Lesson Plan for Counting



Problem-Based Instructional Task Lesson Plan

A CONSTANT RATE OF CHANGE

Based on the corresponding task in

Navigating Through Discrete Mathematics in Grades 6–12, NCTM, 2008

Learning Goals: Represent and analyze functions by using iteration and recursion; Use iteration and recursion to model and solve problems

Title: A Constant Rate of Change – A Recursive View of Some Common Functions – Part 1

Grade Level/Course: 6-12/Algebra or Integrated Math

Estimated Time: One class period – 45 minutes

Pre-requisite Knowledge: Students should have studied linear and exponential functions, although they need only an elementary understanding of these functions.

NCTM Standard(s) (shaded):

|NCTM Content Standards |Number & Operations |Algebra |Geometry |Measurement |Data Analysis & |

|( | | | | |Probability |

|NCTM Process Standards |Problem Solving |

|( | |

Rigor and Relevance Framework (for high school only):

|C |D |

| |X |

|A |B |

Materials Needed

Audio-visual: Elmo (optional)

Manipulatives: none

Technology/Software: graphing calculator

Literature: none

Handouts: “A Constant Rate of Change”, Navigating through Discrete Mathematics in Grades 6-12, pages 162-163

Other: none

LESSON DEVELOPMENT

LAUNCH

Relate an actual situation that needs a solution: “I want to watch recently released movies at home on my TV. I can sign up for movies through my cable company for $20/mo. I also received a promotion from a DVD club that allows me to join the club for $4 and rent each DVD for $3. Which will save me money in the long run, joining the DVD club or paying my cable company? Discuss your thoughts with your group. Can you make a table that would represent the number of DVDs rented and the cost per month?” Select a student to put their table on the board. “Does anyone have a table different from this one?”

You can represent functions in several ways—by using tables, graphs, and equations, for example. In parts 1 and 2 of this investigation, you will examine representations of two fundamental types of functions. Your goal, by the end of the investigation, is to find answers to these questions:

• How can two different types of equations represent each function—the one that you examine in part 1 and the one that you encounter in part 2?

• How does each equation show fundamental properties of the function and its graph?

• How do the equations help you see similarities and differences between the functions that you encounter in parts 1 and 2?

Distribute the student handout – A Constant Rate of Change

EXPLORE

Get students engaged in teams in investigating important mathematics.

Given the student handout, students will find guiding questions in the task where they will have student-to-student communication and a high level of student engagement

Key Ideas

|Key ideas and important points in the |Teacher strategies and actions to ensure that all students recognize and understand the key ideas and |

|lesson: |important points (e.g., ask targeted questions, facilitate mini-summary, point out key problems in the |

| |lesson, etc.): |

|Looking at a recursive view of linear |Have students begin by describing all the patterns that they can find in the function table. Then guide|

|functions and relating this view to |them though a systematic analysis. Students need to look for patterns vertically, down the y-column, |

|the graphical, tabular, and y=…. |then horizontally, from the x-column to the y-column. The x-y horizontal pattern is the more |

|representations |conventional method of analysis and generates a y=…equation. |

Guiding Questions

|Good questions to ask students: |Possible student responses and actions: |Possible teacher responses: |

| | |What will you do? How will you respond? |

|The students will complete the activity sheet |The y value increases by 3 each time. |How does each of these equations show the rate |

|#1-8. As the students are working on the |Intuitively this vertical pattern is NEXT = NOW|of change or slope? |

|graphic organizer, questions the teacher can |+ 3. Start at 4 | |

|ask: “As the value of x is increasing by 1, | | |

|what is happening to the value of y? How is | | |

|this shown in a recursive formula? | | |

|What did you notice when you showed the |Y=4 + 3x | |

|relationship between x and the corresponding y | | |

|value? | | |

|What relationships do you see between the |Constant rate of change is a linear graph and |What makes it a constant rate of change? |

|graph, table, recursive and explicit formulas? |function. | |

| |4 is the starting point, value of y when x=0, y| |

| |intercept, and constant in the explicit | |

| |formula. | |

| |3 is the slope of the graph (over 1 and up 3), | |

| |the amount added to each y to get the next y, | |

| |and the amount multiplied by each x value to | |

| |get the y value. | |

Misconceptions, Errors, Trouble Spots

|Possible student misconceptions, errors, or potential trouble spots: |Teacher questions and actions to resolve misconceptions, errors, or |

| |trouble spots: |

|Because the table starts with x=0 and each x increases by 1, students |Give the students additional tables: |

|may have the misconception that the starting point will always be the |x y x y |

|1st y value given in the table and the rate of change will always be |0 3 1 6 |

|the increase of the y values. |2 11 2 10 |

|Another possible error would be in the situation involving the DVD |4 19 3 14 |

|club. Students need to understand that after the first month, there |6 27 4 18 |

|would no long be a $4 charge for joining the club. There would only |The common misconception in the first table is that the rate of change|

|be a charge of $3 for each DVD. |is 8 instead of 4. The misconception in the second table is that the |

| |y-intercept is 6 instead of 2. |

| | |

SUMMARIZE:

Provide closure and summary for the lesson, typically teacher led. Students are engaged in providing the summary, by giving responses, solutions, or perhaps presentations.

Whole class discussion on the following 3 problems from the student handout:

4. What type of function is represented by the table and by the equations that you have found? Describe the basic characteristics of this function.

6. The slope of a graph of a linear function also shows the constant rate of change of y with respect to x. Describe how the table shows this constant rate of change.

7. Examine the equations in the boxes on the previous page.

• Describe how the slope and constant rate of change are shown in each of the two equations.

• Circle the number in those equations that corresponds to the slope. To show what the circled number represents, draw an arrow from each circled number to the box at the bottom of the page, and enter the number in the box.

• Do you think that one equation shows the slope and constant rate of change more clearly than the other? Explain.

Revisit the Launch question:

Would you pay the cable company or join the DVD club? Why? Justify your answer.

MODIFY/EXTEND

Based on students’ different mathematical understandings (ascertained partly from formative assessment), learning styles, and academic and social needs, proactively plan to:

Take the next step to look at subscripts of function notation, arithmetic sequences, and possibly parametric equations.

Recursive NEXT-NOW representations highlight and deepen students’ understanding of fundamental features of linear change. The NEXT-NOW language can serve as an effective transition to more formal notation for sequences, such as subscript of function notation. NEXT=NOW + 3 translates easily into An+1 = An+3 or A(n+1) = A(n) +3 or An=An-1+3 (NOW-PREVIOUS language).

NEXT =NOW + 3 can describe a linear function, and clearly describes a sequence that generates the next term by adding 3 to the current term – that is, an arithmetic sequence.

CHECKING FOR UNDERSTANDING

(at the end of the lesson, in addition to throughout the lesson as indicated above)

Typical American shoppers borrow money for large purchases like houses, cars, furniture, or entertainments systems. Suppose your family finds a special deal for a new $1,400 HD television: no interest will be charged, and the family can pay the loan back at the rate of $120 per month.

• Make a table showing the relation between number of payments and unpaid balance.

• Make a plot of (number of payments, unpaid balance) data pairs. Discuss how the pattern in the graph matches the pattern in the data table.

• Write rules showing how the unpaid balance changes from one month to the next in the NEXT-NOW form

• Use the letters N for the number of payments made and U for the number of dollars unpaid to write the rule for the situation, U=

• How many months will it take to pay off the HD television? Use your data to support your answer. What was the easiest method used to arrive at an answer?

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?

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