1. TOPIC 2 FOUNDATIONS OF ALGEBRAIC PROBLEM SOLVING
GED® ADULT EDUCATOR MATHEMATICAL REASONING INSTITUTE
FOUNDATIONS OF ALGEBRAIC PROBLEM SOLVING
TOPIC 2

2. Topic 2 - Foundations of Linear Equations & Inequalities

3. LESSON GOALS
A.1.c – Write linear expressions as part of word-to-symbol translations or to represent common settings.
A.2.c – Write one-variable and multi-variable linear equations to represent context.
A.3.a – Solve linear inequalities in one variable with rational number coefficients.
A.3.d – Write linear inequalities in one variable to represent context.
A.6.c – Use slope to identify parallel and perpendicular lines and to solve geometric problems.

4. WORKING WITH THE CONTENT
Think, Pair, Share
Think & Work Alone
Explain how the lesson will work.
You will think individually at times.
You will work with a partner at times.
You will work in small groups at times.
Cooperative Learning In Small Groups

5. Symbols to Words Key Phrase Math Symbols
Sum, Increase, Add, All together, Total
Subtract, Decrease, Difference Minus, Fewer
Times, Multiply, Product
Divide, Per, Quotient
+ (Addition)
- (Subtraction)
x (Multiplication)
÷ (Division)
Begin with discussing the key phrases on slide 5 and how they translate to the math symbols associated with the key words in the phrase.

6. Symbols to Words Activity
Process:
Five posters numbered 1 to 5 with linear expressions written on them
Number off 1 to 5
Start at the poster numbered with your number
15 seconds at each poster to write as many word phrases as you can
Tell the participants that five posters are posted around the room with linear expressions written on them. Have them number off 1 to 5 to determine which of the posters they will start with. Tell the participants that they will have 15 seconds at each poster to brainstorm with your group members as many word phrases as you can to correctly translate the linear expression to a word phrase. When the chime rings move clockwise to the next poster and add any new phrases your can think of to what is already there.
After each group has had 15 seconds at each poster. Have them do a “gallery walk” and look at all the different ways that the class came up with how to write word phrases for each of the linear expressions. Then, ask them to go back to the poster they started with to reunite with their small group for the next part of the activity.

7. Words to algebraic expressions
Process:
Index card with a word phrase
Write the word phrase on your poster
Reach consensus on the correct translation to an algebraic expression
Record translation on poster
Use slide 7 to instruct the participants on the next part of the activity. Have the following written on five different index cards and pass out one card to each group standing at their assigned poster.
1. the difference between twice x and y
2. the difference between the square of x and x
3. the quotient of 10 and the sum of 2 and a number
4. twice the quantity of a number increased by five
5. the quantity of six times a number, all raised to the second power
Allow each group ample time to write the word phrase on their poster and then to brainstorm the correct way to translate it into an algebraic expression. Once they have reached consensus on the correct translation, then record it under the word phrase they wrote on the poster.
When all groups are finished, have the groups rotate clockwise to each of the five posters and spend one minute at each discussing what they see and whether or not they agree with the translation. After all have rotated through, ask them to return to their seats and distribute Symbols to Words and Back! (Handout 1)

8. Try These with a Partner!
For each of the following, write an expression in terms of the given variable that represents the indicated quantity.
The total cost of a mechanic to repair your car if he spends h hours on the job and charges $39 for parts and $45 per hour for labor.
The sum of three consecutive numbers if the first number is n.
Have the participants conduct a Think/Pair/Share with the translations from context to linear expressions with the last four problems at the bottom of Handout 1. Ask them to do them one at a time where they think and come up with an answer on their own, then, they share with their partner. Allow the pairs 3 minutes per problem and call them back to a whole group to share out answers to each problem.

9. Try These with a Partner!
For each of the following, write an expression in terms of the given variable that represents the indicated quantity.
The amount of money in Steve’s bank account if he put in d dollars the first year, $600 more the second year than the first year, and twice as much the third year as the second year.
The first side of a triangle is s yards long. The second side is 3 yards longer than the first side. The third side is three times as long as the second side. What is the perimeter of the triangle in feet?

10. Translating Words to Linear Equations
n + 32 = 40 4x = 36 K - 7 = 15 3w = -15 6/x = 2
A number increased by 32 is equal to 40. Four times a number is 36. Seven less than a number is 15. The product of a number and 3 is -15. Six divided by a number is equal to 2.
Tell the participants that now that we have practiced translating words to algebraic expressions, that we will take it one step farther and translate words and context to linear equations. Remind them that this is GED Assessment Target A.2.c. Using slide 10, talk through the first two translations with the whole group and then ask them to work with a partner to do the last three on the slide.

11. Context to Linear Equations
John called a plumber to fix his broken toilet. In addition to a $50 fee for the visit, the plumber charges $22 per hour. Write an equation that models this situation to determine how many hours the plumber took if John’s total bill was $116.
h = hours the plumber worked h h = 116
Now we will take it another step further and take a situation (context) and translate the information into an equation that can be solved. Introduce this with slide 11 and talk the whole class through how to arrive at the variable assignment and then the equation word phrase by word phrase.

12. Context to Linear Equations
Jane needs $2100 for a vacation for spring break. She plans to save $350 per month for the trip. Write an equation that represents this situation to help Jane determine how many months it will take her to save for the trip at this rate.
m = number of months to save for trip 350m 350m = 2100
Have the participants work in pairs to translate the situation on slide 12 to a linear equation. Then have a few pairs share out their answers and how they thought through the phrases.

13. Context to Multi-variable Linear Equations
A line on a graph represents a ramp that extends from the back of a moving truck to the ground. The line has a slope of -.5 and passes through (8, 0). The y-intercept represents the height of the back of the moving truck. Write an equation with two variables that represents this situation.
y = mx + b y = -.5x + b 0 = -.5(8) + b 0 = -4 + b 4 = b y = -.5x + 4
Use the example on slide 13 to show a situation in which you would use what you know about linear equations to write a multi-variable linear equation to represent context.

14. Linear Inequalities https://www.youtube.com/watch?v=8hhewFQ_K0w
Begin the next part of the lesson by introducing one variable inequalities with this video from YouTube and the Math Dude (7 min. 14 sec.).

15. Inequalities vs. Equations Activity
After the video, distribute Inequalities vs. Equations Activity (Handout 2). Have the participants form groups of three and give them 15 minutes to complete the activity. Once the 15 minutes are up have three groups report out answers to 1, 4, and 5 and then discuss as a whole group the other two answers.

16. Slopes of Parallel & Perpendicular Lines
Use the Khan Academy Video referenced on slide 16 to introduce the content for GED Assessment Target A.6.c. Show the video and ask students to take notes as they watch the video. (Video lasts 9 min. 13 sec.)

17. Try with a partner!
Describe in your own words the relationship of the slopes of parallel lines.
Describe in your own words the relationship of the slopes of perpendicular lines.
Have the participants pair up and answer the following questions from their notes.

18. Try with a partner!
Using what you know about parallel and perpendicular lines and the relationships of their slopes and what you know about writing the equations of lines do the following:
Find the equation of the line that is perpendicular to
y = -4x + 10 and passes through the point (7, 2). Leave your answer in standard form.
Find the equation of the line that is parallel to y = -4x + 10 and passes through the point (7, 2). Leave your answer in standard form.
Have the participants pair up and answer the following questions from their notes.

19. Please come back on time.
Lunch Time!
End the morning session with this slide and tell the participants that they have exactly one hour for lunch and we will begin promptly at 1:00 p.m. for the afternoon session.
Please come back on time.