Lesson 15
Off the Line
These materials, when encountered before Algebra 1, Unit 2, Lesson 15 support success in that lesson.
15.1: Estimation: Coin Weight (5 minutes)
Warm-up
The purpose of an Estimation warm-up is to practice the skill of estimating a reasonable answer based on experience and known information, and also help students develop a deeper understanding of the meaning of standard units of measure. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking yourself “Does this make sense?” is a component of making sense of problems (MP1), and making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).
Launch
Display the image for all to see. Ask students to silently think of a number they are sure is too low, a number they are sure is too high, and a number that is about right, and write these down. Then, write a short explanation for the reasoning behind their estimate.
Student Facing
How much does a nickel weigh?
- Record an estimate that is:
too low about right too high - Explain your reasoning.
Student Response
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Activity Synthesis
Ask a few students to share their estimate and their reasoning. If a student is reluctant to commit to an estimate, ask for a range of values. Display these for all to see in an ordered list or on a number line. Add the least and greatest estimate to the display by asking, “Is anyone’s estimate less than \(\underline{\hspace{.5in}}\)? Is anyone’s estimate greater than \(\underline{\hspace{.5in}}\)?” If time allows, ask students, “Based on this discussion does anyone want to revise their estimate?”
Record the estimates in a place that they can remain until the end of the lesson. In a following activity, students will work to discover the answer.
15.2: Row Game: Equations (15 minutes)
Activity
The purpose of this activity is for students to recognize that multiplying an equation by a value results in the same solution. This will be useful when transforming equations for the elimination method of solving systems of equations in the supported Algebra lesson. Students will work in pairs and each partner is responsible for answering the questions in either column A or column B. Although each row has two different problems, they share the same answer. Ensure that students work their problems out independently and collaborate with one another when they do not arrive to the same answers. Students construct viable arguments and critique the reasoning of others (MP3) when they resolve errors by critiquing their partner's work or explaining their reasoning.
Launch
Arrange students in groups of two. In each group, ask students to decide who will work on column A and who will work on column B. If students are unfamiliar with the row game structure, demonstrate the protocol before they start working.
Student Facing
Work independently on your column. Partner A completes the questions in column A only and partner B completes the questions in column B only. Your answers in each row should match. Work on one row at a time and check if your answer matches your partner’s before moving on. If you don’t get the same answer, work together to find any mistakes.
Solve each equation.
row | column A | column B |
---|---|---|
1 |
\(3x-1=5\) |
\(6x-2=10\) |
2 |
\(4(x+1) = 3x - 12\) |
\(4x+4 = 3(x-4)\) |
3 |
\(14x+10=4x+6\) |
\(7x+5 = 2x+3\) |
4 |
\(6x+3=33\) | \(4x+2=22\) |
5 |
\(4x + 5y = 2 - 4x + 5y\) | \(4x + 9y = 2 + 9y - 4x\) |
6 |
\(2x + 6y = 10\) | \(5x + 15y = 25\) |
Student Response
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Activity Synthesis
The purpose of the discussion is to notice that equations that have been multiplied by the same value have the same solution.
Consider asking students to explain why their solutions might be the same for each row.
For each row, the two equations are equivalent. Sample responses for each row:
- The equation in column B is 2 times the equation in column A.
- In each column, one side of the equation is the distributed form of the same side of the equation in the other column.
- The equation in column B is \(\frac{1}{2}\) of the equation in column A.
- The equation in column B is \(\frac{2}{3}\) of the equation in column A.
- The equation in column B has \(4y\) added to both sides of the equation from column A.
- The equation in column B is \(\frac{5}{2}\) of the equation in column A.
While there may be other valid ways of connecting the equations (for example, in row 4, dividing the equation in column A by 3 and dividing the equation in column B by 2 result in the same equation), encourage students to recognize that there is a single multiple of one equation that results in the other equation.
15.3: What Were They Thinking? (20 minutes)
Activity
The mathematical purpose of this activity is for students to get a chance to follow through on a plan of how someone might reason through a contextual situation that can be modeled by a system of linear equations. As students come to understand solving equations as a process of reasoning about equivalent statements, they benefit from getting to interpret the reasoning of others.
This will benefit students when they attempt to solve systems of linear equations on their own in their Algebra 1 class.
Launch
Provide students access to 2 different types of items that can represent nickels and dollar coins in the first example and plastic bricks and number cubes in the second example.
Student Facing
Read each student’s reasoning and answer the questions.
Jada says, “I know 4 nickels and 7 dollar coins weigh 76.7 grams. I know 4 nickels and 5 dollar coins weigh 60.5 grams. Here’s what else I can figure out based on that:
- 2 dollar coins weigh 16.2 grams.
- 1 dollar coin weighs 8.1 grams.
- 5 dollar coins weigh 40.5 grams.
- 4 nickels weigh 20 grams.
- 1 nickel weighs 5 grams.”
- How did Jada figure out that 2 dollar coins weigh 16.2 grams?
- Why might Jada have done that step first?
- After Jada figured out how much 1 dollar coin weighed, why did she calculate how much 5 dollar coins weighed?
- Once Jada knew how much 5 dollar coins weighed, how did she figure out how much 4 nickels weighed?
Priya says, “I know 9 plastic bricks and 3 number cubes weigh 39 grams, and 7 plastic bricks and 6 number cubes weigh 50.5 grams. Here’s what I can figure out based on that:
- The weight of 18 plastic bricks and 6 number cubes is 78 grams.
- The weight of 11 plastic brick is 27.5 grams.
- The weight of 1 plastic brick is 2.5 grams.
- The weight of 9 plastic bricks is 22.5 grams.
- The weight of 3 number cubes is 16.5 grams.
- The weight of 1 number cube is 5.5 grams.”
- Why is Priya’s second step only about plastic bricks, not number cubes?
- How does it help Priya to have a statement that’s just about plastic bricks, and not number cubes?
- Why might Priya have started by finding the weight of 18 plastic bricks and 6 number cubes?
- After Priya figured out how much 1 plastic brick weighed, why did she calculate how much 9 plastic bricks weighed?
Student Response
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Activity Synthesis
The purpose of this discussion is to help students synthesize and generalize what Jada and Priya were doing as they explored different combinations of objects whose weights they could determine, that would help them find the true weights of the objects. Here are some questions for discussion:
- When did they add or subtract what was on the scales? Why? (When they knew the weight of the same number of items as are on the scale, they could subtract to find the weight of the remaining items.)
- When did they multiply what was on the scales? Why? (When they knew the weight of some items and multiplying would make it match the number of items on the scale. This would let them subtract the weights to find out the weight of the remaining items.)
- What does this have to do with solving systems of equations? (There are two situations with two unknown weights of objects.)
- Can you think of a move that Jada and/or Priya could have made that would not have helped them? (Adding the weights at the beginning to find the combined weight of all the items would not have been useful.)
Ask students to pick one of Jada or Priya’s moves that you thought was crucial, and explain why it was so helpful in finding the weight of one object on their scale.