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Solving Equations and Using Variables as Placeholders
Transcript of Solving Equations and Using Variables as Placeholders
to determine geometric and angle-problem relationships
determine the algebraic relationship between the number of faces, edges, and verticesof a polyhedron
relate the geometric and algebraic representations of the Pythagorean theorem
determine the formula of the area (including surface area) and volume of various complex shapes
determine and formulate the equation of a line
determine the point of intersection of two linear
describe the connections between each algebraic representation and the graphs
solve problems involving the urface areas of prisms, pyramids, and cylinders, and the volumes of prisms, pyramids, cylinders, cones, and spheres
solve problems that arise from realistic situations described in words or represented by given linear systems of two equations involving two variables
The Solving Equations and Using Variables as Placeholders theme aims to provide students with opportunities to develop their ability to make generalizations and deepen their algebraic understanding. Students will explore, develop, select, apply and compare a variety of techniques when solving equations. Students will be initially introduced to variables that serve as unknown placeholders and in the latter grades will use these variables to create equations that define rules for relationships (Edugains, 2010) (The Ontario Curriculum Grade 1-8 Mathematics, 2005) (The Ontario Curriculum Grades 9 and 10 Mathematics, 2005).
Around the Math World
Solving Equations and Using Variables as Placeholders Learning Station
By: Hannah Lee
Key Student Learning
In Grades 7-10 mathematics students will develop and deepen their problem-solving strategies and understanding by:
students will create and solve a diversity of algebraic equations by substituting known values into a formula to find the unknown value
represent linear growing patterns using concrete materials, graph, and algebraic expressions and algebraic equations
simplifying algebraic equations
create equations that define rules for relationships and make generalizations
learning specific components of algebraic equations such as variables, equal sign, constants, operations
solve related algebraic equations that arise from realistic situations
connecting algebraic equations to real life examples such as economics, industry, yield, speed/distance/time, business (Edugains, 2010) (The Ontario Curriculum Grade 1-8 Mathematics, 2005) (The Ontario Curriculum Grades 9 and 10 Mathematics, 2005).
IN GRADE 7, STUDENTS LEARN...
develop and represent a linear growing equation, involving on operation
model everyday relationships involving constant rates, using algebraic equations with variables to represent the changing quantities in the relationship
compare pattern rules that generate a pattern by using various operations to get the next term with pattern rules that use the term number to describe the general term
translate simple mathematical relationship phrase into algebraic expressions, using concrete material
evaluate algebraic expressions by substituting natural numbers for the variables
make connections between evaluating algebraic expressions and determining the term in a pattern using the general term
solve linear equations in the form of ax = c, c = ax, ax + b = c, c = bx + a (where a, b, and c are natural numbers), using a variety of methods
IN GRADE 8, STUDENTS LEARN...
describe different ways in which algebra can be used in everyday situations
translate mathematical relationship statement into algebraic expression and equations
use equations to generalize patterns
evaluate algebraic expressions with up to three terms by substituting the variables
solve simple linear equations using inspection, guess and check, and balance model
IN GRADE 10, STUDENTS LEARN...
solve first-degree equations involving one variable, including equations with fractional coefficients
using a formula, determine the value of a variable with an exponent of one
express the equation of a line in the form y = mx + b (where m is the slope and b is the y-intercept) , given the form ax+ by + c = 0 (where a, b, and c are natural numbers)
manipulate algebraic expressions to understand quadratic relations
solve two linear equations involving two variables with integral coefficients, using the algebraic method of substitution or elimination
choose appropriate algebraic or graphical method to represent real life situations
Connections Across Grades
IN GRADE 9, STUDENTS LEARN...
understand what are inverse operations and how it can used to simplify expressions and solve equations
explain the relationship between algebraic and geometric representations with one variable where the highest exponent of the variables in this equation is up to three
evaluate algebraic expressions involving exponents
identify variables and the relationship between the two variables
IN GRADE 9 (APPLIED), STUDENTS LEARN...
add and subtract polynomials with the same variable that has the highest exponent up to three
multiply a polynomial by a monomial involving the same variable using a variety of tools
solve first-degree equations with non-fractional coefficients, using a variety of tools and strategies
substitute into algebraic equations and solve for one variable in the first degree
IN GRADE 9 (ACADEMIC), STUDENTS LEARN...
add and subtract polynomials with up to two variables
multiply a polynomial by a monomial involving the same variable
expand and simplify polynomial expressions involving one variable
solve equation where the highest exponent of the variables in this equation is one (first degree) using a variety of tools
rearrange formulas with variables in the first degree with and without substitution
compare solutions of algebraic methods to other solution methods
Unto the next adventure....
Connections Across Strands
Concepts related to solving equations and using variables are not only incorporated in the Patterning and Algebra unit, but are also included in the Number Sense Unit, Measurement Unit, Geometry Unit, and Data Management Unit (Edugains, 2010).
IN Later years, STUDENTS LEARN...
Depending on students' choice to continue with math it will determine the degree to which they apply their understanding of concepts related to solving equations and using variables.
Algebra is required to:
solve for length, width, height and perimeter of various shapes
develop equations to solve for the area of composite figures and trapezoids
develop equations to solve for the volume of right prisms with polygonal bases
develop equations to solve for the circumference and area of a circle
determine and solve the formula for the volume and surface area of a cylinder
develop formulas and solve for various shapes such as pyramids, cones, spheres and triangles (Pythagorean theorem)
solve problems involving similar triangles and primary trigonometric ratios
Algebra is required to:
understand estimation, evaluate expressions with decimals, fractions and integers
represent and solve problems involving operations with integers and exponential notation
solve for unknown variables in a proportion
Algebra is required to:
draw conclusions from data
identify trends in data
compare the theoretical and experimental probability
interpret and draw conclusions from data
identify trends based on rate of change of data
Common Challenges and Misconceptions
2) Students may ignore the letter which represents a variable and replace them with numerical values, or regarded the letters as standing for shorthand names. (For example: 3m students may get it confused for 3 meters. But, in algebra, 3m means 3 times some unknown number of meters.). Important to be careful with what letter you use and be explicit that it is a variable (Booth, 2008)
3) students may consider ab to mean the same as a + b (Booth, 2008)
1) Understanding what positive and negative sign means. Failing to tie the negative sign to the term it modifies or to understand how changing or moving a negative sign impacts the equation can cause many problems when learning of equation-solving procedures (Booth, 2008)
4) Equality and the meaning of the equals sign in particular are difficult for students who are in the process of transitioning from arithmetic to algebraic thinking. Students often think of the equals sign as an indicator of the result of operations being performed or the answer to the problem rather than of equivalence of two phrases. One way of helping students understand is making a scale where you can add numbers so both sides of the scale are equal (Booth, 2008)
5) Students think that only the letters x and y can be used for variables
6) The variable is always on the left side of the equation
7) Understanding distributive property and integer operations when solving linear equations
8) Making a mistake when solving the equation.
To overcome this challenge put your answers back into the equations, they should make both equations true when you plug them in (ACCL)
9) students’ misinterpretations of the distributive law
10) Students may have a hard time understanding the difference between adding/subtracting exponents and multiplying/dividing exponents
11) Re-arranging the equation and then substituting it back into itself. This will make everything cancel out (ACCL)
Teacher Resources (Activites)
1) Solving One and Two Step Equation Mazes
The activity is a compilation of mazes that will help students learn to solve simple one and two step equations.
Author Unknown. (2010). Continuum and Connections - Solving Equations and Using Variables as Placeholders. Edugains. Retrieved from http://www.edugains.ca/resources/LearningMaterials/ContinuumConnection/SolvingEquations.pdf
Booth. J.L., & Koedinger, K.R. (2008). Key Misconceptions in Algebraic Problem Solving. Carnegie Mellon University, PACT Center. Retrieved from
Ontario Ministry of Education. (2005). The Ontario Curriculum Grades 1-8, Mathematics. Toronto, ON: Queen's Printer for Ontario.
Ontario Ministry of Education. (2005). The Ontario Curriculum Grades 9 and 10, Mathematics. Toronto, ON: Queen's Printer for Ontario.
System of Equations - Common Mistakes. Retrieved from http://www.epcc.edu/CollegeReadiness/Documents/CM_Systems_of_Equations_20-50.pdf
TeachersPayTeachers. Retrieved from https://www.teacherspayteachers.com/
2) Choose Your Own Adventure
In this assignment you'll explore a mystery country. As you make choices and solve problems you'll receive pieces of information about where in the world you are.
3) Solving Multi-Step Equations Stations Maze
Each group will begin at a station. The students will answer the questions and move to the stations that correspond with their answer. For example, group 5 should begin at station 5. If their answer says to go to station 9, that’s the station they will visit next. They should follow this procedure until that have visited all of the stations. If a group is sent back to a station they have previously visited, they know they have made a mistake and should go back and check their work.
4) Multi-Step Equations Relay Race
In groups of 4 each member does one problem and passes it on to the next member when they complete it
5) Bingo Equation Game
Standard bingo game where the caller calls out equations and students have to solve and match with numbers on their card.
Teacher Resources (Web)
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