To Divide Me is to Multiply Me
Lesson 5 of 14
Objective: SWBAT solve division equations by using the appropriate steps.
The students will be doing a recall activity to start class. I want them to write down everything they remember about multiplication equations in their memory box. Allow students a few minutes to jot down what they remember. Then, have them refresh their memories by taking one minute to look over their notes about solving multiplication equations. After a minute, have them close their notes, and write for one minute more.
Once the students are done writing, have them do an “I have that” activity. Students will recite ideas from their memory box out loud. For example, “I have, the inverse operation for multiplication is division”. If other students have that, they respond “I have that!” and cross it off their list. If they don’t have it, they can add it to their list. This continues until there are no more new ideas.
Tools: To divide me is to multiply me notes (do now)
I will be going over with the students what an inverse operation means and how it applies to an equation. A few key points to remember are that opposite operations cancel each other out which means they equal one. The connection here would be to ask students what property says that if you multiply and number by 1, you get your number back? (SMP 2) Next, equations are like balances and they must be kept equal. If I alter one side, I have to alter the other in the same way. Finally, since we know that there is only one solution to an equation, we must check the solution to support its validity (SMP 6)
I will be going over this for each operation. I will be discussing what it means to be an inverse operation and what it takes to solve equations. This is an important concept for students to understand and the repetition will help them remember the steps.
Solving equations is used in both math and science in high school and it is critical that students understand how to solve them, even one-step equations. Some students do not like to “show their work” because they can do it in their heads. Normally, I completely support doing mental math, but because the bigger concept is to know “how” to solve the equation. I tell the students that we are laying the groundwork for high school. In high school the equations become more complicated and by knowing how to solve the equation will help later on. It’s important that they understand that this is a concept that will be used high school and beyond.
Tools: Inverse operations and equation examples
Modeling Division Equations
I’ve chosen three problems to model. The variable in the problems is located on each side of the equal sign. When I teach the students to solve equations, I begin by reading the equation out loud. For example, I would say x is being divided by 8 to get 4. Sometimes this audio helps students understand what is happening in the equation. I also draw a reflection line through the equal sign and down. This helps students remember that what happens to one side has to happen to the other. (SMP 1) Then I will say, we need to get the variable by itself in order to find its value. So, if I’m dividing x by 8 , what do I have to do to “undo” it? (multiply). In order for the scale to be balanced or the reflection to look the same, I have to multiply 8 on both sides. What happens when I get 8 divided by 8? (they make 1). What happens when I multiply 1 by any number or variable. (I get my number or variable back, it cancels out) So we say they cancel each other out which leaves the variable by itself. This is a little difficult for students to see. It’s important to write the 8 as the coefficient. This way they can see 8 over 8 to see the division even if we are multiplying to undo. Finally, we move to the other side of the equal sign and do the math. In this case, we would take 4 and multiply by 8 which equals 32. Since there is only one solution to an equation, we can substitute in 32 for x and see if it works. This method of checking supports SMP 6 and SMP 3. When we substitute in, the equation becomes and then I say, does 32 divided by 8 = 4? Yes, 4 = 4! During the check, it is important that the students substitute the value for the variable exactly where the variable appears. This doesn’t matter so much with addition and multiplication because they are commutative, but it does not work with subtraction and division. It’s a good habit to start right away and have them substitute in where the variable appears.
Tools: Solving division equation examples.
There are several problems to use to check for understanding. I’m going to have the students work out the problems on white boards and show me their solutions with the check. I like to use the white boards because I can assess the whole class at once. You can do as many problems as needed during this time. If you see that the students are getting them move on. If the students are having difficulties, practice some more.
Tools: white board problems.
Students will be working on a roundtable activity. The first 4 problems are simple, one step equation problems. The last 2 problems will be a little more difficult. I wanted the students to know that equations use not only whole numbers, but fractions and decimals too. The CCSS says that students should solve equations using non-negative rational numbers.
For each problem, the person they pass their paper to will do the check. For example, I solve the equation and get x = 6. I pass my paper and the person who gets my paper will do the substitution to check my solution. If it’s correct, celebrate! If it’s incorrect, then they should peer tutor. (SMP 3)
If students are having a hard time working with the fractions, have them draw a model or use the fraction strips to assist their learning (SMP 5). If they still don’t get it, pull those students out of the group to give them a quick re-teach on using operations with fractions and decimals.
Tools: Roundtable activity
I want to wrap up this lesson and see if students can create a word problem to represent a given equation and then solve the equation. Students will be writing equations soon and I want them to make the connection of how it works. I will not be formulating this idea until a later date. I just want them to play around with the equation and come up with a situation to represent it. Students can share word problems and solutions if time permits.
Tools: Closure example