Students will use input / output tables to help them solve algebraic equations.
Main Curriculum Tie:
Mathematics - 5th GradeStandard 2
Students will use patterns and relations to represent and analyze mathematical problems and number relationships using algebraic symbols.
- Two of Everything by Lily Toy Hong
- Overhead calculator for teacher
- Paper and pencil
Lessons for Algebraic Thinking by Maryann Wickett, Katharine Kharas
and Marilyn Burns
Challenge Math by Edward Zaccaro
Elementary School Mathematics by John A. Van De Walle
Beginning Algebra Thinking for Grades 3-4 and 5-6 by Shirley Hoogeboom
and Judy Goodnow
The Mailbox Intermediate April/May 2001
Disney’s Donald in Mathmagic Land (includes information about
Fibonacci number sequences).
- NCTM E-Standards
- Math Solutions
- Patterns in Mathematics
This site is an interactive activity where students enter in two
numbers and the function is a “mystery.” It is like a function
machine and students can continue to try it with different numbers
until they think they can solve it and then the computer tests them.
Background For Teachers:
The numeric relationship between the frame number and the number of objects
in the frame is a powerful pattern. If a rule can be discovered that can give
the value of any frame in terms of the frame number, the number of objects in
any frame can be determined without building or calculating all of the intermediate
frames in the pattern.
These vocabulary terms should be taught and emphasized during the lesson:
Variable: a letter, symbol, or other placeholder in a mathematical
expression that has an unknown value
Function: a relationship between two variables in which the value
of one variable (often called the output) depends on the value of the other
variable (often called the input)
Equation: a mathematical sentence that contains an equal sign
Intended Learning Outcomes:
3. Reason mathematically.
4. Communicate mathematically.
Invitation to Learn
It is a good idea to give the students plenty of experiences with input/output
tables before having them look for patterns and function relationships. One
way they could do this is to have a race.
Have students come to the board and draw an input/output T-chart with five
rows. Then have each student write down the five numbers you call out on the
input (left) column such as 5, 12, 18, 25, and 40. Remind students that the
input is the number to which an operation rule is applied, and that the output
number is the result of that operation. Next announce a rule such as “add
15” and have each player write the rule at the top of their chart. On
a signal, the students fill in the output column using the rule, then turn and
face the class. When both are finished have the class help you check their work.
They could score one point for each correct answer and five points for whoever
finishes first. Continue playing with different input numbers and different
rules. This will help students feel comfortable with how these types of tables
work. In addition, you could have students who are still in their seats do this
activity in their math notebook at the same time as those who are doing it on
- Begin by reading Two of Everything. Make a T-chart on the board to show
what went into the pot (input value) and what came out (output value).
- Ask what would happen if you put 3 coins in the pot and write “3”
midway down the In column. Students should state that 6 will come out. Continue
by asking what would happen if you put in four coins, then five, ..., ten?
- Ask students to describe to you what patterns they notice in the columns.
- Ask if anyone sees a different pattern. Continue to record responses and
- Lead students to these descriptions: “The Out number is equal to
the In number times two,” or “The Out number is equal to the In
number added to itself,” or “The In plus In equals Out.”
- Explain that an easier way to describe these patterns is to use symbols
(variables) in place of the words. Draw a square above the In column and a
triangle above the Out column.
- Use one of the sentence descriptions, such as “The Out number is equal
to the In number times two.” Ask which symbol represents the Out number
(the triangle) and which symbol represents the In number (the square) and
write the symbolic/algebraic equation:
- Show the other algebraic equations:
- Now refer back to the pot and T-chart and ask the students what would happen
if the pot did something different or followed a new rule. Again make a T-chart,
enter in numbers, and conduct a discussion until students discover the new
rule (function). An example could be:
- Have students guess the next number in the pattern sequence and practice
writing the algebraic equations.
- After the rule/function has been determined, see if students can skip several
numbers and follow the rule for a larger number (e.g., What would the output
be if the In was 30? 150, etc).
- Conclusion: Using organized lists, or tables and variables, helps solve
problems and understand patterns. This is using the power of patterns to predict
beyond the information currently available.
Tell students that they have a “mission possible” and that they
have to figure out the mathematical mystery. But instead of you giving them
the rule and having them figure out the number answers, you are going to give
them the number answers and have them figure out the rule.
Have students prepare to write down a series of numbers and try to figure out
the pattern. Using the overhead calculator and without letting the students
see what you are entering, enter 2 + 5 =; let the students see the 7 then continue
to press the = button several times (they will see 12, 17, 22, 27, etc.).
When several students believe they have figured out the pattern rule allow
them to share. You can also have them observe the number in the ones place.
How long is it before the pattern begins to repeat? Do this activity several
times with different beginning arithmetic problems.
Homework and Family Connections
You could use the Find the Patterns worksheet for homework. Have students make
the tables, find the patterns, write the equations, and answer the questions
Students could also do the extension activity at home and create In/Out charts
with their own rules and fill in several examples on the chart.
Allow the students to work in pairs to choose a rule, write it as an equation,
and generate a T-chart. Brainstorming a list of possible rules might be helpful
so that students who have difficulty deciding on a rule of their own or who
want to make their rules too complicated have some guidelines to follow.
Created Date :
Sep 03 2003 16:32 PM