This page is part of an informal content review of the Mathematik Plus Gymnasium curriculum. This page provides only a summary of content for one grade of the textbook series. The main content review page provides an assessment of the curriculum.
This content summary was posted to the Kto15 email list on Oct 23, 2001. Please see the Education Page of Bas Braams - Links, Articles, Essays, and Opinions on K-12 Education - for related matter.
There are seven Chapters.
The first chapter concerns divisibility and factorization. The pupils know about division, of course, and they have seen the decomposition of numbers in an unsystematic way. This is reviewed. Then some more formal rules are given and illustrated; such as: "if the natural number p is divisible by the natural number r, then also p*q is divisible by r; q may be an arbitrary natural number." The best known rules for divisibility are introduced and justified one-by-one; first divisibility by 10, 5 and 2; then divisibility by 4 and 8, and then by 3 and 9 (11 is left for later). Then come the concepts of common factors, common multiples, largest common divisor and least common multiple. At this stage the recommended algorithm for finding the lcd of two numbers is: obtain all divisors of the smaller number, and try them in descending order on the larger one. Prime numbers and prime factorization follows next, and then Euclid's algorithm for the lcd is exhibited, but there is only a hint at a proof of its correctness. ("The Euclidean algorithm relies on the difference rule: if two numbers are divisible by the same number then also their difference is divisible by this same number.")
The second chapter concerns fractions. In 5th grade the pupils have learned simple arithmetic on fractions. Now they learn how to bring fractions to a common denominator, compare fractions and multiply and divide them. Percents are introduced, and there is extensive practice with arithmetic on fractions, decimal numbers, and percents, also in mixed combinations. A sample exercise (rated as easy):
Inspect the numbers carefully and compute as economically as you can:
(13/15)*(4/9)*(15/26)*(18/8)
(12/17)*(3/11)*(22/6)*(34/36)
Later there is an exercise involving an elaborate shopping list for a pizza "quattro stagione" given for 4 persons and to be executed for 7.
Division of fractions:
Washing powder is sold in packets of 4.5 kg. How many packets does one obtain from 5 ton of powder? (rated medium difficulty.)
A later exercise, rated easy.
Compute "im Kopf" (mental arithmetic):
7.2/0.8 ; 12.1/1.1 ; 0.42/0.06 ; 0.81/0.9 ; 10.24/5.12 ; 51.2/2.56 ; 3.2/0.04 ; 28.9/1.7 ; 1.44/0.12 ; 0.868/0.02
I should mention here that the electronic calculator is not yet used; it will make its first appearance in grade 7. The arithmetic exercises are both of the mental and of the written kind; I think there is somewhat more mental arithmetic than paper-and-pencil arithmetic, but both skills are developed seriously.
Finally, non-terminating periodic decimals are introduced.
Chapter 3 concerns symmetry and reflection, translation and rotation of figures. Pupils must recognize symmetries and carry out the indicated operations on graph paper or using their Geodreieck (geometry triangle). They must recognize congruence. The exercises normally involve pictures, so I am going to leave them to the reader's imagination. Later on in this chapter the various rules for angles of intersecting lines are covered: opposite angles equal, neigbouring angles adding up to 180 deg, parallel lines making equal angles to a third line.
The fourth chapter concerns triangles and congruence. We begin with the classification of triangles (acute, obtuse, isosceles, equilateral, right). Then comes the rule that sum of angles equals 180 deg, with detailed proof. Next are two basic constructions with ruler and compass: a normal divider and an angle bisector. The various special lines in a triangle are shown: the altitudes, the medians, the angle bisectors and the midpoint perpendiculars. For each kind, the three lines of that kind in a triangle intersect in a point; this is stated, but there is no attempt to prove it. (Hints of proofs of this property of the various triplets of special lines will re-appear throughout grades 7-9, but I haven't found a systematic treatment.)
The pupils are taught the various rules for deciding congruence of triangles: three equal sides; two equal sides and equality of the enclosed angle; two equal angles and equality of the enclosed side; two equal sides and equality of the angle opposite the larger side; all with plenty of exercises.
The chapter continues with quadrilaterals and their classification, decomposition into triangles and the rule that angles sum to 360 deg.
The fifth chapter is called Proportionality, but it is really a gentle introduction to functions; the functions being represented by y = a*x and y = a/x -- proportionality and inverse proportionality. Pupils are given a rule of one or the other kind and then must represent such a rule in a graph or a table. Inversely, they look at a graph or table and decide if the data correspond to a proportionality, inverse proportionality, or neither.
The sixth chapter concerns solid geometry. There is some discussion of perspective drawing. Classification of figures: what is a prism, a pyramid, a cylinder and a cone? Recognizing a figure from an unfolded representation.
The final chapter concerns circumference and area of objects composed of rectangles and triangles. Exercise: given a drawing of the floor plan of a house, compute its floor area.
For my assessment of the qualities of this curriculum please refer to the main content review page.
Bas Braams
--
Bastiaan J. Braams
- braams@math.nyu.edu
Courant Institute, New York University
251 Mercer Street, New York, NY 10012-1185