should work in groups or pairs. This will encourage discussion during
the lesson, which will help with understanding the manipulative
representation. Try to match students who are likely to understand the
manipulatives with students who may have trouble initially.
Begin the lesson by distributing the Geology Rocks Equations Activity Sheet and the manipulatives. Have students read the opening
paragraph and Question 1 on their own. Use the first question as a
demonstration of what they will be doing with the questions that
Geology Rocks Equations Activity Sheet
Tell students that the blocks represent the crates (which hold the same number of rocks), and each
chip counter represents a 1-pound rock. For simplicity's sake, assume that the weight of each crate is also zero. Ask students to create a
representation of the problem on their desks using the manipulatives.
Then, ask them to discuss in their groups ways to figure out how many
rocks are in the crates using only this information.
You may want to allow students time to explore and share their
thoughts within their groups about how they are using the manipulatives
to help them solve the problem. After you have given students time to
explore, ask the groups to share their answers with the class. You
could also allow students to create their own representation with the
chips, which will help ensure that each student understands the
problem. After 30 seconds or so, they can pair-share to correct any
problems with their representations and discuss their solutions.
The correct answer to Question 1 is that there are 2 rocks in
each crate. Have students who found the correct solution share their
strategy with the class. If no students found the answer, lead the
class through the process:
1. Mr. Anderson realizes that he can remove 2 rocks from each
side of the pan balance and still have a balanced pan balance. This leaves him with
2. Then, Mr. Anderson realizes that he can remove a crate from
each side of the pan balance and still remain balanced. This leaves him with
Students can represent these situations algebraically. Use boxes instead of x's:
+ + 2 = + 4
This could then be simplified to:
2 + 2 = + 4
Students can remove boxes and numbers just as they did on the pan balance. Teachers often give students manipulatives and expect them to
complete the analogy on their own. However, quite often, students are
not able to make the connection. The teaching is much more effective if
you make the analogy explicit for students. That is, explain explicitly
how the boxes and rocks represent variables and numbers.
Some students may have used a guess-and-check strategy until
they found out the weight of the crate, while others may have
simplified the equation. As an introduction to the problem, this is
fine. As students begin to think-pair-share their strategies to balance
the equation, encourage those who are using guess-and-check to
collaborate with those using algebra.
Guess and check is a valid strategy when beginning with these
equations; it requires solving the problem mathematically using a
step-by-step process. Students are performing the operations multiple
times and finding solutions that do not work, but at the same time they
are gaining practice until they find the solution that does work.
Allow students to work on their own to complete Questions 2
to 5. As you move through the class, ask students to explain to you how
the equation they are building using the manipulatives represents the
problem on the paper. Encourage students to use multiple approaches to
solve each problem—this will help them build the connections between
After students have finished Questions 2 to 5, have them share
within their group how many rocks they think are in each crate. For
each solution, ask the class if they agree. If all groups agree, move
on to the next problem. If someone disagrees have the class re-create
the problem using the manipulatives and walk through the solution steps
together to agree on the solution. Use student volunteers where
possible as you go through the answers.
Activity Sheet Answers
2. x = 4
3. x = 12
4. x = 4
5. x = 7
Once the class has agreed on all the solutions, inform students
that the solutions they created are actually linear equations and can
be represented mathematically with equations, similar to the equations
you showed earlier using boxes.
Ask students to look over each problem and write an equation
they think mathematically represents the initial picture Mr. Anderson
made. Some students may not immediately make the connection that the
boxes stand for unknown values of x and the rocks represent
constants. You should stress this connection as students are working.
To help them with the connection, you can have students use x's or boxes or both. Some students may have an easier time making the connection using the instead of x.
2. 4x + 2 = 18 or 4 + 2 = 18
3. 2x + 12 = 3x or 2 + 12 = 3
4. 2x + 8 = x + 12 or 2 + 8 = + 12
5. 5x + 3 = 2x + 24 or 5 + 3 = 2 + 24
As soon as students have created the equations, discuss them as
a class so they are agreed upon. When students have finished writing
their equations, allow them to complete Questions 6 to 9 on the
activity sheet. Allow students to solve these equations without or
without manipulatives, depending on their own preference.
6. x = 3
7. x = 6
8. x = 4
9. x = 3
To summarize the lesson, have students share with the class any
strategies they came up with that make it quicker to solve the problems
without creating a manipulative representation each time. Students may
have realized that you are subtracting or adding a constant, and then
dividing by the coefficient of x. Discuss with students the
steps involved in solving these equations. You are isolating the
numbers on one side of the equation and the unknown on the other. You
are grouping like terms and solving. Use an example not on the activity
sheet, such as the one below in your discussion:
Ask students to write down on their activity sheet a list of
steps to solve the problems and compare their lists with one another.
Most solutions will require at least 2 steps to solve. As students
begin to more fully grasp the process of 2-step equations, ask them to
solve the equation 5x – 8 = 1x + 2. In solving this
equation, students will be faced with using negative integers and
accepting a solution that is not a whole number (x = 2.5).
Students may find it tough to create manipulative representations of
these harder equations. Allow them to explore and work it out on their
own. This is truly an exploration for students.
Lappan, G., J. Fey, W. Fitzgerald, S. Friel, and E. Philips. 2005. Investigation 3.2: Exploring equality. In Moving straight ahead: Linear relationships. Connected Mathematics 2 Series, Pearson Prentice Hall.