Problem-Based Instructional Task Lesson Plan

TELEPHONE NUMBERS – HOW MANY??

Learning Goals: To develop the skill of systematic counting by thinking about the number of possibilities; To understand and apply the Multiplication Principle of Counting.

Title: TELEPHONE NUMBERS – HOW MANY??

Grade Level/Course: 12th grade/Integrated math

Estimated Time: 40 Minutes

Pre-requisite Knowledge: None

NCTM Standard(s) (shaded):

NCTM Content Standards  / Number & Operations / Algebra / Geometry / Measurement / Data Analysis & Probability
NCTM Process Standards  / Problem Solving / Reasoning & Proof / Communication / Connections / Representation
NCTM Content Standard Goal(s) / Understand meanings of operations and how they relate to one another.

Rigor and Relevance Framework (for high school only):

C / D X
A / B

Materials Needed

Audio-visual:

Manipulatives:

Technology/Software: calculator

Literature:

Other: student handout for the exploration, large paper for writing ideas brought up in the launch

LESSON DEVELOPMENT

LAUNCH

Provide a brief purposeful introduction to the lesson, typically teacher led.

Counting is a basic skill that you learned early and use often. It helps you answer the question, How many? You may be surprised by the wide variety of situations that require counting and the amount of mathematics that has been developed to help solve counting problems. Earlier, we used counting to arrange the reading groups.

What are some situations where a problem addresses “How many???” (Record the participants responses)

In year 1, algebra, we studied how to model population growth. It can affect many variables, from availability of housing and schools to assignment of license plates and telephone numbers. Fundamental to each of these situations is the question, “How many?”

An article was written in the Fairfield Weekly Reader, 1992 –

Important Notice to Fairfield Customers

Effective October 3, 1992, telephone customers establishing new telephone service in Fairfield may be assigned a new telephone number prefix – 469

Existing 472 prefix numbers in Fairfield are currently filled to capacity, making this addition necessary.

New GTE customers in Fairfield will be assigned a new 469 telephone prefix or a 472, if available.

Customers who are currently assigned a 472 prefix will continue with that number.

a)Besides population growth, what other factors might have made it necessary to create new telephone numbers in Fairfield?

b)Does the community you live in have more than one telephone number prefix? If you know the population of a town or city, how could you estimate the number of prefixes?

c)How many different phone numbers can be created using the 472 prefix? How many different phone numbers will be available in Fairfield with the introduction of the 469 prefix?

EXPLORE

In groups, the students will work on the Exploration, answering questions listed on the student handout.

Key Ideas

Key ideas and important points in the lesson: / Teacher strategies and actions to ensure that all students recognize and understand the key ideas and important points (e.g., ask targeted questions, facilitate mini-summary, point out key problems in the lesson, etc.):
Using the Multiplication Principle to find the number of area codes, prefixes, or local numbers possible. / Walk around the room as the groups work through the exploration, making sure that they understand the restrictions on the numbers or letters that can be used in the area codes, prefixes, or local numbers.
Interpretations can vary. Make sure the students can clearly explain their interpretation and why their answer matches their interpretation.

Guiding Questions

Good questions to ask students: / Possible student responses and actions: / Possible teacher responses:
What will you do? How will you respond?
Does the order that the numbers are in matter as the outcome of “How many?” / Yes, they each count as separate phone numbers / If they responded correctly, I would ask “What effect does repetition have on the number of outcomes?”
Why are there possibly different answers for 1a and 1b? For 2(b)(i) and 2(b)(ii)? / We got the same answers. / Push students to think if the answers really are the same. Under what interpretations? Why?
In 2(c) and 3(b), why does not allowing repetition affect the answer? / It didn’t. I don’t know. / If you choose A as the first letter, can you also choose A as the second letter? Why? How does this affect your answer? And so on …

Misconceptions, Errors, Trouble Spots

Possible student misconceptions, errors, or potential trouble spots: / Teacher questions and actions to resolve misconceptions, errors, or trouble spots:
#1 or 2 -When they look at how many different telephone numbers are possible? Determine whether the letters on the same number denote a different possibility or not. / Will a different telephone number be possible if two different letters are used on the same number (ex. Is using an A or a B going to give you 2 different numbers?)

SUMMARIZE:

The groups will first discuss the questions and then we will discuss them as a class:

Students might suggest that they drew blanks or boxes to represent phone numbers, and then filled in the blanks or boxes. They might suggest using systematic lists or tree diagrams.

Ask students why you multiply (and not add) in the Multiplication Principle. Make sure they can explain, in words and with lists and/or diagrams. (See Modify/Extend below.)

MODIFY/EXTEND

For those students that are having trouble with the Multiplication Principle, I would present a small problem where the student can use other strategies (lists, trees, pictures, etc) to find the answer.

Ex. If a student had 3 shirts, 2 pairs of pants, and 1 pair of shoes, how many different outfits could they wear? After visualizing the results, I would bring in the Multiplication Principle and how it works.

The extend would led the students into permutations and combinations.

CHECKING FOR UNDERSTANDING

In Iowa, a license plate had six characters. The first three characters were numbers and the last three were letters. Think about how many different automobile license plates that can be made.

  1. How many different license plates were possible in Iowa?
  2. Suppose that no letters or numbers can be repeated. How many license plates are possible?
  3. Suppose that the three letters were used to represent the county, with JOH used for all license plates issued to resident of Johnson Count. Do you think this is a good plan? Explain.

REFLECTION after teaching the lesson

In general, student achievement increases in classrooms of reflective teachers. Reflecting is not done in a few minutes after class; it is a mindful act done habitually and works well when done with others.

Characteristics:

  • Involves self-deliberation while making sense of one’s teaching
  • Uses past experiences to think about solutions to pedagogical and curricular problems
  • Is done while teaching and after teaching

Teachers will consider questions like:

  • If I teach this Problem-Based Instructional Task again, what would I do the same? Differently? Why?
  • How did I know which students learned mathematics? How can I better assess their learning?
  • What did I do that contributed to student learning? (Be specific, focus on questioning, instructional decision-making, planning, tools used, etc.)
  • How did I support the learning of students who struggle? (Be specific, focus on questioning, instructional decision-making, planning, tools, etc.)
  • How could I revise the lesson to improve student learning of important mathematics?

Telephone Numbers – How Many?? Lesson Plan Page 1 of 4