Additional Algebra Skills Needed to Solve Equations

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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.

Lesson Overview (1 of 4)

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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. http://www.corestandards.org/

Lesson Overview (2 of 4)

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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: http://www.algebralab.org/practice/practice.aspx?file=algebra1_3-3.xml

Online Resources for Absent Students

http://www.algebra-class.com/solving-algebra-equations.htmlGood LearnZillion lessons on this topic:https://learnzillion.com/lessonsets/560-solve-linear-equations-in-one-variablehttps://learnzillion.com/lessonsets/560-solve-linear-equations-in-one-variable

Lesson Overview (3 of 4)

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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:http://faculty.etsu.edu/gardnerr/Galois/history-of-equations.htm

Warm Up

Agenda

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Answers

j

k

l

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.

Agenda:

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

8) Assessment – Exit Ticket - YOU

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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.

6) Guided Practice – practice solving equations – US

7) Independent Practice – practice solving equations – YOU

Agenda

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Mini-lesson #1: Picking a First Move

Launch

Ex. 1 3m + 13 = 5m + 6

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.

Solve using the symbolic method.

Agenda

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There is more than 1 first move from which to choose.

Launch

Ex. 1 3m + 13 = 5m + 6

3m + 13 = 5m + 63m + 13 = 5m + 6

13 = 2m + 6

- 3m - 3mOR

- 5m - 5m

-2m + 13 = 6

Will there be 2 different solutions? Let’s find out!- 6 - 6

2 2 7 = 2m

7/2 = m

- 13 - 13 -2m = -7

-2 -2 m = 7/2

Either 1st move canbe used to get the

same result.

Agenda

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There is more than 1 first move from which to choose.

Launch

Ex. 1 3m + 13 = 5m + 6

3m + 13 = 5m + 63m + 13 = 5m + 6

13 = 2m + 6

- 3m - 3mOR

- 5m - 5m

-2m + 13 = 6

Are there any other first moves?

Turn and Talk: Discuss with your partner. See if you can work together to find all the possible first moves.

Agenda

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Launch

3m + 13 = 5m + 6

3m + 13 = 5m + 6

13 = 2m + 6

- 3m - 3m3m + 13 = 5m + 6

- 5m - 5m

-2m + 13 = 6

These are all the possible first steps. Do they all result in the same solution?

3m + 13 = 5m + 6 3m + 13 = 5m + 6- 13 - 13 3m = 5m + -7

- 6 - 6 3m + 7 = 5m

Turn and Talk: Is there one first move that is better to use? Why do you think that one is better than the others?

Agenda

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Launch

3m + 13 = 5m + 6

13 = 2m + 6- 3m - 3m

3m + 13 = 5m + 6- 5m - 5m -2m + 13 = 6

3m + 13 = 5m + 6 3m + 13 = 5m + 6- 13 - 13 3m = 5m + -7

- 6 - 6 3m + 7 = 5m

Some people might say that the calculations are easier if you do not have to divide by a negative.

- 6 - 6

2 2 7 = 2m

7/2 = m

- 13- 13 -2m = -7

m = 7/2-2 -2

- 5m - 5m -2m = -7

-2 -2 m = 7/2

7 = 2m- 3m - 3m

27/2 = m2

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.

These two first moves are similar.

Agenda

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Launch

2 7/2 = m2

- 6 13 = 2m + 6

3m + 13 = 5m + 6- 3m - 3m

- 6 7 = 2m

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.

In general, a preferable order of moves would minimize the need to calculate with negative numbers, fractions, or decimals.

Agenda

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Mini-lesson #2: Shortcut for a fractional coefficient.

Launch

Ex. 2 8 + ¼b = 5 Solve for b. Check your answer.– 8

¼b = -3– 8

Why is subtracting 8 a better first move than subtracting ¼b?

¼ ¼ 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.

Agenda

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How to cancel the fractional coefficient :

Launch

8 + b = 5– 8

b = -3– 8

b = -12

8 + b = 5– 8 – 8

b = -3

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 .

4( ) ( )4

b = -12-3 ÷ =

-3 =

- = -12

Practice: Solve for the variable, substitute to check

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Agenda

1.) -6 + x = -5

3.) 0 = 4 +

2.) x + 9 = - x + 12

nThis is the same as

Answers

Practice

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Agenda

Mini-lesson #3: Distributing a negative sign.

Ex. 3 14 = – (p – 8)

2(x – 5)

– p + 8– 8 – 8

14 =

2x – 106 = -p

-1 -1 -6 = p

Let’s review the distributive property:

-(p) = -1(p) = -p

-(-8) = -1(-8) = +8

Wait, what happened?

Now, solve for p.

Practice: Solve for the variable, substitute to check

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Agenda

4.) -8 = –(x + 4) 5.) 12 = – (-6x – 3)

6.) –(y – 2) + = 3(y + 1)

Answers

Practice: Independent Class work

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AgendaAnswers Next blank section

Practice: Class worksheet

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AgendaGo to AnswersExit Slip

Practice: Class worksheet

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AgendaGo to AnswersExit Slip

Practice: Class worksheet

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AgendaGo to AnswersExit Slip

Exit Slip

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Agenda

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