Lesson 10

Arrange students in groups of 2–3. 

Begin the discussion by polling the class for their lengths and widths, recording these in a table for all to see. Select groups to share how they figured out different combinations. For example, some may guess and check, draw a sketch and label two opposite side lengths before calculating the other two side lengths, or use the formula for perimeter.

Next, invite students to predict what would happen if we plotted all the points. After students share at least 4, display a graph with the data from the class table plotted alongside the equation \(2x+2y=50\). Here is an example of what the axes might look like along with \((10,15)\) plotted. By drawing in the horizontal and vertical segments for the point, a rectangle with perimeter 50 is created with the two axes.

An important takeaway for students from this activity is that linear equations don’t all look like \(y=mx+b\). Depending on the situation or information given, other forms of equations may be more useful.

If time allows, here are some more questions for discussion:

• “What does the slope of the line through these points mean?” (The slope is -1, which means that if the width increases by 1 unit, the length decreases by 1 unit and vice versa.)
• “If we drew a line through all the points, what does the vertical intercept of the line mean?” (When the width is 0 units, the length of the rectangle is 25 units. The vertical intercept of this line doesn’t make sense in this situation since a rectangle cannot have a width of 0 units.)
   

In previous lessons in this unit, students have analyzed multiple situations that lead to equations of the form \(y = mx + b\), including positive, negative, and 0 slope. In the previous lesson, they saw that vertical lines cannot be described by equations of this form and saw a geometric situation that could be represented by an equation of the form \(Ax + By = C\). This form of a linear equation is examined in greater detail here as students consider combinations of numbers that keep a total cost constant and write an equation that will be used again in the next activity. In this problem, the solutions are limited to non-negative integers and the set of all solutions is finite. In the next activity, the solutions comprise all real numbers and the set of solutions is infinite.

There are many equations students could use to describe the cost of apples in dollars, \(a\), and the cost of oranges in dollars, \(r\). Identify students who write \(a + 2r = 10\), and ask them to share during the discussion. There is no reason to solve for one variable in terms of the other, in this case, because a graph has not been requested and neither variable is a “natural” candidate for dependent versus independent. In this activity, students need to explain their reasoning (MP3).

Adjust this activity to 10 minutes. For the last question, instruct students to only find one combination of apples and oranges that is true and one that is false.

The activity narrative references a “geometric situation” from a previous lesson. Due to the sequence in IM 6–8 Math Accelerated, that geometric situation is the warm-up of this lesson and this lesson is the introduction to equations of the form \(Ax+By=C\) as a form of linear equation.

Arrange students in groups of 2. Give students 3–5 minutes of quiet think time to answer the first question and think about the others. Have partners compare solutions and discuss the remaining questions. Follow with a whole-class discussion.

Students may write an equation like \(a + r = 10\) where \(a\) is the cost of the apples and \(r\) is the cost of the oranges. While this is technically correct, it leaves extra work figuring out whether or not a solution to this corresponds to some actual number of apples and oranges. Ask these students what they are trying to figure out (numbers of apples and numbers of oranges that cost $10). Then suggest that they use their variables to represent the number of apples and the number of oranges rather than their cost. 

The equation \(a + 2r = 10\) (or some equivalent) is an example of a naturally arising equation of the form \(Ax + By = C\), introduced in the previous lesson. In a graphical representation of this situation, either variable (oranges or apples) could be plotted on either axis. If students mention slope, make sure to ask them how they are plotting the variables. This will be taken up in greater detail in the next activity where the graphical representation will be central. 

Invite students from different groups to share what they discovered with their partners by asking:

• “How many combinations that cost $10 did you find in total? How do you know you have found them all?” (6 combinations, these are all the possibilities of whole numbers that would make the sum 10.)

• “Did you notice any patterns that helped you find the combination?” (There are many patterns; the one we would like students to notice is that buying 1 less orange means you can buy 2 more apples.)

• “What would the graph of the equation you wrote look like?” (Some may realize that the graph would contain the six points that represent the six combinations for $10. Help them to see that these points would all lie on a line, given by the equation \(a + 2r = 10\). The non-integer points on the line would represent a fractional number of pieces of each fruit.)

• “According to the equation you wrote, if you bought \(\frac{1}{2}\) orange and 9 apples you would spend $10. Do you think this situation is realistic when buying fruit in a store?” (Probably not, you would only buy whole pieces of fruit.)

Students write an equation representing a stated relationship between two quantities, and use the equation to find pairs of numbers that make it true and pairs of numbers for which it is not true. By graphing both sets of points, students see that the graph of a linear equation is the set of its solutions, that is, the points whose coordinates make the equation true.

Students should be encouraged to use rational number coordinates and points in all four quadrants and on the two axes in their graphs, and to see that solutions exist all along the line, between and beyond the pairs they explicitly choose, making for an infinite number of solutions.

Some students might notice that the points where \(x=0\) and \(y=0\) are easy to quickly find and that connecting these two points yields the line containing all the other solutions. They may then try to read solutions from the graph instead of using the equation to more accurately calculate the coordinates of solutions. While this is a valuable observation in terms of quickly drawing the graph of the linear equation, students should be guided to realize that the equation enables a level of precision that reading from the graph often lacks. Precision can be improved through the use of technology in the form of graphing calculators and applications.

As students find points, both satisfying the relationship from the first question and not satisfying the relationship, monitor for these choices and invite these students to share during the discussion.

• Points in the first quadrant
• Points on the axes (there are only two of these!)
• Points in the first and/or fourth quadrants
• Points with non-integer coordinate values

Students should have access to geometry toolkits, especially rulers and graph paper.

Display for all to see the following statement: “You have two numbers. If you double the first number and add it to the second number, the sum is 10.”

Poll the class for “predict how many pairs of numbers make this statement true?” Display the answers for all to see.

Instruct students to pause their work after creating their graph and then to check their graph with you. Identify students who struggle with writing the equation or with finding and graphing solutions. Check that the graph and the equation match since some students might write the equation as \(x + 2y = 10\) instead of \(2x + y = 10\). Also check that students are not restricting \(x\) and \(y\) values to non-negative integers. 

Students might write \(x + 2y = 10\). Ask these students what \(x\) and \(y\) represent and to pick a couple of values of \(x\) and \(y\) so that the double of \(x\) and \(y\) sum to 10. Ask them to see if these are solutions to their equation. They most likely will not be. Suggest that these students reconsider their equation.

Select students to share their examples of points in the coordinate plane that make their equation true, sequenced in the following order:

• Points in the first quadrant
• Points on the axes (there are only two of these!)
• Points in the first and/or fourth quadrants
• Points with non-integer coordinate values

If it does not come up in student work, ask them if the equation could be true if \(x = \text-3\) (yes, \(y = 16\)) or if \(y = \text-4\) (yes, \(x = 7\)). What if \(x = 3\frac{1}{2}\)? (Yes, \(y = 3\).) Through the discussion questions, bring out that there are an infinite number of solutions to this equation and that, taken collectively and plotted in the plane, they make up the line of all solutions to the equation \(2x+y = 10\). 

Connect student solutions to the graphical representation of the scenario by asking: 

• “For an equation with two variables, a pair of numbers that makes the equation true is called a solution of the equation. Where in your graph do you see solutions of the equation that you wrote? Are all possible solutions on your graph?” Bring out here that the set of all solutions to the equation is a line with \(y\)-intercept 10 and slope -2. So an alternative way to write its equation, other than \(2x + y = 10\), is \(y = \text-2x + 10\). 

• “Where in your graph do you see pairs that are not solutions of the equation you wrote?” (All points not on the line.)

• “Based on your observations, what is the relationship between the solutions of an equation and its graph?” (For this equation, the set of solutions is a line.)

• “What does the graph tell you about the number of solutions of your equation?” (There are an infinite number.)

• “Does the line contain all possible solutions to your equation? How do you know?” (Yes, for each value of \(x\), there is exactly one value of \(y\) giving a solution.)

In summary, articulate the idea that the solutions are precisely the points on the line, and no more.

Ask students to compare solutions to the apple and orange activity with solutions in this activity. “Would a graph of all solutions to the apples and oranges problem look the same as the graph in this activity?” It would not because you can only purchase non-negative whole number values of apples and oranges. Make sure students understand that in this activity it makes sense to draw a line connecting all points because the coordinates of all the points on the line are solutions to the equation. 

A solution to an equation with two variables is an ordered pair of values that makes an equation true, and the graph of the equation is the set of all solution pairs \((x,y)\) plotted as points in coordinate plane: for \(2x + y = 10\), this set of solutions is a line with slope -2 and \(y\)-intercept 10.