Secondary III Mathematics
year of examples, a  bridge and pause between
junior and senior high school mathematics.

First Draft - December 6th, 2006

These lesson ideas or plans  secondary III mathematics represent a personal suggestion. So they do not follow the curriculum or course design of any school district..  Mathematics course design  needs to be kept lean and simple in order not to alienate students by requiring too much.. Mathematics course design needs to provide lessons and lesson plans that are easily understood and repeated, and also effective in the classroom. The aim is to cover the essential skills,  not impose an excessive amount of mathematics on students and their teachers.

The aim is provide a useful knowledge of mathematics that students in secondary I and II may see a reward for their efforts. The aim is also to consolidate or provide examples and context for further, more abstract, studies in senior high school mathematics.  In secondary I and II, students may wonder why there is or should be so much emphasis on measurement, fraction and algebraic skills and sense. The third year of high school mathematics, secondary III, may be a year of examples provided to engage students and to suggest that mathematics has many applications. We try to provide a year which leaves a favorable last or inviting impression of mathematics for students at the end of their interest in mathematics and for students with the will and patience to continue. The aim is to provide arithmetic, algebra and geometry. skills, a context for them,  that ongoing students may remember or terminal students may use. 

Let secondary III mathematics be a pause and a bridge between junior and senior high school studies in the subject. 

• Parts A and D  of  Secondary III mathematics may consolidate and extend the arithmetic, algebraic and geometric skills and sense met in secondary I and II.
• Part B of of secondary III mathematics may give or explore example after example of mathematics in action in consumer, business and science or technology related situations that they are likely to meet in the near or distant future.  The examples may be selected to give a practical foundation for consumer, business and work-related activities and also to lay a foundation for further studies. Examples for the former can vary from location to location but examples for the latter may or should be chosen carefully.
• Part C may extend statistical sense and skills.
• Part E may  mix 2 and 3 dimensional geometry.
• Part G may give spatial construction exercises - building shelves and tables - for hands on experience of solid geometry.
• Part F secondary III  may serve as preparation for secondary IV mathematics and beyond. Secondary IV and beyond may point to the role of logic if not axioms (assumed patterns) in the further development of mathematics. Prior to this,  the third year of secondary mathematics may provides students with a set of skill set in arithmetic, algebra and geometry which leads to repeatable, reproducible and therefore verifiable results, independent of the need for formal dependence on logic.

The foregoing selection of topics is not final. There is room for change and improvement. The foregoing represents a first draft of a bridge and pause between junior and senior high school mathematics.

Secondary III mathematics may mostly serve as a year of examples which consolidate and even extend arithmetic, algebra and geometry skills and concepts. Student skills and confidence may be based on arithmetic, algebraic and geometric methods which lead to repeatable, reproducible and thus verifiable results in the examples or situations presented.

Instruction is an iterative and cumulative affair in which students need to be reminded of key or missing skills and concepts annually. Abilities and comprehension need to be maintained through practice and repeated message from teachers of how all fits together. Ergo

• Secondary I is the year of fractions, percentages and decimals representations included.
• Secondary II is the year of algebra - the direct and indirect use of formulas and proportionality equations. 
• Secondary III is the year of examples and potential applications - material to engage students, material that has a semblance of usefulness (consumer math, map reading, navigation, construction, three dimension drawing) with a focus on being able to follow methods, step by step, in a repeatable and reproducible, and thus skill and confidence building manner.
• Secondary IV and V are years of logic,  proofs, trig, functions, quadratics and solving linear equations etc to prepare for  calculus, science, technololgy,  technical trades and business. Technology and technical trades may require trig and complex numbers - graphing and map or plan reading skills in 2 and 3 dimensions.. Business or accounting may require calculus to understand formulas.

Part A. Consolidate secondary I and II

The third year of high school mathematics, secondary III,  could be the year of arithmetic, algebraic and geometric examples in which students first consolidate and then extend fraction and algebra skills and sense introduced in secondary I and II,

What is taught before calculus and after secondary I (year of fractions) and secondary II (year of algebra)  may vary from school district to school district. That being said, the  third and further years of secondary school need to maintain and develop the fraction and algebra skills and sense developed in secondary I and II. See the associated lesson plans and see site sections on solving linear equations and on fractions, ratios, rates, units and proportionality. The third year of high school mathematics could aim to reward students for their earlier patience in learning to do or apply arithmetic, algebraic and geometric skills in a repeatable, reproducible and thus a verifiable manner.  The role of implication rules B if A in mathematics is delayed to the fourth year of mathematics, an arbitrary decision, not necessarily optimal.

Secondary I, that is the first year of secondary school mathematics is say the year of fractions. In it, students consolidate and extend their measurement and fractions skills and sense, and that may include calculations with prime numbers, and identification of least common multiples and greatest common divisors with the aid of prime decomposition of the whole numbers in numerator and denominators of fractions. Compound fractions may be introduce as alternative notation for the division of one fraction by another.  Students may see the additive properties of area to calculate the areas of complicated regions in the plane from partition into simpler sub-regions with calculable areas: rectangles, quarter and semi-circles, triangles, trapezoids and parallelograms.

Secondary II., is the year of algebra. Here students learn to use formulas and proportionality relations directly and indirectly, backwards and forwards. The emphasis is or could be on the systematic development of algebraic thinking skills, a comprehension of the shorthand role of letters and symbols in describing calculations (arithmetic) which may or may not be done, and the role of words in describing numbers, amounts and quantities (and the shorthand symbols that may stand for them) as known or not, and variable or constant in one sense or another.

Students in the first  two years of high school may come with a weak to non-existence command of the times table (addition table too) and with a weak to non-existence fraction sense and abilities.  The most important service of first year mathematics in high school is to consolidate fraction sense and skills. See Solving Linear Equations with Stick Diagrams if your students have a weak command of fractions or if you want to develop algebraic thinking skills.

Part B. Examples and More Examples

 Secondary III may be known as the year of examples and applications

Examples to engage students by showing or developing roles and applications for arithmetic, algebraic and geometric skills, as is or extended, follow.  Students should see the forward and backward use of all formulas there-in. A lot of variety is possible. Leave some time for further parts of this course.

• Road Slopes:  Height gain or loss versus horizontal or actual distant travelled.
• Constant speed travel.  Calculating and graphing distance versus time.
• Simple Interest examples in the which the interest is spent or put aside in place of re-investment.
• Simple growth examples in farming where only a fraction of the seeds produced by a crop is replanted to generate next years crop, while the rest is put aside or consumed.
• Calculating number of weeks, days and hours between two dates on the Calendar.
• Compound growth and decay examples with money, radioactive material and varying populations (people, fish, wildlife). 
• Constant Rate Situations (work, speed, production).
• Direct Variation and Direct, Inverse, square, cubic, inverse square and inverse cubic examples.
• Joint Proportionality of work done to number of machines or workers, and duration of their use (length of time worked).  Plus indirect use of this proportionality to calculate duration of work or number of machines or workers from work done or required. The latter gives duration of work being jointly proportional to the work required with an inverse proportionality dependence on the number of workers or machines available.
• Partial Variation or modified proportionality Examples in which the value of one variable,  total equals the initial value plus a further amount proportional to the change in a second variable.  Examples may be given by taxi rides, telephone, monopolies and utilities set-up charges. Here students will be using and graphing straight lines y = ax+ b, calculating the parameters a and b from problem data, and comparing alternatives.
• Cost of Living Examples:  Cost of living alone or with a friend. Effect on budgets and how to budget. Explore cost of food and lodging. Payment of taxes included.
• Examples of multiple proportionality or multiple ratios in cooking and also (?)  in consumption and production.
• Scale factors in 2 and 3 D. How length, areas, volumes and quantities proportional to the latter depend on scale factors. How to use the latter relations forwards and backwards, that is to find the scale factor and then to use it. Give applications to 2D maps and diagrams and 3D models.
• Construction of maps using scale factors - connection to dilatation about a point without and with coordinates. 
• Construction of Plans using scale factors - connection to dilatation about a point without and with coordinates.  The plans may be made  on or with a map, or without.
• Arrows and Navigation: Use of maps for navigation and the description of displacements using lengths and directions (angles) or using components (change in horizontal and vertical coordinates).   Introduce arrows or vectors for the graphical description of movements and their head to tail addition. Show how to add movements or arrows graphically and with coordinates.

Teachable Moments: If an application of mathematics is topical due to a local or global events, use it.

Part C. Statistics

Develop statistics and survey skeptism and sense: Present measure of central tendency in social and technical situations; that is in surveys and in instrument calibration.

W: I am not fond of statistics. Go elsewhere for advice on what to include here

Part D. Extend - Number Sense and Skills

Arithmetic: cubes, Exact calculation or representation of of cube roots  using prime decomposition, what are real numbers - rational and irrational.  

In arithmetic, students may review what is a square root and learn about cube roots. With a knowledge of only real numbers, the square root of negative numbers cannot be defined while each positive number has two square roots, a negative one and a positive one.  The negative one is the additive inverse (that is the negative) of the positive one, and the latter is called the principal square root.  The graph of  x = y² can be used to develop the foregoing.  Students may calculate decimal approximations of square roots with the aid of a calculator. In the case of cube roots, the graph of  x = y³ indicates that each real number x has a unique cube root, a root with the same sign as x.  Calculators again can be used to obtain decimal approximations.

• The Pythagorean theorem with the Chinese square dissection proof
• Backward and forward use of the Pythagorean theorem. 
• Why the square root of a prime number (2?) is not a fraction after the discussion of the Pythagorean theorem.  How to construct the length sqrt(2) geometrically from the isoceles right triangle with two sides of length 1..Other irrational numbers (pi?) and roots of other primes.
• Exact Representation of square and cube roots of whole numbers using prime factorization of latter.
• A fraction has a finite decimal expansion when and only when the prime decomposition of its denominators contains only twos and fives. Otherwise, a fraction has an infinite periodic decimal expansion.
• Indicate how infinite decimal expansion represent a sequence of approximations for the coordinate of a point  on the real number line. So 0.9999 (9 repeating) represents a sequence of approximations to the number 1 - show or state how the error in these approximations decreases - see site Number Theory  lesson on this [matter]
•  Show how to convert an infinite periodic decimal expansion into a fraction with integral (whole number) numerators and denominators.  Here we may use arithmetic with infinite decimal expansion with or without explanation of why the operation works.  See next item.

Part E. A mix of 2D and 3D geometry

The skills and knowledge are not essential for the core components of secondary IV mathematics.

 Introducing projective drawings methods and views; and link to art and technical drawing classes on paper or on screen; the volume or capacity of solids and containers, the direct and indirect calculation and measurement of volume and density; the discoveries of Archemedes; the latter may include  formulas for volumes with physical methods to verify; 

• 2 and 3D Geometry:  What is Area?, What is Volume? Develop idea of covering regions and solids with small squares and cubes to approximate what should be their area or volume, and say if taking smaller and smaller squares or cubes converges a single real number then that number is taken to be the area or volume of the region or solid in question.  Give formulas for volumes of boxes (parallelepipeds),  prism and cylinders (V = base areas time height).  Review formulas for area of plane regions that may serve as a base,  

  Include here (?) The measurement or description of lengths, areas and then volumes as whole and then fractional multiples of unit lengths, unit areas and unit volumes,  the effect of change of units on these unit quantities and multiples there-of.  The unit area could be a unit square. The unit volume could be a unit cube.  Add to the foregoing the approximation of lengths, areas and volumes with finite and infinite decimal expansions. Explore the consequences of change of scale (unit length) on the description of lengths, areas, volumes and on the description of linear, 2D and 3D densities.

• Geometry - Physical checking or confirmation of consequences of volume formulas: Show  physically how the volume or capacity of a cone is one third that of a circular cylinder with same height and based. Show via a physically example  how the volume or capacity of a semi-sphere plus the volume of a cone equals the volume of a circular cylinder when all have the same height and same base area. See Example below
  
• Drawing 3D objects in the plane with the aid of various projections: the need for two or more views for full information: Technical Drawing, Perspective Drawing in art, and Computer Graphics may provide a context or motivation for developing and describing different view of solids. One applied project may be to draw or design, a computer support table or just a counter, or a set of shelves from a large piece of plywood or press-wood.  The question here is how does draw a 3D object in a way that others can construct it. Examples of solid objects may be used to illustrate concepts.  The foregoing may count as another example and overlap with exercises in art and in technical drawing.

Volume Examples

A cone with the same base (or top) area as a cylinder  has a third of the volume of the cyclinder when both have the same height. To fill the cyclinder to the brim or top using the cone, one has the fill the cone three times. That can verified in a class. If the height of the cylinder and cone equals the diameter. radius R of the cyclinder, then students may verify that the volume of a solid hemisphere of diameter D = 2R plus the volume of the cone equals the volume or capacity of the cylinder. Here it may easier to take a solid ball, cut it in two hemispheres  and use its diameter D to provide the inner dimensions of the cone and cylinder. Place the hemisphere in the cyclinder. Then take a  cone filled to its brim with water and pour its contents on top of the hemi-sphere in the cylinder. The water should reach the top of the cylinder and hemisphere. One could do a similar activity with a sphere in place of a hemi-sphere if the H = D and not 2R, but water poured on top the sphere tightly fitted in the cyclinder would not reach the space underneath the sphere in the cyclinder because its path is blocked by the sphere - Workaround: put half the water in first.

Physical Verification of Formulas: The calculation of volume or capacity from the product of base area times height can checked or tested  in the mathematics or physical science. The foregoing shows how formula for the volume of a sphere can be related to formulas of volumes of cylinders and cones.  Prior to testing formulas for volumes, we may test formulas for  area calculations for circles or disks: For example, take a piece of paper or carboard with with a constant thickness and area per square unit (centimeter or inch) and verify that the weight or mass of a disk of radius R of the material is pR2 times the weight of a square unit.

Part F.  Introduce Logic in Mathematics

The ability to follow a multi-step process in a repeatable and reproducible manner, modulo some accidents, is a sign that the students master further multi-step operations in and outside of arithmetic. That is the skill or intelligence we seek. Start emphasizing in it in arithmetic. Calculators betray students by allowing them to skip a first example of a multi-step process in which accuracy is demanded at each and every step. The last topic, statistics,  should be exploited as much as possible to develop and reinforce fraction skills and sense. 

Secondary IV and V mathematics school and college mathematics  may introduce and expand upon the role of logic and assumptions (assumed patterns) in codifying mathematics and provide logic-based foundation and structure for pure and some applied mathematics. 

Secondary IV and V mathematics school and college mathematics  may introduce and expand upon the role of logic and assumptions (assumed patterns) in codifying mathematics and provide logic-based foundation and structure for pure and some applied mathematics. 

The Challenge: Students who have learnt mathematics as a collection of  given or logically developed method,  with repeatable and reproducible results may question the need for a logical development or codification of the subject.  There-in lies an opportunity to describe the un-ruled origins of mathematics and the ad hoc ways in mathematical methods were found and give the students another model for reason.  There-in lies the challenge of presenting the axiomatic codification in a convincing fashion to students.

Part G: Spatial Construction Exercises  (Optional)

The hands-on or manipulative nature of these exercise may engage the boys.
Purchase a rectangular piece of plywood or press-word and have it cut into rectangles A to E as shown. Piece E can be thrown away. Pieces B and C are identical.
Attach the pieces together as shown using 15 braces and 60 short screws.
Tools required: screwdriver and electric drill. There is some flexibility in deciding the dimensions of the pieces A, B, C and D.  Students could make a scale model from a piece of paper.  Note: The middle piece D of the supporting H (formed from A, B and D) is shorter than end-pieces A and B.  Making all three the same height leads to imbalance problems on uneven floors.
Other Plywood  Projects Book Shelves	Computer Table
The question of how much paint is required to cover this furniture or other three dimensional objects points to a practical reason for calculating surface area.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science