SMP#8 Look for and express regularity in repeated reasoning. Students can connect to a variety of Standards for Mathematical Practice in this activity, try to focus on SMP#8. The accompanying worksheet provides some structure to the process of relating the recursive representation of the pattern to the explicit representation of the pattern. Though it is difficult to see students struggle, take the time to allow them to process what they are doing and ask questions of themselves and each other to discover the pattern and make generalizations. Encourage students to demonstrate the following characteristics of mathematically proficient students. Characteristics Notice if and when calculations are repeated. Look for general methods Activity 3 Developing the Concept Materials Part III of the “Saving Pennies” handout (pages 17) Implementation Groups should complete Part III of the “Saving Pennies” handout. Part III requires students to build the symbolic form of the relationships they’ve examined in Parts I and II. Students will first look at the recursive relationship relating the current term to the previous term. Students will then build an explicit relationship, relating the independent variable (in this case the number of days) to the dependent variable (in this case, total number of pennies). Using a Jigsaw approach, form new groups and have students compare answers. Complete a whole class final debrief of problems 1-10 from Part III of the “Savings Pennies” handout. Extension Assign either problem 4, 5 or 6 to the student groups. The group members should create a scatter plot of their assigned problem on the attached grid. ( page 23) After completing the scatter plot, instruct students to sketch a curve which contains the points on the scatter plot. Groups should display their scatter plots. Ask students to share their observations about the behavior of the displayed curves. Note: This is simply to make connections; students are not expected to be able to graph an exponential function at this point. Formative Assessment Provide students with several additional situations for which they will write an exponential expression. This could be done on individual white boards, with students at boards around the room, or simply on their paper for teacher
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Your parents would like to save money for you as a graduation gift. They have given you the two options below. Which option would you choose? Justify your choice. Option 1: 1 penny on the first week, then doubled each week Option 2: $100 each week Saving Pennies Part III: As your fund-raising plan, you decide to put 5 pennies in your box to encourage friends and family to contribute and set a goal to double your total number of pennies each day. Why would you choose to double your pennies each day, rather than add 100 pennies each day? What is the constant ratio that will define your exponential model? Write a rule for this sequence to determine the total number of pennies in the next day. Complete the table below so that you can model your plan for the class officers. Number of days (d) Total number of pennies (in expanded form) - T(d) T(d) 0 T(0)=5 1 2 3 4