Mathematical Activities - Solving for two unknowns four ways. (8-12 Grade)



In this investigation students explore equations by solving problems of two equations with two unknowns four different ways: graphing, substitution, addition, and determinants and communication what an equation and graph represent.

Mathematical Processes - Problem solving, representation, proof and reasoning, communications, and connections

(How mathematics inquires - process, skill, methodology)

Equations can .

Concepts and facts -

  • A general problem solving procedure can be used with others to solve a problem and solutions can be demonstrated to others.


  • Solve a problem using a problem solving heuristic.
  • Use different representations to communicate, by demonstrating solutions.

Specific outcomes -

  1. Problem solving heuristic includes general steps for solving mathematical problems
  2. Use graphs to represent equations and mathematic relationships of variables.
  3. Solve two equations with two unknowns using four methods: graphing, substitution, addition, and determinants.
  4. Communicate with peers of the same and opposite sex mathematical ideas.

Mathematical content - number value, geometry, measurement, algebra, data analysis, statistics, and probability

(What mathematics explains - enduring understanding, big ideas, generalizations)

Graphs are a graphic representation of a relationship between variables.

Concepts and facts -

  • Variables are conditions that can be changed and that can affect outcomes.
  • Variables can represent, size, shape, temperature, amount, volume, rate, ...


  • Solve and demonstrate the solutions to problems using graphing, substitution, addition, and determinants.

Specific outcomes -

  1. Solve a problem using graphing, substitution, addition, and determinants.
  2. Identify and explain attributes for graphing (x, y, y intercept, x intercept, slope, y=mx + b) as they apply to different problems.
  3. Develop a procedure for solving problems by graphing, substitution, addition, and determinants.
  4. Demonstrate method for solving problem by graphing, substitution, addition, and determinants; and generalize them to problems of a certain type.
  5. Identify variables and describe how they operate to effect other variables. (operational definition).

Habits of mind - values, attitudes

(Attitudes and values that contribute to mathematical success)

Curiosity, open-minded, skepticism, persistence, .

Concepts and facts -

  • Both sexes are equally capable of learning and teaching math.
  • Diverse populations can achieve similar results.
  • Cooperation brings together diverse ideas for the benefit or all.

Outcome - cooperation and planning

  • Develop attitudes and skills for cooperation.
  • Develop planning skills.

Specific outcomes -

  1. Cooperations looks, sounds, and feels like:

Activities - to provide sufficient opportunities for students to attain the targeted outcomes.

Possible Activity Sequence

  1. Divide students into four groups with equal numbers of girls and boys in each group. Consider separating girls and boys who usually associate with each other and students at the same skill levels and other diverse groups.
  2. Assign each group one of four methods for solving two equations with two unknowns: graphing, substitution, addition, and determinants
  3. Tell students to use their textbooks and supplementary books to learn how to use this method. They should work together to apply the method to various problems in the books and to make sure that everyone in the group can solve the equations using the given method.
  4. Once students feel confident about using the method being learned by their group, the teacher should check their problems to be sure they understand it.
  5. When the teacher assesses the students understand the method, the group can begin to design its demonstration, which will demonstrate the method to other students.
  6. The demonstration should have four part: purpose, materials, procedure, and proof and reasoning for confidence of the demonstrated method. Each student in the group needs to make a contribution to the demonstration. The demonstration can be as long or short as the group feels is necessary to demonstrate the method and justify it's methodology. Each group member should be able to demonstrate their method.
  7. The teacher should assess each group’s demonstration to see how the method will be communicated and make copies of the demonstration for each student in the group.
  8. Divide students into four new groups with equal numbers of girls and boys in each group. Each group should have one or two students who can do one of each of the four demonstrations. Again separate girls and boys, skill levels, and other diverse groups.
  9. Each student in the group should demonstrate his or her method for solving equations to the other group members.
  10. The students should make sure that all of the members of the group can perform the four methods. When the groups feel they are confident about the methods, the game begins.
  11. Each student in the group should choose a number from 1 to 4 (or up to the number of students in the group), which will serve to identify the student from each group who will be competing. Choose a number, and the student from each group with that number should come to the board. Read the problem, and the students must write and solve the problem. The problem must be solved by the method specified by the teacher. The groups earn four, three, two, or one point for solving the problem first, second, third, or fourth, respectively. No points are given to a group that solves the problem incorrectly. All groups are required to keep score. The winning team may be given some reward, although the reward should not be overemphasized.
  12. On the day after the game is played, give the students a quiz on which they must demonstrate their abilities to solve the equations.


Resources and materials

Lab notes, graphing calculator, graphing data sheet,

Pedagogical ideas

The jigsaw model asks students understand a method of solving equations, communicate their understanding to her or his peers, and to attempt to help the other group members to understand it and plan a demonstration for the method to the other peers. This enables every class member to have an area of expertise and empowers all students to be valuable and capable among their peers, which helps improve students’ self-efficacy with math and social communications with peers.

Reduce stereotyping: Students often view math more as a male subject or other racial preferences rather as gender and racial neutral. Also many begin to doubt their competence in math and can too easily believe they can’t do math. Some students, including females, tend to learn math better when taught using groups rather than the traditional competitive classroom methods of teacher demonstrating and lecturing. Making this a truly cooperative lesson would further reduce the competition. Removal of the game or change it to make it a competition against accuracy rather time will also encourage more students.

Asking each student to prepare as an expert to demonstrate his or her method to his or her peers can provide motivation and focus.

Focus questions -

Scoring guides

Problem solving heuristic and strategy

Solve equations using graphing, substitution, addition, and determinants

Communication of mathematical ideas.

Habits of mind for cooperation - listen, don't interrupt, respond to others ideas, use others ideas as example or as showing an exception

Treatment of all students is not biased and equal.




Activity: Order objects from hot to cold by touching them

Materials:  sample problem for each group


Put students into work alike groups according to: graphing, substitution, addition, and determinants

Give each group their problem:

  1. Graphing - Ralph started a lawn service business and is going to pay him self two dollars an hour more than what he is paying Chris. What equation will describe both salaries? What would the wages be for different hour amounts? What would a graph of this relationship look like?
  2. Substitution -
  3. Addition - and
  4. Determinants



Materials:  resources for each problem type: 1. Graphing - Lab notes, graphing calculator, graphing data sheet, 2. substitution - 3. Addition - and 4. Determinants


  1. Graphing - Describe what each of the following are and the relationship to your problem: x, y, y intercept, x intercept, slope, y=mx + b, and create a procedure to demonstrate how to use graphing to solve real world problems and relate each of these ideas to a graphing method.
  2. Substitution -
  3. Addition - and
  4. Determinants




Create other problems and describe ways to decide which of the four procedures might be most appropriate.

  1. Graphing -
  2. Substitution -
  3. Addition - and
  4. Determinants


Lab Notes

Graphing Method
x + y = 4
x – y = 4
y = x + 4
y = x – 4

Solution: (4, 0)

x | y
0 | 4
1 | 3
2 | 2
3 | 1
4 | 0

Substitution Method

3x + 5y = 3
x + 4y = 8
3(8-4y) +5y = 3
24 – 12y + 5y = 3
-7y = -21
y = 3
x = -4
Solution: (-4, 3)

Determinants Method
3x – 2y = 7
3x + 2y = 9
D = 3 -2 = 6 – (-6) = 12
3 2

Dx = 7 -2 = 14 – (-18) = 32
9 2

Dy = 3 7 =27 – 21 = 6
3 9

x = Dx/D = 32/12 = 8/3

y = Dy/D = 6/12 = ½
Solution: (8/3, ½)

Addition Method

x + y = 2
3x – 2y = 0
2x + 2y = 4
3x – 2y = 0
5x = 4
X = 4/5
4/5 + y = 2
y = 10/5 – 4/5
y = 6/5
Solution: (4/5, 6/5)


Dr. Robert Sweetland's Notes ©