##
* *Reflection: Staircase of Complexity
Creating Equations to Solve Problems - Section 4: Closure

In the student_work shown I had many scaffolds in place to ensure that students could be successful. I have found that this graphic organizer helps students to slow down their thinking. They must first define their variables, then solve the problem, and then finally ensure that their answer makes sense. You can see from the student work that two of three students were able to represent the situation mathematically and take both constraints into account when solving the problem. One student (middle example of student work), showed some understanding by finding two ages that were 5 years apart but that did not sum to 31. This is, however, a jumping off point for that student going into the second day of work with these algebraic representations.

*Staircase of Complexity: Mathematizing a Situation*

# Creating Equations to Solve Problems

Lesson 1 of 13

## Objective: SWBAT use linear equations to represent and solve problems.

#### Warm Up

*8 min*

I plan to start class by giving students a few minutes to solve these warmup problems on their own. While students are working, I will monitor their progress and watch out for students that are lacking fundamental skills. I will ask my students to justify each step in their solution process. I may also ask students to post their answers on the board, if I am concerned about the errors that students are making as they solve.

As students complete the warm-up, I will also look to see how many students are plugging their solutions back into the original equation. In raising this issue, I will remind students of MP6 and the importance of habits that support precision. At the same time, part of being precise is determining if your answer makes sense. This approach to precision is going to be a major focus of today's class and classes to come.

*expand content*

#### Opening

*5 min*

After the Warm Up, I will distribute the opening task. I will give students a few minutes to look through the question and work. Then, I will use a Non-Verbal Cue (thumbs up or down) to determine how many students think that the answer is correct. Once I have a sense of the level of students' confidence, I will have students do a Think-Pair-Share to discuss this.

While they work I will listen to student conversations. After, I will have one or two students share out their answers with the whole class. I hope that students will volunteer the idea that if they multiply Kaiden's age by 3 and Leah's age by 5 they will get the same answer of 30 (**MP2**). This shows that the solution is correct.

as we conclude this opener, I will explain to students that they are going to be solving problems like this one today. But, a large emphasis of the class will be put on how to check your solutions to make sure that they make sense in the context of the problem.

*expand content*

#### Practice

*20 min*

For today's practice, I developed a series of questions that will allow students to construct equations to solve problems:

I have also attached a graphic organizer that I use with my students which has been very helpful in the past. The organizer serves a few purposes:

- Slow students down - Students need to take their time in constructing a let statement which will represent the unknowns in the problem. This setup is the most important part of the process and needs to be emphasized.
- Organization - student's work should be organized in some way and this graphic organizer gets them in the habit. It is amazing to me that once students have used this for a few days and the organizer is taken away they still have their work setup in a way similar to this.
- Emphasis on checking -The check should not just be something that you do if there is time. Students need to actively determine how to make sure that their answer makes sense in the context of the problem.

Using this document, I will do the first example with students so that they can see how the graphic organizer is used. I will require that all of my students use the graphic organizer for the first few days. After that, if they no longer want it they don't need to take it. But, I will provide one for any students who want to continue to use it.

After introducing the organizer, I plan to let students work with their partners on the remaining exercises and continue to emphasize the setup and the check.

*expand content*

#### Closure

*7 min*

This Ticket Out the Door will help you to determine if students are able to construct the let statement and solve the problem. The scaffold of telling students the unknowns has been taken away and students need to construct the two expressions on their own. This formative assessment will offer valuable feedback as you prepare the next lesson on using equations to solve problems.

#### Resources

*expand content*

Cary,

Thanks for the feedback! Most of my lessons are "home grown" but I have used the modules on EngageNY for references from time to time. There are some really nice rigorous questions to draw from there.

| 9 months ago | Reply

This lesson is perfect! I spend a lot of time on MP1 and getting my students to persevere in solving any problem. I am just wondering if you teach from Eureka Math or Engage NY? Thank you so much!!

| 9 months ago | Reply*Responding to Nicole Needleman*

Hi Nicole. Nice to hear from you again! Time is ALWAYS an issue in any course. As you have noticed, the course that I designed is heavily reliant on laying a solid foundation in understanding functions, expressions, and equations (units 1,2, 8 and9) as that takes up 1/3-1/2 of the course. My philosophy was, if students had that foundation then they could handle quadratic, exponential, etc. because in the end they are all just input/output and students understand the concept of function and have looked at many modeling situations using expressions and equations. Now for reality...you still have content that must be "covered" in a certain amount of time. Consequently, you may have to skip some lessons and that is okay. As far as the balance between 'old school textbook teacher' and 'new age yeah CCSS! teacher' it really comes down to the fact that you are trying to have your students grapple with non-routine problems that will build their stamina with time on task and their perseverance (along with the other 7 MPs!). Each day you can ask yourself if your students are doing a majority of the work, the thinking and the reasoning? If yes then you are on your way to being a 'yeah CCSS! teacher' because textbooks typically do not ask students to do much thinking and reasoning and students watching you do math (i.e. work problems on the board) is not them learning how to think mathematically. I wish I could offer you the silver bullet but you need to do what is best for your students and balance that with your professional obligation to your colleagues. Keep checking in...always good to hear the feedback!

| 10 months ago | Reply

Hi James! I'm a tenth year teacher but I honestly feel like a first year. :o/ I've found your units very thought provoking and have used them this entire year thus far. I'm concerned about timing. I decided to teach your unit on understanding equations right after your unit on linear functions. And my coworkers are not using your materials (well, they've borrowed some of the items I've saved on our drive) but we all need to more or less be on the same page with our pacing. I like your unit on expanding equations but am wondering how you fit all of this in....I am nervous to dive deeper into equation world at this time in the year and am wondering if I should just go straight to solving systems of equations.....I'm caught somewhere between being an 'old school textbook teacher' and a 'new age 'yeah CCSS!' teacher.. please help.

| 10 months ago | Reply

Hi Nicole,

You could certainly teach unit 9 before you teach unit 8. I did teach units 8 and 9 before I did unit 2 at one point in time. It seems to me that the way these units are listed are out of order from the way I taught them. Will have to take a look at why.

| 11 months ago | Reply

Hi James! I'm currently using your material from Unit 2 and find it very effective. Thank you! I'm not sure which unit to try next however. I need to stay consistent with my colleagues and need to cover your unit 8 and 9 but it seems perhaps I should reverse the order of them? What do you recommend? I can tell by your groupon question in the Linear Functions Unit that at one point you taught units 8 and 9 before linear functions.....

| 11 months ago | Reply*expand comments*

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- LESSON 1: Creating Equations to Solve Problems
- LESSON 2: More Solving Problems with Equations
- LESSON 3: Creating Inequalities to Solve Problems
- LESSON 4: The Exercise Plan (Day 1 of 2)
- LESSON 5: The Exercise Plan (Day 2 of 2)
- LESSON 6: Solving Literal Equations
- LESSON 7: Literal Inequalities
- LESSON 8: Rewriting Formulas
- LESSON 9: Printing Presses: Solving more difficult Linear Equations
- LESSON 10: The Groupon Question: Solving Two Variable Equations
- LESSON 11: More with Solving Two Variable Equations
- LESSON 12: Solving Two Variable Inequalities
- LESSON 13: Modeling with Two Variable Equations and Inequalties