Summary
This activity gives a fun introduction to two-step equations.
Materials
Additional Resources
Books
Algebra Tiles for the Overhead Projector, by Hilde Howden; ISBN 0-914040-42-1
Background for Teachers
Solving two-step algebraic equations is a concept used throughout
all of algebra. In order to be successful, students must understand
equality and variables (which may be taught using the Pyramid Equality
lesson), as well as the order of operations. They should be able to
add and subtract integers, as well as understand the concept of zero
pairs of tiles. This occurs when a negative x tile and a positive x tile
are together, which create a sum of zero. For example, -2 + 2 = 0.
Similarly, they should understand that addition and subtraction are the
inverse operations of each other, just as multiplication and division are
the inverse operations of one another.
Students must also understand exponents. The expression x^2
means x squared, or x times x.
Algebra tiles provide a useful way to introduce algebra operations
to students of all ages. Students use the tiles as numbers to replace the
variables, which provides a visual image of the equations. This will
make the transition to paper and pencil much easier to understand.
Using the manipulatives will also aid in retention of the concepts.
Instructional Procedures
Invitation to Learn
Write the following question on the board: If tickets to a high
school football game cost $4 per person, explain in words, numbers,
or pictures how you can calculate how much money it will cost your
family to go to the game?
Give students a few minutes to work alone, and then pair them up
to share their strategies. Discuss as a class. (Students should realize
that the number of people per family would cause the cost of the
tickets to vary.
Instructional Procedures
Day 1: Introduction to Algebra Tiles
- Prior to this lesson, give each student two sheets of Algebra Tiles
Cut-Outs on green and red cardstock if you are not using the
commercial tiles. The students should cut them out at home,
put them in a resealable plastic bag, and return them to school
ready to be used. You may also have the students prepare the
tiles in school prior to the lesson.
- Using the transparency Algebra Tiles only, show the students
the tiles and name them. The small squares are units, or ones;
one of them stands for 1, and 5 of them stands for 5. The long
rectangles are each an x, and therefore represent a variable. The
large squares represent x2, x in other words, a variable multiplied
by itself.
- Explain that green tiles indicate addition and red tiles indicate
subtraction. In other words, green tiles are positive and red tiles
are negative, but it is not necessary for students to understand the
term negative as they will not be used in the lesson.
- Make a collection of pieces using all three types of tiles. For
example, use one x2 tile, three x tiles, and 5 ones. Assemble the
pieces on the overhead projector, then write down its name: x2 +
3x + 5. This collection represents an algebraic expression. (See
figure one.)
- Have students create their own expression of tiles using a
maximum of 10 tiles. They should then draw their expressions
in their math journals and write down the name of the expression
using numbers and symbols. Remind the students of the
commutative property; it does not matter what order the addition
occurs: x2 + 3x + 5 is the same as 5 + x2 + 3x.
- To introduce the idea of addition, have the students combine
their expressions with a partner. The students should draw
their new expression and show the addition using numbers
and symbols. For example, the above expression of x2 + 3x + 5
combined with 2x2 + 8 would equal 3x2 + 3x + 13. (See figure
two.)
- To illustrate the idea of subtraction, have the students experiment
with removing tiles. Using their expression from step 5, instruct
them to remove three of any type of tiles. Ask, how can you
write this new equation and answer? Students should draw
their expression and represent it with numbers and symbols.
Ask students to remove 2 more tiles from their new expression.
Again, they should draw their new expression and represent it
with numbers and symbols.
- Repeat the process of pairing up, adding, and subtracting if
desired. The students do not need to use the red tiles for
subtracting at this point.
- After this brief introduction to the tiles, the students are now
ready to solve linear equations.
Solving Two-Step Equations with Algebra Tiles
- Have students take out a white piece of paper and draw a vertical
line down the center. Ask them to model the equation x + 7 = 10
by placing one positive green x tile and seven positive green unit
tiles on the left side of the line, and ten positive green unit tiles
on the right side of the line. (See figure 3.)
- Explain to the students that in order to maintain equality of the
sides, each action must be performed on both sides! Their goal
is to isolate the variable, that is, to make sure the variable stands
alone on one side of the equality sign.
- Ask the students what needs to occur for the variable, or the
positive green x tile, to stand-alone. After responses, remove seven tiles from each side of the equation. Now the x tile and 3
units remain, therefore signifying that x = 3. (See figure 4.)
- Repeat the process with a new equation, 5x 1 = 9. Have
students set up the equation on their own (5 positive green x tiles
and 1 negative red unit on the left; 9 positive units on the right).
Now how can they get the variable to standby itself? They will
need to use the zero pair to remove the one from the left then
add that positive unit to the right. In other words, adding a
positive green unit tile on the left will remove itself and the 1,
and another positive green unit tile also needs to be added to
the right to maintain equality. This creates 5x = 10. Now the
problem becomes much like the pyramid problem the students
must evenly distribute the units on the right among the x tiles on
the left to determine the value of x, which is 2. (See figure 5.)
- Pass out a Can You Use Algebra Tiles? Worksheet for each
student. Have them work on the problems in groups of two or
three, making sure to draw their tiles for each equation.
- Correct and summarize strategies. Continue with the tiles, and
move on to the paper and pencil method when ready. Students
should remember the concept of equality: what you do on one
side must be done on the other.
Day 2: Solving Two-Step Equations with Paper and
Pencil
- Review the Can You Use Algebra Tiles? Worksheet from Day 1.
Look at problem one: 2x 4 = 10. Ask, how can I solve this
problem without using the tiles? Remember, I must maintain
equality!
- Wait for responses. Then teach the two-step paper and pencil
method while demonstrating with the above problem.
Step One: Add or subtract the inverse operation on each side of
the equation.
2x 4 + 4 = 10 + 4. (After simplifying, 2x = 14.)
Step Two: Take the inverse (divide) from both sides of the
equation. This will affect the number directly beside the
variable.
2x / 2 = 14/2. (After simplifying, x = 7.)
- Try problem two from Can You Use Algebra Tiles as a class.
Allow students ownership in finding the answer.
- Work in groups of two or three to solve the rest of the problems
using the paper and pencil method.
Strategies for Diverse Learners
- Provide extra algebra tiles instruction to students with special
needs.
- Allow advanced learners to bypass the algebra tiles when ready
and move to paper and pencil.
Extensions
- Try the activity named Algebraic Equations Gizmo on
explorelearning.com to translate English sentences into
equations and equations into English sentences.
Family Connections
- Use the activity Algebraic Equations Gizmo on explorelearning.
com to translate English sentences into equations and equations
into English sentences.
- Try an input-output game. Students create an equation and
a list of possible variables and solutions. Now cover up your
equation and see if your family can figure it out based on the
other information!
Assessment Plan
- Can You Use Algebra Tiles? Worksheet. You may also revisit
this worksheet after teaching the paper and pencil method
of solving two-stop equations and have students solve each
equation without using tiles.
- Lets Do the Two-Step! Worksheet to be completed after step 18
in the instructional procedures.
- Have students explain the steps to solving equations with
algebra tiles and with paper and pencil in their math journals.
Bibliography
Leitze, A.R., & Kitt, N. A. (2000 September). Using homemade algebra tiles to develop
algebra and prealgebra concepts. Mathematics Teacher, 93, 462-466, 520.
Algebra for all is possible by using algebra tiles as concrete models
in the classroom. This article describes how to use homemade tiles
to reach a broader group of students for successful algebra thinking.
Provides concepts appropriate for this approach.
Leinenbach, M., & Raymond, A.M. (1996). A two-year collaborative action research study
on the effects of a hands-on approach to learning algebra. ERIC Source (ERIC
ED398081). Retrieved November 30, 2006, from http://www.eric.ed.gov
A hands-on approach to algebra enhances students confidence,
interest in, and ability to solve and retain understanding of algebraic
equations. This article describes a two-year research project focused
on two phases and data collection.