Lesson 2; Keeping the Equation Balanced
Lesson Overview
This lesson is the first of a sequence of eight lessons where students learn to work with equations that have variables on each side. In this lesson, students recall a representation that they have seen in prior grades: the balanced hanger. The hanger is balanced because the total weight on each side, hanging at the same distance from the center, is equal in measure to the total weight on the other side.In the warm-up, students encounter two real hangers, one balanced and one slanted, and notice and wonder about what could cause the hangers’ appearance. This leads into the first activity where students consider two questions about a balanced hanger: first, whether a change of the number of weights keeps the hanger in balance, and second, how to find the unknown weight of one of the shapes if the weight of the other shape is known. Students learn that adding or removing the same weight from each side is analogous to writing an equation to represent the hanger and adding or subtracting the same amount from each side of the equation. They reason similarly about how halving the weight on each side of the hanger is analogous to multiplying by 12 or dividing by 2. In both the hanger and the equation, these kinds of moves will produce new balanced hangers and equations that ultimately reveal the value of the unknown quantity.In the second activity, students encounter a hanger with an unknown weight that cannot be determined. This situation parallels the situation of an equation where the variable can take on any value and the equation will always be true, which is a topic explored in more depth in later lessons.As students use concrete quantities to develop their power of abstract reasoning about equations, they engage in MP2.
Warm Up
What do you notice? What do you wonder?
Ask your self
Ask students to share their ideas. Record and display the responses for all to see. In the interest of time, you can ask if anything students wondered was a “why” question, meaning the question begins with the word why. Refer to MLR 2 (Collect and Display).If not brought up during the first part of the discussion, ask students why they think the left hanger is unbalanced while the right hanger is balanced. Students should understand that a hanger will only balance if the weight of the unknown objects in both socks is the same. If they are not the same, then the heavier side is lower than the lighter side.
Lauching the Assignment
The purpose of this task is for students to understand and explain why they can add or subtract expressions from each side of an equation and still maintain the equality, even if the value of those expressions are not known. Both problems have shapes with unknown weight on each side to promote students thinking about unknown values in this way before the transition to equations.While the focus of this activity is on the relationship between both sides of the hanger and not equations, some students may start the second problem by writing and solving an equation to find the weight of a square. While students are working, identify those using equations and those not using equations to answer the second problem during the whole-class discussion.
Launch
Display the problem image for all to see. Tell students that this is a hanger problem similar to the one in the warm-up, only instead of the weights hidden inside socks, each block type represents a different weight. Give 5 minutes of quiet work time followed by a whole-class discussion. If using the digital activity, introduce the hanger problem to set the context and connection to the warm-up. Give students individual work time to figure out the weights and use the applet to check their work.
Applet Usage/Question
This picture represents a hanger that is balanced because the weight on both sides is the same.
Question 2
Building on the previous activity, students now solve two more hanger problems and write equations to represent each hanger. In the first problem, the solution is not an integer, which will challenge any student who has been using guess-and-check in the previous activities to look for a more efficient method. In the second problem, the solution is any weight, which is a preview of future lessons when students purposefully study equations with one solution, no solution, and infinite solutions. The goal of this activity is for students to transition their reasoning about solving hangers by maintaining the equality of each side to solving equations using the same logic. In future lessons, students will continue to develop this skill as equations grow more complex culminating in solving systems of equations at the end of this unit.As students work, identify those using strategies to find the weight of one square/pentagon that do not involve an equation. For example, some students may cross out pairs of shapes that are on each side (such as one circle and one square from each side of hanger A) to reason about a simpler problem while others may replace triangles with 3s and circles with 6s first before focusing on the value of 1 square. This type of reasoning should be encouraged and built upon using the language of equations.
Launch
Arrange students in groups of 2. Give 5 minutes of quiet work time followed by partner discussion. Let students know that they should be prepared to share during the whole-class discussion, so they should make sure their partner understands and agrees with their solution.If students use the digital activity, the applet provides a way for students to check solutions. Encourage students to work individually (most likely they will need paper/pencil to work these problems) and then check their thinking using the digital applet. After students have had 5 minutes to work alone and with the applet, give them time to discuss their thinking with a partner before the whole-class discussion.
Pass
A triangle weighs 3 grams and a circle weighs 6 grams.
Hey, are you ready for more?
What is the weight of a square on this hanger if a triangle weighs 3 grams?
Summary
Select previously identified students to share their strategies for finding the unknown weight without using an equation. Ask students to be clear how they are changing each side of the hanger equally as they share their solutions.Next, record the equations written by students for each hanger and display for all to see in two lists. Assign half the class to the list for Hanger A and the other half to the list for Hangar B. Give students 1–2 minutes to examine the equations for their assigned hanger and be prepared to explain how different pairs of equations are related. The goal here is for student to use the language they developed with the hangers (e.g., “remove 6 from each side”) on equations.For example, for Hanger A, you might contrast 3+6+6+6=4x+6 with 21+x=6+5x. Possible student responses:
- Removing an x from each side of the second equation would result in the first equation.
- x=3.75 grams makes both equations true.
- You can subtract 6s from the sides of each equation and they are still both true.
Homework
Here is a hanger that is in balance. We don’t know how much any of its shapes weigh. How could you change the number of shapes on it, but keep it in balance? Describe in words or draw a new diagram.
"Summary 2"
The purpose of this discussion is to have students revisit the warm-up and connect it to the activities, reflecting on why the hanger is an appropriate and helpful analogy for an equation.Ask these questions:
- “In the warm-up we wondered why one hanger was slanted, whether there were weights in one blue sock that made it heavier than the other, whether the crooked hanger would straighten out if another sock was added to the other side (add any other pertinent things your students wondered). How would you answer these questions now?”
- “What is an equation? What does the equal sign in an equation tell you?” (An equation is a statement that two expressions have the same value. The equal sign tells you that the expressions on either side must have the same value, however that value is measured—as a count of objects, a measurement like 10 miles or 6 seconds, or numbers without units.)
- “What features do balanced hangers and equations have in common?” (Both representations have sides that are equal in value, even if the actual value of a side is unknown. Each side can contain numbers we do not know in the form of either shapes or variables. Changing the value of one side of a hanger or equations means changing the value of the other side by the same amount.)
- “You saw an example of a hanger where the unknown weight could not be determined. Can you design your own hanger like this one? How would you think about the weights needed on each side?” (If students completed the extension, you might ask them to also design a hanger with no solution.)