![]() ![]() The goal of this discussion is for students to identify the growth factor for each sequence and learn that sequences with a growth factor are called geometric sequences. In more advanced courses (like this one), ratio is sometimes used as a synonym for quotient. In earlier courses, students may have learned that a ratio has two or more parts. For example, students may want to say that the pattern of the third sequence is “divide by 2 each time.” This is true, but the growth factor is \(\frac12\), because it is the number you multiply by to get the next term. For example, in the second list, the growth factor is 4 because \(8=4\boldcdot2\), \(32=4\boldcdot8\), \(128=4\boldcdot32\), and \(512=4\boldcdot128\).Įmphasize that the growth factor is defined to be the multiplier from one term to the next said another way, the quotient of a term and the previous term. Tell students that this “same thing” is called the growth factor or common ratio, and that we will use the first of these. “In the last sequence, if you divide any term by the previous term, you always get \(\frac34\).”.“In the first sequence, you always multiply a term by 3 to get the next term.”.Encourage students to use the word term, and to be specific when they describe what is happening, for example Students may describe how the “same thing” is happening with consecutive terms. If the idea of each consecutive term in a sequence growing by the same factor or having a common ratio does not come up during the conversation, ask students to discuss this idea. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If possible, record the relevant reasoning on or near the sequence. Record and display their responses for all to see. These two sequences also give opportunity to point out that we still use "growth factor" even when the terms are decreasing.Īsk students to share the things they noticed and wondered. If they notice that in the first two sequences, each pair of consecutive terms has the same quotient, they could inspect the quotients in the last sequence. The purpose of including these sequences is to encourage students to notice and make use of structure (MP7). The last two sequences may present a challenge since the growth factor is less than 1. They might first propose less formal or precise language, and then restate their observation with more precise language in order to communicate more clearly. When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). (Students likely encountered it in an earlier course when they studied exponential functions.) Students notice and describe that each sequence is characterized by the same type of relationship between consecutive terms. The purpose of this task is to re-introduce growth factor. By thinking about them and responding, students gain entry into the context and might get their curiosity piqued. Often, the goal is to steer the conversation to wondering about something mathematical that the class is about to focus on. The purpose is to make a mathematical task accessible to all students with these two approachable questions. After students have had a chance to write down their responses, ask several students to share things they noticed and things they wondered. The prompt to students is “What do you notice? What do you wonder?” Students are given a few minutes to write down things they notice and things they wonder. This is the first Notice and Wonder activity in the course. ![]()
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