Direct and Partial
Variations


Mathematics 10

by
Ray Mah
Rossignol School
Ille à la Crosse, Saskatchewan
1993

Introduction

This unit is taught upon completion of the unit on linear functions. Direct and partial variations are applications of linear functions.

Foundational Objective:
To use the knowledge of linear functions and equations to solve problems involving direct and partial variation.

Learning Objectives:
Curriculum Guide (p. 24)
19. To identify, describe, and interpret examples of direct variation in real world situations.
20. To solve proportions involving direct variation.
21. To solve problems involving direct variation.
22. To identify partial problems involving partial variation.
23. To solve problems involving partial variation.

See Curriculum Guide, p. 124-127 for instructional notes, examples and ideas for adaptive dimensions.

References:
Holtmath 10 c1987,
Addison-Wesley Mathematics 10, c1987
Math Matters 10, Nelson Canada, c1990 (Multi-text approach is recommended)
This is a Yes: Concept Attainment. Sheryl Mills. Saskatoon: Saskatchewan Professional Development Unit and Saskatchewan Instructional Development and Research Unit, 1991

Instructional Strategies:
(See Curriculum Guide, p. 248-250). In addition to lecturing, practice, and drill, the following instructional strategies are very suitable:

Assessment and Evaluation:
(See Curriculum Guide, p. 9-16 and p. 73-92)

It is recommended that a variety of methods is used. An excellent collection of templates for assessment and evaluation is included in the curriculum guide.

Objective 19:

To identify, describe, and interpret examples of direct variation in real world situations.

Lesson Plan:

1. Review of prerequisite skills
2. Develop working definitions of new terms
3. Work out some examples
4. Practice

1. Review of prerequisite skills:

2. Definitions:

(NOTE: Concept attainment works well here)

3. Worked example:

Wild rice retails at $5.00 per 500g bag. Construct a table of values relating cost and number of bags purchased. Graph the variation. Is it a direct variation? What is the constant of variation?

Graph of the variation

Table of Values

Bags
   0    1    2    3    4   
Cost $
   0    5    10   15   20

NOTE: Since the graph is a straight line starting at the origin, it is a direct variation. The constant of variation is 5. The equation of the line is y = 5x

Summary: The type of variation can be identified by the graph of the relation. The graph of a direct variation is a straight line and starts at the origin. The slope of the line is the constant of variation. The equation of a direct variation is of the form y = mx, more commonly written as y = kx.

4. Practice:

Objective 20:

To solve proportions involving direct variation.

Lesson Plan:

1. Review of prerequisite skills
2. Develop working definitions of new terms and symbols
3. Worked examples
4. Practice

1. Review of prerequisite skills

2. Develop working definitions of new terms

3. Worked Example 1:

The following table is a direct variation. Determine the constant of proportionality.

x y y/x
	1	
	5	
	1/5	
	2	
	10	
	2/10=1/5  

	The constant of proportionality 1/5.

Partially Worked Example 2:

Solution:

a. The amount of oxygen needed varies directly as the amount of propane present.
Let G = propane gas and P - propane gas.
G u P or G = kP

The oxygen to propane ratio is 5 to 1. We can set the proportion as 5:1 = G:10kg
or
form the equation 5/1 = G/10kg. Solving for G, we find that 50 kg of oxygen is needed.
(Ask the students to provide alternate methods, eg, table of values, graphically)

b. Assign part b, perhaps as group work.

4. Practice

Chemistry problems involving masses and moles of molecules are excellent for practice.

Objective 21:

To solve proportions involving direct variation.

Systematic problem solving techniques should be emphasized. It is important to identify the independent and dependent variables. Once the relating variables are identified the problem may be solved as direct variation or as a proportion. Some students may want to follow a flow chart such as the one below:

Practice questions are readily available in math and science texts. Ask each student to concoct a number of problems for the class to solve. Make sure that they are direct variations.

Objective 22:

To identify partial problems involving partial variation.

Lesson Plan:

1. Review of prerequisite skills
2. Develop working definitions of new terms
3. Work out some examples
4. Practice

1. Review of prerequisite skills

2. Develop working definitions of new terms

******************************
A question to ask:

In a direct variation, what is the effect on the second variable if the first is doubled?

Is the effect the same in a partial variation?

******************************

Examples of partial variations:

These are not partial variations:

3. Worked example:

Sales 0 1000 5000 10 000 15 000 20 000
Commission 0 20 100 200 300 400
Basic income 500 500 500 500 500 500
Total income 500 520 600 700 800 900

Extension:

4. Practice

Objective 23:

To solve problems involving partial variation.

(SUGGESTION: Post the flow chart on the wall for a few days.)

Worked example:
Hockey jerseys are priced at $50.00 each plus $1.50 per letter. How much will Jerry Desjarlais' jersey cost if his last name is sewn on the back?

Understand the problem:

Given: $50.00/jersey and $1.50/letter
Find: Cost of jersey with DESJARLAIS on it.

Form a variation equation:

Cost varies directly as number of letters
C u L

Form an algebraic equation:

C = kL
Add the constant term
C = kL + 50

Compute k and substitute:

k = cost per letter
k = $1.50
C = 1.50L + 50

Solve for y algebraically or graphically:

Algebraically
C = 1.50L + 50, L = number of letters in DESJARLAIS = 10
By substitution we obtain C = 1.50*10 + 50 = 15 + 50 = 60

Verify and answer the question:
Cost of letters = 10*1.50 = 15.00
Cost of jersey	          = 50.00
Total cost	          = 65.00

Practice questions


Reprinted with permission from: This is a Yes: Concept Attainment. Saskatoon. Saskatchewan Professional Development Unit and Saskatchewan Instruction Development and Research Unit.


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 :
Stewart Resources Centre,
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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