STAGE ONE: DESIRED RESULTS

ESTABLISHED GOALS: The students should leave the classroom with clear understanding of Series and Parallel circuits in relation to total resistance – i.e. their concepts, mathematical derivation and mathematical relationships of multiplication, division, squares and square roots and the idea behind reciprocals. The state standards of Vermont DoE are some previous (7.8.-20 & -21) with new ones that are more stressed throughout this lesson. All of them are listed below:

• Standard 7.8: Functions and Algebra Concepts

MHS: 19 Solves and models problems by formulating, extending, or generalizing linear and common nonlinear functions/relations.)

And makes connections among representations of functions/relations (equations, tables, graphs, symbolic notation, text).

MHS: 20 Demonstrates conceptual understanding of linear relationships and linear and nonlinear functions (including f(x) = ax2, f(x) = ax3, absolute value function, exponential growth) through analysis of intercepts, domain, range and constant and variable rates of change in mathematical and contextual situations.

MHS: 21 Demonstrates conceptual understanding of algebraic expressions by evaluating, simplifying, or writing algebraic expressions; and writes equivalent forms of algebraic expressions or formulas (d = rt—> r = d/t or solves a multivariable equation or formula for one variable in terms of the others).

• Standard 7.10: Mathematical Problem Solving and Reasoning—Applications

MHS: 30 Demonstrate understanding of mathematical problem solving[2] and communication by:[4]

• Approach and Reasoning—The strategies and skills used to solve the problem, and the reasoning that supports the approach;

• Execution—The answer and the mathematical work that supports it;

• Observations and Extensions—Demonstration of observation, connections, application, extensions, and generalizations;

• Mathematical Communication—The use of mathematical vocabulary and representation to communicate the solution; and

• Presentation—Effective communication of how the problem was solved, and of the reasoning used.

[2] Problem-solving situations are mathematical problems that reflect the levels of mathematic in the Grade Level Expectations.

[4] See Vermont High School Level Mathematics Portfolio Scoring Guide for additional information.

UNDERSTANDINGS:

.

–       Series circuits have resistors lined up one after the other one

-they have the same Voltage “V” (generated by a battery) and the same Current “I” flowing through all of them

-then the total Resistance “R” in Series circuits is just R=V/I

-but all partial resistors with R₁,R₂,R₃…R_n

are in a line that can be “squeezed” into one point with still the same I and V

-thence for R we sum up (R=R₁+R₂+R₃+…+R_n)

–       Parallel circuits have resistors positioned parallel, positioned parallel to one another

-although they have the same V, the I “flow” is separated into independent ones, each going through one of the parallel resistors

-thus the total Current is just the sum:

I = V/R₁+V/R₂+V/R₃+…+V/R_n = V×(1/R₁+1/R₂+1/R₃+…+1/R_n) = V×R where R is the total Resistance

-but then we can divide (“cancel”) both sides by V and we get 1/R=1/R₁+1/R₂+1/R₃+…+1/R_n

-further simplifying (by using the idea of reciprocals) we get the total Resistance:

R=1/(1/R₁+1/R₂+1/R₃+…+1/R_n)

ESSENTIAL QUESTIONS:

.

What are the different ways of combining more resistors in a circuit?

How would we simplify them to a few basic kinds?

And how can the Current flow – what are the paths it can travel through?

How would we try to calculate the total Resistance?

And why would we want to do it?

i.e.: again, here the teacher should make it clear that the general application for this is when electricians set up new circuits (e.g. building a new house); and at the same time, the teacher should stress the fact that understanding of this lesson is crucial for further development in this unit (bulb interchanging)

What kind of a relationship from the Ohm´s Law are we going to be looking at in order to actually calculate the total R?

(i.e.: I, V, R relationship, yet thinking without the Power “P”)

Is the Voltage same all through the circuit?

Is the Current same all through the circuit?

If not, how is it split across all resistors (the same way or with a certain ratio)?

How could we find a general expression for R?

KNOWLEDGE:

NB: the first three are previously covered and now applied to our topic

.

-Ohm´s Law (information from the page 19, or 57, in the resource1 #3)

-safety precautions for electrical circuits

-Electromotive Force “EMF” (the force causing the flow of electrons and ions / the current flow) & batteries (electrochemical cells converting stored chemical energy into electrical) are plugged in a circuit, thus generating constant V (same magnitude and same polarities)

-resistance in Series circuits is summed up

-resistance in Parallel circuits is calculated through a reciprocal sum formula (reciprocal sum of reciprocals)

SKILLS:

.

By the end of this lesson, students should clearly understand why we calculate total R differently for Series and Parallel circuits. They should be confident about identifying, distinguishing and calculating properties for such circuits

Also, students should have gained fair skills of multiplication, division, use of fractions and composite numbers, cancelation rule, simplification of an equation by taking its reciprocals.

As mentioned in the Lesson Plan #1, most of these concepts are not new to students and that is why we can confidently cover all of them just in one lesson. The concepts are being “recycled” for our application in circuits. For most of the students, the only new idea will be “swapping an equating upside-down” (reciprocals) in order to get the value for the denominator variable.

COMMON MISCONCEPTIONS:

.

While revising the safety precautions during the first part of the first classroom meeting, it is crucial to stress the common misconception that Voltage kills – Voltage does not kill at all but the Current it generates does. 0.1-0.2A is enough of I for an injury and 0.5A is usually fatal.

Also, some students may try to memorize incorrectly the total R as the sum of reciprocals (1/R₁+1/R₂+1/R₃+…+1/R_n) because it looks more intuitive. It is important to tell the students that they are almost right but that way they would have calculated only 1/R. Explain them why the correct formula is reciprocal sum of reciprocals [1/(1/R₁+1/R₂+1/R₃+…+1/R_n)]

ESSENTIAL VOCABULARY:

.

– prior vocabulary:

voltage

current

resistance, resistor

power

Ohm´s Law

battery

EMF

circuit

– circuits: series & parallel

– sum & reciprocals

STAGE TWO: ASSESSMENT EVIDENCE

.

-solving classroom examples (e.g. some of the questions 1, 2, 14, 16, 18, 31, 36 on the resource2 #1 from http://www.allaboutcircuits.com/pdf/worksheets/dc_sp.pdf)

NB: the resource worksheet contains many good questions that are out of the knowledge for our  class and therefore I selected only a few relevant examples

-answering homework questions (e.g. some of the questions 3, 5, 6, 7, 13, 17, 19, 23, 37 on the same resource2 #1)

-answering true false statements and providing their reasoning

-calculating with reciprocals for both classroom work and homework (e.g. some of the questions 20, 21, 30, 32 from the resource2 #1)

-participating in the final quiz requiring both mathematical skills and contextual reading skills

(again, see the resource2 #1 for the last and most comprehensive questions 38 and 40)

.

-ongoing classroom evaluation based on student reactions, questions and answers

-participation and reactions of students to brainstorming discussion on the beginning of the second classroom meeting (questions would be for example: “Do you think holiday lights are an example of parallel or series bulbs in a circuit?”, “Do you think the bulbs in the parallel circuit or the series circuit will burn brighter?”, “If you remove a bulb in your parallel circuit, with the other bulb(s) still light?”, “If you remove a bulb in your series circuit, with the other bulb(s) still light?” – retrieved from the resource2 #2, page 8 )

-students start the last (third) classroom meeting with a quick write up of the general R formula for Series and Parallel circuits (then, the teacher would be able to quickly set back on track those students who might be confused about the different concepts)

-solving the homework problem created by the class itself on the end of the second classroom setting: students stand up and arbitrarily create a circuit by holding their hands (either in series or parallel way) so that once they are done, everybody shares the month of his/her birthday that would actually represent the resistance value

.

Can students easily identify and see differences among series and parallel circuits (to determine these relations of resistors combined in a circuit, I recommend the question 4 on the resource # 1)?

Have the students shown any progress and development of their ideas?

Are the students doing their tasks responsibly and actively participating in the class?

Do the students make calculations intuitively without trial-error approach, i.e. trying to calculate either through division or multiplication and guessing more likely answer (thence, some numerical examples should include numbers too big or too small – to uncommon – to be observed in the natural world)?

Are the students able to solve more complex tasks such as the pointed questions from the resource2 #1 numbered 20+?

Do students have fun, enjoy the class and are they engaged?

STAGE THREE:  LEARNING PLAN

(LESSON SEQUENCE + LEARNING ACTIVITIES, “WHERETO”)

.

This lesson is placed towards the second half of our unit. Therefore, there has been a fair amount of information covered by this moment. Before starting to synthesize the ideas from the Ohm´s law and electrical circuits into one application, we would need to revise these main ideas. I would ask the class what we know about current, voltage and resistance at first. As they would react promptly (hopefully), I would write a few notes on the blackboard. If nobody would respond, I would try to restate the questions for more specific ones, i.e. what are the signs and units for… After they would also comment on the mathematical relationship among I, V, R, we would quickly do the same for the power as well. We should end up with something like this after a few (circa 5) minutes:

Also, I would recall an example from the Ohm´s Law lesson when we knew R and V and wanted to get P. In that case, we replaced I=V/R in P=VI and simply calculated P=V²/R. The students would be shown a useful application for knowing R and V.

Afterwards, I would ask them about what they remember from the electrical circuits’ lesson. Hopefully, I would hear a few ideas of continuous unbroken path, source of energy and load/device needed to be powered. Again, if they would not seem to understand what the main ideas from that lesson were, I would reformulate the question for “What were the three conditions for an electrical circuit?” I would not start with this less open-ended question though since I would like them to point out the safety precautions as well. It is crucial to stress that Voltage does not kill but the Current it generates does. Students should remember that 0.1-0.2A is enough of I for an injury and 0.5A is usually fatal.

Now, recall that students were (so far) thinking of only one resistor and they were building simple circuits. This lesson should help them to expand that knowledge and apply what they know in theory from the Ohm´s Law. For the next 10 minutes or so, I would ask the students questions about possibilities to connect more resistors into one circuit (see the first six Essential Questions in the Stage 1). From this brainstorming, I hope to get some general understanding what series and parallel circuits might be.

Then we would move to a traditional lecture-style part of our classroom meeting. These twenty minutes would be spent with thorough explanation of mathematics behind series and parallel circuits. The teacher should basically go through all the points outlined in the Understandings section (Stage 1), while constantly checking on students´ comprehension. Looking into their eyes and trying to see confusion are only some actions to take. Teachers should also allow students to interrupt their lecture at any time of struggle. Also, a great metaphor I would certainly use to help with thinking about reciprocals would be: “Alex, there is a light right above your head. And Joanna, do you see that fan right above your head? Well, if I turn my head upside down (or if I actually stand on my hands), it looks like the light is under Alex´s head now. But at the same time, the fan is under Joanna´s head too. Isn´t that great? No matter what way I look at it, the relations among these objects are kept the same. Either both the light and the fan are above, or they are both below. So if you have fractions, you can turn them around as long as you turn the both sides at the same time. I mean, a/b=c/d is the same as saying b/a=d/c. So (1/R=some stuff) is (R=1/that stuff).”

The last 40 minutes would be spent with solving some simple non-numerical classroom examples, starting general diagrams of series and parallel circuits:

__________________________________________________________

and continuing through questions 1, 2, 14 from the resource #1. The question 14 is to be left for the end of this classroom meeting because it suddenly introduces the other order of resistor (which is actually the most common among all real examples). There is a combination of series and parallel circuits at the same time. Homework problems would be assigned as well (questions 3, 5, 6, 7 from the resource #1).

.

The first twenty minutes of this classroom meeting are allocated for students to check on any problems from the homework set. If some of the homework questions would be non-problematic, we would not necessarily go through them. I would also walk around the classroom and see which students seem to have any confusion about the topic. Again, I would try to classify them into three groups (firstly mentioned in the previous Lesson Plan): “word-to-word memorizers”, “conceptual-realizers” and “misunderstood-thinkers”. My aim by the end of this lesson unit would be to have as many students as possible in the middle group. For that, I would personalize the solution of homework problems for problematic students by including them in a dialogue with me. I would try to make them finding solutions instead of me solving something in front of them.

Afterwards, we would pick up on the idea where we ended the last time, a combination of a series-parallel circuit. We would have seen in the homework that sometimes it was easier to think of resistors R₁, R₂, R₃, etc. as some numbers. Thus, we expand our general conceptual understanding  of resistors and include numbers too (firstly having simple examples such as the one below requiring only fractions, later including computations with calculators too):

It is important to point out the simplifying method for calculating the total R – students should identify the whole circuit as a union of some “sub-circuits” which are either series or parallel. They can be dealt individually at first. On the end, one would always end up with a simple series or parallel circuit. For example, the graphic example above can be looked at as R₁ & R₂ being one parallel circuit R_A, and similarly R₂ & R₃ being another parallel circuit R_B. Then both R_A & R_B are actually just a simple series circuit.

And of course, we would continue with exemplary solutions for classroom problems. I recommend continuing with the questions 16, 18, 21 from the resource2 #1 as they follow our intellectual progress the best way. Notice that the last one introduces an integration of our well-known Ohm´s Law and lets us to calculate other variables besides resistance. I suggest that the students would work on these problems in groups of three. Then they could help each other to understand some concepts they might still struggle with, and they would enjoy learning in a positive environment.

In order to keep students engaged and make their classroom experience enjoyable, I believe that it is worthy to spend the last 15 minutes with a personal “get-to-know-each-other-a-little-better” game. It will have been already a few weeks these high school freshmen were in a class together and thus I believe it is appropriate to make them stand up and arbitrarily create a circuit by holding their hands. Students should do so either in series or parallel way so that once they are done, some form of a circuit is created. However, try to keep it not too complicated as the ordering would be used for something important in a minute. We would have known each other names by that time for sure, but I am positive that we would have no idea about our birthdays. Since some the students may consider such information inappropriate, only the month of one´s birthday would be crucial. Yes, we would say the month out loud one by one. Of course, if someone chooses to share a day as well, do not stop him/her. While everybody shares the month of his/her birthday and something else he/she wants too (so that I have more time), I would note down a diagram of all students and their numbers represented by the months of their birthdays (January as 1, February as 2 and so on). Naturally, I would be included in the circuit as well so that I become a part of our social classroom community on the same level. But in order to maintain an authoritative position, I would be the one guiding them through the game and keeping discipline.

On the end, this diagram with numbers for each of the students (and the teacher) would become a homework assignment with a simple task: “Find out the total resistance of our circuit”. Also, some other questions would be included: 17, 19, 20 from the resource2 #1.

.

Naturally, we would start this classroom meeting by collecting the homework. I would ask the students to raise their hand if they think they correctly solved the homework assignment we made on the end of the previous session (by creating a human circuit). Then I would present detailed solution for our homework problem and try to see how many students raise their hand this time. If too many of them drop down, it means that there is a problem in the class with breaking down the circuit into more simple parts. In that case, more attention and slower explanation would be necessary. We would also go through the homework problems and since they were about the same topic, I could start re-teaching the class using those examples. I suppose that all should not take more than 25 minutes.

Then we would move onto the last step of our ladder – we would try to compute the most difficult question I dare to give the ninth graders at this moment. It would involve five resistors and it could be question 31 from the resource2 #1. Also, I want the students to have not only mathematical skills but also contextual understanding of what is going on (e.g. question 36: “What will happen to each resistor’s voltage and current in a circuit if a resistor R₁ fails to open?”).

This class would be our revision class as well since I want to prepare the students for an exam following. Hence the last third should be spent revising the material from the previous classroom meetings, including both theory and computation. Students should have no problems by now to apply their knowledge and solve questions 13, 23, 32 from the resource2 #1. I would make sure that students try to work through the questions in groups of three, while I would walk around the classroom to help them on individual basis. After approximately 20 minutes, we would find solutions together. I would appoint a student for each question (or it´s part) to read his/her answer with reasoning out loud. Then the other students would get an opportunity to check on their own performance and self-assess themselves. Hopefully, they would realize what kind of questions they have trouble with and they could concentrate on those before the exam.

Also, besides the encouragement to go over all questions we have done so far, I would assign a short homework. The questions 30, 37 from the resource2 #1 would be the best copies of the new material taught in this class (recall 31 and 36) and the question 4 from the resource2 #1 would help them to visualize series and parallel circuits.

.

This would be the examination time. Students would come to the class and get a chance to clarify any ideas about series and parallel circuits. I would make sure that the question 4 was understood and solved properly as I want to include a similar graphic approach on the exam.

The students would get to answer questions 13, 38, 40 from the resource2 #1 and one other example. The questions 13 and 18 are clearly conceptual and they are seeking for students´ understanding of this topic, while the question 40 would test them on computational skills in the highest extent (there are 7 different resistors). Notice that the diagrammatic approach there is not usual but more realistic – one we would have got a chance to touch or hands in the homework question 4.

The other example I mentioned would seek for their understanding of the application for calculating the total resistance in a circuit. The question “What is the total electrical power in the circuit below”:

could be approached with a tedious way through the formula P=VI. A student would calculate the total R in order to get the total current flown through the circuit (V=IR, so I=V/R) and then multiply the current with voltage. Yes, that is a correct approach but not the easiest one. Why not to look at the both formulas V=IR & P=VI and think about what we know, what we are looking for and we can find easily. We know V and we want P. So P=VI should work the best if we could replace I with something we can find easily. The total resistance R in this case saves the day because I=V/R (from V=IR) and so P=VI=V(V/R)=V²/R. This method saves us from calculating six partial currents and summing them up. The basic idea of integrating these two formulas into one was first introduced in the Ohm´s Law lesson and so I do not expect too many students to be surprised.

Nevertheless, I would allocate for this exam one hour at most, so there is time on the end of the classroom meeting to reflect upon the exam (the last question especially) and the lesson as a whole too. We would conclude that this lesson was useful because we were suddenly able to compute the total power usage without knowing a single piece of information about the magnitude of the current flown through a circuit. It is easy to find the resistance of a device at home and usual households have potential difference (voltage) of 120V. Therefore, the most active, knowledgeable and eager students could be encouraged to calculate the power usage of some easily identifiable circuit at home.