Multiply with fractions as an approximation

Suppose we want to do
27 * 77

(30-3)*(70+7)

= 30(1-3/30)70 (1+7/70)

= 30*70 (1-1/10)*(1+1/10)

= 2100 * (1- 1/100)= 2100-21= 2079

It won’t always be this nice.

27*73 = 2100 (1-1/10)(1+3/70)

(1-1/10)(1+3/70)= 1-1/10+3/70-3/700

-1/10+3/70 = (-7+3)/70 = 4/70

4/70 * 2100 = 4 *30 = 120

2100/700 = 3

so we have

2100 – 120-9 = 1980-9= 1971

This approach helps teaches rules for using parentheses and leads to algebraic thinking.

Students might be guided towards this in a project.

In some cases, we can change the fractions upstream to get an approximation.  We can also drop the fraction product term.

27*73 = 2100 (1-1/10)(1+7/70-4/70)

We can at first ignore the 4/70 and then add it back in as a correction.

Project for algebra or calculus class.  Treat a term like the -4/70 as h.  Work out rules for h as a varying term.

More advanced project:  Trace the thinking of Fermat, Lagrange and Cauchy in thinking of such increments.  Also look up the letters they used for such increment variables.

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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