The goal of this lesson is for students to understand that we can generally approach \(p+x=q\) by subtracting the same thing from each side and that we can generally approach \(px=q\) by dividing each side by the same thing. This is accomplished by considering what can be done to a hanger to keep it balanced.
Students are solving equations in this lesson in a different way than they did in the previous lessons. They are reasoning about things one could “do” to hangers while keeping them balanced alongside an equation that represents a hanger, so they are thinking about “doing” things to each side of an equation, rather than simply thinking “what value would make this equation true.”
- Interpret hanger diagrams (orally and in writing) and write equations that represent relationships between the weights on a balanced hanger diagram.
- Use balanced hangers to explain (orally and in writing) how to find solutions to equations of the form $x+p=q$ or $px=q$.
Let's use balanced hangers to help us solve equations.
- I can compare doing the same thing to the weights on each side of a balanced hanger to solving equations by subtracting the same amount from each side or dividing each side by the same number.
- I can explain what a balanced hanger and a true equation have in common.
- I can write equations that could represent the weights on a balanced hanger.
A coefficient is a number that is multiplied by a variable.
For example, in the expression \(3x+5\), the coefficient of \(x\) is 3. In the expression \(y+5\), the coefficient of \(y\) is 1, because \(y=1 \boldcdot y\).
solution to an equation
A solution to an equation is a number that can be used in place of the variable to make the equation true.
For example, 7 is the solution to the equation \(m+1=8\), because it is true that \(7+1=8\). The solution to \(m+1=8\) is not 9, because \(9+1 \ne 8\).
A variable is a letter that represents a number. You can choose different numbers for the value of the variable.
For example, in the expression \(10-x\), the variable is \(x\). If the value of \(x\) is 3, then \(10-x=7\), because \(10-3=7\). If the value of \(x\) is 6, then \(10-x=4\), because \(10-6=4\).