For Twelve to Fourteen Year Olds

A. Geometric Formula Evaluation

The first objective is to provide a show work format for students to follow in evaluating formulas for areas, perimeters and volumes - geometric quantities all. Formulas for areas of m by n rectangles in the special case where m and n are whole number may be implied by counting principle that a m columns of n squares will gives m $\times$ n squares. Following that, the product of dimensions, rectangular area formula

Area = [width] $\times$ [Length]

may be given and used. The formula for the area of a square is a special case.

In mid-level skill development, the aim is rigour not necessarily in the derivation or justification of formulas, but in their use. Rigour in skill mastery is provided by geometric formula evaluation show-work format given for doing and recording all steps in the use and evaluation to provide written work that can be seen and approved or corrected. The show-work format defines what we mean by skill in a visible, repeatable, reproducible and verifiable or correctable manner.

Formulas for the areas of right triangles may be implied by observing each right triangle is one half of a larger rectangle. But formulas for area of further triangles, parallelograms, trapezoids and circles may be given, explained briefly where possible, used and even confirmed empirically with the aid of measurements. The formula for a perimeter or circumference of a cicle may be given as well. In this, the mathematical derivation of circle area and perimeter formulas requires a mastery of higher level mathemtics in the form of calculus. Physical derivations or confirmations of the formulas is possible with in the bounds of calculation and measurement errors.

Initially at this level, the algebraic-geometry reasoning sufficient to imply formulas for areas of general triangles, parallelograms and trapezoids may be algebraically to complex for students to follow. But in time, after students are introduced to algebra and after they are shown how to find lengths, areas and measures by recognizing and adding sublengths, subareas and submeasures, the reasoning will be followed. Iniitally, the aim is to build a consistent web of rules, patterns and practices which can be used one at a time, one after another, and eventually combined in chains of figuring or reasoning to imply results, each other or new rules, patterns and practices, all in an empirically consistent manner. The higher mathematics, Euclidean objective of identifying a minimal set of rules and patterns whose assumption provides a base for deriving the others is not for young students. In the first instance, instruction may put rigourous practice ands rigourous skill development first, and theory second. In the latter, rigour may develop over time as more and more examples of deductive reason, informal and formal, are met. Before that arithmetic, algebra and logic skills ought to be mastered separately.

B. Evaluation of Geometric Formula - Show Work

Formula evaluation may be put with arithmetic, algebra or geometry, as we like.

Steps

1. Write the geometric formula neatly.
2. Draw or sketch the diagram, and on it indicate the values of the letters or quantities in the formula.
3. Substitute the latter values in the formula,
4. After substitution, simplify as much as possible without the aid of a calculator.
5. Lastly, if wanted, evaluate the simplified expression with or without a calculator.

Rectangle Area Example:

Solution:

\begin{eqnarray*} \def\cm{\mbox{ cm}} \mbox{Area } A &= W \times L \qquad & \quad \mbox{Write formula} \\ &= [12 \cm]\times[5 \cm] &\quad \mbox{Substitute Values}\\ &= [12 \times 5] \cm^2 & \quad \mbox{Simplify} \\ &= 60 \cm^2 &\end{eqnarray*}

The evaluation does and records steps in an in observable and verifiable manner. Here again equal signs are present and vertically aligned.

A 12 cm by 5 cm rectangle may be seen as by

1. 12 columns of 5 square squares or

2. 5 rows of 12 squares

Formatting Advantages: The above format for formula usage or evaluation provides a model for students to follow not for rectangle area evaluation, and also for the evaluation of formulas triangle, trapezoidal, parallelogram and circle area and perimeter. There-in lies a model for showing work and for showing and recording comprehension in mathematics, science and further quantitative arts and disciplines, where formula evaluation questions.

More Explanation and Examples

• 1 Written work formats for developing and showing Skill

• 2 Another Rectangle Area Formula Example
  

• 4 Circle Area Formula Example

The first link above leads to a more detail look at the show work format.

C. Area of Right Triangles

A
 o               
 |              
 |  x             
 |   .           
 |               
 |b     x        
 |               
 |        x      
 |90 deg           
 +-----------o
C     a      B

with perpendicular sides of length a and b, respectively is [ab]/2

Geometric Proof: Complete the rectangle determined by the perpendicular sides of the right triangle. The hypotenuse of the original triangle equals the hypotenuse of a rotated copy of the original --- a triangle obtained by rotating the original 180 degrees. The area ab of the rectangle is the sum of the areas of the triangles = twice the area of the original right triangle.

A

 +-----------o 
 |           | 
 |  x        | 
 |   .       |   
 |           |   
 |b     x    |    
 |           |     
 |        x  |    
 |90 deg     |       
 +-----------o

C     a      B

So the area A of a right triangle with base a and height b is $A = \frac 12 ab $

D. Measures by addition, subtraction and approximation

Perimeters, Areas and Volume via Decomposition:

In mathematics, science, money matters, contruction and cooking classes, students may be shown how to calculate total lengths, mass, weight, volume, cost, area by addition of sublengths, submasses, subwieghts, subvolumes, individual costs, and subareas. Calculations may done step by step in a way that allows for calculation data and steps to be seen and checked. Further, lengths, masses, weights, volumes , costs, areas given by the difference of others may be obtained by subtraction. Talking about and illustrating the foregoing in class sets the stage for calculating areas through addition and subtractions. Moreover, once students algebra skills are sufficient, the foregoing sets the stage for deriving formulas for areas of triangles, parallelogram and trapezoids instead of just giving them.

The covering of floors, walls and roofs with carpets, paints and shingles etc may provide examples where recognizing and exploiting this decomposition or division of perimeters, areas and volumes in smaller pieces, the measure of which is easily compute by formula. The foregoing may have some take home value.

Physical Approach: To confirm and check area calculations by formulas or approximate for small regions, one draw the region in question on a piece of paper or carboard with with a constant thickness and density - mass per square unit. Cut out the region and measure its weight or mass. The latter quantity divided by the density will approximate the area of the region, and serve as a check on the other calculutations - if any. Such an approach or exercises may lifts formulas off the page and connects to mechanics or physics. Meeting difficulties or finding the limitations of this approach provides off papers experience of the quantitative kind.

E. More on Formula Usage

Students may be given formulas for areas of triangles, parallelograms and trapezoids to use and apply with data and steps written as done to record the reasoning for immediate or later confirmation or correction. Brief algebraic geometric explanations can be given with the message that the algebra in them will understandable this site algebra starter lessons. Before the systematic development of algebra skills and concepts, the shorthand role of letters and symbols will beyond the reach of most. The exercise of deriving or justifying formulas for triangles, parallelograms and so forth from the method for calculating the area of a rectangle resembles and compounds the exercise of obtaining areas of figure decomposable into shapes - non-overlapping subregions whose areas are easier to find.

Areas of Triangles - height over base

Then with numerical examples we show or confirm the alternative calculation \begin{eqnarray*} A & =& \frac12h(b + c) \\ & =& \frac12h w \\ \end{eqnarray*} gives the same result. Whence again

Triangle Area =$\frac12$ [height][based]

The argument here might be given or repeated after a discussion of the distributive law in algebra, or after a discussion of how areas of larger regions may be obtained by totaling the area of subregions.

3 Triangle Area Formula Example

Areas of Triangles - height not over base

Triangle Area =$\frac12$ [height][based]

The argument in all or part here might be also given or repeated after a discussion of the distributive law in algebra, or after a discussion of and illustration of how area may be calculated by subtraction.

Formulas for Areas of Parallelograms and Trapezoids

Area Formulas for Parallelograms: Draw a diagram to imply that the area of parallelogram should be the same as the area of a rectangle with the same base length and height.

Formula for Area of a Trapezoid.

F. Inter-Related Volume Formuals

• 5 Box Volume Formula Example
  
• 9 Volume of Cone
  
• 10 Volume of Pyramid
  
• 11 Volume of Sphere
  
• 12 Cone Cylinder Sphere Lesson Idea
  
• 13 Naming Identifying Formulas with Words

Hands-On Experiments.

Student operational command of formulas may follow from two physical examples.

1. Observe physically that the volume or capacity of a cone is one third that of a circular cylinder with same height and based.

   A cone with the same base [or top] area as a cylinder has a third of the volume of the cylinder when both have the same height. To fill the cylinder to the brim or top using the cone, one has the fill the cone three times.

2. Observe physically 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. If the height of the cylinder and cone equals the diameter. radius R of the cylinder, 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.

   Alternative: 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 cylinder 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.

The foregoing shows how formula for the volume of a sphere can be related to formulas of volumes of cylinders and cones. In science labatories or with equipment borrowed from one, volume calculations for small solids may be checked by water displacement and overflow methods with the aid of graduated cylinders. Likewise, capacity calculations may verified by measuring with graduated cylinders the amount of water necessary to fill a container. Quantitative skills may be further developed by showing how volumes and capacities may be found by addition and differences of measureable volumes. The two physical examples implies empirical relations between formulas and shows how knowledge of one volume formula may confirm or imply others. In primary and secondary mathematics, the use of physical reasoning may reinforce skill and concept development. Students may be told that physical reasoning may suggest formulas, but pure mathematician have more confidence with formulas that are mathematically derived without the use of physical reasoning. The latter message will leave room for thought - if heard.

G. 3D Geometric Construction Exercise - Optional

Spatial Sense and its representation: 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.
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 Construction 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