few_shots: 
Example:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_neg_0

6
standard_description: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
problem_activity: Task
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$x$"}, {"_value": "$f(x)=128\\cdot \\left({\\frac {1}{2}}\\right)^x$"}, {"_value": "Successive quotients"}]}, {"td": [{"_value": "0"}, {"_value": "128"}, {"_value": "----"}]}, {"td": [{"_value": "1"}, {"_value": "64"}, {"_value": "$\\frac{f(1)}{f(0)}=\\frac {64}{128} = \\frac{1}{2}$"}]}, {"td": [{"_value": "2"}, {}, {"_value": "$\\frac{f(2)}{f(1)}=$"}]}, {"td": [{"_value": "3"}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}]}]}]}]}
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$x$"}, {"_value": "$f(x)=128\\cdot \\left({\\frac {1}{2}}\\right)^x$"}, {"_value": "Successive quotients"}]}, {"td": [{"_value": "0"}, {"_value": "128"}, {"_value": "---"}]}, {"td": [{"_value": "2"}, {"_value": "32"}, {"_value": "$\\frac{f(2)}{f(0)} = \\frac {32}{128} = \\frac{1}{4}$"}]}, {"td": [{"_value": "4"}, {}, {"_value": "$\\frac{f(4)}{f(2)}=$"}]}, {"td": [{"_value": "6"}, {}, {}]}, {"td": [{"_value": "8"}, {}, {}]}]}]}]}
Let $f(x)=a \cdot b^x$. Let $x_0$ be any particular $x$-value. Show that if $x_0$ is increased by a constant, $\Delta x$, the successive quotient  $$\frac{f(x_0+ \Delta x)}{f(x_0)}$$ is always the same no matter what $x_0$ is.
Is b) an example of the result of c)? Explain.


aligns: no