# Circuit Problems

Purpose

The purpose of this unit is to engage the student in applying their knowledge and skills of algebra to solve problems within the context of electrical circuits.

Achievement Objectives
NA5-7: Form and solve linear and simple quadratic equations.
NA5-9: Relate tables, graphs, and equations to linear and simple quadratic relationships found in number and spatial patterns.
Specific Learning Outcomes

Students develop their skills and knowledge on the mathematics learning progression Using Symbols and Expressions to Think Mathematically, in the science learning area, the physical world.

Description of Mathematics

Students will generalise the properties of current electricity and use their skills and knowledge of algebra to solve circuit problems.

Note to teachers: This unit includes practical activities that can be carried out using standard science lab equipment, or with wires, multi-meters, batteries and component resistors that can be purchased at electronics stores. Whether multi-meters or ammeters and voltmeters are used, make sure that any current reading is carried out in series, and voltage reading in parallel. Ammeters (and multi-meters set to a current reading) are easily damaged if placed incorrectly. Check that ammeters and voltmeters are connected correctly with a very short momentary connection (the touch test) to see that the needle moves clockwise and not past the end of the scale.

Activity

### Structure

This cross-curricular, context based unit has been built within a framework that has been developed with input from teachers across the curriculum to deliver the mathematics learning area, while meeting the demands of differentiated student-centred learning. The unit has been designed around a six session focus on an aspect of mathematics that is relevant to the integrating curriculum area concerned. For successful delivery of mathematics across the curriculum, the context should be meaningful for the students. With student interest engaged, the mathematical challenges often seem more approachable than when presented in isolation.

The first session is an introductory activity that is aimed to spark the imagination of students, to introduce the need for a particular idea or technique in mathematics that would enable them to explore deeper into that context. It is expected that rich discussion may be had around the context and around the nature of the mathematics involved.

The following five sessions are each based around a model of student-centred differentiated learning.

• There is a starting problem to allow students to settle into the session and to focus on the mathematics within the chosen context. These starting problems might take students around ten minutes to attempt and/or to solve, in groups, pairs or individually.
• It is then expected that the teacher will gather the students together to review the problem and to discuss ideas, issues and mathematical techniques that they noticed during the process. It may be helpful to summarise key outcomes of the discussion at this point.
• The remaining group of activities are designed for differentiating on the basis of individual learning needs. Some students may have managed the focus activity easily and be ready to attempt the reinforcing ideas or even the extending ideas activity straight away. These could be attempted individually or in groups or pairs, depending on students’ readiness for the activity concerned. The students remaining with the teacher could begin to work through the building ideas activity together, peeling off to complete this activity and/or to attempt the reinforcing ideas activity when they feel they have ‘got it’.
• It is expected that once all the students have peeled off into independent or group work of the appropriate selection of buildingreinforcing and extending activities, the teacher is freed up to check back with the ‘early peelers’ and to circulate as needed.

### Introductory activity

(to motivate students towards the context/integrated learning area and to inform teachers of each student’s location on the learning progressions):

The standard unit of power, the rate at which energy is transformed, is the watt. One watt is equivalent to one joule per second. Energy companies bill electrical usage in units of kilowatt hours (kWh). The average annual household electricity usage is 7600 kWh. How many joules of electrical energy does the average daily household use?

In this activity, the teacher(s) will be able to locate their students on the learning progression Using Symbols and Expressions to Think Mathematically by observing how students use the given units of measurement to inform them of the mathematical relationship between quantities. This activity integrates mathematical skills and knowledge with the science learning area. In the activities that follow, students are encouraged to explore the relationships between currents in and voltages  across different parts of a circuit.

Mathematical discussion that should follow this activity involve:
Household usage accounts for 13% of the total electricity usage in the New Zealand.  What is the total annual electricity usage in the New Zealand?
Why might energy companies measure energy in kWh rather than J?
Why might energy companies measure power in kW or GW rather than W or Js-1?

### Session One

Focusing on problem solving

Focus activity

When the Smith family was at work and school for six and a half hours, they left enough lights and appliances running to draw a current of 30 A. An ampere, A, measures the amount of charge running through the meter every second. The power supplied to the house is at 230 V. A volt, V, is a measure of the amount of electrical energy transformed per unit charge. How much electrical energy did the Smith family’s house use in their absence that day?

Discussion arising from activity:

• Can we use the definitions of current and voltage to find a way of measuring the current and voltage to find the power used?
• How can we work out the electrical energy used given electrical power used and the time it is used over?
• Can we use the definitions of current and voltage to find a way of measuring the energy used in a given time?

Building ideas

Power, measured in W, is calculated from the rule P = VI:

1. What do the symbols, P, V and I stand for?
2. Use this rule to find the power output of a heater that operates on 230 V and draws a current of 7.8 A.
3. Use this rule to find the voltage of a 2000W heater that draws a current of 10 A.

*Check that all your solutions are given with the appropriate units.

Reinforcing ideas

Power, measured in W, is the rate at which energy is used. Power is the product of the voltage across and current through an electric supply electric circuit or any appliance using electricity.

1. Write this rule as an algebraic equation using P for power, V for voltage and I for current.
2. Use your rule to find the power output of a torch that uses a single AA ‘battery’ of 1.5 V and draws a current of 0.2 A.
3. Use your rule to find the voltage of a 2000W heater that draws a current of 8.7 A.
4. What is the total electrical energy transformed by the heater if it is switched on for 2 hours?

*Check that all your solutions are given with the appropriate units.

Extending ideas

Find the difference in energy usage of an oil heater that runs for 4 hours, drawing a current of 8.5 A on mains electricity over a 425 W panel heater that is left running for 12 hours.
*Show all your working and check that solutions for all of your calculations are given with the appropriate units.

### Session Two

Focusing on using algebra to generalise the results of a practical investigation.

Focus activity

Electrical current, in amperes (A) is defined as the rate of flow of charge. It is measured with an ammeter placed in series in a circuit. Look at the circuit below: 1. Identify the ammeter in this circuit.
2. If the ammeter was used to measure the current in the resistor R1, where would it be placed?
3. If the ammeter was used to measure the current in the resistor R2, where would it be placed?

Discussion arising from activity:

• Why are ammeters placed in series in the circuit?
• What do you expect the relationship between the readings on the ammeter in these three positions to be? (ie the same, different, related in some way – and if so why?)

Building ideas 1. Build the circuit above, measuring the current supplied (A1) by three 1.5 cells connected in series, to two 50 Ω resistors connected in parallel.
2. Measure the current through each of the resistors (A2 and A3).
3. Describe the relationship between A2 and A3

Reinforcing ideas Current is the rate of flow of charge. The two resistors in the circuit above are the same size.

1. If the current in A1 is 5 A, what will the readings on each of A2 and A3 be?
2. If the current in A1 is 0.2 A, what will the readings on each of A2 and A3 be?
3. If the reading on A2 is 2 A, what will the readings on each of A1 and A3 be?
4. If the reading on A2 is 0.15 A, what will the readings on each of A1 and A3 be?
Extending ideas 1. Build the above, using three 1.5 cells connected in series, to two resistors chosen from a selection of 50 Ω and 100 Ω resistors and connected in parallel so that A2 = 2A3.
2. Copy and label the circuit diagram with the size of the resistors used and the currents through each.
3. Predict the total current supplied (A1 ) by three 1.5 cells.
4. Measure the total current supplied (A1 ) by three 1.5 cells.
5. Explain the similarities/differences between the currents you found for activities 3 and 4.

### Session Three

Focusing on using graphical techniques to identify and describe a linear relationship found in a practical investigation.

Focus activity

The current in a circuit is controlled by the size of the load, the resistance of the components in the circuit. Resistance, R (in Ω) is the gradient of the linear relationship; voltage against current. Find the resistance of the circuit described by the graph below. Discussion arising from activity:

• How did you ensure accuracy when you found the gradient of this graph?
• Give the unit Ω in terms of volts and amperes.
• To gather the data for this graph, a variable power supply was used. Give suitable instructions for carrying out this experiment.
Building ideas
1. Set up the following circuit. 2. Vary the voltage to complete the table below.
 V (V)0.5 1.0 1.5 2.0 2.5 I (A)
3. Construct the graph of of voltage against current.
4. Describe the relationship between voltage and current.

Reinforcing ideas

1. Set up the following circuit, to measure the data necessary to graph voltage against current.
2. Vary the voltage to obtain at least four data points. 1. Find the gradient of the line of best fit from your graph of voltage against current.

Extending ideas

1. Investigate the circuit shown in the diagram below to find the relationship of voltage in terms of current. 2. Write the general rule for this relationship using the symbols V, I and R.
3. Find the power output of the resistor in this circuit.

### Session Four

Focusing on using algebraic techniques to solve problems involving voltage, current and resistance.

Focus activity

The relationship between current, voltage and resistance of an electrical component, or of a circuit, is known as Ohm’s Law. It is:
V = I R
Use this rule to find the current drawn by a 50 Ω kettle connected to mains electricity (230 V).

Discussion arising from activity:

• What is the power output of this kettle?
• Explain the energy transformation(s) when the kettle is switched on.

Building ideas

Use Ohm’s Law, V = IR, to solve the following problems:

1. Find the voltage across a 60 Ω resistor which is drawing a 2.5 A current.
2. Find the voltage across a 60 Ω resistor which is drawing a 7.5 A current.
3. Find the voltage across a 120 Ω resistor which is drawing a 2.5 A current.
4. Find the voltage across a 150 Ω resistor which is drawing a 2.5 A current.

*Check that all your solutions are given with the appropriate units.

Reinforcing ideas

Use Ohm’s Law, V = IR, to solve the following problems:

1. Find the voltage across a 500 Ω resistor which is drawing a 7.5 A current.
2. Find the current drawn by a 500 Ω resistor connected to a 12 V supply.
3. Find the current drawn by a 500 Ω resistor connected to a 12 V supply.
4. What is the resistance of a component connected to a 9.0 V supply that draws a 3 A current?

*Check that all your solutions are given with the appropriate units.

Extending ideas

Use Ohm’s Law, V = IR, to solve the following problems:

1. Find the voltage across a 500 Ω resistor which is drawing a 7.5 mA current.
2. Find the current drawn by a 500 Ω resistor connected to a 3000 mV supply.
3. Find the current drawn by a 0.75 kΩ resistor connected to a 12 V supply.
4. What is the resistance of a component connected to a 9.0 V supply that draws a 300 mA current?

*Check that all your solutions are given with the appropriate units.

### Session Five

Focusing on using algebraic techniques to solve problems involving power, voltage, current, and resistance.

Focus activity

Use the rules V = IR and P = VI, to find the relationship between;

1. Power, voltage and resistance.
2. Power, current and resistance.

Discussion arising from activity:

• Describe how increasing the load (resistance) of a mains electrical circuit would affect the current drawn.
• Describe how increasing the load (resistance) of a mains electrical circuit would affect the power output.

Building ideas

A heater with resistance of 1500 Ω is connected to a 230 V supply.

1. Use Ohm’s law, V = IR, to find the current drawn by the heater.
2. Find the power output of the heater.
3. Find the energy output of the heater over one hour.

*Check that all your solutions are given with the appropriate units.

Reinforcing ideas

1. Find the current drawn by a 1000 Ω resistor connected to a 230 V supply.
2. Find the power output of a 1000 Ω resistor connected to a 230 V supply.
3. Find the current drawn by a 2000 W heater connected to a 230 V supply.
4. Find the resistance of a 2000 W heater connected to a 230 V supply.

*Check that all your solutions are given with the appropriate units.

Extending ideas

Use the relationships you have derived to:

1. Find the power output of a 2.5 kΩ resistor connected to a 230 V supply.
2. Find the resistance of an 1800 W heater connected to a 230 V supply.
3. If an 1800 Ω heater is replaced by a 600 Ω heat pump, find the relative energy savings (in kWh) over one hour of use connected to mains (230 V).

*Check that all your solutions are given with the appropriate units.