Minds On
Painter's Savings Graph
What information do you know by checking this graph?
Press ‘Answer’ to reveal the information.
- It is a line graph with the plotted points (1,1) (2, 4) (3,7) (4,10) (5,13) (6,16) and uses a straight line to connect all the points at an upward angle.
- The title of graph: Painter’s Savings.
- The X axis is labelled “number of day,” starts at 0 and goes up by 1s to 20.
- The Y axis is labelled “amount of money ($)” and starts at 0 and goes up by 1s to 20.
How could you describe this graph as a pattern?
Consider the following:
- Is there a pattern?
- If yes, what kind of pattern is it?
- How do you know?
Mr. Green's Graph
What do you notice about this graph?
Press ‘Answer’ to reveal the information.
- It is a line graph with the plotted points (1,150) (2,140) (3,130) (4,120) (5,110) (6,100) and uses a straight line to connect all the points at a downward angle.
- The title of the graph is Mr. Green’s Pencils.
- The X axis is labelled “days,” starts at 0 and goes up by 10s to 200.
- The Y axis is labelled “number of pencils,” starts at 0 and goes up by 1s to 20.
How could you describe this graph as a pattern? Consider the following:
- Is there a pattern?
- If yes, what kind of pattern is it?
- How do you know?
Action
Graphing a linear pattern
Growing and shrinking patterns can be linear patterns, but not all growing and shrinking patterns are linear.
If a pattern is linear, that means that each term increases (grows) or decreases (shrinks) by the same value every term.
Graph 1 from the Minds On would be linear since all the points increase by the same value.
A graph that does not increase or decrease by the same values would be considered non-linear.
How do you graph a pattern?
The graphs are simply a translation of an existing pattern.
- The term number, or the number that tells the position of a number in a pattern, is represented on the x-axis. This is the line along the bottom of the graph.
- The term value, or the number that is actually listed in the pattern, is represented on the y-axis, which is the vertical line in the graph.
Examples:
For example, the pattern may be: 1, 4, 7, 10, 13, 16…
Let’s start by translating this pattern to a table of values.
| Term number | Term value |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 7 |
| 4 | 10 |
| 5 | 13 |
| 6 | 16 |
The first term number is 1 with a term value of 1, making an ordered pair (1,1). That means it will be plotted as 1 on the X axis and 1 on the Y axis.
Term 2 has a term value of 4. That means it will be plotted as 2 on the X axis and 4 on the Y axis.
Term 3 has a term value of 7. That means it will be plotted as 3 on the X axis and 7 on the Y axis.
Determine the coordinates for the last 3 terms. How will you plot them on the graph?
Select the correct answer.
Task 1: Creating a table of values from a graph
A table of values has two columns titled term number and term value. The term number is the number that identifies the order of a term in a pattern, and term value are the numbers that appear in the pattern and are changing with each new term.
The term number is graphed along the x-axis. The term value is graphed along the y-axis.
Complete the following table of values and questions based on the graphs in the Minds On section. Record your thoughts for Graphs’ Info in your notebook or using the following fillable and printable document. You may also use a method of your choice.
Alternatively, you may wish to create your own values which follow a specific rule.
| Term Number | Term Value |
|---|---|
Is this pattern growing or shrinking? How do you know?
- What is the pattern rule?
- What is the number the term values are growing or shrinking by each time?
| Term Number | Term Value |
|---|---|
Is this pattern growing or shrinking? How do you know?
- What is the pattern rule?
- What is the number the term values are growing or shrinking by each time?
Press the ‘Activity’ button to access the Graphs’ Info.
Press ‘Answers’ to access the solutions to check your work.
Graph 1: Painter’s Savings
| Term Number | Term Value |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 7 |
| 4 | 10 |
| 5 | 13 |
| 6 | 16 |
The pattern is growing. I know because the line goes up to the right and the numbers in the term value column are getting bigger.
The pattern rule is Term Number × 3 − 2 = Term Value
The term values are growing by 3 each time.
Graph 2: Mr. Green’s Pencils
| Term Number | Term Value |
|---|---|
| 1 | 150 |
| 2 | 140 |
| 3 | 130 |
| 4 | 120 |
| 5 | 110 |
| 6 | 100 |
The pattern is shrinking. I know because the line is going down from the first term number.
The pattern rule is Term Number × (−10) + 160 = Term Value
Task 2: Creating a graph from a pattern
Here is an example of a pattern.
There is one triangle in the first term.
In the second term, a triangle gets added on top and to the right.
In the third term, a triangle gets added on top and to the right.
More triangles are added in the same pattern for the following 2 terms.
Explore the Triangles Pattern and go through the following tasks.
- Step 1. Create numerical values for each term by counting the number of triangles.
- Step 2. Record, digitally, orally, or in print, these values into a table of values.
- Step 3. Number the x and y axis on a graph, starting at 0 and plot all of the points. You can create a paper, or digital graph, or create an audio recording which includes a detailed discussion about your graph.
- Step 4. Connect all the points in your graph with a line.
Complete Triangles Pattern in your notebook or using the following fillable and printable document.
Press the ‘Activity’ button to access the Triangles Pattern.
Press ‘Answer’ to access a solution to check your work.
| Term Number | Term Value |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 5 |
| 4 | 7 |
| 5 | 9 |
Task 3: Linear or not linear?
Student Success
Think-Pair-Share
Record your thoughts on the following patterns using a method of your choice.
Decide if the following patterns are linear or not linear. If possible, discuss your ideas with a partner.
- 3, 5, 7, 9, 11…
- 2, 4, 8, 14, 22…
- 100, 97, 92, 85, 76…
- 50, 45, 40, 35, 30…
- 27, 24, 21, 18, 15…
Press ‘Answers’ to access the solutions to check your work.
3, 5, 7, 9, 11…
- Linear because the term value is growing by 2 each time (the same amount).
2, 4, 8, 14, 22…
- Non-linear because the pattern increases by multiples of 2- different numbers each time.
100, 97, 92, 85, 76…
- Non-linear because the pattern decreases by 3, 5, 7- different numbers each time.
50, 45, 40, 35, 30…
- Linear because it’s decreasing by 5 each term (the same amount).
27, 24, 21, 18, 15…
- Linear because it’s decreasing by 3 each term (the same amount).
Note to teachers: See your teacher guide for collaboration tools, ideas and suggestions.
How did you decide if the patterns are linear? There are a few different strategies!
After thinking about it on your own, you can press ‘Answer’ to reveal some possible strategies.
You can:
- Graph the patterns. If it creates a straight line, it is linear. If it creates a curved line, it is not linear.
- Determine the pattern rule. If the pattern rule involves a change of the same amount for each term, it is a linear pattern! If the pattern rule involves a non-constant change in the amount for each term, it is not linear.
Consolidation
Task 1: Creating and graphing patterns for a real-world scenario
Use the following criteria about two runners. For each runner, create a pattern rule, a table of values, and a graph, using a method of your choice.
- Each pattern that you will create will describe a person training to run a marathon. The runners start by running a certain distance on day 1 (term 1). Then they add or subtract the same amount of time to their run each day.
- Person 1 starts by running a longer amount of time than person 2 but ends with a shorter time or lower number than person 2.
Task 2: Think about graphing
Answer the following questions using a method of your choice.
- How can graphing help you to identify that there is a pattern?
- What strategy is most helpful in determining if a pattern is linear? Why?
Reflection
As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.
I feel...
Now, record your ideas using a voice recorder, speech-to-text, or writing tool.
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