Lesson 10

Only three problems are given to allow time to discuss different values of \(a\) and \(b\) for each problem. It may not be possible to share every student’s responses, given limited time. Consider gathering only two or three different sets of values per problem. Each problem was chosen to elicit a different understanding of integer operations, so as students share theirs, ask how the information in the equation impacted their choices for \(a\) and \(b\).

This warm-up is intended as an opportunity to review operations on positive and negative numbers.

Display the problems all at once. Give students 1 minute of quiet think time per problem and ask them to give a signal when they have at least one set of values for each question. Follow with a whole-class discussion.

Students may have forgotten that the quotient of two negative numbers is positive.

Ask students to share their values for \(a\) and \(b\) for each problem. Include at least one set of values for each problem where \(a\) or \(b\) (or both) are negative. Record and display their values for \(a\) and \(b\) for all to see. Ask students how they decided on their values based on the information given in the equation. To involve more students in the conversation, consider asking:

• “Did anyone choose the same values?”
• “Who can restate ___’s reasoning in a different way?”
• “Did anyone choose different values?”
• “Does anyone want to add on to _____’s reasoning?”
• “Do you agree or disagree? Why?”

The purpose of this activity is to compute the slopes of different lines to get familiar with the formula “subtract \(y\)-coordinates, subtract \(x\)-coordinates, then divide.” Students first compute slopes for some lines with positive slopes, and then special attention is drawn to the fact that a line has a negative slope. Students attend to what makes the same computations have a negative result instead of a positive result.

Have partners figure out the slope of the line that passes through each pair of points.

1. \((12,4)\) and \((7,1)\) (answer: \(\frac35\))

2. \((4,\text{-}11)\) and \((7,\text{-}8)\) (answer: 1)

3. \((1,2)\) and (\(600,3)\) (answer: \(\frac{1}{599}\))

4. \((37,40)\) and \((30,33)\) (answer: 1)

Ask students to share their results and how they did it. Students may say they just found the difference between the numbers; making this more precise is part of the goal of the discussion for this activity. Ask students to complete the questions in the task and share their responses with a partner before class discussion. Provide access to graph paper and rulers.

Students may struggle with operations on negative numbers and keeping everything straight if they try to take a purely algorithmic approach. Encourage them to plot both points and reason about the length of the vertical and horizontal portions of the slope triangle and the sign of the slope of the line, one step at a time. Sketching a graph of the line is also useful for verifying the sign of the line’s slope.

It is common for students to calculate a slope and leave it in a form such as \(\frac {\text{-}3}{\text{-}5}\). Remind students that this fraction is representing a division operation, and the quotient would be a positive value.

When using two points to calculate the slope of a line, care needs to be taken to subtract the \(x\) and \(y\) values in the same order. Using the first pair as an example, the slope could be calculated either of these ways: \(\displaystyle \frac{4-1}{12-7}=\frac35\) \(\displaystyle \frac{1-4}{7-12}=\frac {\text{-}3}{\text{-}5}=\frac35\) But if one of the orders were reversed, this would yield a negative value for the slope when we know the slope should have a positive value.

It is worth demonstrating, or having a student demonstrate an algorithmic approach to finding the negative slope in the last part of the task. Draw attention to the fact that keeping the coordinates “in the same order” results in the numerator and denominator having opposite signs (one positive and one negative), so that their quotient must be negative. It might look like: \(\displaystyle \frac{11-2}{1-8}=\frac{9}{\text{-}7}=\text{-}\frac{9}{7}\)

Ask students how sketching a graph of the line will tell them whether the slope is positive or negative. They should recognize that when it goes up, from left to right, the slope is positive, and when it goes down, then the slope is negative. Using a sketch of the graph can also be helpful to judge whether the magnitude of the (positive or negative) slope is reasonable. 

The goal of this activity is for students to recognize information that determines the location of a line in the coordinate plane, and to practice distinguishing between positive and negative slopes. In this activity, one partner has a design that they verbally describe to their partner, who then tries to draw it. The purpose of this activity is to provide an environment where students have to describe or interpret the slope and locations of several lines. (Students are not expected to communicate by saying the equations of the lines, though there is nothing stopping them from doing so.) Students take turns describing and interpreting by doing this two times with two different designs.

Monitor for students who use language of slope and vertical or horizontal intercepts to communicate the location of each line. Invite these students to share during the discussion. There are many other ways students might communicate the location of each line, but the recent emphasis on studying slope and intercepts should make these choices natural. 

The two designs in the blackline masters look like this:

You will need the Info Gap: Making Designs blackline master for this activity.

Thanks to Henri Picciotto for permission to use these designs.

Tell students they will describe some lines to a partner to try and get them to recreate a design. The protocol is described in the student task statement. Consider asking a student to serve as your partner to demonstrate the protocol to the class before distributing the designs and blank graphs.

Arrange students in groups of 2. Provide access to straightedges.

From the blackline master that you have copied and cut up ahead of time, give one partner the design, and the other partner a blank graph. Arrange the room to ensure that the partner drawing the design cannot peek at the design anywhere in the room. Once the first design has been successfully created, provide the second design and a blank graph to the other student in each partnership.

After students have completed their work, ask students to discuss the process of communicating how to draw a line. Some guiding questions:

• "What details were important to pay attention to?"
• "How did you use coordinates to help communicate where the line is?"
• "How did you use slope to communicate how to draw the line?"
• "Were there any cases where your partner did not give enough information to know where to draw the line? What more information did you need?"

Students might notice that the lines with the same slope can be described in terms of translations (for example, line \(b\) is a vertical translation of line \(a\) down two units). This is an appropriate use of rigid motion language which recalls work done earlier in this unit. Students might also describe \(b\) as parallel to \(a\) and containing the point \((0,3)\). Finally, some students might use equations to communicate the location of the lines. All of these methods are appropriate. Keep the discussion focused on describing each line using slope and coordinates for each individual line on its own.