1. Additional Algebra Skills Needed to Solve Equations

2. 21st Century Lessons – Teacher Preparation
Please do the following as you prepare to deliver this lesson:
Spend AT LEAST 30 minutes studying the Lesson Overview, Teacher Notes on each slide, and accompanying worksheets.
Set up your projector and test this PowerPoint file to make sure all animations, media, etc. work properly.
Feel free to customize this file to match the language and routines in your classroom.
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3. Lesson Overview (1 of 4) Lesson Objective
OBJECTIVE: Students will be able to efficiently solve equations by thoughtful selection of first moves, eliminating fractional coefficients and distributing negative signs.
LANGUAGE OBJECTIVE: Students will discuss with a partner potential solution moves in order to better understand the reasoning for selecting a particular first move.
Lesson Description
This is the second in a series on basics of solving equations. This lesson covers some more sophisticated ideas involved in solving equations. Students explore selecting a first move where they come to understand the value in scanning and assessing options before taking action to find the most efficient means of solving. They will develop skill in distributing a negative sign using distributive property and in eliminating fractional coefficients by multiplying by the denominator of the fraction. These skills enable students to add sophistication to their equation solving skills.

4. Lesson Overview (2 of 4) Lesson Vocabulary
Distributive property, negative, coefficient, constant, denominator
Materials
independent class work, homework, exit slip, powerpoint, calling sticks
Common Core
State Standard
8EEc7b - Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

5. Lesson Overview (3 of 4) Scaffolding
The frequent turn-and-talk strategy used throughout the lesson is a method utilized to aid student understanding by giving them time to think and to hear the thinking of others besides the teacher. This is a great strategy for both ELL students and for students with learning differences. The work is scaffolded with many opportunities for guided practice and color coding for each new move.
Enrichment
Students seeking additional challenges will find challenging work on both the class work and homework worksheets. Here is some challenging online practice:
Online Resources for Absent Students
Good LearnZillion lessons on this topic:

6. Lesson Overview (4 of 4) Before and After
The work of solving equations has been built upon through previous grades and also the ratios and proportions 8th grade unit. In 6th grade for example students have solved basic equations and they have created equivalent expressions using the distributive property and combining like terms. The work is solidified in 7th grade using numbers in any form (decimals, fractions, and negative numbers) and relying more heavily on the properties of the operations. This lesson follows algebra work looking at expressions and graphs and the previous lesson titled Introduction to Solving Equations covers basics of solving equations. Later lessons will move into solving systems of equations and reasoning about the shape and characteristics of the graph of a line by looking at an equation, often requiring manipulations first – manipulations that this lesson provides the skills for.
Topic Background
A nice history of solving equations can be found here:

7. Warm Up j l k m 5(n + 6 + 2p) = -3(2x + 4) = Evaluate. Simplify.
OBJECTIVE: Students will be able to efficiently solve equations by thoughtful selection of first moves, eliminating fractional coefficients and distributing negative signs.
LANGUAGE OBJECTIVE: Students will discuss with a partner potential solution moves in order to better understand the reasoning for selecting a particular first move.
Evaluate.
Simplify. j l
5(n p) = k m
-3(2x + 4) =
(Time on this slide – 5 min) Time passed 5 min
In-Class Notes
Click red box to see the first answer and enable other answers with clicks.
Students should be instructed to follow the directions. This activity and the correct answers can be a reference for students as they work on guided practice and independent work.
As the correct answers are revealed students should make changes as needed to their work so that the Do Now is a valid reference.
Preparation Notes
These warm-up problems were carefully chosen because these skills are required pre-requisites for some of the work of this lesson. Specifically knowing how to divide fractions by fractions will be helpful for the mini-lesson on eliminating fractional coefficients. Distributive property basic understanding will be necessary to understand and accomplish the objective of the mini-lesson about distributing a negative sign.
Agenda
Answers

8. Agenda: 1) Warm Up – basic skills review - YOU
OBJECTIVE: Students will be able to efficiently solve equations by thoughtful selection of first moves, eliminating fractional coefficients and distributing negative signs.
LANGUAGE OBJECTIVE: Students will discuss with a partner potential solution moves in order to better understand the reasoning for selecting a particular first move.
1) Warm Up – basic skills review - YOU
2) Mini-Lesson #1 – Picking a First Move - ME
3) Mini-Lesson #2 – Shortcut for a Fractional Coefficient – ME
4) Guided Practice – practice solving equations – US
5) Mini-Lesson #3 – Distributing a Negative Sign – ME
6) Guided Practice – practice solving equations – US
(Time on this slide – 1 min) Time passed 9 min
In-Class Notes
provide a brief overview of the agenda, no need to spend a lot of time here.
7) Independent Practice – practice solving equations – YOU
8) Assessment – Exit Ticket - YOU

9. Launch Mini-lesson #1: Picking a First Move 3m + 13 = 5m + 6
Ex. 1
3m + 13 = 5m + 6
Solve using the symbolic method.
Did you get m = 7/2 or 3½ for a solution?
What was your first move?
Turn and Talk:
Take turns speaking with a partner to share your first move. Was it the same? If not, ask your partner why he or she chose that move first.
(Time on this slide – 6 min) Time passed 15 min
In-Class Notes
Students solve this problem individually and without comment. Teacher circulates and pays attention to each student’s process.
After 3 minutes or more of individual work click to reveal the solution.
Then have students process their method by speaking with another student.
Preparation Notes
First goal: you can assess each student’s comfort with solving equations.
Second goal: you can assess the method each student uses. Do they subtract the variable terms or constants first? Do they always subtract the term on the left from the term on the right (or vice versa) or are they making a decision about which to do first? These preferences will tell you a lot about the level of proficiency and flexibility for each student.
Agenda

10. Will there be 2 different solutions? Let’s find out!
Launch
There is more than 1 first move from which to choose.
Ex. 1
3m + 13 = 5m + 6
3m + 13 = 5m + 6
3m + 13 = 5m + 6 OR
- 3m
- 3m
- 5m
- 5m
13 = 2m + 6
-2m + 13 = 6
- 6
- 6
- 13
- 13
Will there be 2 different solutions? Let’s find out!
7 = 2m
-2m = -7
(Time on this slide – 2 min) Time passed 17 min
In-Class Notes
this can be fairly fast, if students understand each step clearly.
Preparation Notes
You may wish to have students write out both methods.
You may ask students to pause at 13 = 2m + 6 and -2m + 13 = 6 to notice how these equations are similar. Can one be manipulated to look like the other? How? (if students are ready for this)
Either 1st move can
be used to get the
same result. 2 2 -2 -2
m = 7/2
7/2 = m
Agenda

11. Are there any other first moves?
Launch
There is more than 1 first move from which to choose.
Ex. 1
3m + 13 = 5m + 6
3m + 13 = 5m + 6
3m + 13 = 5m + 6 OR
- 3m
- 3m
- 5m
- 5m
13 = 2m + 6
-2m + 13 = 6
Are there any other first moves?
(Time on this slide – 2 min) Time passed 19 min
In-Class Notes
This is a think, pair, share opportunity. Have students think for 30 seconds or more before discussing.
Take only a few suggestions.
Preparation Notes
The idea is to have students see that you can begin with the variable terms or the constants. Since there are four terms there are four first moves. So why would you want to begin with one and not another? Help students to find and ponder this question.
Turn and Talk: Discuss with your partner. See if you can work together to find all the possible first moves.
Agenda

12. Launch 3m + 13 = 5m + 6 3m + 13 = 5m + 6 - 13 - 13 - 6 - 6
(Time on this slide – 2 min) Time passed 21 min
In-Class Notes
Take time on this slide. Either give students time to consider the possibilities or give them time to work the problem 4 ways. If you have the time it would be beneficial to have students solve the problem four times, using a different sequence each time.
Possible student response:
Some equations just “look right” – for example 3m + 7 = 5m because there is just one term after the equal sign. A student who says this may need more examples with equations in different forms. 13 = 2m + 6 does not have this form but represents the result of a good first move.
Students may not realize that 3m = 5m + -7 and -2m + 13 = 6 will require dividing by a negative. Notice this need to develop an awareness and make sure it is explicit discussed in the next slide.
Turn and Talk: Is there one first move that is better to use? Why do you think that one is better than the others?
These are all the possible first steps.
Do they all result in the same solution?
Agenda

13. Launch
3m + 13 = 5m + 6
3m + 13 = 5m + 6
- 13
- 13
- 6
- 6
3m = 5m + -7
3m + 7 = 5m
- 5m
- 5m
- 3m
- 3m
-2m = -7
7 = 2m -2 -2 2 2
These two first moves are similar.
m = 7/2
7/2 = m
3m + 13 = 5m + 6
3m + 13 = 5m + 6
- 5m
- 5m
- 3m
- 3m
13 = 2m + 6
-2m + 13 = 6
- 6
- 6
- 13
- 13
(Time on this slide – 2 min) Time passed 23 min
In-Class Notes
Ask students to spend time considering these sequences and finding how each pair being discussed is similar and how each is different. Again, students will have a higher level of engagement if they have solved the problem for themselves using each of the four opening moves.
7 = 2m
-2m = -7 2 2 -2 -2
7/2 = m
m = 7/2
Agenda
Some people might say that the calculations are easier if you do not have to divide by a negative. You can avoid this if you do not create a negative with your first move.
Some people might say that the calculations are easier if you do not have to divide by a negative.

14. Launch
This order of moves is the way most solutions will be presented in examples. Although it is important to realize that there are many possible first moves.
7/2 = m 2
- 6
13 = 2m + 6
3m + 13 = 5m + 6
- 3m
7 = 2m
In general, a preferable order of moves would minimize the need to calculate with negative numbers, fractions, or decimals. 2
(Time on this slide – 1 min) Time passed 24 min
In-Class Notes
Ask students to identify what the sequence is.
Preparation Notes
Sequence of moves: subtract the smallest variable term from the largest first, isolate the variable by subtracting the constant next and then dividing by the coefficient. By subtracting the smallest from the largest I am avoiding a negative coefficient.
Agenda

15. Launch Mini-lesson #2: Shortcut for a fractional coefficient.
Ex. 2
8 + ¼b = 5
Solve for b. Check your answer.
– 8
– 8
Why is subtracting 8 a better first move than subtracting ¼b?
¼b = -3
We know that ¼ is attached to the b by multiplication and the way to undo a coefficient is to divide by the coefficient. But there is a faster way to undo this coefficient because it is a fraction.
(Time on this slide – 1 min) Time passed 25 min
In-Class Notes
This mini-lesson is about fractional coefficients. Before you begin, ask students, “What is the fractional coefficient in this equation?” You may wish to review the word “coefficient”.
See if any students feel comfortable working with fractional coefficients before beginning.
Agenda

16. Launch How to cancel the fractional coefficient : 8 + b = 5 8 + b = 5
– 8
– 8
– 8
– 8
b = -3
b = -3
4( ) ( )4
b = -12
-3 ÷ =
-3  =
(Time on this slide – 3 min) Time passed 28 min
In-Class Notes
The blue box is a reminder of how to divide by a fraction and the connection to multiplication.
Isn’t the sequence on the right faster and easier? (more efficient?)
Preparation Notes
This mini-lesson is meant to build on the strengths that students have in dividing by fractions. If your students weren’t reminded of their strengths in this area from the warm-up they may just like to know that mutliplying by the denominator is a powerful key that eliminates fractional coefficients.
Remember that when you divide by a fraction you multiply by the reciprocal. So if you multiply both sides by 4 you will cancel the .
-  =
-12
b = -12
Agenda

17. Practice: Solve for the variable, substitute to check
1.) x = -5
2.) x + 9 = - x + 12
3.) 0 = 4 +
This is the same as n
(Time on this slide – 5 min) Time passed 33 min
In-Class Notes
As students are working you will want to circulate to inquire about students’ first moves. Ask “What was your first move? Why did you pick that move first?” or “How did that first move help you?” or “Is there another first move that would be more helpful?”
Remind students to check their work by substituting their solution back into the original equation to make sure it makes a true statement.
Answers
Agenda

18. Practice Mini-lesson #3: Distributing a negative sign. 14 = – (p – 8)
Ex. 3
14 = – (p – 8)
Let’s review the
distributive property:
Wait, what happened?
14 =
– p + 8
– – 8
-(p) = -1(p) = -p
2(x – 5)
6 = -p
-(-8) = -1(-8) = +8
2x – 10
Now, solve for p.
-6 = p
(Time on this slide – 2 min) Time passed 35 min
In-Class Notes
help students to see that a negative sign outside of a parentheses is shorthand for -1.
Similarly, in the last step a –p can be divided by -1 because –p is -1 x p.
Agenda

19. Practice: Solve for the variable, substitute to check
4.) -8 = –(x + 4)
5.) 12 = – (-6x – 3)
6.) –(y – 2) = 3(y + 1)
(Time on this slide – 5 min) Time passed 40 min
In-Class Notes
Make sure students understand how to check their work.
Answers
Agenda

20. Practice: Independent Class work
(Time on this slide – 4 min) Time passed 44 min
In-Class Notes
Click the purple button to see the other blank class worksheet sections.
Click the red button to see the teacher’s key with answers.
Agenda
Answers
Next blank section

21. Practice: Class worksheet
(Time on this slide – 4 min) Time passed 48 min
Exit Slip
Go to Answers
Agenda

22. Practice: Class worksheet
(Time on this slide – 4 min) Time passed 52 min
Exit Slip
Go to Answers
Agenda

23. Practice: Class worksheet
(Time on this slide – 4 min) Time passed 56 min
Exit Slip
Go to Answers
Agenda

24. Exit Slip Agenda (Time on this slide – 4 min) Time passed 60 min
In-Class Notes
Collect and analyze to ensure student understanding.
Agenda

25. 21st Century Lessons The goal…
The goal of 21st Century Lessons is simple: We want to assist teachers, particularly in urban and turnaround schools, by bringing together teams of exemplary educators to develop units of high-quality, model lessons.  These lessons are intended to:
Support an increase in student achievement;
Engage teachers and students;
Align to the National Common Core Standards and the Massachusetts curriculum frameworks;
Embed best teaching practices, such as differentiated instruction;
Incorporate high-quality multi-media and design (e.g., PowerPoint);
Be delivered by exemplary teachers for videotaping to be used for professional
development and other teacher training activities;
Be available, along with videos and supporting materials, to teachers free of charge via the Internet.
Serve as the basis of high-quality, teacher-led professional development, including mentoring between experienced and novice teachers.

26. 21st Century Lessons The people… Directors:
Kathy Aldred - Co-Chair of the Boston Teachers Union Professional Issues Committee
Ted Chambers - Co-director of 21st Century Lessons
Tracy Young - Staffing Director of 21st Century Lessons
Leslie Ryan Miller - Director of the Boston Public Schools Office of
Teacher Development and Advancement
Wendy Welch - Curriculum Director (Social Studies and English)
Carla Zils – Curriculum Director (Math)
Shane Ulrich– Technology Director
Marcy Ostberg – Technology Evaluator