Lesson 11

In this warm-up, students review how to apply the distributive property to rewrite expressions that involve division, preparing them to do so in the next activity in the lesson.

Arrange students in groups of 2. Give students a moment of quiet time to work on the first two questions and then time to discuss their responses with their partner before moving on the last two questions.

Expect some students to give \(2x-10\) or \(4x-5\) as an answer to the first question. To illustrate why these are incorrect, take an example like \(\frac{4+6}{2}\). Explain that we know that 10 divided by 2 is 5, but if we divide only the 4 or only the 6 by 2 we won’t get 5. Alternatively, remind students that fraction bars can be interpreted as division, so each expression can be rewritten as, say, \((4x-10) \div 2\), and we can apply the distributive property.

The signs of the numbers in the second expression might be a source of confusion. Students might be unsure if the expression should be \(\text - \frac12 -25x\), \( \frac12 + 25x\), or another expression. Encourage students to substitute a number into the original expression, then try it in each potential answer. To explain why \(\text- \frac12 + 25x\) is correct, appeal to the distributive property again.

Invite students to share their equivalent expressions and how they reasoned about them. Display their expressions for all to see. Make sure students see that rewriting the expressions as sums or differences involves distributing the division (or applying the distributive property on division). 

If not mentioned in students' explanations, point out each division could be thought of in terms of multiplication. For example, \(\frac {4x-10}{2}\) is equivalent to \(\frac12 (4x-10)\), because dividing by a number (in this case, 2) gives the same result as multiplying by the reciprocal of that number (in this case, \(\frac12\)). Applying the distributive property of multiplication to \(\frac12 (4x-10)\) enables us to rewrite this product as a difference.

This activity reinforces the understandings that students began to develop in an earlier lesson about the connections between the structure of two-variable linear equations, their graphs, and the situations they represent.

Students first practice relating the parameters of an equation in slope-intercept form to the features of the graph and interpreting them in terms of the situation (MP2). Next, they practice making a case for how they know that a graph represents an equation given in standard form.

Some students may argue that substituting the \((x,y)\) pair of any point on the line gives a true statement, suggesting that the graph does match the equation. Or they may reason about the points on the graph in terms of almonds and figs and come to the same conclusion. For example, \((8,3)\) and \((11,1)\) are points on the line. If Clare buys 8 pounds of almonds and 3 pounds of figs, or 11 pounds of almonds and 1 pound of figs, the price is $75.

Ask these students how they would check whether the points with fractional \(x\)- and \(y\)-values (which are harder to identify precisely from the graph) would also produce true statements when those values are substituted. Use this difficulty to motivate rearranging the equation into slope-intercept form.

The work in this activity requires students to reason quantitatively and abstractly about the equation and the graph (MP2) and to construct a logical argument (MP3).

Display the two graphs in the task statement for all to see. Tell students the graphs represent two situations they have seen in earlier activities.

Arrange students in groups of 2. Give students a minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder about the graphs. Give them another minute to discuss their observations and questions with their partner before moving on to the task.

Focus the discussion on students' explanations for the last question. If no one mentions that \(6x+9y=75\) can be rearranged into an equivalent equation, \(y=8\frac13 -\frac23 x\), point this out. (Demonstrate the rearrangement process, if needed.)

Ask students if we can now see the \(8\frac13\) and the \(\text- \frac23\) on the graph and if so, where they are visible. To help students connect these values back to the quantities in the situation, ask what each value tells us about almonds and figs. Make sure students see that the \(8 \frac13\) tells us that if Clare bought no almonds, she can buy \(8\frac13\) pounds of figs. For every pound of almonds she buys, she can buy less figs—\(\frac23\) pound less, to be exact.

Previously, students have studied the structure of equations concretely and contextually. In this activity, they shift to reasoning symbolically and abstractly about linear equations in two variables.

Students are prompted to match equations in standard forms to pairs of slopes and \(y\)-intercepts. They could go about making the matches in various ways. Here are a few likely strategies, from less reliant on structure to more reliant on structure:

• Substituting the coordinates of the given \(y\)-intercepts into each equation and see which one produces true equations.
• Calculating the \(y\)-values when \(x\) is 0 and \(x\) is 1. The former gives the \(y\)-intercept. Subtracting the latter from the former gives the slope.
• Graphing each equation (either by hand or using technology) and analyzing the graph.
• Rearranging the equation into slope-intercept form, \(y=mx+b\), and identifying the \(m\) and \(b\).

Some students may notice patterns from manipulating and graphing equations in the past few lessons. For instance, they may observe that when \(A\), \(B\), and \(C\) in \(Ax + By = C\) are positive, the graph always slants down from left to right and therefore has a negative slope.

Others may notice that when isolating \(y\) in an equation in standard form, the constant term in the resulting equation is \(\frac{C}{B}\) and the coefficient of \(x\) is \(\frac{\text- A}{B}\), and that these values tell them the vertical intercept and the slope. In addition to making use of structure (MP7), students who make and apply these observations also practice expressing regularity through repeated reasoning (MP8).

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

As students work, monitor the strategies they use. Encourage those who approach the task by substituting values into the equations to consider rearranging equations or otherwise using the structure of the equations.

Keep students in groups of 2. Consider asking students to take turns finding a match and explaining their strategy to their partner. 

Students will likely use the strategy of rewriting the equations in slope-intercept form. Common mistakes here include isolating \(x\) rather than \(y\), changing the sign of only one term when dividing by a negative number, and dividing only one of two terms by the coefficient of \(y\). (For these last two mistakes, remind students of the work in the warm-up).

Students who recognize that the slope of a line with equation \(Ax+By=C\) is \(\frac{\text- A}{B}\) and that the \(y\)-intercept is \(\frac{C}{B}\)may also write the wrong signs, or get a ratio reversed. Students who use this strategy are likely shortcutting the process of isolating \(y\). Asking them to isolate \(y\) for one equation can help them to identify errors.

Select students to present their strategies. Sequence the order of presentation as listed in the Activity Narrative. After each strategy is presented, ask if other students approached it the same way and consider discussing questions such as:

• "How would you describe the matching process or the strategy? Was it fairly efficient or laborious? Was it prone to errors?"
• "If the list of slopes and \(y\)-intercepts were not available for you to choose from, would you still be able to determine the slope and \(y\)-intercept?" (Yes, except those who rely on the first strategy—using the coordinate values of each \(y\)-intercept on the list to test the equations.)
• (For students who use graphing technology:) "Would you still use graphs to make the matches if the graphing needed to be done by hand?"

Highlight that it is helpful and efficient to use the structure of an equation to get insights about the properties of its graph. At this stage, it is not essential that students recognize that the slope of an equation of the form \(Ax+By=C\) is \(\frac{\text- A}{B}\) and that it crosses the \(y\)-axis at \(\frac{C}{B}\). Students should, however, recognize that solving for \(y\) involves a predictable process and the resulting equation makes the slope and \(y\)-intercept visible.