Lesson 2

Go to page 9 of the textbook to refresh your memory about arithmetic sequences and how you can use the general term to identify the properties of these sequences.

 

Reread “Example 1” and then try “Your Turn” on pages 10 to 11 of the textbook. If you have any questions, contact your teacher. Answer

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Use Graphing and Analyzing Equations to review the properties of linear functions. Select “Linear Functions,” and then select “Graphing and Analyzing Equations.” In the first slide, see if you can discover a definition for linear function. Then work through slides 1 to 7. Focus on how a linear function is represented in a table and in a graph.

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Now watch Determining the Equation of a Line: Given the Graph. This interactive lesson will review how you can determine the equation of a line in slope-intercept form when given information from a graph.

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The domain of a relation consists of all values of the independent variable. The range of a relation consists of all values of the dependent variable. Use Domain and Range to check your knowledge of domain and range.

 

Work through the following pages: (1) Domain and Range (2) Domain and Range from Graphs

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Use Percent to review concepts relating to percentage. In the first topic you will review the definition of percent. In the second topic you will practise determining equivalent percents, fractions, and decimals. In the final topic, you will have the opportunity to play a game that tests not only your knowledge of percents, but also your reflexes! To begin, select “Percent (Food)” under the “Number” column, and then start with the first topic.

Go back to the Are You Ready? questions and try them again. If you are still having difficulty, contact your teacher.

Problem Question

 

How are arithmetic sequences related to linear functions?

 

Materials

 

You will need

• a ruler
  
• Archimedes' Spiral Template
• a pencil

Hypothesis

 

Form a hypothesis to answer the problem question. Do you remember what a hypothesis is?

 

Procedure

 

Step 1: Print a copy of Archimedes' Spiral Template, provided in the Materials list.

 

 

Step 2: Identify the points along the spiral that intersect the radii (or spokes) of the circle. Mark these with small dots.

 

 

Step 3: Use your ruler to measure the length from the centre to the first point of intersection with a spoke as you wind from the centre to the outer edge. Repeat for each intersection point as you continue from the centre and radiate outward. Once you have 12 data values, record your measurements to the nearest millimetre in a table like this one. Then proceed to Analysis.

 

Radius	1	2	3	4	5	6	7	8	9	10	11	12
Distance to Intersection

 

 

Analysis

1. Study your measurements. What kind of sequence do your measurements form? What is the common difference?
   
   
2.  
   1.  Determine an expression of the form tn = a + (n − 1)d for the sequence. Simplify the expression.
      
      
   2. What does n represent in this expression?
      
      
   3. What does tn represent in this expression?
      
      
   4. Does the simplified form of the expression remind you of any other expression?
      
      
3. Plot a graph showing tn versus n.
   
   
4. Do you think the points should be connected? Why or why not?
   
   
5. Describe the type of function that is represented by the graph?
   
   
6. What property of the sequence corresponds to the slope of the graph?

How do your results compare to your initial hypothesis?

 

Save your results in your course folder.

 

You will use your results again later in the lesson.

You already saved Module 1 Glossary Terms in your course folder. In this lesson you will define these terms, and maybe others, in your copy of Module 1 Glossary Terms:

• linear function
• simple interest

Try This 1

 

Digital Vision/Thinktock

 

Imagine that you are investing $1000 in a savings bond that pays 4% simple interest at the end of every year.

1. Calculate the simple interest you would earn each year.

    

2. Complete a table like this one.
   
   
    

   Year	1	2	3	4	5	6
   Total Value of Investment ($) at End of Each Year	1040	1080

   
   
   
3. Graph the year versus the value of the investment.
   
   
4. Use the graph to find the total value of the investment after 10 years.
   
   
5. After how many years will the total value of the investment be equal to $1480?

Save a copy of your answers in your course folder.

 

Share 1

 

Discuss your results with a classmate or in a group. In your discussion, address the following points:

• How does this situation represent an arithmetic sequence? How does the situation represent a linear function?
  
  
• Is the graph more accurately represented with or without a line drawn through the points? Give reasons to support your answer.

If required by your teacher, save notes from your discussion in your course folder.

Go to pages 14 to 15 of the textbook and work through “Example 4,” which demonstrates the relationship between an arithmetic sequence and a linear function. Read the problem and focus on the wording. Then read the solutions with particular focus on “Method 2: Use a Graph.” As you read, look for answers to the following questions:

• What is the significance of the phrase “plus $42/hr or portion of an hour”?
  
  
• When is it appropriate to model a problem using an arithmetic sequence?
  
  
• What property of the graph corresponds to the common difference of the arithmetic sequence?

Self-Check 1

 

Check your understanding of arithmetic sequences by completing Arithmetic Sequences Quiz.

Complete questions 7, 13, 16, 17, and 19 on pages 17 to 19 of the textbook. Show your work to support your answers. You may find you need to review relevant parts of this lesson as you work through the problems. Answer

Open the Lesson 2 Assignment you saved in your course folder at the start of this lesson. Print one sheet of Polar Grid Paper. Complete the assignment.

 

Project Connection

In Module 1 Project: Financing Your Dreams, complete Project Connection 2: Budgeting.

 

Save all of your work in your course folder. You will build on your work in each lesson.

Going Beyond

 

You have already seen arithmetic sequences applied in several different real-world contexts. These contexts include simple interest, Archimedes' Spiral, and auditorium seating. In fact, you can model situations using an arithmetic sequence whenever you discover a pattern of repetitive addition.

 

You will now investigate perpetual calendars, another application of arithmetic sequences. Can you figure out what the common difference is when an arithmetic sequence is used to model a calendar?

 

Part A

1. Find a calendar for the current year. For the first two months of the calendar year, construct a table that shows which days of the year fall on a Thursday. Your table might look like the following:

    

    

   Thursday	Day of the Year
   1st	3
   2nd	10
   3rd
   4th
   5th
   6th
   7th
   8th

   
   
2. Graph the data from the table. Show the independent values (left-column data) on the horizontal axis and the dependent values (right-column data) on the vertical axis.
   
   
3. Determine the general term, tn, that relates the day of the year to the Thursday position.

Part B

Check the Internet to learn more about perpetual calendars. Try these questions to help you begin your investigation:

• If January 1 occurs on a Thursday this year, what day of the week will January 1 land on next year? Does the answer depend on any other factors?
  
  
• When do leap years occur? How can you determine if a particular year is a leap year?
  
  
• How many one-year calendars does a perpetual calendar consist of?
  
  
• What are the steps used to calculate the day of the week for a particular date?
  
  
• What are each of the World Calendar, the International Fixed Calendar, and the Pax Calendar?

Your search might lead to other questions.

 

Save your work in your course folder.