of this method
Russian Peasant method of multiplication
Ancient Egyptian method
John Napier “Napier’s Bones”
For more on Rudolf Steiner’s “Robber Sum” and other methods of teaching arithmetic see here
page 84
Here are some solutions to “All the Fours”.
If 4/4 = 1 then it is possible to make any number. 5 = 4/4 + 4/4 + 4/4 + 4/4 + 4/4
Better would be 5 = 4 + 4/4
4 x 4 = 16 so 9 could be 4 x 4/√4 + 4/4
Try to find the most elegant solutions.
! is the factorial sign, so 4! = 4 x 3 x 2 x 1 = 24
√ is the “square root” sign, so √16 = 4, √25 = 5, √4 = 2
These signs could be used like this: 8 = 4 x √4 or 6 = 4!/4
page 86
O4: Arithmogons and other puzzles
page 88
More Arithmogons: Examples of subtraction and multiplication [View]
More ideas
Arithmogons and pyramids are available in the Resources section
Blank arithmogons and pyramids are available in the Resources section
Hungarian puzzles are available in the Resources section
Japanese puzzles are available in the Resources section
page 95
These puzzles are based on resources from CIMT Maths Enhancement Programme
O5: Teaching multiplication tables
page 102
Truth tables for odd and even for all four operations, and strings of operations [View]
page 103
Loop Cards [View]
O6: Divisibility rules
Vedic maths and Trachtenberg speed methods. You and your students might be interested in shortcuts for calculations. Sometimes they make the concept of multiplication clearer, but sometimes they complicate and confuse.
A Russian pacifist, Jacow Trachtenberg developed a complete set of arithmetic shortcuts, which rely on doubling, halving, addition and subtraction, in a Nazi concentration camp during the World War II. His book of arithmetic shortcuts “Trachtenberg speed system of basic Maths” was published after the end of World War II in the 1940s. Read about his amazing story. You could try the “Trachtenberg speed system” iPhone app
Bharti Krishna Tirthaji, a Hindu nationalist, wrote “Vedic Mathematics” in 1957, and it was published in English in 1965. It often uses an algebraic method such as (a + b)(c + d) = ac + bc + ad + bd which, as vertical method, makes a pattern I x I. Some good examples of the method can be found on YouTube. The origins of Vedic Maths are, however, disputed.
See also Chisanbop (Korean finger counting), and the mental abacus