Alg1 Lesson 2.2: Understanding System Solutions

Lesson 2.2: Understanding

Solutions to Systems of

Equations

Obj: I can find 2 numbers that satisfy a system and verify the solution.

EQ: How can I evaluate a point to verify that it is a
solution to a system?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Poll

How do you plan to contribute to your group today?

• Keep other members on task.

• Ask peers and/or teacher questions about the topic.

• Write expressive notes in my journal.

• Keep retrying the slides until I demonstrate mastery.

Part 1: Warming Up

Find two numbers that have a sum of 10.

What is the difference of those two numbers?

Are they the only two numbers that have this sum and difference?

Try to find another pair of numbers with the same sum and difference.

Dropdown

There are exactly 6 possible combinations of two numbers that have a sum of 10. ​​​

Select answer

1+9, 2+8, 3+7, 4+6, and 5+5 are the only possible combinations of two numbers that have a sum of 10. ​

Select answer

Dropdown

10.5 + -0.5 is a sum of 10. ​

Select answer

There are infinitely many combinations of two numbers that have a sum of 10. ​

Select answer

Multiple Choice

Which equation to model these requirements: “Two numbers whose sum is 10”

• x + y = 10
• 2x + 3y = 10
• x - y = 10
• xy = 10

• x/y = 10

Multiple Choice

Which equation to model these requirements: “Two numbers whose difference is 6.”

• x + y = 6

• 2x + 3y = 6

• x - y = 6

• xy = 6

• x/y = 6

Multiple Choice

What is the intersection of the graphs x + y = 10 and x - y = 6?

• (4, 6)

• (6, 4)
• (8, 2)
• (2, 8)
• (9, 1)

Shapes Puzzle

Circle =

Sun =

Square =

Triangle =

Drag and Drop

Circle = ​

Placeholder

Sun = ​

Placeholder

Square = ​

Placeholder

Triangle = ​

Placeholder

Drag these tiles and drop them in the correct blank above

7

3

5

1

2

4

6

8

9

0

Poll

Which of the following kind of puzzles do you like the best?

• Jigsaw

• Word Search Crossword Sudoku

• Trivia

• Riddles Brain Teasers

• Tetris

  Block Matching

Part 2: Solving a System of

Equations Numerically

Formulate and Solve

Find a pair of numbers that have a sum of -3 and a difference of -7.

Drag and Drop

A pair of numbers whose sum is -3: ​

Placeholder

A pair of numbers whose difference is -7: ​

Placeholder

Drag these tiles and drop them in the correct blank above

m + n = -3

m - n = -7

mn = -3

m + n = 7

mn = -7

Fill out the tables in your journal.

m + n = -3

m

process

n

-6

-6 + n = -3 = 6-3=3

3

-5

-4

-3

-2

-1

0

1

m - n = -7

m

process

n

-6

-6 - n = -7 = 6 - 7 = -1

-1

-5

-4

-3

-2

-1

0

1

Drag and Drop

At what values of m and n are the tables the same?

m = ​

Placeholder

and n = ​

Placeholder

Drag these tiles and drop them in the correct blank above

5

-2

1

-1

-3

3

-5

2

Part 3: Summary and

Practice

Note

It is possible that students will get tired of the calculations and argue that creating a graph gives the solution to a system much more quickly than generating tables of values. This is an important insight, and recognizing it puts students in the perfect position to engage with Lessons 2.3 and 2.4, which explore a graphing approach.

Unless specifically asked, you may use any method in solving system problems that involves paper, pencil, and/or Desmos/TI Nspire.

Handout 2.2: Practice Solving Systems of Equations

Numerically and Graphically

Solve each system of equations by either graphing or constructing a table. Identify which method you chose, and explain why.

1. Equation 1: y = 2x + 2

Equation 2: y = −x +8

2. Let K and T be the current ages of siblings Katie and Thomas. Katie is
currently twice the age of Thomas. In 6 years, Katie will be 4 times Thomas’s
current age. How old are Katie and Thomas?

(a) Write a system of equations to describe the situation.

(b) Find the solution to the system and answer the question.

3. Let x and y represent the tens digit and ones digit of a two-digit number, respectively. The sum of the digits of a two-digit number is 9. If the digits are reversed, the new number is 27 more than the original number. What is the original number?

(a) Write a system of equations to describe the situation.

(b) Find the solution to the system, and answer the question.

4. Mindy’s high school class has a total of 150 students, and they are
planning a field trip. The school has small buses and vans to transport all students. Each van can seat 8 students, and each bus can seat 14
students. There are only 15 drivers, so the total number of vehicles that
they can take must be 15. How many of each vehicle will Mindy’s class
need to transport everyone?

(a) Write a system of equations to describe the situation.

(b) Find the solution to the system, and answer the question.

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the wheel as a class and spend a minute or so
discussing our answers.

Poll

What do you plan to do next?

• Summarize my notes.

• Paraphrase my notes.

• Do Deltamath.

• Exit Ticket on Canvas.

• Try these slides again.