Source: Wiggins, G., & McTighe, J. Understanding by Design. Merrill Prentice Hall: 1998.
For further information about Backward Design refer to http://www.ubdexchange.org/
Title: Calendar Math Subject/Course: Algebra
Designer(s): Pam Joseph
Topic: Algebra
Grade(s): 7-8
In this activity, students will use a fun mathematical puzzle to demonstrate solving a simple,
linear equation. It is an easy math trick, but an excellent way to demonstrate algebraic
applications.
Stage 1 – Desired Results
Established Goal(s)
Georgia Performance Standards-Mathematics grades 7-8:
M7A1 Students will represent and evaluate quantities using algebraic expressions.
a. Translate verbal phrases to algebraic expressions.
b. Simplify and evaluate algebraic expressions, using commutative, associative,
and distributive properties as appropriate.
c. Add and subtract linear expressions.
M7A2 Students will understand and apply linear equations in one variable.
a. Given a problem, define a variable, write an equation, solve the equation, and
interpret the solution.
b. Use the addition and multiplication properties of equality to solve one- and
two-step linear equations.
M8A1 Students will use algebra to represent, analyze, and solve problems.
a. Represent a given situation using algebraic expressions or equations in one
variable.
b. Simplify and evaluate algebraic expressions.
c. Solve algebraic equations in one variable, including equations involving
absolute values.
d. Interpret solutions in problem contexts.
M8A2 Students will understand and graph inequalities in one variable.
a. Represent a given situation using an inequality in one variable.
b. Use the properties of inequality to solve inequalities.
c. Graph the solution of an inequality on a number line.
d. Interpret solutions in problem contexts.
M8A3 Students will understand relations and linear functions.
a. Recognize a relation as a correspondence between varying quantities.
b. Recognize a function as a correspondence between inputs and outputs where
the output for each input must be unique.
c. Distinguish between relations that are functions and those that are not functions.
d. Recognize functions in a variety of representations and a variety of contexts.
e. Use tables to describe sequences recursively and with a formula in closed form.
f. Understand and recognize arithmetic sequences as linear functions with
whole-number input values.
h. Interpret the constant difference in an arithmetic sequence as the slope of the
associated linear function.
i. Identify relations and functions as linear or nonlinear.
j. Translate among verbal, tabular, graphic, and algebraic representations of functions.
M8A5 Students will understand systems of linear equations and use them to solve
problems.
Source: Wiggins, G., & McTighe, J. Understanding by Design. Merrill Prentice Hall: 1998.
For further information about Backward Design refer to http://www.ubdexchange.org/
a. Given a problem context, write an appropriate system of linear equations.
b. Solve systems of equations graphically and algebraically, using technology as
appropriate.
c. Interpret solutions in problem contexts.
Understanding(s) Students will understand
that...
1. in real-world, familiar experiences, one may
assign variables to different concepts, make an
equation and solve it using algebra.
U
Essential Question(s)
1. What is an example of something
familiar from everyday life that uses
algebra?
2. How does one assign variables to
concepts in order to make an algebraic
equation?
22Q
Students will know...
Students will be able to...
that when assigning variables to concepts, you connect algebra to a real-world experience
must begin with one variable, then attach the
using a calendar math puzzle.
necessary components to that variable, e.g., n is
the first number, then n+ ?or n X ? is the
second number and so on depending on what is
stated in the problem.
K
Source: Wiggins, G., & McTighe, J. Understanding by Design. Merrill Prentice Hall: 1998.
For further information about Backward Design refer to http://www.ubdexchange.org/
Stage 2 – Assessment Evidence
Performance Task(s) Summary in G.R.A.S.P.S. form
Students will be assessed in the following ways:
 Practicing solving the puzzle with each other;
 Evaluation of student-designed puzzles;
 Observation as they play their own puzzles.
T
Key Criteria:
Key criteria are the ability to explain how to assign variables, how to set up the equation, and how to solve
the equation.
Other Evidence
Teacher observation, answers, explanations
Source: Wiggins, G., & McTighe, J. Understanding by Design. Merrill Prentice Hall: 1998.
For further information about Backward Design refer to http://www.ubdexchange.org/
Stage 3 – Learning Plan
Learning Activities Consider the W.H.E.R.E.T.O. elements.
1. First, I will draw a giant calendar on the board. I will also hand out copies of the calendar to the
students.
2. I ask them to choose 4 days (and not to tell me what they are) that form a square in the calendar such
as:
18
19
25
26
3.
4.
5.
6.
I ask them to sum the 4 days and tell me only the sum of the days, not the actual days.
I tell them that I will now be able to tell them what the 4 days are.
I ask “How does this puzzle work?”
I say” You know how people want to see a use for algebra with something familiar to us in everyday
life. This puzzle with a common calendar uses algebra.
7. I would use the 4 numbers as the example. Let’s say they gave me the 4 numbers from # 2 and only tell
me that the sum is 88.
8. I demonstrate assigning variables to the different days and writing and solving the equation:
Let’s call the first number n. Then you know that the next number would be n + 1 and the next number is n
+ 7. The last number would be n + 8. Explain why the 1, 7 and 8 are these numbers.
9. Now add the 4 numbers and make the sum equal 88:
n + n + 1 + n + 7 + n + 8 = 88
10. Simplify the equation by adding like terms:
4n + 16 = 88
11. Ask- How would this be solved? First, subtract 16 from both sides. Then divide both sides by 4. This
gives us the first number, n. Then add 1, 7 and 8 for the other numbers.
12. Let the students practice solving the puzzle with each other.
13. Demonstrate creating a different puzzle.
14. Explain how they will make their own puzzle.
Extension
Alternate method: Subtracting 16 mentally is not easy, so one may factor 4 from the left side of the
equation:
4(n + 4) = 88
Now, divide both sides by 4:
(n + 4) = 22
Subtract 4 from both sides:
n = 18
This is easier to do mentally. Divide by four and then subtract 4.
Algebra makes all of this possible!
References:
http://math.rice.edu/~lanius/Lessons/cnotes.html
http://math.rice.edu/~Lessons/calen.html