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AMI Assignments (for Inclement Weather Days)

General Information about AMI Assighments (for Inclement Weather Days)  

General Information about AMI Days (AMI stands for "Alternative Method of Instruction")

Students may be given paper copies of the AMI assignments (and/or assignments from our text books) for our AMI Days.  These assignments can be completed for credit in the event we are out of school for inclement weather during the school year.  Parents/students may contact me with any questions regarding assignments by email at joshua.ward@pottsvilleschools.org


AMI Day 1 (1/22/20)  

How many seconds have you been alive?  (Show all your work/calculations, and check your answer on Google.)


AMI Day 2 (3/17/20)  

“Think About It” Question

If we double the width of a rectangle and we half the length, then what happens to the area?  What happens to the perimeter?

*Students Will…

1.  Discuss the problem with someone (if possible), and explore possible answers.

2.  Show your work on paper.   (Students work may include several examples illustrated, a table of data, a picture, written explanation, or in other possible ways.)

3. Write your final answer, and include a written explanation of what you discovered.

(Hint:  You could draw a rectangle, and label one side the width and a side next to it as the length.  Then, just make up some measurements for the width and the length (like width = 3 ft. and the length = 7 ft.).  Next, draw a second rectangle with a width twice as long as your first rectangle and a length half as long as your first rectangle.  Then, calculate the area and the perimeter of each of your rectangles and compare.  Remember:  Area of a rectangle = the length times the width, and the Perimeter of a rectangle is the sum of the sides (all 4 sides added up), or you could use the formula: perimeter = 2 times the length + 2 times the width.)


AMI Day 3 (3/18/20)  

Answer the following questions in number of years, number of days, number of minutes, and number of seconds:

1. How long is one million seconds?

2. How long is one million minutes?

3. How long is one billion seconds?

4. How long is one billion minutes?

5. How long is one trillion seconds?

6. How long is one trillion minutes?


AMI Day 4 (3/19/20)  

“The Fast and the Furious”

If you traveled from Pottsville to Fort Smith (90 miles) at an average speed of 40 mph, then what would your average speed have to be on the way back so that your average speed for the whole trip is 60 mph?  (Show all your work, steps, and calculations you use to arrive at your final answer.  This problem is not as straight forward and easy as it might seem.  The first time I was given this problem, I got it wrong.  I thought it should be really easy, and I thought all I had to do was get the rates of speed to average out to 60 by making my answer 80 mph for the trip back, BUT that's not right.)  

Hint:  Okay, so this was a problem we did earlier in the year.  We worked on several problems like it, but it has been a while, so let me try to help you out. First, remember to identify the variables in any word problem, and in this case, we have "Distance," "Rate of Speed," and "Time."  I think the best way to approach this problem is to set up a table.  If you decide to approach the problem this way, then I recommend creating the table with 3 columns: "Trip There," "Trip Back," and "Whole Trip/Round-Trip"; and then include 3 rows, which would include each of the three variables involved in the question: "Distance," "Rate of Speed," and "Time."


AMI Day 5 (3/20/20)  

“2 Fast and 2 Furious”

Two cars leave from Memphis headed for Oklahoma City on Interstate 40. The first car leaves at 4 p.m. and the second car left at 6 p.m. The second car passed the first car at 10 p.m. The second car’s average speed was 35 mph faster than the average speed of the first car.

What was the average rate of speed of each car from the time each one left Memphis until 10 p.m.? (Show all of your work.)

Hint:  

Use the formula:

distance = (rate of speed)(time in hours)

You can use the formula to write two separate equations, one for each of the two cars.  You don't know the distance, and even though you will be able to figure it out, you don't need it, in order to answer the question.  You can set up a system of linear equations both in the form of d = rt.  You should be able to plug in the time (in hours) for each car.  Then, you can let the rate of speed for the first car be "x" (since we aren't given either car's speed).  Then, you can call the rate of speed for the second car "x + 35" (since we don't know it's speed, but we know it's 35 more than the first car).  Now, use "substitution method" for solving a system of linear equations to solve for x, and then you will know the average rate of speed of both cars.


AMI Day 6  

1.  Use Pythagorean Theorem to find the distance of a diagonal line segment between the following two points:  (4, 5) and (-2, -3)

(Hint: Remember, Pythagorean Theorem is a+ b2 = c2 for a "right triangle," and "a" and "b" are the two "legs" of the right triangle, which are the two sides which come together to form the 90° right angle, and "c" is the "hypotenuse," which is the longest side of the right triangle and the side opposite of the right angle.)

2.  Use the Distance Formula to find the distance of the same line segment between the same two points:  (4, 5) and (-2, -3)

(Hint:  Remember, the Distance Formula is as follows:

d = distance
(x_1, y_1) = coordinates of the first point
(x_2, y_2) = coordinates of the second point

3.  Since we're using two different ways to find the exact same distance, you should arrive at he same answer both ways.  Please explain how Pythagorean Theorem and the Distance Formula are related, and why they can both be used to find the distance between two points.

 


AMI Day 7  

1. Describe the different number sets (Real Numbers, Imaginary Numbers, Rational Numbers, Irrational Numbers, Integers, Whole Numbers, Natural Numbers, and also "Prime numbers.")

2. Give examples of each set of numbers to help define each set of numbers as distinct and different from the other number sets.  (You can give several examples for each, and then also give several "non-examples" for each number set.)


AMI Day 8  

Name at least 3 Career Options (preferably careers you are interested in possibly pursuing in the future), and then do some research to discover some ways mathematics is used in these careers.  Include at least one page that explains how mathematics is used in each of the three careers of your choosing.


AMI Day 9  

Consider the following scenario and how it could be used/applied to help students with mathematic calculations using the fundamental operations of mathematics including adding, subtracting, multiplication, and division.  

For the purposes of this, we must first agree to accept the following generalized truths:

Friend = +  (A "friend" is a "positive" or "a plus")

Enemy = –  (An "enemy" is a "negative" or "a minus")

Now, please use your reasoning skills to fill in the blanks in the most logical way (giving consideration to what is most reasonably and generally true, and excluding the exceptions, which may come to mind):

(Hint:  Two of the answers will be "friend," and two of the answers will be "enemy.")

1. The "friend" of my "friend" is my ________ .

2. The "enemy" of my "friend" is my ________ .

3. The "friend" of my "enemy" is my ________ .

4. The "enemy" of my "enemy" is my ________ .

Now, translate this into mathematical symbols, using the following:

+ for "friend"

– for "enemy"

× for "of" ("of my")

= for "is" ("is my") 

This can now be used as a chart of signs, which works for multiplication of two numbers.  

Can this chart of signs be used also for division of two numbers?  If not, explain why not.  If so, show examples to demonstrate this.

Can this chart of signs be used for evaluating expressions with numbers, when two positive/plus or minus/negative signs are shown consecutively (when two of these signs are shown between two numbers; for example 2 – (-3) or 4 + -5), so that the two consecutive signs could be replaced with only one sign, either + or – ?

Why is it this chart can NOT be used, when combining two separate numbers or terms?  (Example:  -2 – 3)


AMI Day 10  

1.  If you were to stand 20 feet away from a wall, and if you moved towards the wall precisely half the distance you were from the wall at that time, and then you continued to repeat this process until you reached the wall, then how many times would you have to do this (how many "moves" of half the distance would you have to make) before you actually reached the wall?

(Hint:  Assume you were somehow able to precisely measure the "half distance" each time, and then perfectly accurately move that exact distance towards the wall each time.)

*PLEASE MAKE SURE YOU EXPLAIN YOUR ANSWER.


AMI Day 11 (Note: Students may be given the option to do this particular assignment on different topics to count towards more AMI days, if they would like.)  

Research an area of mathematics and write about its history, where it originated and who was responsible for the development of that specific area of mathematics.  (Please include at least one page.  You can include the history of more than one area of mathematics, if you would like.  (Also, if you would like to do this same assignment for more than just one day of AMI, then please let me know, and that may be an option.)


AMI Activity (Worth 5 AMI Days; Day 12, 13, 14, 15, & 16)  

Algebra Activity

*Each student needs pencil, ruler, graphing calculator, one piece of notebook paper to show work, and one piece of graphing paper (preferably a copy with the x/y axis already on it, numbered and labeled).

Instructions:

1.    Choose one single digit whole number, then choose a second single digit whole number, and write the two numbers down in the order you choose them.

(Teacher will assign each student a number of 1, 2, 3, or 4, and this will be the quadrant where they will graph their first point.)

2.    The first number you chose will be the x-coordinate, and the second number you chose will be the y-coordinate of the first point you will graph. If necessary, make one or both of your numbers negative, so that they will represent a point in the quadrant you have been assigned. Now, graph your point and label the point with its coordinates (x,y).  

 

3.    Now, choose a positive integer less than 10, and graph it as a point on the y-axis. Label the point with its coordinates (x,y).

 

4.    Next, choose a negative integer greater than -10, and graph it as a point on the y-axis. Label the point with its coordinates (x,y).

 

5.    With a ruler, connect your first point and your second point to form a straight line. Extend the line across the graph paper, placing arrows at both ends to indicate the line continues on forever in both directions.

 

6.    With a ruler, connect your first point and your third point to form a straight line, and extend the line across the graph paper, placing arrows at both ends to indicate the line continues on forever in both directions.

 

7.    Use the slope formula (refer to reference sheet, if necessary) to find the exact slope of the first line you graphed. Show your work, and express the slope in simplified fraction form.

(If you need help with this, use the calculator, and press

“MATH”, “ENTER”, “ENTER.”)

Note: Make sure if your line slopes upward as you look from left to right, that the slope of the line is positive; and if your line slopes downward as you look from left to right, that the slope of the line is negative.

8.    Since you now know the slope of your first line, and you already know the y-intercept of the line, write the correct equation of your first line in slope-intercept form (y = mx + b). Write the equation of the line next to one of the two arrows of your first line on your graph.

 

9.    Use the slope ratio (m = rise/run) to find the exact slope of the second line you graphed. On your graph, show dashes or dots as you count the run and then the rise from your first point to your third point (the y-intercept with negative value). Then, measure the slope of your second line with the slope ratio from the third point to the first point. Show them both as simplified fractions.

(If you need help with this, use the calculator, and press

“MATH”, “ENTER”, “ENTER.”)

Note: Make sure if your line slopes upward as you look from left to right, that the slope of the line is positive; and if your line slopes downward as you look from left to right, that the slope of the line is negative.

10.    Since you now know the slope of your second line, and you already know the y-intercept of the line, write the correct equation of your second line in slope-intercept form (y = mx + b). Write the equation of the line next to one of the two arrows of your second line on your graph.

11.    Now, take the coordinates of the very first point you graphed, and substitute them into the equation of your first line. Then, simplify both sides to verify the ordered pair really is a solution to your first equation. (Remember, this means both sides of the equal sign should end up with the same number, making the equation a true statement.)

12.    Next, take the coordinates of that same point (the very first point you graphed), and substitute them into the equation of your second line. Then, simplify both sides to verify the ordered pair really is a solution to your second equation. (Remember, this means both sides of the equal sign should end up with the same number, making the equation a true statement.)

13.    Take the coordinates of your second point (the y-intercept with positive value), and substitute those coordinate values into the equation of your first line. Then, simplify both sides and state whether or not the ordered pair is a solution to your first equation.

*Write a sentence explaining why it is or is not a solution based on your results. Then, write a second sentence explaining why it is or isn’t a solution based on how the graph of the point and the graph of the line are related.

 

14.    Take the coordinates of your second point (the y-intercept with positive value), and substitute those coordinate values into the equation of your second line. Then, simplify both sides and state whether or not the ordered pair is a solution to your second equation.

*Write a sentence explaining why it is or is not a solution based on your results. Then, write a second sentence explaining why it is or isn’t a solution based on how the graph of the point and the graph of the line are related.

15.    Take the coordinates of your third point (the y-intercept with negative value), and substitute those coordinate values into the equation of your first line. Then, simplify both sides and state whether or not the ordered pair is a solution to your first equation.

*Write a sentence explaining why it is or is not a solution based on your results. Then, write a second sentence explaining why it is or isn’t a solution based on how the graph of the point and the graph of the line are related.

 

16.    Take the coordinates of your third point (the y-intercept with negative value), and substitute those coordinate values into the equation of your second line. Then, simplify both sides and state whether or not the ordered pair is a solution to your second equation.

*Write a sentence explaining why it is or is not a solution based on your results. Then, write a second sentence explaining why it is or isn’t a solution based on how the graph of the point and the graph of the line are related.

 

17.    Re-write your first equation, and substitute only the y-coordinate value of your first point into the equation. Then, solve the equation for x. Show all of your work. (Before you start, think about what the solution will have to be based on the results of your previous work.)

 

18.    Re-write your second equation, and substitute only the y-coordinate value of your first point into the equation. Then, solve the equation for x. Show all of your work. (Before you start, think about what the solution will have to be based on the results of your previous work.)

 

19.    Now, write a new one-variable equation by using the “substitution method” (for solving a system of linear equations). To do this, look at each of your linear equations, and notice the right side of each equation is an expression “in terms of x,” and each expression is equal to the same variable “y.” So, set these two expressions equal to each other. (Before you start, think about what the solution will have to be based on the results of your previous work.)

Note: Remember, in algebra and in every aspect of life (at least I can’t think of an exception), when two different things are both equal in value to a third thing, then the two different things must be equal in value to each other.

 

20.    Graph a third line that is parallel to your first line.

 

21.    Next to your third line, write the correct linear equation for that line. What do you notice, when you compare the slopes of your first line and your third line?

22.    Now, graph a fourth line that is parallel to your second line.

 

23.    Next to your fourth line, write the correct linear equation for that line. What do you notice, when you compare the slopes of your second line and your fourth line?

 

24.    Finally, determine exactly where your third line will intersect your fourth line.