Lesson 7

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).

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 _____? Is anyone’s estimate greater than _____?” If time allows, ask students, “Based on this discussion, does anyone want to revise their estimate?”

Then, reveal the actual value and add it to the display.

Ask students how accurate their estimates were, as a class. Was the actual value inside their range of values? Was it toward the middle? How variable were their estimates? What were the sources of the error?

The purpose of this activity is to give students an opportunity to practice checking whether a particular value is a solution to an equation, and to recall properties of operations and equality that preserve the solution set of an equation.

This will be useful when students consider when and why equations have the same solution in the associated Algebra 1 lesson.

Monitor for students who:

• check the value of \(x\) in only one expression and reason about the second expression

• identify properties by name such as commutative, distributive, and associative

• identify properties informally such as “changing the order doesn’t matter” or “you can add first and then multiply or multiply first and then add” or “you can multiply three or more terms in any order” or “you can add the same thing to each side”

• manipulate the expressions

• graph one or both expressions

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

Ask students to talk to their neighbor about some things they can look for, to see if two equations are likely to have the same solution or different solutions. If needed, draw students’ attention to which equations had the same solution and which had different solutions, perhaps by displaying them for all to see, grouped by whether the equations were equivalent or not.

Call on students to share their thinking and make an informal list displayed for all to see. It’s fine if students do not name all possible properties, or only describe them informally. They will have more opportunities to encounter them formally in their Algebra 1 course.

Remind students that equivalent equations have the same solution set. Use the language of equivalent interchangeably with “have the same solution,” and monitor for students to pick up on this vocabulary.

The purpose of this activity is for students to practice generating equivalent equations. The structure of taking turns and justifying your thinking to your partner supports students to check their work as they go, and the class has a chance to check for equivalence when the total score is recorded.

Students also have to justify, whether formally or informally, why each move keeps the equation equivalent. This will be helpful for students in an associated Algebra 1 lesson when they must determine what moves are acceptable to make to an equation without changing its solution.

Monitor for students who solve the equation first and then work to create new equations with the same answer, and for students who simply create new equivalent equations from the given equation. Monitor for students who add or multiply by the same number on each side of an equation and students who use properties of operations (distributive, commutative, associative).

Arrange students in groups of 2. Explain to students that they are going to play a cooperative game in which the class tries to come up with as many different equations with the same solutions as they can. You give them an equation, and their job is to come up with other equations with the same solution as the original equation. The partner’s job is to check that the new equation is equivalent to the original by listening to their partner’s reasoning and making sure they agree. Each partner should create their own equations before moving to the next question. 

At the end of each round, you will ask them to share the equations they came up with, and keep track of how many different equations the class came up with. Set a goal for the second round of coming up with 4 more equations than the first round, and continue to set meaningful challenge goals with each round.

Calling on previously identified students to share strategies may help the class be more successful.

Display these equations, one at a time:

1. \(x = 5\)
2. \(x + 1 = 0\)
3. \(2(x + 1) = 10\)
4. \(\frac12x + \frac32 = \frac72\)

Remind students that today they did a lot of thinking about equivalent equations and moves they could do to equations that wouldn’t change the solution.

Make a semi-permanent display of “moves that won’t change the solutions to equations.” Sort the list by moves that are done to each side, and moves that are done to one side. Use students’ language, adding formal language if that is an emphasis in your school.

Possible list:

• Add the same value to both sides.

• Subtract the same value from both sides.

• Multiply both sides by the same value (but not zero!). Divide both sides by the same value (but not zero!).

• Change the order of terms (on one side) being added or multiplied (commutative property).

• Change the grouping of terms (on one side) being added or multiplied (associative property).

• Distributive property: \(a(b + c) = ab + ac\).