Lesson 15

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).

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.

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.

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.

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.

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:

1. The equation in column B is 2 times the equation in column A.
2. In each column, one side of the equation is the distributed form of the same side of the equation in the other column.
3. The equation in column B is \(\frac{1}{2}\) of the equation in column A.
4. The equation in column B is \(\frac{2}{3}\) of the equation in column A.
5. The equation in column B has \(4y\) added to both sides of the equation from column A.
6. 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.

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.

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. 

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.