8th Grade Parents:
Students are beginning Chapter 6, Exponential Equations and Functions.
The sequence of numbers 2, 4, 8, 16, 32, 64,… grows exponentially while the sequence 2, 4, 6, 8, 10, 12,… grows linearly. If you remember the ancient story regarding the chess board in which 2 cents are placed on the first square, 4 cents on the second square, 8 cents on the third square and so on, the last square will have almost $200 million billion on it (2^64 cents is approximately $1.8 x 10^19). By contrast, 2 cents placed on the first square, 4 cents on the second square, 6 cents on the third, and so on will result in $1.28 on the last square. In general, a sequence grows exponentially (or geometrically) if its rate of growth is proportional to the amount of the quantity present – i.e., if each number in the sequence comes from multiplying its predecessor by the same factor. A sequence grows linearly (or arithmetically) if its rate of growth is constant – i.e., if each number in the sequence comes from adding the same factor to its predecessor.
Like money in a compound-interest account, populations (whether of people or bacteria) tend to grow exponentially, and like money earning simple interest, food production tends to increase only linearly. The early-nineteenth-century British economist Thomas Malthus put these two observations together and concluded that poverty and famine were unavoidable. THE ARGUMENT IS FLAWED AND CAN BE ATTACKED ON A NUMBER OF POINTS.
Listed below are the students’ textbook sections aligned with Khan Academy. Over this 3 day weekend, I would advise the students go to Khan Academy, Algebra I, and the chapter entitled Rational Exponents and Radicals, and peruse the beginning videos and practice problems
NOTE: Students will be reviewing ideas/skills from Khan Academy, 8th grade, Numbers and Operations
Student Textbook (Khan Academy > Algebra I > Rational Exponents and Radicals)
6.1 Properties of Square Roots
6.2 Properties of Exponents
6.3 Radicals and Rational Exponents
Student Textbook (Khan Academy > Algebra I > Exponential Growth and Decay)
6.4 Exponential Functions
6.5 Exponential Growth
6.6 Exponential Decay
Student Textbook (Khan Academy > Algebra I > Sequences)
6.7 Geometric Sequences
7th Grade Parents
Students are beginning Chapter 5 – Similarity and Transformations
5.1 Identifying Similar Figures
- Two figures are similar if corresponding side lengths are proportional and corresponding angles have the same measure
5.2 Perimeters and Areas of Similar Figures
- If two figures are similar, then the ratio of their perimeters is equal to the ratio of their corresponding side lengths
- If two figures are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths
5.3 Finding Unknown Measures in Similar Figures
5.4 Scale Drawings
January 28, 2019
The Discreet Charm of the Graph
Deep in the Amazon rainforest, a tough, river-wise woman boats down tributaries home to blood-sucking fishes and swarming mosquitoes to stop at forest huts rarely graced by anyone outside their few isolated inhabitants. She is not a character from the Middle Ages. She lives today. Who is she? A doctor perhaps? A foreign aid worker? Not even warm. She is peddling creams, perfumes, and cosmetics for the Avon company.
Back at the New York headquarters, suited executives analyze their worldwide war against dry skin employing techniques invented by a man to whom one can safely say they have never given any thought. International in blue, domestic in red, one may imagine, graphs compare the year-by-year erection of Avon’s profits in each sector. Their annual report analyzes the company’s cumulative return, net sales, business unit operating profit, and pages of other data utilizing all sorts of fancy graphs, bar graphs, and pie charts.
A merchant presenting data in this way in the Middle Ages would have been greeted with a blank stare. What is the meaning of these colorful geometric figures, and why do they appear on the same document with all those Roman numerals? Macaroni and cheese had been invented (a fourteenth-century English recipe survives), but not the idea of marrying numbers and geometric figures. Today, graphical representation of knowledge is so familiar that we hardly think of it as a mathematical device; even the most math phobic executive at Avon could tell that an upward slanted line on the profits graph is a happy thing. The invention of the graph was a vital step on the path to a theory of place.
The marriage of numbers and geometry is one concept the Greeks got wrong, a spot on the road where philosophy got in the way…