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THE UNKNOWN // FIELD NOTES // WEEK 20

Week 20: Difference of squares, cones, and scientific notation

Field Book
72 to 74
Due
Test 17 due
Method
Paper first, every step
One pattern this week is worth memorizing outright: two perfect squares with a minus sign between them always factor the same way, every single time, no searching required. Scientific notation shows up in the same week, which is fitting, because both are about recognizing a shape instead of grinding through the arithmetic.

What you have to be able to do

The procedures

Difference of two squares
  1. Check the shape: two perfect squares, a minus sign between them, nothing else.
  2. It factors into the sum times the difference of their roots.
  3. This never has a middle term, and it never works when the sign is a plus.
Pyramids and cones
  1. A pyramid is one third of the prism with the same base and height.
  2. A cone is one third of the cylinder with the same base and height.
  3. So find the matching prism or cylinder volume and divide by three.
Scientific notation
  1. Move the decimal point until exactly one nonzero digit sits in front of it.
  2. Count how many places you moved. That count is the exponent on 10.
  3. Moved left, the exponent is positive. Moved right, it is negative.

Worked

WORKED
Factor x2 − 49.
x2 is a perfect square and 49 is a perfect square, with a minus between them.
Roots are x and 7, so it is the sum times the difference.
ANSWER: (x + 7)(x − 7)
WORKED
Write 47,000 and 0.00031 in scientific notation.
47,000: move the point 4 places left to get 4.7.
0.00031: move the point 4 places right to get 3.1.
ANSWER: 4.7 × 104 and 3.1 × 10−4

Terms

TermMeaning
perfect squarea number or term that is something squared, like 25 or x2
scientific notationone nonzero digit, a decimal, and a power of ten
coneone third of the cylinder with the same base and height

Check yourself

1. Factor x2 − 36
2. Write 0.0052 in scientific notation
3. A cylinder holds 60 cubic units. What does the matching cone hold?
reveal answers
(x + 6)(x − 6) · 5.2 × 10−3 · 20
This page is the summary, not the lesson. The full working, every step of every example, is on the board in class. Come here to remember what the week was about. Go there to learn how it is done.
-- THE UNKNOWN -- week 20, difference of squares, cones, and scientific notationalgebra 1 · oda