Name ______
9th Grade Summer Packet
Due Thursday, September 8th 2011
Summer Packet Contents (Total Points: 100)
- Problem Solving (20 points)
- Mathematical Connections (30 points)
- Mini-Projects (20 points)
- Mind Benders (20 points)
- Folder, presentation, organization (10 points)
Scoring Rubric
Problem Solving / · Four mathematical topics will be reviewed, each worth 5 points.
· Work will be graded based on completion of problems.
Mathematical Connections / · You will choose six mathematical topics that you have learned
· For each mathematical topic, you will find a picture or an image (from the Internet, your neighborhood, your household, etc.) that applies to that particular topic
· Under each picture or image, you will write 5-6 sentences describing how the picture relates with the math topic you selected
Mini-Projects / · You will be given two mini-projects where you will observe a real world application of mathematics or on the Internet.
Mind Benders/
Reasoning and Making Conjectures / · You will choose two Mind Benders to solve from a list of difficult situations.
· You will be graded on TRYING a Mind Bender and explaining how you found your solution.
This box for teachers only
Section / Grade / Total Possible
Problem Solving (5 points each) / /20 points
Mathematical Connections (5 points each) / /30 points
Mini-Projects (10 points each) / /20 points
Mind Benders (10 points each) / /20 points
Presenter (10 points) / /10 points
Total Grade:
PROBLEM SOLVING (PART 1)
PATTERNS
While waiting for a bus, Todd decided to play with consecutive numbers (whole numbers that increase in order without skipping, such as 5, 6, and 7). His work is shown below:
a. Write the next three entries.
b. Describe any patterns you notice in the answers.
c. Can three consecutive numbers add up to 60? If so, find the numbers. If not, explain why not.
d. Can three consecutive numbers add up to 8? Again, explain why or why not.
Joel decided to try a new diet. He followed the diet for four months
and kept track of the results on the graph shown at right. Write a
story about his diet.
PROBLEM SOLVING (PART 2)
SOLVING EQUATIONS
Solve the following equations for the indicated variable. Show all of your work.
Solve for x: -10x + -20 = 50 Solve for y: 8 + 8y = 24
Solve for w: 16 - 2w = 1 + w - 4 Solve for x:
MacKensie solved the equation below but Stephan says that she made some mistakes. Who is right?
Explain how you know.
Solve each of the equations below for y. For each equation, state the growth and the y-intercept.
PROBLEM SOLVING (PART 3)
LINEAR RELATIONSHIPS
Study the tile pattern below.
a. Draw Figure 3 and Figure 4. Explain how the pattern grows.
b. Write an equation (rule) for the number of tiles in the pattern.
c. Explain how the growth factor appears in your equation.
Given the following two clues, show all four representations (tile pattern, table, rule, graph) of the pattern.
PROBLEM SOLVING (PART 4)
QUADRATIC RELATIONSHIPS
Albert wants to factor but now he’s stuck.
What should he do next?
Be sure to finish factoring .
Factor each:
a. b.
c. d.
Examine the parabola and the quadratic rules below. For each rule that does not match, explain how
you know it doesn’t.
a.
b.
c.
Find the roots for: Quadratic Formula
(use the Quadratic Formula)
y = x2 + -9x + -5
REAL WORLD MATH
You will find a “Real World” object and write 5-6 sentences on how it applies to a
specific mathematics topic. In total, you will select SIX mathematical topics.
Directions:
1.) Find a picture on the Internet or take a picture using a camera of something in
your home, neighborhood or community that involves a specific math topic.
2.) Print out, or attach this paper to a sheet of computer or construction paper.
3.) Under the picture, write 5-6 sentences that describes how the image relates to a
specific math topic.
4.) Write the name of the math topic in large letters above the picture.
Example:
Equations / Proportions / System
Area Model / Growth Factor / Rule
Generic Rectangle / Figure 0 / Table
Graphs / Starting Point / Variable
Distributive Property / Factoring / Vertex
Quadratic Equations / Solving Equations / x-axis
Linear Equations / Tile Pattern / y-axis
Slope / Parabola
Roots and Zeroes / Point of Intersection
Questions that will help you write your 5-6 sentences:
How is math used?
Why is math used?
Is math important for this specific picture?
Who uses math for this image?
What does someone need to know before using this math topic?
What other math topics can apply to this picture?
MINI-PROJECT #1
Don’t Fence Me In
The Central Park Authority is designing a field where horses have the freedom to roam and graze. Because of the city budget cuts, they can only afford 300 feet of fencing and the area must be in the shape of a rectangle. What is the biggest area that the horses can have?
PART 1: What are possible designs? Amount of fencing can only be 300 feet.
Remember that Length · Width = Total Area
and that the total of the lengths and widths should be 300.
That is,
2 · Length + 2 · Width = Total Perimeter
Find the total area, width or length for the dimensions below. Show your work below. Make up your own if some are missing.
SHOW WORK:
Width / 25 ft / 100 ftLength / 125 ft / 25 ft
Total Area
PART 2: Use a visual to find the
maximum area.
Create a Width vs. Area Graph using
the table below. Label all of your axes
and make a complete graph.
PART 3: Justify your answer.
What is the biggest area that the Central Park horses have with their 300 feet of fencing? Describe why your answer makes sense and why it is the best answer choice. This should be TYPED UP and 5-6 sentences.
MINI-PROJECT #2
Firework Paths
A town is celebrating the end of a year's worth of harvesting and Aribell decides that she wants to launch a firework off of the ground during the celebration. She predicts that the following will happen: the firework reaches a maximum height of 80 feet from the ground after it is in the air for 3 seconds and lands on the ground 3 seconds later.
Draw a picture of this situation including...
· the starting point of the firework
· maximum height
· time it takes for the firework to
land on the ground
· any other important details
· label the x-axis (time)
· label the y-axis (height)
· label your scale on both axes
Using your sketch, about how high is the firework at 5 seconds? (estimate)
Jorge plans to launch a second firework (that's just like Aribell's) three
seconds after Aribell launches hers. When will Jorge's firework hit the ground?
Sketch this firework's path on the same graph as Aribell's firework.
CREATING A GRAPH FROM A RULE
Lorena wants to help plan out the fireworks show for the town's party. She wants to launch a firework from the top of a building that is 160 feet tall.
She knows the following:
· the initial velocity of the firework is 92 feet/second
· gravity pushes the firework down at a rate of -16 feet/second2.
Lorena writes the rule where y = 160 + 92t – 16t2
t is the amount of time that the firework is in the
air and h is the height of the firework.
Use this rule to sketch the height vs. time
relationship of the firework on the graph:
(use a calculator!)
then using the Quadratic Formula...
· find the roots of the parabola
How long will Lorena's firework be in the air? What is the maximum height of the firework?
SKETCH A GRAPH FROM A TABLE
Taylonn launches a firework from the top of a building.
Using a computer and a radar gun that measures height,
he was able to get the following information about the
firework:
Based on the table, how high is the building
that Taylonn launched the firework from?
Draw a sketch of the height vs. time relationship of this firework.
Taylon figures out that the rule that best describes the path of the firework is. If he decides to move the firework to the top of a 50 foot building, how would the rule change? Sketch the path of this firework on the graph that you used above.
FINAL SKETCH
Sketch the firework paths that Aribell, Jorge, Lorena and Taylonn each created on the graph below. Include any important equations, roots, labels, titles and vertices. Use a legend to show the height vs. time relationship for each firework.
MIND BENDERS
Choose TWO of the situations below. Show as much work as possible. You will be graded based on strategies, thinking and reasoning. Even if you try something that does not work, you should still include that work.
· Casper the Rabbit
Kari is training his pet rabbit Casper to climb up a flight of 10 steps. Casper can only hop up 1 or 2 steps each time he hops. He never hops down, only up. How many different ways can Casper hop up the flight of 10 steps? Provide evidence to justify your thinking.
· Flashlight and the Bridge
One of my friends gave me this problem. Try as I might, I cannot seem to get the answer she says she does. Well, here goes:
Four people want to cross a bridge. Only two people may cross at a time.
Aribell takes 1 minute to get across.
Taquana takes 2 minutes to get across.
Daquone takes 5 minutes to get across.
Rashard takes 10 minutes to get across.
Here is the catch, if two people cross the bridge together, they must walk at the pace of the slower one. So if Aribell and Rashard cross, they take 10 minutes. Also, it is night. Each trip requires a flashlight. There is only one flashlight. They are not allowed to toss the light over the river.
My friend said that she can get them across in 17 minutes flat. How is this possible?
· Crossing the River
A certain number of adults and a certain number of children need to cross a river. A small boat is available that can hold one adult or one or two children (i.e., three possibilities: 1 adult in the boat, 1 child in the boat, or 2 children in the boat). Everyone can row the boat. How many one-way trips does it take for all of them to cross the river if there are:
1. 10 adults and 2 children?
2. 12 adults and 2 children?
3. 10 adults and 5 children?
4. A adults and C children? (Does your formula work for all possible
values of A and C?)
· The Locker Problem
Imagine you are at a school that has 100 lockers, all shut.
Suppose the first student goes along the row and opens every locker.
The second student then goes along and shuts every other locker beginning with locker number 2.
The third student changes the state of every third locker beginning with locker number 3. (If the locker is open then the student shuts it, and if the locker is closed the student opens it.)
The fourth student changes the state of every fourth locker beginning with number 4.
Imagine that this continues until the 100 students have followed the pattern with the 100 lcokers. At the end, which lockers will be open and which ones will be closed?
· Broken Eggs
A farmer is taking her eggs to market in her cart but she hits a pothole, which knocks over all the containers of eggs. Though she is unhurt, every egg is broken. So she goes to her insurance agent, who asks her how many eggs she had. She says she doesn’t know but she remembers some things from various ways she tried packing the eggs. She knows that when she put the eggs in groups of two, there was one egg left over. When she put them in groups of three, there was also one egg left over. The same thing happened when she put them in groups of four, groups of five, or groups of six. But when she put them in groups of seven, she ended up with complete groups of seven with no eggs left over.
Your Task
Your task is to answer the insurance agent’s question. In other words – what can the farmer figure out from this information about how many eggs she had? And is there more than one possibility?
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