Learn the Trachtenberg method and the ideas around it.
Begin with the method, try a rule, then build confidence through worked examples. These guides also put Trachtenberg beside other mental calculation approaches, so you can choose what helps you.
Start with the method
The Trachtenberg method, explained
Understand the system, see why the ×11 pattern works, and decide where the rules are useful.
Trachtenberg method multiplication
Walk through the ×11 and ×12 rules, including carries, and check each result by place value.
How to learn the Trachtenberg method
A realistic way to learn one rule at a time, practise it, and know when you are ready to move forward.
Practise the arithmetic
Multiplication table tricks
Understand checkable routes through the 6, 7, 8 and 9 times tables.
How to multiply in your head
Use decomposition, a nearby base or a digit rule with worked examples.
Mental math methods
Compare compensation, multiplication facts, rule-based methods and an abacus.
For parents and teachers
How to teach multiplication
A step-by-step order from equal groups to fluent facts, by grade, with help for children who struggle.
The multiply by 11 trick
2-digit, 3-digit and longer numbers, why it works, and a practice set with answers.
Mental math tricks for kids
Nine easy tricks and one magic trick, each with the reason it works.
Multiplication games for kids
Seven free games with cards, dice or paper, plus a free online ×11 game.
Kumon alternatives for math
Tutoring, free online courses, workbooks and apps compared honestly.
Explore further
The Trachtenberg book
Find the full title, what it covers and how to check an edition or PDF.
Who was Jakow Trachtenberg?
The engineer from Odessa whose rules for mental calculation began in prison.
Abacus mental math and soroban
See how bead positions represent numbers and how that differs from Trachtenberg rules.
Trachtenberg vs Vedic math
Read two worked multiplication procedures and see what each asks you to do.
Our Trachtenberg app
See the guided practice, devices, pricing model and current family-focused audience.