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Mixing Candies
Task
A candy shop sells a box of chocolates for \$30. It has \$29 worth of chocolates plus \$1 for the box. The box includes two kinds of candy: caramels and truffles. Lita knows how much the different types of candies cost per pound and how many pounds are in a box. She said,
If $x$ is the number of pounds of caramels included in the box and $y$ is the number of pounds of truffles in the box, then I can write the following equations based on what I know about one of these boxes:
 $x+y=3$
 $8x + 12y + 1 = 30$
Assuming Lita used the information given and her other knowledge of the candies, use her equations to answer the following:

How many pounds of candy are in the box?

What is the price per pound of the caramels?

What does the term $12y$ in the second equation represent?

What does $8x + 12y + 1$ in the second equation represent?
IM Commentary
This task assumes students are familiar with mixing problems. This approach brings out different issues than simply asking students to solve a mixing problem, which they can often set up using patterns rather than thinking about the meaning of each part of the equations. Students have a difficult time distinguishing between the coefficients (dollars per pound) and the terms (total dollar value of a given amount of particular kind of candy) in the second equation.
The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.
This task is linked to Standard for Mathematical Practice #7, “Look for and make sense of structure.” The student is asked to interpret the structure of the equations in the context of the task. This type of task prepares students to move between a real life situation and writing their own equations in the future by identifying the connection between the individual terms in the equations and the context of the task. Students can step back from the task and see parts of the equations as single objects that mean something in the context of the problem. They can also think about using this structure for a purpose to solve future problems with these equations in this context and make strategic manipulations of the equations. One example of such a manipulation would be solving the first equation for x. By making this manipulation one can see that the number of pounds of caramel in the box is always 3 minus the number of pounds of truffle in the box.
Solution
The box contains 3 pounds of chocolates, since the total number of pounds of the caramels and truffles, represented by $x + y$, equals 3.
It appears that the second equation is based on the cost of a box, since everything equals 30. If that is true, then the caramels cost \$8 per pound; you can tell because 8 is multiplied by the number of pounds of caramels in the equation that relates the number of pounds of each kind of candy to the cost of a box.
$12y$ represents the value of the truffles. Since $12y$ is in the equation that relates the number of pounds of each kind of candy to the total value of the box, the truffles must cost \$12 per pound, and that multiplied by $y$, the number of pounds of truffles, will give their dollar value.
This represents the total value of the box of chocolates: the value of the caramels added to the value of the truffles added to the fixed cost of \$1.
Mixing Candies
A candy shop sells a box of chocolates for \$30. It has \$29 worth of chocolates plus \$1 for the box. The box includes two kinds of candy: caramels and truffles. Lita knows how much the different types of candies cost per pound and how many pounds are in a box. She said,
If $x$ is the number of pounds of caramels included in the box and $y$ is the number of pounds of truffles in the box, then I can write the following equations based on what I know about one of these boxes:
 $x+y=3$
 $8x + 12y + 1 = 30$
Assuming Lita used the information given and her other knowledge of the candies, use her equations to answer the following:

How many pounds of candy are in the box?

What is the price per pound of the caramels?

What does the term $12y$ in the second equation represent?

What does $8x + 12y + 1$ in the second equation represent?
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