Minds On
Variables
A variable is an unspecified value often represented using a letter or a symbol. When we notice a variable appear multiple times in the same question, we know that it represents the same number. For example, n + n = 6.
In this question, the variable “n” can only represent 3.
Sometimes variables are not represented with letters but symbols. Choose one of the following options and determine if you can solve the questions.
Record your thinking digitally, orally or in print.
Option 1
Option 1 is a list of variables represented as letters in a series of equations with a riddle to solve.
x + x + x + y + z = 31
y + y + y + y + y = 15
v + y + w + w + w = 28
z + z + y + w + w = 23
y + y + x + y + y = 20
w + y + y + x + z = 24
x + y + z + w + y = ??
Option 2
Option 2 is a riddle using symbols to solve the hidden puzzle.
There are seven equations that use pictures as variables. The first equation is sun plus sun plus sun plus boat plus clover equals 31. The second equation is boat plus boat plus boat plus boat plus boat equals 15. The third equation is house plus boat plus cloud plus cloud plus cloud equals 28. The fourth equation is clover plus clover plus boat plus cloud plus cloud equals 23. The fifth equation is boat plus boat plus sun plus boat plus boat equals 20. The sixth equation is cloud plus boat plus boat plus sun plus clover equals 24. The seventh equation is sun plus boat plus clover plus cloud plus house equals an unknown.
Action
The unknown
Regardless of whether you use letters or symbols, when you are using a variable you are representing an unknown number.
It is a placeholder that represents an unknown that can change.
Task 1: Three patterns
Examine the following patterns.
Table of values activity
We will now record each of the patterns into a table of values.
Answer the following questions to fill out the table of values. You can record the patterns in the following fillable and printable Three Patterns Table of Values document or use another method of your choice.
| 1. For each of the patterns, what is changing? (multiplier) |
|---|
| Pattern 1: |
| Pattern 2: |
| Pattern 3: |
| 2. For each of the patterns, what is staying the same? (constant) |
|---|
| Pattern 1: |
| Pattern 2: |
| Pattern 3: |
Press the ‘Activity’ button to access Three Patterns Table of Values.
When you are ready, press the ‘Answers’ button to reveal the table of values for each pattern.
Pattern #1
- multiplier: 2 squares
- constant: 3 circles
- pattern rule: term number multiplied by 2 + 3
| Pattern #1 | |
|---|---|
| Term number | Term value |
| 1 | 5 |
| 2 | 7 |
| 3 | 9 |
| 4 | 11 |
| 5 | 13 |
| 10 | 23 |
| 0 | 3 |
Pattern #2
- multiplier: 2 squares
- constant: 1 triangle
- pattern rule: term number multiplied by 2 + 1
| Pattern #2 | |
|---|---|
| Term number | Term value |
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
| 5 | 11 |
| 10 | 21 |
| 0 | 1 |
Pattern #3
- multiplier: 1 pentagon
- constant: 2 squares
- pattern rule: term number + 2
| Pattern #3 | |
|---|---|
| Term number | Term value |
| 1 | 3 |
| 2 | 4 |
| 3 | 5 |
| 4 | 6 |
| 5 | 7 |
| 10 | 12 |
| 0 | 2 |
In each pattern, the 0 term value is important to know because it tells you what the constant is.
Task 2: Planning a virtual event
Let’s explore the idea of planning an online event, such as a virtual celebration.
- The fixed or constant cost of using an online platform is a flat fee of $75.00.
- The variable or multiplier cost per person is $0.25 cents.
Questions
Answer the following questions about the virtual event.
If you’d like, you can answer these questions using the following fillable and printable Task 2: Planning a Virtual Event document. You can also complete these questions in your notebook or using another method.
Let’s explore the idea of planning an online event, such
as a virtual celebration.
|
|||
|---|---|---|---|
| 1. Use the information to create an algebraic expression. | |||
| 2. How could you use the information provided to show the cost of 10 people, 50 people, 100 people and 500 people attending the celebration? | |||
| 10 people | 50 people | 100 people | 500 people |
Press the ‘Activity’ button to access the Task 2: Planning a Virtual Event.
When you are ready, press the ‘Answers’ button to reveal the answers for Task 2.
- $75.00 + 0.25p
-
-
If p = 10
$75.00 + 0.25p
= $75.00 + 0.25 (10)
= $77.50
If p = 50
$75.00 + 0.25p
= $75.00 + 0.25 (50)
= $87.50
If p = 100
$75.00 + 0.25p
= $75.00 + 0.25 (100)
= $100
If p = 500
$75.00 + 0.25p
= $75.00 + 0.25 (500)
= $200
-
- The Term Number would be the number of people attending.
- The Term Value would be the total cost of planning the online event.
-
Term Number Term Value ($) 0 75 1 75.25 2 75.50 3 75.75 4 76 - The pattern rule is $75 + n (0.25)
Consolidation
Task 1: Investigate the sequence
Examine the following table of values:
| Term Number | Term Value ($) |
|---|---|
| 0 | 45 |
| 1 | 50 |
| 2 | 55 |
| 3 | |
| 4 | 65 |
| 40 | 245 |
A student correctly determined that the nth term of the sequence {45, 50, 55, …} is 45 + (5n).
Questions
Now, answer the following questions. Record your responses digitally, orally or in print.
Using this information, what would be the 3rd term?
The student then determined that the 40th term number would be 245. Is this correct? What do you think the student did to get this calculation?
Use the information provided to check.
Justify the student’s answer using mathematical terminology learned in this learning activity (variable, constant, multiplier).
Can you think of a real-life scenario for this problem?
Task 2: Explore an expression
Choose a number on the following hundreds chart and replace the value with the variable s.
Questions
Now, answer the following questions. Record your responses digitally, orally or in print.
- Why does the expression s + 10 describe the number directly below s?
- Record the two numbers below s in terms of s.
- What algebraic expression would result from adding s to the two numbers directly below it?
- Write an equivalent expression for your answer to question 3.
- Write two equivalent expressions for adding s to the two numbers on each side of that square.
Bringing it all together
- How does using a table of values help you to determine the multiplier and the constant?
- When extending a table of values, is it helpful to find the pattern rule before or after completing the table? Explain.
Reflection
As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.
I feel...
Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.
When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.