Lesson 16

This warm-up prompts students to compare four systems of equations. It gives students a reason to use language precisely (MP6) and gives the opportunity to hear how they use terminology and talk about characteristics of the items in comparison to one another. To allow all students to access the activity, there is at least one item with an obvious reason it does not belong. Encourage students to move past the obvious reasons and find reasons based on mathematical properties.

Arrange students in groups of 2–4. Display the systems of equations for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as coefficient or constant. Also, press students on unsubstantiated claims.

In this activity, students find intercepts and slopes for pairs of equations. They should notice that the equations in each pair are multiples of one another and therefore are equivalent equations. Students look for and make use of structure (MP7) when they notice the relationships between the pairs of equations.

Ask students what they remember about \(x\)- and \(y\)-intercepts. One useful thing to remember is that the \(x\)-intercept of a graph occurs where the \(y\)-coordinate is zero. (Demonstrate using a graph, if needed.) So, to find the \(x\)-intercept of a graph representing an equation, you can replace \(y\) with 0 and solve the equation for \(x\). (And vice versa to find the \(y\)-intercept.)

Ask students how they can use two points like \((1,0)\) and \((0,8)\) to find the slope of a line. (Using a formula like \(m = \frac{y_2 - y_1}{x_2 - x_1}\), or a less-formal version like “subtract the y’s, subtract the x’s, then divide.” For this example the slope is -8 since \(\frac{8 - 0}{0-1} = \text{-}8\).)

The purpose of the discussion is to show that equations that are multiples of one another will have graphs with the same intercepts and slope. Select students to share their solutions and methods for finding the intercepts and slope as well as things they notice. While students may notice many things, ensure it is mentioned that in each pair of equations, one is a multiple of the other.

Some questions for discussion:

• "How would the graphs of each pair of equations look?" (Each graph would look like a single line since the two equations are equivalent. Time permitting, demonstrate this using graphing technology.)
• "Select a pair of equations. Find another equation whose graph would have the same slope and intercepts as the pair already given." (Another multiple of either equation will work.)

In this activity, students multiply an equation to get a specified coefficient to a target value. In the associated Algebra lesson, students will need this skill for solving systems of equations by elimination.

Ask students to recall the meaning of the term coefficient. Display the equation \(6x - \frac{2}{3}y = 18\) and ask students to find the coefficients.   (The coefficient of \(x\) is 6. The coefficient of \(y\) is \(\text{-}\frac{2}{3}\). Regarding the 18, sometimes this is called “the constant term.”)

The purpose of the discussion is to understand strategies for multiplying an equation to get a target coefficient for a given variable.

Some questions for discussion:

• "How did you know whether to multiply by a positive or negative number?" (If the coefficient already had the same sign as the target coefficient, I would multiply by a positive number. If not, I would multiply by a negative number.)
• "Kiran says, 'To get from \(5x\) to \(7x\) I know I want to divide by 5, then multiply by 7, but I don't know what number to multiply by to get there in one step.' What can you tell Kiran to help him?" (Dividing by 5 and multiplying by 7 is the same as multiplying by \(\frac{7}{5}\).)

• “How can you check if you multiplied the original equation by the right number?” (If the new target coefficient is correct, and if the equation I created is equivalent to the original one, then I know I multiplied the right number. I also know that I multiplied each term correctly.)