to model the problem. What geometry might they be able to use? Can they set up an equation? activity adheres to Guideline 7: Provide options for recruiting interest ,Checkpoint 7.2: Optimize relevance, value and authenticity. SMP#1 Make of problems and persevere in solving them. While students can connect to a variety of Standards for Mathematical Practice in this activity, try to focus on SMP#1. It is natural to want to point students in the correct direction, but the class has just considered a similar situation and students should be given a chance to choose a similar method or devise their own plan. Encourage students to demonstrate the following characteristics of mathematically proficient students. Providing a rubric for this might be a way to informally capture student progress towards proficiency. Activity 1 Materials Part I of the “Saving Pennies” handout (pages 17) Implementation Place students in groups based on pre-assessment data. These groups could be based on ability, observed methods of solving problems, student learning preferences, or other grouping techniques. Groups should complete Part I of the “Saving Pennies” handout. Note that no scaffolding is provided for the students. Challenging students to complete unscaffolded problems provides opportunities to build their proficiency with the Standards for Mathematical Practice. At this point, you are not looking for correct answers, but rather the realization that doubling (tripling, etc) grows much faster than adding a specific amount each week. Possible teacher questions to facilitate (only use these questions if needed): For question #2: What if tomorrow I double the number of pennies you already have in your box? Do you want me to do this again the next day? Why? What would happen? Possible extension question for students who need enrichment and/or are moving faster: Do you think it is reasonable to ask students to carry around a box of one cubic foot? Hopefully students will consider the weight of the box in their response.
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 and shortcuts. Notice regularity in the way values are rearranged or manipulated to simplify and provide structure. Maintain oversight of the process while attending to the details. 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 24) 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 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 observation.
Username: Algebra Password: student Optional graphing technology Resources (must be available to all stakeholders) The two below are good places to get more information about the Fibonacci Sequence and the Fibonacci Cascade as they relate to the Golden Ratio. In the second link, be sure to scroll down a bit to the part about the Golden Cascade to relate to one of the patterns in the “Relationships” pre-assessment task.
Follow the Pattern Directions: Examine the cards in your envelope. Each card has two lines on it. The second line is a pattern for which you need to determine the next two terms. Remember to consider adding, subtracting, multiplying, or dividing when comparing one term to the next in order to determine your pattern. Once you’ve determined the next two terms in the given pattern, locate the card that has these two terms as the first line, and then match them up. Continue by completing the pattern on the second line of this matching card. When you are finished, you will have a chain of patterns. 25, 37, 49, 61, … 12, 48, 12, 3, , … 8575, , , 7, , … -17, -38 , 2, , 73, 85 81, 27, 9, 3, … 27, 40, 53, 66, … , , -10, -15, , … -3, , , , 1, 10, , 11, , … 79, 92 -25, 175, -1225, … , 88, 67, 46, 25, 4, … 2, 1, , , , … Relationships Directions: Your envelope contains two cards larger in size: linear and not linear. These will be the categories in which you will group the remaining cards. Examine the rest of the cards in your envelope. Each has a pattern that you must place in the linear or not linear category.
Linear Not Linear 25, 37, 49, 61, … 10, , 11, , … 81, 27, 9, 3, … -25, 175, -1225, … , , 7, , … 88, 67, 46, 25, 4, … , -10, -15, , … 1, 4, 9, 25, 36, … , , , , 2, , 26, 28, 32, 38, 46, … 1, 2, 6, 24, 120, … 1, 1, 2, 3, 5, 8, 13, … 1, 1.618, 2.618, 4.236, 6.854, 11.09, … Linear vs. Non-linear Relationships Pre-Assessment Record S = Needs Support P= Proficient A = Advanced Student Name Identify Next Terms Identify Linear Identify Non-Linear S P A S P A S P A