Mathematics 10

1993

S 105.10

A group of teachers who had piloted the course in 1992-93 were invited to a two and a half day workshop in August, 1993 at the STF. The teachers worked alone or in pairs to develop a plan for a section of the course.

Jim Beamer, University of Saskatchewan, and Lyle Markowski, Saskatchewan Education, Training and Employment, acted as resource persons for the workshop.

Throughout the sample units, the following symbols are used to refer to the Common Essential Learnings:

CCommunication

CCTCritical and Creative Thinking

ILIndependent Learning

NNumeracy

PSVSPersonal and Social Values and Skills

TLTechnological Literacy

Section One

Section Two: Properties of Polygons

Section Three

Section Four:The Pythagorean Theory

Section Five: Trigonometric Ratios

- To identify and apply common properties of triangles, special
quadrilaterals,
and n-gons. (10 05 01). Supported by learning objectives 1, 2, 3, 4, 8,
9, and 10.

- To apply the special quadrilaterals to real world situations. (10 05 02).
Supported by learning objectives 6 and 7.

- To develop an understanding of Pythagoras' Theorem, the primary
trigonometric
ratios, and their applications. (10 05 03). Supported by learning objectives
11 to 17.

- To define and illustrate terms related to the study of angles.

- To solve diagrammatic and word problems involving these angles.

** Note:** Activity 2 could be done before or after Activity 1
or left out. Activity 3 is intended as an adaptive dimension.

- Students may use the following activity sheet to review, define, and
illustrate angular terms. Following this, the students may be assigned
suitable questions involving these terms. (IL, N, CCT) Examples from Geometry
- Jurgensen 1985 edition could include:
**Page Numbers and suitable questions:**

- page 13 classroom exercises 1-38

- page 14 written exercises 1-38

- page 31 classroom exercises 1-18

- page 32 written exercises 1-32 challenge 35-38

- page 13 classroom exercises 1-38
- Students may be broken into groups of 3 or 4 in which they discuss occupations involving a knowledge of angles (see pages 120-121 of mathematical curriculum guide). Presentations may be as suggested or a guest speaker may be invited into class to discuss how his/her occupation involves angles or to demonstrate to students any special equipment that occupation may use to measure angles. (PSVS, C, CCT, N)

- informal questioning

- anecdotal records

## Term Illustration Definition

* angle

* acute angle

* right angle

* obtuse angle

* straight angle

* reflex angle

## Term Illustration Definition

* complementary angles

* supplementary angles

* adjacent angles

* vertical angles

* congruent angles

* central angle of a regular polygon

- To define and illustrate terms related to polygons.

- To classify quadrilaterals and polygons.

- To construct types of parallelograms.

- To state and apply properties of parallelograms.

- The students may be grouped into 3 or 4 for the following concept mapping
activity. Each group will require a large sheet of bristol board and an
envelope containing two or three of each of the following polygons of varying
sizes or shapes: triangles (scalene, isosceles, equilateral, acute, right,
obtuse); quadrilaterals (with no sides congruent both convex and non-convex,
trapezoids, isosceles trapezoids, parallelograms, rectangles, rhombuses
and squares); pentagons, hexagons, and octagons (at least one of each of
these should be regular).
Students would be required to set up a classification system for all above shapes, using the properties of the shapes (see A-2.03 curriculum guide instructional notes page 122). When completed, each group should be requested to explain their classification system. Geometric properties may be summarized on board - consistencies and inconsistencies of each system may be discussed. (CCT, C ,PSVS,)

- Students need to study the properties of quadrilaterals and more
specifically
parallelograms. Students may complete a chart such as the following or
page 171 - Jurgensen. Encourage students to draw sketches to experiment
with each characteristic, or use MIRA to test some. (IL, TL)

Alternately use the chart on page 171, Jurgensen 1985 as an assessment chart.

- The teacher may wish to use the following Venn diagram to help explain
the properties of quadrilaterals from general to specific.

- Student discussions could be used to explore methods of informally
constructing
parallelograms, rectangles, rhombuses, and squares. Paper folding and MIRA
are options as suggested in the curriculum guide, page 122, as is the mechanical
method using straight edge, rulers, and set squares. If formal constructions
were covered previously (see page 118 of curriculum guide), then this may
also be used. Students are encouraged to come up with other methods. (C,
PSVS, CCT)

- Students may be encouraged to search for examples of polygons in nature.
For example, the hexagons of honeycombs, pentagons on shells of turtles,
polygons on body of giraffes, etc. Polygons are also used in construction,
machinery, jewelry, etc. Presentations may be made in the form of posters
where diagrams may be hand-drawn or pictures from magazines or as oral
presentations of applications of polygons in specific occupations.
Alternatively,
students may wish to create picture-art in which all elements of the diagram
are made from polygons or compass. Students may be encouraged to find out
how to construct kaleidoscopes. See media house catalogue for available
videos.

- On-going student activity

- Activity or assessment sheet following

- Homework

- Presentation Concept mapping

- group assessment

- presentation assessment

- group assessment

PROPERTIES OF POLYGONS WORKSHEET

## Term Illustration Properties and/or Characteristics

* polygon

* regular polygon

* convex polygon

* non-convex polygon

* triangle

* acute triangle

* right triangle

* scalene triangle

* obtuse triangle

## Definition and/or Term Illustration Properties and/or Characteristics

* isosceles triangle

* equilateral triangle

* quadrilateral

* parallelogram

* rectangle

* rhombus

* square

* trapezoid

* isosceles polygon

Back to the table of contents

## SECTION THREE

## Objectives:

(7, 8, 9, 10 from Curriculum Guide)

- To determine the sum of the measures of the interior and exterior
angles of a convex polygon of n sides.

- To determine the measure of a central angle in a regular n-gon.

- To determine the measures of the interior and exterior angles of regular
n-gons.

- To determine the number of diagonals in a polygon of n sides from each
vertex.

- Protractor
- straight edge

- Students may work in groups of 2 to complete exercises 1 and 2 (attached),
and prepare a report to the class. Alternatively, the teacher may have
groups meet and debate their results before presentations are made to the
larger body. (PSVS, C, CCT, N)

- Exercise 3 may then be completed in groups or as the teacher feels
appropriate.
Students may require some help to fill in the row dealing with n-gons.
(Exercise 3) With guided discussion, some students may develop the sum
of interior angle formula. (IL, CCT, N, C)

- Students will need more practice in questions related to above. Geometry
Jurgensen 1985, pages 80-84. (IL, N)

- Presentation

- Group Assessment

The teacher may choose to administer a quiz or test at this point of this unit.

Draw the central angle -C and one external angle -E for each of the following
regular polygons. Measure these and complete the table below. Write a short
conclusion at the bottom of the page.

Conclusion:

m "greater than"C m "greater than"E

A

B

C

D

E

Measure one interior angle (a) and one exterior angle (b) of each of the following regular polygons.

Since all the interior angles of a regular polygon are equal and all the exterior angles of a regular polygon are equals.

Find the sum of interior angles of:

A)

B)

C)

D)

E)

Find the sum of exterior angles of:

A)

B)

C)

D)

E)

The following information can be arranged into a table format.

Figure out the following information when looking at polygons with 3, 4, 5, 5, 7, 8, 9, 10 sides. Number of Diagonals (from same vertex) Number of Triangles Sum of Interior Angles measurment of int. <'s of regular polygon measurement of ext. <'s of regular polygon m. of central Angle sum of all the exterior angles

- The students may require some practice in determining the square root
of teacher specified numbers using the calculator. (TL, N, IL)

- The teacher may introduce the Pythagorean Theorem giving several examples
on the board by altering the position of the right angle in the diagram.
The teacher may have students estimate the value of the sides before determining
the values using calculators. (CC, IL, N) (See sample pages on Pythagorean
Theorem attached). Additional questions may be found in:

- Addison-Wesley, Math 10 (1987) - pages 380-391
- Principles and Process 9(1988) - pages 106-107
- Geometry Jurgensen (1985) - pages 252-259
- selective Math Is 4 - pages 37-42
- Principles and Process 10 - pages 56-57

- Quiz or Test

The Pythagorean theorem states:

In a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides.

Examples: (On Board)

(Diagrams are not drawn to scale)

b) Which of the following are not right triangles? Give a reason.

Knowledge of right triangle and Pythagorean Theorem

3 periods

- The students may require practice in labeling the right angle, triangles
in terms of hypotenuse, side adjacent and side opposite with reference
to one of the two acute angles. (IL, CCT)

Example: *ABC

In both diagrams the hypotenuse remains the same, however, the sides adjacent and opposite are interchanged when the reference angle is changed.

- Students should become familiar with determining the sine, cosine,
and tangent values by use of calculator (or tables if necessary). (TL,
N, IL)

- A more formal introduction of the trigonometric ratios may be used,
in terms of hypotenuse, sides adjacent and opposite. A sample introduction
on trigonometric ratios is included. Examples should lead into word problems
and applications of these trigonometric ratios. Angles of evaluation and
depression should be included. (CCT, C, N, IL, PSVS, TL)

Sample exercises may be found:

- Holt Math 10 (1987), pages 332-340
- Holt, Teachers Resource Manual (1987) - Master 13-1 and 13-2
- Math Is 4, pages 332-340
- Principles and Process 10 (1992), pages 462-480

- The teacher may use the concepts of similar triangles when introducing
trigonometric ratios.

Comprehensive Test on unit of Angles and Polygons may be given.

- Sin function Sin A =
__opposite A__

hypotenuse

- Cos function Cos A =
__side adjacent A__

hypotenuse

- Tan function Tan A =
__side opposite__

side adjacent

In *PQR let us find Sin P, Cos P, Tan P

- Sin P =
__opposite A__=__3__hypotenuse 5

- Cos P =
__side adjacent A__=__4__

hypotenuse 5

- Tan P =
__side opposite__=__3__side adjacent 4

In *ABC

Sin C =

Cos C =

Tan C =

- Geometry, Jurgensen, et all, 1985, Houghton Mifflin.

- Geometry, Clemens, et al, 1984, Addison-Wesley Publishing Co.

- Principles and Process 9, Ebos, et al, 1988, Nelson.

- Principles and Process 10, Ebos, et al, 1992, Nelson.

- Holt Math 10, Bye, Dale, et al, 1987, Holt, Rinehart and Winston.

- Holt Math 10, Teachers Resource Manual, 1987, Holt, Rinehart and
Winston.

- Math Is 4, Ebos & Tuck, 1979, Nelson.

- Math 10, Addison-Wesley (1987).

This unit comes from the The Stewart Resources Centre which
provides
library resources and teacher-prepared materials for teachers in Saskatchewan.
To borrow materials or obtain a free catalogue listing unit and lesson
plans contact :

Sask. Teachers' Federation,

2317 Arlington Avenue,

Saskatoon, SK S7J 2H8;

phone 306-373-1660; fax 306-374-1122,

e-mail src@stf.sk.ca.

http://www.stf.sk.ca/

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