Breaking Up Vectors

Name(s): Raymond Castillo                                                                      

 

Title of lesson: Breaking up Vectors

 

Length of lesson: 55 minutes.

Description of the class:

                     Name of course: Pre-calculus

                     Grade level: 10-12

                     Honors or regular: Either

 

Source of the lesson:

            Original

 

TEKS addressed:

C3.(A): use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data;

 

C3.(D) solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas.

C6.(A) use the concept of vectors to model situations defined by magnitude and direction; and

 

C6.(B) analyze and solve vector problems generated by real-life situations.

 

Overview
The students will work through a teacher orchestrated activity describing how vectors can be broken into x and y components.
 
II. Performance or learner outcomes

            Students will be able to:

                        Describe vectors as magnitudes and angles.

                        Use trig functions to determine the x and y components of vectors.

                        Use trig functions to determine the a and b sides of a right triangle.

                        Relate the components of a vector to the sides of a triangle.

III. Resources, materials and supplies needed

             Chalkboard or overhead slide.

             Protractors.

 

 

IV. Supplementary materials, handouts. (Also address any safety issues

Concerning equipment used)

            

 

                                               

Engage

Teacher does:

 

Probing questions

Students do: (expected answers and misconceptions

 

The teacher must have set up a vector/force table. Students can come see it on their way into class, or if a video camera feed can be projected on the screen, this would be good too. One of the weights should be aligned at zero degrees, while the others should be at uncommon angles. Display small cards, telling the weights on each.

 

 

 

 

 

 

 

 

In the previous lesson, we learned what vectors were, and what happened if they were combined.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Why does the ring in the center not move? Are the forces all the same?

But the weights are different? How can they be the same?

 

What do you remember about finding the magnitude or length of the resultant? How did you do it yesterday?

 

 

 

Students should look at the apparatus on their way into class

 

 

 

 

 

 

 

 

 

 

 

Students might say something about the forces being equal.

 

 

 

 

Students should say that they measured it.

Time: __10 minutes_

 

 

 

Explore

 

 

Òsuppose you didnÕt have a ruler, or graphpaper. All you had was the details of the vector, and some scratch paper to draw on? Everyone take out a piece of paper. And a protractor.Ó

 

DO the same sketching on board or overhead.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Add to the drawing:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


At this point the teacher will handout a paper that explains and shows what Sine, Cosine and Tangent are.

 

 

Teacher will then explain what each function means, paying special attention to what adjacent, and opposite means. Introduce the mnemonic: SOHCAHTOA.

 

 

Approx. Time__20_mins

 

 

 

 

 

 

 

 

 

 

If I give you this vector, say 50 @ 30 degreesd. Sketch a pair of axis, and then sketch the vector.

 

 

 

 

 

 

 

 

What does the arrow look like? Anything familiar? Possibly from geometry class? Specifically, what does it look to be forming with the x-axis.

 

What does the vector form with the x axis and the verticle line?

What kind of triangle?

 

 

 

 

 

 

What do we know about right triangles?

 

 

 

 

What sides are a, b, c?

So go ahead and label the sides, a, b, c.

 

 

 

 

What use are the sides a and b if we only know what C is? Can we find A and B with only C?

 

Remember from Algebra, if we have 2 unknowns, how many equations do we need to solve for those unknowns.

SO we canÕt find A and B without t something else?

 

Well, what else do we know about the vector besides the magnitude, or in this case, what else besides the hypotenuse?

 

IÕm sure for the longest time youÕve wondered what those tan and sin and cos buttons are on your calculator. Well, they are trigonometric functions that we can use to.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Students should draw something like:

 

 

 

 

 

 


Students may say that it looks like a right triangle.

 

 

 

 

 

IF they donÕt know, ask What if we showed the vertical line that marks the tip of the vector?

 

A triangle or a right triangle.

 

 

 

Students may say something about the angles, one being 90, the other two add up to 90. You are looking for students to say something about a^2+b^2 = c^2

 

 

 

 

 

 

 


Students may try to say that we can work backwards with the equation.

 

 

2 equations.

 

 

 

 

No.

 

 

The angle.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                   

Explain

 

 

Teacher Does

Probing Questions

Student Does

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Go ahead and use your calculator to solve these equations.

 

 

 

 

 

 

 

 

 

 

If students are getting wrong answers make sure their calculators are set to degrees and not radians

 

 

 

 

 

 

 

ÒIf you know you are correct, go ahead and put the values on the corresponding side of the triangle

 

 

 

Now that we have these special functions, we can use the two pieces of information about the triangle to figure out all the other. Namely we want to figure out the two other sides of the triangle.

Looking on your equation sheet and your labeled triangle, which equation could we use to find the vertical side?

 

Which one to find the horizontal side?

What are we solving for?

 

 

 

 

 

 

 

 

 

 

 

 

 

Do the numbers you got for the answer make sense? Think about the numbers used, and look at the sketch.

 

 

 

 

 

IS there a way to check to see if we are right? Since we found sides A and B, on a right triangle, what can we use to find side C, or the hypotenuse.

 

 

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Sin o =o/h

 

 

Cos 0 = a/h

Adjacent side and opposite side. (a and B)

 

Students solve the 2 equations.

 

 

 

 

 

 

 

 

Students should recognize that the answers make sense, or not.

 

 

 

 

 

 

A^2 + b^2 = C^2

Students should check their answers. They shouldnÕt be exact, since trig functions are approximations.

 

 

Students will label the sides on their sketch.

 

 

 

 

Time: 20min

 

 

 

Elaborate

 

 

Teacher Does

Probing Questions

Students Do

 

 

 

 

 

 

 

 

 

 

 

 

 

 

So what does it mean when we find the vertical and horizontal lines on that triangle, in respect to the vector that we started with?

 

If you remember the resultant lesson from yesterday, what would be the resultant of the vertical and horizontal lines? The A and B sides?

 

Student may know that vectors can be broken into x and y components.

If not, this will be explored in the next lesson.

 

Allow students to work through this. Their resultant should be the hypotenuse (C) side.

Time: 5 minutes_

 

 

 

Evaluate

 

 

Teacher Does

Probing Questions

Student Does

 

Handout a list of problems, such as the one used in class: 120@20degrees

Add in some that are at weird angles: 8@232degrees. Tell them to find the x and y components.

 

 

Students will work on these problems with the remaining time, or as a homework assignment

Time: 30 minutes