This activity introduces students to basic algebraic equations using counters, cups, and drawings.
- A two-pan balance scale with weights
- 3-ounce cups
- Plain piece of paper
- Beans or counters (25 for each pair of students)
Hands-On Math by Frances M. Thompson
Safari Park by Stuart J. Murphy (MathStart Series)
Background for Teachers
To get a feel for what students know about algebra, have students write their
response to this question in their math journals: “What do you think it
means to think algebraically?” Have students share their responses either
orally or by turning in their responses for you to review. This will inform
you what students know about algebra at this point in their schooling. Students
may not realize what they know or that they have actually been thinking algebraically
for several years. This lesson can help your students see the connection between
arithmetic and algebra.
Intended Learning Outcomes
2. Become mathematical problem solvers.
Invitation to Learn
Show the students the balance scale. Show them how both sides need to have the
same amount of weight on them to balance. Demonstrate how adding or removing
objects from one or both pans affects the balance. Explain that mathematical
equations must be equal on both sides to balance.
- Give each pair of students one piece of paper, one cup, and 25 counters.
Have the students fold their paper in half, open it up and draw a line down
the middle. Explain that this paper will act as a balance scale and each side
of the paper has to remain the same to be balanced.
- Have the students hide four counters under their cup on the left half of
their paper. Ask how many counters would need to be on the right side to make
their paper scale balance? (4)
- Now have them empty the cup and put it on the left side of the paper with
3 counters next to it. Put 8 counters on the right side of the paper. Tell
the students that we will pretend that the correct number of counters are
under the cup to make the paper scale balance. To be able to find out how
many counters are under the cup, you could remove the same amount of counters
from each side (thereby keeping it balanced) until all the beans you can see
on the left are gone (what remains under the cup will equal what remains on
the right side).
- Students should remove three counters from the left and in turn remove
three from the right. What remains shows what is "hiding" under
the cup (5). To verify this, have students put 8 counters on each side. Cover
up 5 of them with the cup and then do the same procedure as before. When they
lift the cup, they will see 5 counters on each side.
- Write the equation for this problem: x + 3 = 8.
- Write the equation for what happened:
- Now give the students other equations to solve (you may have to model a
few more first). Here are some sample problems:
- x + 5 = 12
- x + 8 = 20
- 24 = 14 + x
- 15 = x + 3
- Have students share their results and write number sentences on the board
to record the steps they used.
- To move from the concrete to pictorial practice, you will now have students
draw the counters, balance scale, and cup to solve equations.
- In their math journals, have students draw a similar type diagram:
- Remind students that the "cup/unknown/variable" has to be all
alone on one side to reveal what is beneath (it's value). Ask them to
get rid of or cross out the circles on the left and a corresponding amount
on the right. Then have them write the equation they just completed, including
the value of x:
|x = 5 = 9
| - 5 = - 5
|x = 4
- You may do several more problems like this before moving on.
- Now give students basic one-step equations to solve and have them do it
without counters, cups, or drawing. Emphasize that you are doing "inverse
operations" so that when problems look like this: x -- 3 = 10,
the students will know to add. It isn't recommended that you try to
do this with the counters or the drawing though, just with the equations.
This can lead into solving equations with larger numbers too such as: 112
+ x = 245, or x -- 3,000 = 5,000.
Some students may need to continue to use the counters and cups throughout the
entire activity. Make them available for that purpose. Also, a literature book
that gives students practice with equations and finding unknowns is Safari
Park by Stuart J. Murphy.
Homework & Family Connections
Send home a cup with beans and the paper used in class with each student. Have
them show their parents the activity they did in class and explain how to solve
for missing addends.
Give students several problems to solve equations with variables. Many textbooks
have lessons and practice problems like this or you could create your own.