Solving Linear Equations by Balancing (Lesson 3.3)
Learning Targets
Use comparative reasoning with constant and variable terms to find solutions to linear equations.
Understand the solution to an equation as the equalizer that makes two variable expressions equal.
Justify the process of solving a linear equation symbolically or with a semi-concrete representation (such as a bar model).
| Tasks/Activity | Time |
|---|---|
| Activity | 20 minutes |
| Debrief Activity with Margin Notes | 15 minutes |
| QuickNotes | 5 minutes |
| Check Your Understanding | 10 minutes |
Activity: Whose Gift is Better?
Lesson Handouts
docxpdfAnswer Key
pdf
Experience First
While yesterday’s lesson had students using context to isolate the impact of the variable term, today students return to the idea of balancing and relational reasoning to solve equations that could have variables on both sides. Todd and Tiffany’s gift each have a known component (cash) and an unknown component (gift card value). In questions 1 and 2, students decide if the allocation of cash and gift cards is fair, which here we define as the two twins getting a gift of equal monetary value. Students may have their own opinions about the value of a gift card over cash, but ultimately, the value of each gift depends on how much money is on each gift card! By knowing that grandma spent the same amount of money on each gift, students can reason that the extra gift card must have the same value as Todd’s extra $15. That’s the only way the gifts could have the same value! In question 4 students use similar relational reasoning. Tiffany has 5 more dollars but Todd has 4 extra boxes of candy. How is it possible that the two gifts are really “equal”? Only if the 4 extra boxes of candy have the same value as the $5, meaning that each box of candy is worth $1.25. In question 5, Tiffany gets 5 more dollars than Todd but they both get the same number of socks (assume they are identical socks of equal price). Is it possible that the gifts have equal value? No! The socks are identical and “balance” each other out, but Tiffany got $5 more. There is nothing that Todd got that can make up for those extra $5. There is no value of socks that could make the two gifts equal (again, assuming the socks are identical and of equal price). Monitoring Questions:
Formalize Later
We introduce the bar model today as a problem solving technique that helps students visualize the solution to an equation. Bar models can be used when variables are on one side or both sides of the equation, but are best introduced with positive coefficients and positive constant terms. The visual allows students to see what each variable term balances with or is responsible for on the other side of the equation. The bar model works on length, so two expressions that are equivalent will have the same length. Actually seeing how the variable terms and constant terms line up with each other is powerful for isolating the impact of the variable and determining its value. You or your students may find the bar model to be tedious to draw each time. The goal is not that students always draw out a bar model every time they see a linear equation, but we do want to avoid rushing to an algorithm students do not yet understand. The use of a visual model allows for differentiation. Some students may need to actually draw the bar model only 2 or 3 times before they can start thinking about the principle in their heads (4 extra x’s must make up for the 20 extra unit lengths, each x must be 5). Other students can use the bar model as a reasoning tool for much longer before they switch to the mental model. Since students generalize strategies at their own pace, students should feel free to use any model for as long as it is useful for them.