Download Unit/Title: Super Evil Mystery Numbers Date(s): 9/30/11 – 10/3/11

Unit/Title: Super Evil Mystery Numbers
Date(s): 9/30/11 – 10/3/11
Lesson Objective(s):



Use word problems to understand how to solve equations that have fractions and variables on
both sides
Develop techniques to deal with fractions in an equation
Solve equations where variable appear on both sides of the equation
TEKS Addressed:
§111.32 Algebra I (a) Basic Understandings (2) Algebraic Thinking and Symbolic Reasoning
§111.32 Algebra I (a) Basic Understandings (4) Relationships between equations and functions
§111.32 Algebra I (b) (A.3) Foundations for Function (A)
§111.32 Algebra I (b) (A.4) Foundations for Function (B)
§111.32 Algebra I (b) (A.7) Linear Functions (B)
§111.32 Algebra I (b) (A.7) Linear Functions (C)
Vocabulary:
Solution Set: The set of values that make an equation true. Denoted S = {answers}
Empty Set (for no solution): The set containing nothing. Denoted S = {} or S = Ø
Identity (for infinite solutions): An equation that is true for every value of the variable.
Lesson Notes:
 The focus is on equations with variables on both sides and fractional equations.
 For Variables on both sides, you will have to discuss and do at least 2 additional problems in
the summary of the lesson. These examples show interesting solutions…ie infinite and none.
For infinite solutions the solution set is S = R (set of all reals). The equations are said to be
identities.
 For Fractions, you will make sure students understand the notation, how to deal with
fractions, and how to get rid of them! Be sure to emphasize that answers must be exact!

1. Students work on warm up for 5 minutes and then turn into manila folder.
2. Students will grade HW 1.11 (The Distributive Property) and review any questions (go over
distributing a negative)
20-30
minutes
2. Super - Evil Mystery Numbers  In Groups…
a) Everyone will find the answers to A – F.
b) When everyone is done, members in the groups will compare answers.
c) Once everyone agrees on the answer to A-F, students will answer part 2 and show
the teacher their sum before moving on to part 3.
d) If the sum is correct (I usually give the kids a sticker), if the sum is not correct,
send them back to figure out what went wrong.
e) Encourage groups that get done early to try to come up with a good challenge
problem in part 3. In the past, I have offered bonus points to groups that come up
with good problems. It is during this lesson that I will give the students the
guidelines for the bonus
1. Must be structured like one of the mystery number problems (written as a
word problem)
2. Must be on a sheet of paper with title and heading.
3. Must have the corresponding equation and a solution.
4. Everything must be correct and the problem has to be challenging!
f) There is one with a fractional answer in this set!  good hint to give the kids!
3. Mystery Numbers  Answers
PART 1
A. 23
B. 17
C. 7
D.
37
/2 = 18 ½
E. 8
F. 7
PART 2
Sum = 80.5
30-40
minutes
4. Mystery Numbers  Whole Class Discussion…
Convert each into an equation. I would break the discussion into two sections as
follows.
Equations with variables on both sides.
Write the equations. Tell students to simplify by doing any necessary distribution
and combine any terms on any given side. (this is not necessary with the given
examples) Once each expression in the equation is simplified, then use APOE and
SPOE to move all of the variables to one side of the equation. Solve as usual.
A) 3x – 8 = 2x + 15
B) x = 68 + -3x
C) x + 56 = 9x
D) 4x + 25 = 6x – 12
Additional Examples
#1. 13x - 5x – 5 = x + 7 + x
8x – 5 = 2x + 7
6x – 5 = 7
6x = 12
x=2
#2. 3(x – 2) + 3x = 4x - 6
3x – 6 + 3x = 4x – 6
6x – 6 = 4x – 6
2x – 6 = -6
2x = 0
x = 0  students will mistake an answer of 0 with no answer at all
#3. 12x + 36 = 2(6x – 10)
12x + 36 = 12x – 20
36 = -20  uh no!
S=Ø
#4. 5x – 3(x + 2) = 2x - 6
5x - 3x – 6 = 2x – 6
2x – 6 = 2x – 6  as soon as both sides are identical, students should stop,
going further (which some do) will just yield 0 = 0
S=R
The equation here is an identity.
Avoiding fractions.
E) ½ x + ¼ = 6
F) 2/3(x – 4) = 2
Discourage students from converting the fractions into decimals unless they are
terminating decimals. Never convert the fractions in an equation into a decimal if it
is a repeating decimal!
Students can always solve these just like regular equations, you will just add,
subtract, multiply and divide by fractions.
If you want you can teach your students the “avoiding fractions until the end”
method. Essentially you use the multiplication property of equality and multiply
both sides of the equations by the least common denominator. The problem with this
is that students will not apply MPOE correctly. For example
E. (correct way) …gets rid of all fractions
4( ½ x + ¼ ) = 4 * 6
Distribute to get
2x + 1 = 24
2x = 23
x = 23/2
E. (the wrong way)
4*½x+¼=4*6
It is a common error to only multiply one term and not the entire side.
F. (correct way) … just get rid of the denominator
2
/3(x – 4) = 2
3* [2/3(x – 4) = 2]
2(x –4) = 6
2x – 8 = 6
2x = 14
x=7
F. (wrong way)
3* 2/3(3*x – 3*4) = 3*2
sometimes they multiply too much!
Additional examples:
#1. ½ x + 1/3 x = 5
#2. 5/6 (x – 2) = 4
Homework: HW 1.12: Solving Equations with Fractions and Variables on Both Sides
Pgs 157 -158: 7, 10, 14, 23, 25, 26, 32, 34, 37, 40
-Unit 1.2 Quiz Corrections, if needed