Lesson 10

The purpose of this warm-up is to elicit the idea that certain elements of equations are visible in graphs representing them, which will be useful when students match graphs to possible equations in a later activity. While students may notice and wonder many things about these graphs, how you can see 5 and 2 on the graphs are the important discussion points.

Display the equations and graphs for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the equations and graphs. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the connection between the 5 and 2 in the equations and the graphs does not come up during the conversation, ask students to discuss this idea.

The form and parts of an equation can give us information about the graph of the equation: about the slope, the intercepts, or the position relative to the axes. Some forms allow us to predict features of a graph more easily.

• Slope-intercept form \(y=mx+b\) helps us identify the slope and \(y\)-intercept of the graph right away.
• The form \(ax+by=c\) can help us identify the \(x\)- and \(y\)-intercepts. When either variable has a value of 0, we can find the other value fairly quickly.
• When the constant term is 0 (as in the form \(y=mx\)), the graph goes through the origin.
• When the equation is \(y=\) a number, the graph is a horizontal line. When the graph is \(x=\) a number, the graph is a vertical line.

Demonstrate some ways of thinking about this task. Choose one of the equations and ask students how they would narrow down which graph could represent it. (Presumably, students don't have access to graphing technology for this activity.) For example, consider \(x+y=6\). Here are a few approaches that students might suggest, or that you might bring up if no one suggests:

• Make a quick sketch by plotting a few points. Some pairs of values that make \(x+y=6\) true are \((0,6)\), \((1,5)\), and \((4,2)\). Once these are plotted we can see that F is the only graph that could pass through all these points.
• Related to the previous strategy but quicker once you are familiar with it, what would the \(x\)-intercept of the graph need to be? At the \(x\)-intercept, the coordinates are \((\text{something},0)\). The \(y\)-coordinate would have to be 0. If \(x+0=6\) then \(x\) is 6. The \(x\)-intercept is \((6,0)\). This rules out every graph except for D and F. Then, you could reason about the \(y\)-intercept.
• If we rewrite the equation to isolate \(y\), we get \(y=\text-x+6\). So we are looking for a graph whose slope is -1 and whose \(y\)-intercept is \((0,6)\).

In this practice activity, students analyze whether a few different equations could be represented by a graph, propose a new equation that could be represented by other graphs, and then interpret the meaning of the intercepts of a linear equation representing a context.

Here are some questions for discussion:

• “How can you use an equation in the form \(ax+by=c\), like \(x-y=12\) or \(2x + 4y = 16\) to find the \(x\)- and \(y\)-intercepts of the graph?” (When you substitute 0 into either variable, you can find the other value fairly quickly.)
• “How can you tell whether an equation has a graph that goes through the origin, like A or C?” (When the constant term is 0 (as in the form \(y=mx\)), the graph goes through the origin.)
• “When an equation involves only one of the variables, what do you know about its graph?” (When the equation is \(y=\) a number, the graph is a horizontal line. When the graph is \(x=\) a number, the graph is a vertical line.)
• “What must be true about the coordinates of a \(y\)-intercept?” (Its \(x\)-coordinate is 0.)
• “What must be true about the coordinates of an \(x\)-intercept?” (Its \(y\)-coordinate is 0.)