The goal of this Warm-up is to practice using an equation to model a situation characterized by exponential decay.
One strategy for choosing the right equation is to evaluate each equation for some value of t. For instance, students may choose 1 for the value of t, evaluate the expression, and see if the result makes sense for the amount of money after 1 week. Though this is a valid approach, encourage students to think about what is happening in the situation and to see if it is correctly reflected in the way each expression is written.
For instance, in Option A, the term 31t in the expression would mean “a third of the time in weeks” (rather than a third of the gift money), suggesting that this expression doesn’t accurately reflect the situation. Similarly, in Option C, 31⋅180t would mean 31 being multiplied by 180 repeatedly (t times), which also does not represent the situation.
Jada receives a gift of $180. In the first week, she spends a third of the gift money. She continues spending a third of what is left each week thereafter. Which equation best represents the amount of gift money g, in dollars, she has after t weeks? Be prepared to explain your reasoning.
g=180⋅(32)t
Ask students for the correct response, and prompt them to explain why the other ones are not correct.
Make sure students understand that if a third of the balance is spent every week, two-thirds of the balance is what remains every week, so the value of b is 32, and the amount available after t weeks would be 180⋅(32)t.
Some students may choose Option B because of the words 31 of what is left in the prompt. Ask them what fraction of the quantity is left if 31 of it is spent. If students are still not convinced, ask them how much of the 180 is spent in week 1? How much is left? Use your equation to verify that 120 is left.
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The goal of this Warm-up is to practice using an equation to model a situation characterized by exponential decay.
One strategy for choosing the right equation is to evaluate each equation for some value of t. For instance, students may choose 1 for the value of t, evaluate the expression, and see if the result makes sense for the amount of money after 1 week. Though this is a valid approach, encourage students to think about what is happening in the situation and to see if it is correctly reflected in the way each expression is written.
For instance, in Option A, the term 31t in the expression would mean “a third of the time in weeks” (rather than a third of the gift money), suggesting that this expression doesn’t accurately reflect the situation. Similarly, in Option C, 31⋅180t would mean 31 being multiplied by 180 repeatedly (t times), which also does not represent the situation.
Jada receives a gift of $180. In the first week, she spends a third of the gift money. She continues spending a third of what is left each week thereafter. Which equation best represents the amount of gift money g, in dollars, she has after t weeks? Be prepared to explain your reasoning.
g=180⋅(32)t
Ask students for the correct response, and prompt them to explain why the other ones are not correct.
Make sure students understand that if a third of the balance is spent every week, two-thirds of the balance is what remains every week, so the value of b is 32, and the amount available after t weeks would be 180⋅(32)t.
Some students may choose Option B because of the words 31 of what is left in the prompt. Ask them what fraction of the quantity is left if 31 of it is spent. If students are still not convinced, ask them how much of the 180 is spent in week 1? How much is left? Use your equation to verify that 120 is left.