In this lesson, students use the concept a pan balance model to understand equivalency and solve linear equations. IMP uses the context of a "mystery bags" that are filled with gold and placed on the pan balance. All mystery bags weigh the same amount when they are place don the scale. Known weights are also placed on the scale and students must use what they know about keeping the balance level in order to figure out how many ounces of gold are in each bag. Students may be familiar with a similar model and may already be able to solve simple equations. The focus here is both on the "why" of solving equations and getting students to justify their and answers by constructing a viable solution.
The task in this activity is taken from IMP's Year 1 Textbook (2009), pages 252-253.
You can begin class with a visual like the one in the Resources section for students to figure out. This is a fairly easy problem to figure out, so place an emphasis here on how students know the weight of each bag. Ask them to prove it to you. Then let them get to work individually on the remaining problems.
There is only 20 minutes allocated in this section for students to work. You want to leave time in this class not only for discussion, but also for students to go back to the problems and write algebraic equations. Therefore, pay close attention to time during this lesson!
As students work, keep an eye out for the following issues:
Have students report out their answers on Questions 2 through 7. Pay special attention to Questions 6 & 7. In Question 6, there are 6 bags and 13 ounces on one side and 6 bags and 14 ounces on the other side. This is impossible! Elicit ideas from students about what could be happening - perhaps the scale is unbalanced or the bags of gold are not all the same. In Question 7, the bags of gold could come out to any weight because the number of bags and the amount of weight on each side is the same.
Next, ask students if they could describe, in words, the procedure that they used to to find out how much gold is in a mystery bag no matter what the scale looks like. Keep track of what they say on a SmartBoard or white board (Mystery Bags_video narrative).
Now, return to the original scale from the start of class and ask students how they could represent this scale using algebra. If students are stumped, you might suggest using a letter (like B) to represent a mystery bag. Next, ask students to return to Questions 2 through 7 in small groups and write an equation for each one.
Bring the class back together and ask students to share out equations. You might start with one like Question 5, where there are different amounts of bags and weights on each side. Ask students to match up the steps they wrote earlier for figuring out how much each bag weighed with their algebraic equations. You might write 11B + 65 = 4B + 100 on the board next to their steps and ask them how they can represent the steps algebraically. You might say, "Earlier you said you took the same amount of mystery bags off of each side of the scale. How can you show that using algebra and this equation? How can you write about removing bags from each side? How many did you take away?" With students participating, show the rest of the steps.
Try to have students raise the idea or question about what order to work in. You might demonstrate that it does not matter if the weight or the mystery bags are removed first. This is a good opportunity to write the same equation side by side and show the steps in a different order.
Remember, that an important piece of this standard is students being able to justify their steps and create a viable argument. To that end, take each step slowly and make sure students are able to explain why they doing it.
Students will continue this work in tonight's homework. More Mystery Bags is an appropriate follow up assignment on page 254 of the IMP Year 1 textbook (2009). At the end of class you can introduce the assignment to students and if there is time, let them get started on the work. Part of the assignment asks students to create their own equations and describe the pan balance set up. Make sure they do this part of the assignment as you will use it at the start of the next lesson as an Opening activity.
This material is adapted from the IMP Teacher’s Guide, © 2010 Interactive Mathematics Program. Some rights reserved.