Lesson 11

The purpose of this warm-up is for students to practice solving an equation for an unknown value while thinking about a coordinate pair, \((x,y)\), that makes the equation true. While the steps to solve the equation are the same regardless of which value of \(x\) students choose, there are strategic choices that make solving the resulting equation simpler. This should be highlighted in the discussion.

Encourage students to not pick 0 for \(x\) each time.

Collect the pairs of \(x\)’s and \(y\)’s students calculated and graph them on a set of axes. For each equation, they form a different line. Have students share how they picked their \(x\) values. For example:

• For the first problem, choosing \(x\) to be a multiple of 7 makes \(y\) an integer.
• For the last problem, picking \(x\) to be a multiple of 3 makes \(y\) an integer.

In the previous lesson, students studied the set of solutions to a linear equation, the set of all values of \(x\) and \(y\) that make the linear equation true. They identified that this was a line in the coordinate plane. In this activity, they are given graphs of lines and then are asked whether or not different \(x\)-\(y\) coordinate pairs are solutions to equations that define the lines. This helps students solidify their understanding of the relationship between a linear equation and its graph in the coordinate plane.

Consider asking students to work on the first 5 problems only if time is an issue. Since this activity largely reinforces the material of the previous lesson, it is not essential to do all 8 problems.

Arrange students in groups of 2. Students work through the eight statements individually and then compare and discuss their answers with their partner. Tell students that if they disagree, they should work to come to an agreement.

Some students may try to find equations for the lines. There is not enough information to accurately find these equations, and it is not necessary since the questions only require understanding that a coordinate pair lies on a line when it gives a solution to the corresponding linear equation. Ask these students if they can answer the questions without finding equations for the lines.

Display the correct answer to each question, and give students a few minutes to discuss any discrepancies with their partner. For the third question, some students might say yes because there is a solution to the equation for line \(n\), which has \(x = 0\), namely if \(y = 0\) as well. For the fifth question, make sure students understand that the line \(\ell\) meets the \(x\)-axis even if that point is not shown on the graph.

Some key points to highlight, reinforcing conclusions from the previous lesson as well as this activity:

• A solution of an equation in two variables is an ordered pair of numbers.
• Solutions of an equation lie on the graph of the equation.

Students are given linear equations—some of which represent proportional relationships—in various forms, and are also given solutions to their partner’s equations in the form of coordinates of a point. The student with the equation decides which quantity they would like to know, \(x\) or \(y\), and requests this information from their partner. They then solve for the other quantity. The activity reinforces the concept that solutions to equations with two variables are a pair of numbers, and that knowing one can give you the other by using the value you know and solving the equation. Students also have a chance to think about the most efficient way to find solutions for equations in different forms.

You will need the I’ll Take An \(x\) Please blackline master for this activity.

Consider demonstrating the first step with a student. Write the equation \(y = 5x - 11\) on one slip of paper and the point \((1, \text- 6)\) on another slip of paper. Give the slip with the coordinate pair to the student. You can ask for the \(x\) or \(y\) coordinate of a point on the graph and then need to find the other coordinate. Ask the student helper for either \(x\) or \(y\). (In this case, asking for the \(x\) coordinate is wise because then you can just plug it into the equation to find the corresponding \(y\)-coordinate for the point on the graph.) Display your equation for all to see and demonstrate substituting the value in and solving for the other variable. Alternatively, ask students how they might use the information given (one of the values for \(x\) or \(y\) to find the other given your equation.

Arrange students in groups of 2. One partner receives Cards A through F from the left side of the blackline master and the other receives Cards a through f from the right side. Students take turns asking for either \(x\) or \(y\) then solving their equation for the other, and giving their partner the information requested.

Students play three rounds, where each round consists of both partners having a turn to ask for a value and to solve their equation. Follow with a whole-class discussion.

The discussion should focus on using the given information to efficiently find a solution for the equation. Consider asking students:

• “How did you decide whether you wanted the value of \(x\) or the value of \(y\)?” (One might be more efficient to solve for: for example, with card B asking for \(x\) makes sense while with card d the arithmetic to perform is similar whether asking for \(x\) or for \(y\).)
• “Which equations represent proportional relationships? How do you know? Which do not?” (C and F are proportional because they can be written as \(y=kx\), although this is hidden at first glance in C.)
• “Once you have identified one solution to your equation, what are some ways you could find others?” (Use the constant rate of change to add/subtract to the solution you know, solve the equation for \(x\) or \(y\), choose values for one variable and solve for the other, for proportional relationships you could find equivalent ratios.)

Point out that all of the equations in this activity are linear. They are given in many different forms, not just \(y = mx + b\) or \(Ax + By = C\).

In the previous activity, the system of equations was represented in words, a table, and a graph. In this activity, the system of equations is partially given in words, but key elements are only provided in the graph. Students have worked with lines that represent a context before. Now they must work with two lines at the same time to determine whether a point lies on one line, both lines, or neither line.

Arrange students in groups of 2. Give students 1 minute to read the problem and answer any questions they have about the context. Tell students to complete the table one row at a time with one person responding for Clare and the other responding for Andre. Give students 2–3 minutes to finish the table followed by whole-class discussion.

Display the graphs from the task statement. The goal of this discussion is for students to realize that points that lie on one line can be interpreted as statements that are true for Clare, and points that lie on the other line as statements that are true for Andre. Ask students:

• “What is true for Clare and Andre after 20 minutes?” (Clare has 15 signs completed and Andre has 20 signs completed.)
• “What, then, do you know about the point \((20, 15)\) and the equation for Clare's graph?” (The point \((20, 15)\) is a solution to the equation for Clare's graph.)
• “What do you know about the point \((20, 20)\) and the equation for Andre's graph?” (The point \((20, 20)\) is a solution to the equation for Andre's graph.)

Invite groups to share their reasoning about points A–D. Conclude by pointing out to students that, in this context, there are many points true for Clare and many points true for Andre but only one point true for both of them. Future lessons will be about how to figure out that point.