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Key to Geometry: Lines and Segments
Key to Geometry introduces students to a wide range of geometric discoveries as they do step-by-step constructions. Using only a pencil, compass, and straightedge, students begin by drawing lines, bisecting angles, and reproducing segments. Later they do sophisticated constructions involving over a dozen steps. When they finish, students will have been introduced to 134 geometric terms and will be ready to tackle formal proofs. Book 1 of Key to Geometry
64 pages, Spiral-bound
First published June 1, 1972
About the author
McGraw-Hill Education
23.7k books151 followersMcGraw-Hill Education traces its history back to 1888 when James H. McGraw, co-founder of the company, purchased the American Journal of Railway Appliances. He continued to add further publications, eventually establishing The McGraw Publishing Company in 1899. His co-founder, John A. Hill, had also produced several technical and trade publications and in 1902 formed his own business, The Hill Publishing Company.
In 1909 both men agreed upon an alliance and combined the book departments of their publishing companies into The McGraw-Hill Book Company. John Hill served as President, with James McGraw as Vice-President. 1917 saw the merger of the remaining parts of each business into The McGraw-Hill Publishing Company, Inc.
In 1909 both men agreed upon an alliance and combined the book departments of their publishing companies into The McGraw-Hill Book Company. John Hill served as President, with James McGraw as Vice-President. 1917 saw the merger of the remaining parts of each business into The McGraw-Hill Publishing Company, Inc.
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February 18, 2023This is an interesting option for practical geometry. It varies quite a bit from the more commonly suggested Hall and Stephens text, in a few critical ways:
[1] The Key to Geometry series uses *only* straightedge and compass for all work in all eight books. Measuring is done solely on the basis of congruency, never on use of measurement standards (in, cm, etc). Hall and Stephens includes measurement by ruler and protractor as part of its work.
Which is better? Depends on the student, to some extent, but also on your objectives in teaching practical geometry.
A child who has extensive practice in using rulers to measure and draw lines with precision will be fine either way. (Charlotte Mason homeschooling families may have mastered this through sloyd already)
For students not as adept with a drawing a straight line (e.g. holding the tool firmly, positioning your hands to avoid blocking yourself), K2G #1 and #2 are going to fill the gap while building understanding of flat plane geometry -- extensive practice using straightedge and compass before progressing to more complex construction. This is what our son needs... nothing more frustrating than a hopping compass or slipping ruler!
I like the K2G approach because it emphasizes logic alone... this dovetails nicely with a study of ancient history, adds an element of "oh gosh now I understand why both arithmetic and standardized measurement systems are so helpful," and doesn't hinge on good eyesight for all those tiny marks on a ruler or protractor. ;) #old
K2G also avoids that bizarre "tenths of inches" situation with Hall and Stephens, which is a minor thing that nevertheless irritates me to no end. =D
[2] K2G takes a somewhat inquiry-based approach to learning what geometry terms mean. I have very mixed feelings about this!
One thing I appreciate about Hall and Stephens is the explicit instruction from page 1, e.g. clarity around what exactly is a point, understanding a line as a point in motion, etc. It's borderline open and go for the teacher.
[3] K2G uses late 20th / early 21st century American English. Its design has a lot of whitespace so there is no visual overhwelm.
Hall and Stephens is a significantly older text. Each page is quite dense and the linguistic style is archaic.
I like Hall and Stephens quite a bit, but for a student who has anxiety about (or active dislike of) mathy topics, I think it could generate more friction than K2G.
[4] K2G is a series of eight consumable workbooks. I felt the price was reasonable, especially since we have only one child, but large families may prefer the one time textbook purchase of a hard copy of Hall and Stephens.
Contents of Book 1:
... drawing lines through points
... drawing triangles with straightedge and three defined points
... drawing quadrilaterals with straightedge and four defined points
... same same with pentagons
... all possible line segments connecting vertices of a polygon
... points of intersection
... straight line is defined by 2 and only 2 points
... triangle defined by 3 points
... naming conventions of lines, line segments, rays, triangles
... basic playing around with building triangles given arcs or circles to start
... basic playing around with building polygons given number of points
... nomenclature of angles: intersection of rays; vertex
... definition of congruent figures and practice with same
Hard to say more until we're in the weeds!
[1] The Key to Geometry series uses *only* straightedge and compass for all work in all eight books. Measuring is done solely on the basis of congruency, never on use of measurement standards (in, cm, etc). Hall and Stephens includes measurement by ruler and protractor as part of its work.
Which is better? Depends on the student, to some extent, but also on your objectives in teaching practical geometry.
A child who has extensive practice in using rulers to measure and draw lines with precision will be fine either way. (Charlotte Mason homeschooling families may have mastered this through sloyd already)
For students not as adept with a drawing a straight line (e.g. holding the tool firmly, positioning your hands to avoid blocking yourself), K2G #1 and #2 are going to fill the gap while building understanding of flat plane geometry -- extensive practice using straightedge and compass before progressing to more complex construction. This is what our son needs... nothing more frustrating than a hopping compass or slipping ruler!
I like the K2G approach because it emphasizes logic alone... this dovetails nicely with a study of ancient history, adds an element of "oh gosh now I understand why both arithmetic and standardized measurement systems are so helpful," and doesn't hinge on good eyesight for all those tiny marks on a ruler or protractor. ;) #old
K2G also avoids that bizarre "tenths of inches" situation with Hall and Stephens, which is a minor thing that nevertheless irritates me to no end. =D
[2] K2G takes a somewhat inquiry-based approach to learning what geometry terms mean. I have very mixed feelings about this!
One thing I appreciate about Hall and Stephens is the explicit instruction from page 1, e.g. clarity around what exactly is a point, understanding a line as a point in motion, etc. It's borderline open and go for the teacher.
[3] K2G uses late 20th / early 21st century American English. Its design has a lot of whitespace so there is no visual overhwelm.
Hall and Stephens is a significantly older text. Each page is quite dense and the linguistic style is archaic.
I like Hall and Stephens quite a bit, but for a student who has anxiety about (or active dislike of) mathy topics, I think it could generate more friction than K2G.
[4] K2G is a series of eight consumable workbooks. I felt the price was reasonable, especially since we have only one child, but large families may prefer the one time textbook purchase of a hard copy of Hall and Stephens.
Contents of Book 1:
... drawing lines through points
... drawing triangles with straightedge and three defined points
... drawing quadrilaterals with straightedge and four defined points
... same same with pentagons
... all possible line segments connecting vertices of a polygon
... points of intersection
... straight line is defined by 2 and only 2 points
... triangle defined by 3 points
... naming conventions of lines, line segments, rays, triangles
... basic playing around with building triangles given arcs or circles to start
... basic playing around with building polygons given number of points
... nomenclature of angles: intersection of rays; vertex
... definition of congruent figures and practice with same
Hard to say more until we're in the weeds!
July 17, 2013
This is a nice, gentle way to introduce foundational concepts of Geometry. This is not a replacement for a rigorous textbook, but I like to use it the summer before we begin Geometry in earnest.
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