Truth and Equations
Students begin the lesson by digging into what it means for an equation to be true or not true. They expand previously-held understandings of equations by thinking about the assumption that equations are always true. Students learn that a letter standing in for a number is called a variable. Students learn that, for an equation with a variable, a value of the variable that makes the equation true is called a solution of the equation. They find solutions to equations by using tape diagrams or reasoning about the meaning of "solution" once an equation is written.
This lesson is where "next to" notation is introduced (for example, \(10m\) means \(10 \boldcdot m\)).
- Comprehend the word “variable” to refer to a letter standing in for a number and recognize that a coefficient next to a variable indicates multiplication (in spoken and written language).
- Generate values that make an equation true or false and justify (orally and in writing) whether they are “solutions” to the equation.
- Use substitution to determine whether a given number makes an equation true.
Let's use equations to represent stories and see what it means to solve equations.
- I can match equations to real life situations they could represent.
- I can replace a variable in an equation with a number that makes the equation true, and know that this number is called a solution to the equation.
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 \(10-6=4\).
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