Multiple Representations

Translating Real-Life Situations into Narrative, Table, Formula, and Graph

CAPS Grade 10 Mathematical Literacy

This document explores the concept of Multiple Representations. It emphasizes translating a real-life situation into four formats: narrative, table, formula, and graph.

The Plumber Scenario

"A plumber charges a call-out fee of R200 and then R150 per hour for labour."

1. The Narrative

Example: "A plumber charges R200 call-out + R150 per hour."

Key Components:

  • Fixed Value: R200 (call-out fee)
  • Variable Rate: R150 (per hour)

2. The Table

Hours 0 1 2 3 4
Cost (R) 200 350 500 650 800

Calculations: 1h: 200+150=350, 2h: 350+150=500

3. The Formula

Cost = 200 + 150 × Hours

Where: 200 = fixed cost, 150 = cost per hour

Example (10 hours): 200 + 150×10 = R1700

4. The Graph

Points: (0,200), (1,350), (2,500), (3,650)

(0,200)
(1,350)
(2,500)
(3,650)

Straight line starting at R200 (y-intercept)
Slope: R150 per hour

Identify Components

3 Questions
1 What is the fixed cost (call-out fee)?
R150
R200
R350
R500
2 What is the variable rate per hour?
R150
R200
R250
R300
3 Total cost for 4 hours?
R600
R650
R700
R800
0/3

Why This Matters for Exams

A single exam question may require you to:

  • Identify variables from the narrative
  • Complete a table based on the information
  • Write a formula for the relationship
  • Draw or interpret the graph
  • Predict future values using the formula or graph

Multiple Representations Challenge

A mechanic charges R300 call-out fee and R200 per hour. Cost for 4 hours?
Total = Fixed + (Rate × Hours)

Interpreting the Graph

Steepness (Slope)

The steepness indicates the rate of change. A steeper line means a higher cost per hour.

In our example: Slope = R150 per hour

Y-Intercept

Where the line starts on the vertical axis represents the fixed cost.

In our example: Starts at R200 (call-out fee)

Common Mistake: Forgetting to start the graph at the fixed cost (y-intercept). If there is a call-out fee, the graph cannot start at zero.

Multiple Representations

4 Questions
1 Taxi: R25 base + R12/km. What is the formula?
Cost = 12 × km
Cost = 25 + 12 × km
Cost = 12 + 25 × km
Cost = 25 × km
2 Cost for 10 km using the taxi above?
R120
R145
R250
R370
3 A graph starts at R50 on the y-axis. This is the:
Rate of change
Fixed cost
Variable cost
Total cost
4 Table: Hours 0,1,2,3 | Cost: 100,200,300,400. Formula?
Cost = 100 × hours
Cost = 100 + 100 × hours
Cost = 100 + hours
Cost = 200 × hours
0/4

Practice: Painters

"A painter charges R400 call-out fee and R180 per hour."

Table (Hours 0-3):

Hours 0 1 2 3
Cost 400 580 760 940

Formula:

Cost = 400 + 180 × Hours

Final Assessment

8 Questions
1 Plumber: R250 call-out + R120/hour. Fixed cost?
R120
R250
R370
R130
2 Cost for 5 hours for the plumber above?
R600
R750
R850
R950
3 Graph starts at R150 on y-axis. This is:
Slope
Fixed cost
x-intercept
Origin
4 Table: Hours 0,1,2,3 | Cost: 50,80,110,140. Formula?
50×h
50+30×h
30×h
50×30×h
5 "R100 rental + R20 per day" - variable rate is:
R100
R20
R120
R200
6 A steeper graph means:
Lower rate
Higher rate
Fixed cost
Zero cost
7 Cost = 200 + 150×h. Cost for 6 hours?
R900
R1000
R1100
R1200
8 Point (0,200) represents:
Cost at 0 hours (fixed)
Cost at 200 hours
Hours at cost 0
The slope
0/8

Conclusion

Mastering Multiple Representations is a vital skill in Grade 10 Mathematical Literacy. By understanding how to move between narrative, table, formula, and graph, you gain deeper insight into real-life mathematical situations.