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Grade 9:  The Learning Equation Math

21.02 Mathematical Modeling

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Patterns

Refresher pp 28-29
 

SeekAWord2ech.class Author

Prerequisite Skills:

 

Key Terms             

modeling, table

 

Learning Outcomes:      Review     Enrichment      Key Terms     Prerequisite Skills

The student will:

Solving a Problem

Develop a Table or Spreadsheet

Example 1: You are about to toss a coin and six-sided dice.  What are the chances of getting heads and a number less than 4.  Draw a chart to show the 12 possibilities for a coin and  six-sided dice tossed.  Click on the Check button to confirm your chart set up and Answer to confirm your solution.

correct.gif (1227 bytes)

Example 2: What will the last digit be in the 20th power of 2 be?

Let's make a spreadsheet and start entering samples.

Power of  2 Standard Name
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256

Do you see the pattern emerging?  The pattern of the last digit of the standard name is:

2, 4, 8, 6, 2, 4, 8, 6,...

The powers of 2 that are multiples of 4 always end in 6.    Therefore the 20th power of 2 must also end in 6.

Example 3: Susan, Irina, Traci, and Debbie are a pilot, dentist, a doctor, and a writer. Use the clues to match each person with her profession.

  • Debbie is a friend of the doctor.
  • Susan and the writer sail with Traci.
  • Neither Susan nor Trina has patients.

Step #1 {Make a Chart}

Pilot Dentist Doctor Writer
       
       
       
       

Step #2 {Fill in the chart from the information given.}

  • Debbie is a friend of the doctor.  Debbie cannot be the doctor.
  • Susan and the writer sail with Traci.  Susan and Traci cannot be the writer.   There for the writer has to be either Debbie or Trina.
  • Neither Susan nor Trina has patients.  Susan and Trina cannot be the doctor or the dentist.  There for the doctor and dentist must be Traci and Debbie.  Since Debbie cannot be the doctor, she must be the dentist and Traci must be the doctor.   If Debbie is the dentist, then Trina must be the writer.   That leaves Susan - she must be the pilot.
Pilot Dentist Doctor Writer
Susan Susan Debbie Susan
  Trina Susan Traci
  Debbie Trina Trina
    Traci  

Step #3 {Finish with a sentence.}

Therefore, Susan is the pilot, Debbie is the dentist, Traci is the doctor, and Trina is the writer.

 

Using a Graph to Find a Pattern

Graphs and spreadsheets are powerful visual tools in mathematics.  The graph to help answer one of the previous examples: "What will the last digit be in the 20th power of 2 be?"

powergraph.gif (21801 bytes)

Do you see the pattern emerging?  The pattern of the last digit of the standard name of the powers of 2 are:

2, 4, 8, 6, 2, 4, 8, 6,...

The powers of 2 that are multiples of 4 always end in 6.   Therefore the 20th power of 2 must also end in 6.

Did you find the pattern easier to see with the graph?

 

Algebraic Model

Expressions are useful symbolic relationships in mathematics that do not include equality or inequality  signs.  The following interactive components demonstrate simplifying algebraic expressions.
Enter an expression and click the "Simplify button.  The "Random" button will generate expressions for you.

 

There are a number of internet sites that provide algebraic models to help you learn mathematics. Launch the following site, and enter "3x - 2y - 3x" below "TYPE YOUR PROBLEM" and click "SOLVE".

 

Equations are useful symbolic relationships in mathematics that include an equal sign.  The Pythagorean Theorem is one of these useful equations:

a2 + b2 = c2

If we know 2 of the sides we can calculate the third.  For example if we are laying on the ground 8 m from a flag pole, and we know the flag pole is 10 m high, we can calculate our distance to the top of the flag pole.

a2 + b2 = c2

82 + 102 = c2

164 = c2

c approximate 12.8 m

The top of flagpole must be 12.8 m from where we are laying.

Enter an equation along with the variable you wish to solve it for.
Click the Solve button.
Solve   for 

 

Once again, there are a number of internet sites that provide algebraic models to help you learn to solve equations.  Launch the following site, and enter "8^2 + 10^2" on the left side and "c^2" on the right side of the "=" sign below "TYPE YOUR PROBLEM".   Enter "c" for the variable and change "-100" to zero. Click "SOLVE".

 

Using a Calculator

Enter your age:
Your age in Dog Years is:

Enter a variety of ages in Enter your age.  How many people years make a dog year?

 

Review        Learning Outcomes       Enrichment        Key Terms        Prerequisite Skills

Grade Seven Problems

The Canadian School Math Page

Word Problems - Ask Dr. Math - lots of examples showing a variety of methods to solve.

 

Enrichment        Learning Outcomes       Review        Key Terms        Prerequisite Skills

 

Lever Principle

This applet shows a symmetric lever with some mass pieces each of which has a weight of 1.0 N. The lever arms can be read off from the coloured rectangles; one rectangle corresponds to 0.10 m. The lever is in balance, when the applet is started.

You can attach a new mass piece or put it to another place with pressed mouse button. In a similar way you can remove a mass piece by clicking on it.

A lever is in balance, if the total leftside torque is equal to the total right side torque.

source:  Lever Principle

 

Pulley System

You can raise or lower the load with the mouse. If you click on the mouse button, a spring balance will appear showing the tension in the string. You can change the weight of the load and the hanging pulley(s) by using the appropriate boxes. Inputs higher than the spring scale limit (10 N) are automatically changed.

Source:  Pulley System

Your Weight On Other Worlds

 

 

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Comments to:  Jim Reed
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