Lesson 2

This Math Talk invites students to use what they know about fractions, decimals, and the meaning of percent to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students calculate prices that involve a percent increase and write an equation to generalize the calculation.

Finding different percents of the same value (200) is also an opportunity to reason repeatedly and look for and make use of structure (MP7, MP8). 

Students are likely to approach the problems in different ways. They may:

• Convert each percentage into a fraction and multiply the fraction by 200. For example, they may think of 25% as \(\frac14\), 12% as \(\frac{12}{100}\) or \(\frac{3}{25}\), and 8% as \(\frac{8}{100}\) or \(\frac{2}{25}\).
• Convert each percentage into a decimal and multiply it by 200.
• Notice that 1% of 200 is 2, and that any percentage of 200 can be found by multiplying the percentage by 2. For example, 25% of 200 is \(25 \boldcdot 2\), and \(p\)% of 200 is \(p \boldcdot 2\) or \(2p\).

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?"

If one of the strategies shown in the Narrative is not mentioned, consider sharing it with students. 

This activity gives students an opportunity to examine multiple quantities and relationships in a geometric context, and to use letters to represent quantities.

As students analyze the number of faces, vertices, and edges in several Platonic solids and try to identify relationships, they practice looking for structure (MP7). Some of the relationships could be represented by inequalities. One particular relationship can be represented by equations, setting the stage for upcoming work on equivalent equations.

If work time is coming to an end and no students are able to find an equation that relates the parts of the Platonic solids, suggest that students try adding the vertices and faces in each row.

Ask students to keep their books or devices closed.

Display the images of the three Platonic solids. If physical polyhedra are available, consider displaying them as well. Ask students: "In what ways are the three figures alike? In what ways are they different?"

Students may say that the figures are alike in that:

• They are all three-dimensional figures.
• Each Platonic solid has only one kind of polygon for its faces (triangle for the tetrahedron, square for the cube, and pentagon for the dodecahedron).
• The faces of each figure seem to be regular polygons (or polygons whose sides are equal in length).
• They all have an even number of faces, vertices, and edges.

They may say that the figures are different in that:

• Each figure has a different polygon for its faces (a triangle for the tetrahedron, a square for the cube, and a pentagon for the dodecahedron).
• They all have a different number of faces, vertices, and edges.

If students refer to edges and vertices as “lines” and “points," ask if they remember the “math names” for these things. Review the terms "vertices," "edges," and "faces" as needed.

Tell students that they will now investigate the relationships between the faces, vertices, and edges in each polyhedron.

Some students may get the terms "vertex," "faces," and "edges" confused. As students work on the activity, check to make sure that they understand what should be counted.

Some students may see the relationship between vertices, edges, and faces, but be unsure of how to express that relationship using an equation. If students can say in words something like, “you always get two more,” ask them to try writing an equation that might be correct. Then suggest that they test the equation for one of the solids. If it doesn't work, ask them to make changes to the equation until it works.

Invite students to share their observations about the quantities and relationships in the table. Some of the hypotheses students make about the relationships might not be true for all Platonic solids. For now, it is sufficient that they are supported by the values in the table.

Next, elicit the relationship between the quantities that could be represented by \(V + F - 2 = E\). Record and display all correct equations for all to see. If students produce only one correct equation, introduce a variant such as \(V+F-E=2\) or \(V+F-E-2=0\). Ask students whether these equations all represent the same relationship and how they know. Students can simply show that each equation captures the pattern in the table. It is not necessary for them to articulate why the equations are equivalent, as they will have many opportunities to do so in upcoming lessons.

In this activity, students write equations to represent quantities and relationships in two situations. In each situation, students express the same relationship multiple times: initially using numbers and variables and later using only variables. The progression helps students see that quantities can be known or unknown, and can stay the same or vary, but both kinds of quantities can be expressed with numbers or letters.

Students may translate “Mai earned \(m\) dollars, which is 45 more dollars than Noah did” as \(m + 45 = v\), not paying attention to where the plus sign should go. As with other problems throughout this unit, encourage students to try using numbers in their equation to see if the equation really says what they want it to say.

Focus the discussion on students' observations about how the two sets of equations are alike. Then, ask how the equations within each set are different. If students mention that some quantities are known or are fixed and others are not, ask them to specify which ones are which.

Highlight the idea that sometimes we know how quantities are related, but the value of each quantity may be unknown or may change. We often use letters to represent those unknown or changing quantities. 

There might be times, however, when we use letters to represent quantities that are known or are constant. Doing so may help us focus on the relationship rather than the numbers. Tell students we will look at examples of such situations in upcoming activities.

This activity gives students another opportunity to represent a relationship with numbers and letters, to reason repeatedly, and to see more clearly that equations are helpful for generalizing a relationship. 

From the given descriptions, students are aware that there are four relevant quantities in the car purchase situations. In each situation, the value of at least one quantity is not given, creating a need for students to name it, either using a word or a phrase (for instance, "total price" or "original price") or to use a variable (for example, \(T\) or \(p\)). Some students might choose to use a symbol. Monitor for the different ways students represent these quantities.

Ask students if they have had to pay sales tax when making a purchase and, if so, to briefly explain how sales tax works.

Explain to students that a car purchase also involves a sales tax. Car buyers pay not only the price of a car, but also a tax that is a certain percentage of the car price. Car dealerships also often charge their customers various fees. 

Tell students that they will now write equations to describe the relationship between the price of the car, the tax, a fee, and the total price. Emphasize that it is not necessary to evaluate any expressions or perform any computations. 

Arrange students in groups of 2. Give them a few minutes of quiet work time and then a minute to discuss their responses with a partner. Follow with a whole-class discussion.

Some students may be taken aback by the prompt to write an expression relating four quantities. If they have trouble getting started, suggest that they simply calculate the cost of buying a $9,500 car, taking care to show their work.

One part of the first question gives the total price of purchase rather than the original price of the car. If students use the given value as an original price, ask them to double check the given information.

In the last question, students may struggle to represent \(r%\) algebraically. Some students may multiply the price by \(r\), others may write \(0.0r\). Urge students to look at their work in the first question. Ask them: "By what number did you multiply the car price? What operation turns the number 6 into 0.06?"

Select students whose equations are equivalent but in different forms to share their responses. Record and display them for all to see. Then, draw students' attention to the first and last equation in each question. 

From the first question, those equations might be: \(T=9,\!500 + 0.06(9,\!500) + 120\) and \(T=p+0.06p+ 120\). Ask students:

• "In the first equation, what quantities do we know?" (the original price of the car, the tax rate, and the miscellaneous charges) "When might it be useful to write an equation like this?" (when we know all relevant quantities except for one)
• "In the other equation, what quantities do we know?" (the tax rate and the miscellaneous charges) "When might it be useful to write an equation like this?" (when we want to have a kind of formula for finding the total price for a car of any price, assuming the tax rate and miscellaneous charges are fixed at 6% and $120 respectively)

For the second question, the equations might be: \(T=9,\!500+\frac{r}{100}(9,\!500)+m\) and \(T=p+\frac{r}{100} (p)+m\). Ask students:

• "In the first equation, what quantities do we know?" (the original price of the car) "When might it be useful to write an equation like this?" (when we want to know the total cost of a $9,500 car but don't know the tax rate or other fees)
• "In the other equation, what quantities do we know?" (none) "Why might it be helpful to write an equation like this?" (It helps us see the relationship of all the quantities that are relevant in the situation. It is flexible in that it can be used to find the total cost for any car price, tax rate, and miscellaneous charges.)

Emphasize that we might choose to use letters to represent quantities that vary or those that are constant, depending on what we want to understand or know.