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How Much Folate?
Task
Sara's doctor tells her she needs between 400 and 800 milligrams of folate per day, with part coming from her diet and part coming from a multivitamin. Each multivitamin contains 50 mg of folate, and because of the inclusion of other vitamins and minerals, she can only take a maximum of 8 tablets per day.

 What are the possible combinations of $n$, number of vitamin tablets taken, and $a$, the amount of dietary folate, which will give Sarah exactly the minimum of 400 mg of folate each day? Express your answers in a table.
 What are the possible combinations of vitamin tablets and dietary folate which give the maximum of 800 mg of folate each day?
 Now use your tables from parts (i) and (ii) to express your answers as a system of three inequalities. Create a graph of $a$ versus $n$, and compare your graph to your tables from parts (i) and (ii).
 Suppose instead of a multivitamin, Sara is given a powdered folate supplement she can add to water. She can drink any amount of the supplement she wants per day, as long as she does not exceed 400 mg per day. What are the possible combinations of folate she can ingest from her diet and from the powder? Graph your solution set. How does this graph compare to the graph from part (a)?
IM Commentary
This task a could be used as an introduction to writing and graphing linear inequalities. Part (a) includes significant scaffolding to support the introduction of the ideas. First, a chart is used as a means for considering the possible combination of tablets and folate in Sara's diet. Students next construct a system of inequalities, and this provides an opportunity to show how solutions recorded in a chart can be used to develop algebraic notation, and how such notation can then be used to find solutions.
Part (b) demonstrates that, in some situations, writing down all possible combinations is not feasible. Then, students are asked to use algebraic notation and create inequalities in order to find the solution set and graph it.
This task also requires that students consider the reasonableness of their solutions within the given context. For instance, in part (a), students need to be able to see that although the solution region for their system is shaded, only whole numbers of vitamins make sense for Sara to take each day. Asking students to first create a chart supports them in making this distinction. Alternatively, a reasonable discussion about the plausibility of having fractions of multivitamins might be of some benefit, though should ultimately culminate with at least the distinction between the two interpretations.
A variation on this problem could be to give it as a group activity, and to leave out the information that each tablet contains 50 mg. In this case the instructor can ask the students what additional information they would need to proceed or students could be asked to find out what a common dosage of folate is. The goal would be to purposely present a problem with missing information, and require the group to determine what information is needed to proceed.
Solution

 We are given that $a$ is the amount of folate Sara should get from her diet, and $n$ is the number of vitamin tablets taken, and we want to make a table showing combinations of how she can get the minimum of 400 mg of folate. Because her vitamin tablets contains 50 mg of folate each, we arrive at the following table (there are many possible answers for how this table will look):
$n$, number of tablets 0 1 2 3 4 5 6 7 8 $50n$, total folate (in mg) from $n$ tablets 0 50 100 150 200 250 300 350 400 $a$, total dietary folate (in mg) 400 350 300 250 200 150 100 50 0  We want to make a table showing combinations of how she can get the maximum of 800 mg of folate. Because her vitamin tablets contains 50 mg of folate each, we arrive at the following table (there are many possible answers for how this table will look):
$n$, numbers of tablets 0 1 2 3 4 5 6 7 8 $50n$, total folate (in mg) from $n$ tablets 0 50 100 150 200 250 300 350 400 $a$, total dietary folate (in mg) 800 750 700 650 600 550 500 450 400 
To express our answers from (i) and (ii) above using inequalities, we see that a single column in the table from part (i) contains a solution to $50n+a=400$. We also see that each column in the table from part (ii) contains a solution to $50n+a=800$.
Considering what $n$ and $a$ represent, this provides a basis for creating our inequalities: $n$ represents the number of tablets Sara will take, and so $50n$ represents the total amount of folate Sara receives from her vitamins. As $a$ represents the total amount of folate Sara ingests from her diet, we see that $50n+a$ represents the total amount of folate Sara receives per day.
Now, since we want the total amount of folate, given by $50n+a$, to be between the minimum of 400 mg and the maximum of 800 mg, we arrive the following linear inequalities, $$50n+a\geq 400$$ $$50n+a\leq 800.$$ Considering the additional information that she cannot take more than 8 tablets per day, we finally have our system of 3 inequalities: $$50n+a\geq 400$$ $$50n+a\leq 800$$ $$0\leq n \leq 8.$$ Note we may abbreviate the first two as $400\leq 50n+a\leq 800$. If we graph this system of inequalities, we arrive at the following graph, where the shaded region is the system's solution set.
However, taking a fraction of a vitamin does not make much sense, and so we are only interested in the whole numbers on the $n$axis, as we saw in our tables. Therefore, our solution set in the context of this problem is specifically the red vertical lines within our region that occur at each wholenumbered $n$value.
For example, if Sara decides to take 3 tablets a day, then she needs to include between 250 mg and 650 mg of folate in her diet, which we see are the two values that corresponded to $n=3$ in our tables.
 We are given that $a$ is the amount of folate Sara should get from her diet, and $n$ is the number of vitamin tablets taken, and we want to make a table showing combinations of how she can get the minimum of 400 mg of folate. Because her vitamin tablets contains 50 mg of folate each, we arrive at the following table (there are many possible answers for how this table will look):

We now consider if Sara is taking a powdered folate supplement, which she can ingest in any dosage. Our variable $n$ is no longer viable, as Sara is not taking her folate in tablet form. Instead, let $s$ be the amount of folate she receives from the supplement, and let $a$ still represent the amount of folate she receives from her diet. While we cannot possibly list all the possible combinations of supplement and dietary folate, we can algebraically express our solution set.
Using part (a) as a guide, we see that $a+s$ represents the total amount of folate Sara ingests in a day, and we want this amount to be between 400 and 800 mg. Therefore, our system of inequalities is $$a+s\geq 400$$ $$a+s\leq 800$$ $$0\leq s\leq 400.$$ Note that the third inequality is obtained from the fact that Sara can receive a maximum of 400 mg from the supplement.
Graphing $a$ against $s$, we arrive at the following graph of our solution set.
Comparing this graph to the graph from part (a), we see that this time, our entire shaded region is our solution set, and we are not limited to only considering whole numbers because of the change from taking a certain number of tablets to taking a certain number of milligrams.
How Much Folate?
Sara's doctor tells her she needs between 400 and 800 milligrams of folate per day, with part coming from her diet and part coming from a multivitamin. Each multivitamin contains 50 mg of folate, and because of the inclusion of other vitamins and minerals, she can only take a maximum of 8 tablets per day.

 What are the possible combinations of $n$, number of vitamin tablets taken, and $a$, the amount of dietary folate, which will give Sarah exactly the minimum of 400 mg of folate each day? Express your answers in a table.
 What are the possible combinations of vitamin tablets and dietary folate which give the maximum of 800 mg of folate each day?
 Now use your tables from parts (i) and (ii) to express your answers as a system of three inequalities. Create a graph of $a$ versus $n$, and compare your graph to your tables from parts (i) and (ii).
 Suppose instead of a multivitamin, Sara is given a powdered folate supplement she can add to water. She can drink any amount of the supplement she wants per day, as long as she does not exceed 400 mg per day. What are the possible combinations of folate she can ingest from her diet and from the powder? Graph your solution set. How does this graph compare to the graph from part (a)?
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