Some topics feel so foreign because it’s hard to remember a time when “5 x 7” looked like your teacher forgot how to write “+”. We have many (all?) multiplication facts memorized, but your child is just starting this journey.
There are three big phases to understanding multiplication:
Initial understanding of the multiplication process based on skip counting.
Using multiplicative thinking to solve problems about equal groups, areas, and arrays.
Final mastery of the multiplication facts.
Beginning in Kindergarten your child has started skip counting. In fact, skip counting by 2s will be a key skill that will help unlock many multiplication ideas. Encourage your child to use their fingers to count the number of skips (and don’t forget the first number!). Children love to show how smart they are. At this stage, your student can fluently tell you facts like 1 x 8 = 8 and 7 x 1 = 7. Encourage them to show off!
As they begin to investigate other rows in the multiplication table, teachers introduce different multiplication ideas. Here are the three types of problems your child will master this year:
Equal groups. For example, Eva bought 5 bags of apples, and each bag has 8 apples. How many apples did she buy?
Areas. What is the total area of a rectangle that is 6 inches wide and 10 inches long?
Arrays. Mr. Atu’s classroom has 5 rows of desks. Each row has 7 desks. How many desks are in Mr. Atu’s classroom?
These applications help students know “why do we have to learn this?!” In addition, well-designed problems will help students practice facts involving 2s, 5s, and 10s.
Examples of three types of multiplication situation.
5 groups of 4 apples have 20 total apples
This array uses shelves to organize the “rows”. There are 3 shelves with 10 books for a total of 30 books.
The area model is a great way to learn multiplication and practice multiplication of decimals and fractions. Here the rectangle is 10 inches wide and 2 inches long, so it’s area is 20 square inches.
Interactive games like this one help students become fast and accurate with multiplication.
By the end of grade 3, states require students to memorize all the facts in the 10×10 or 12×12 multiplication table. With practice and lots of self-testing, students can master all these facts.
Keep working hard for Friday’s quiz! It will have questions on both the big ideas: calculating areas (integrals) and finding rates of change (derivatives) with a mix of natural logs, exponential functions and u-substitution.
Using Fundamental Theorem of Calculus & U-substitution with Integrals
Using the Power Rule Backwards
This video is the foundational piece for using the power rule backwards to solve integrals. You can use the power rule like normal to check your answers.
Review: Overview of Derivative and Anti-Derivative
Integral of one Term
Solving eu
Trig Function Integrals
Review: Finding Integrals with Anti-Derivative
Previous Videos
Big picture explanation
This video is good at explaining these ideas conceptually.
Derivatives of Natural Logs
Derivatives of Log Properties
Derivatives of ex
Derivatives of ln, logs, and ex
Derivatives of Units 5 & 6 in One Summary
Proofs of ln and ex Derivatives
Review of Limits: If finding limits seems challenging, you can always review this video.
Heart of Calculus: Applications of Derivatives & Integrals
Mean Value Theorem (formal definition):
Contrasting Average Value and Average Rate of Change
Watch by Sunday:
Mean Value Theorem (informal definition):
Implicit Differentiation is the key to understanding Related Rates problems. Please review both these videos and write down questions with timestamps (like, 8:02) that we can review Sunday.
Optimization Practice
If you can’t solve these questions, then please review the four videos below.
An airplane is flying in a horizontal, straight-line path. The speed of the airplane is 100 meters per second, and its altitude is 1000 meters. What is the rate of change of the angle of elevation, , when the horizontal distance from a reference point P on the ground is 2,000 meter?
A ball is expanding at a rate of 0.25 inches per minute. How is the volume changing?
A cylinder is increasing its height at 2 centimeters per minute. How is its volume changing?
New Overview
This gives the overview of how we use derivatives to find the rate of change for position and the rate of change for speed.
More great Crash Course content! This video shows how to use integrals as the inverse operation as the derivative.
Tuesday we will talk though this limit definition of the derivative one more time. We are going to highlight examples of f(x) and how the structure shows that we are finding f‘(x) at a certain point. I want to talk about the concept so the fluency makes more sense.
You are seeing this definition where f(c) is already solved, so f(c) equals a number like, 1000. Then, we see the structure of the function because f(x + c) is never solved. It just shows how f(x) works. Here are the solution steps we will dive deeply into.
The remainder of this page has definitions and tools that we will review up until the test on Thursday.
Tools/Definitions from the Last Quiz
Definition of a continuous function.
Functions have limits if the left and right limits go to the same place
limit of a piece-wise function
When you have time for a 10-minute video. Here is another voice (Sal Khan!) to help make the piece-wise/continuity/approaching ideas make more sense.
The limit is critical in calculus because it gives us access to the idea of how we can find the rate of change at one point. For most of the next month you will find the rate of change of a graph at one point. You will think about the rate of change as a tangent line to the graph and you will use this limit definition. Be sure to actively take notes during this video to think about how you can identify the function and point where you are finding the rate of change.
Power Rule:The rule we use all the time when we have polynomials (but not rational functions). Here is the simplest example:
Product rule:This may require some memorization, but what is the derivative of (f(x))(g(x))?
We say something like “Derivative of the first times the second plus the derivative of second times the first.”
Because we now have the quotient rule, I am going to encourage us to keep this idea of derivative of one times the other. The product rule has the sum of these two types. The quotient rule has the difference of these two types (and a denominator).
Quotient Rule:For rationale functions we can’t use the Power Rule and the Chain Rule is too complicated. We use this rule for functions like:
These functions have derivatives using one rule:
Chain rule to calculate a derivative: This may require some memorization, but what is the derivative of (f(g(x))?
Because the function on the inside is 2x-9 and the function on the outside is ( )2. The derivative of the outside function is 2( ). Power Rule!! The derivative of the inside function is 2. That means altogether we get 2(2x-9)x2.
Derivatives of sinusodial function: What is the derivative of sin? (If you memorize one, then you know the derivative of cos is similar but has a different sign.)
Integration is Antidifferentiation
Indefinite integral (antidifferentiation): Using the power rule, product rule, and chain rule backwards requires the persistence to check your work over and over again. (Also, remember the last 2 problems we did emphasizing adding in the constant term “+c”)
Bigger Problems
Find the critical points of a function: This is why we take derivatives. The process is to take the derivative and set it equal to 0.
Write the intervals over which the function is increasing or decreasing: Another reason to take the derivative. If the derivative is positive, then the rate of change is positive. If the derivative is negative, then the rate of change is?
Horizontal asymptote of a graph: You can solve these with limits if you are interested in extremely large or extremely negative values. In the middle of a function, you use critical points. Question: Given a function, how will you find the local minimum and local maximum values?
antiderivative using power rule (If f'(x)=6x, what is f(x)?)
Today let’s go over any of the topics/explanations that you think are troubling. Then, let’s discuss the homework problems that do not seem possible. Note that the calculator is not available but you can quickly sketch graphs by plotting points and calculating critical points.
Word Problems
Scroll through this video to find related rates word problems which use implicit differentiation to solve word problems you will see throughout this semester.
Welcome back! I am guessing that the only thing that may be confusing for the rest of year is math. If you ever need math help (or any other subject), you can always reach out using the options below. In the meantime, keep finding books you like so you can continue to build your reading skills while you have fun!
Please describe everything you are working on this week. Choose one way to share this information.
If you ever want help with any test or project, just click through these questions to share what you need. I will read through your answers and provide support.
Secondly, if you just want to explore along some of the topics we have been discussing, please check out this Google site. These steps explain how to create your own Google logo using software. You can consider submitting this after you make it perfect.
If you have questions about a sentence or a word, please use the comment section below. (You may need to log in first, but it’s free to do.)
Materials & Activities from Previous Weeks
You can reach out to me any time if you need any help this week. You can use the comments below or upload a file here. Next week, we will be very strategic about preparing for your tests before Winter Break.
Upload any file you want to share here.
Art History Resources
Here are some art resources that you can use to read for information while learning about types of art you may not see in your classes.
UVa’s Art Museum has a variety of media (types of art) and their homepage is definitely artistic!
Please click on the link below and read each section under the “Contents” heading. This 1993 exhibit introduces some of the plastic qualities of African works of art, along with a basic orientation regarding the artworks’ function and significance in their culture. When you have finished reading, return to the homepage and click on the “Art of the African Mask” link at the bottom of the page to view the collection. This exhibit will introduce you to a few different types of traditional African masks. Please note that the term mask refers to the entire costume, and what most students traditionally consider the mask – which is actually the headpiece – often sits atop the head rather than on the face.
Now that we are getting close to a season with less intense academics I want to start getting back to the art topics we started with in the beginning. Today, please look at the six art contests listed here and bring notes that describe: What art is due, When is it due, One thing you like about the contest, One thing you do not like:
Here is a seventh contest. This one is STEAM because it has science you know with art and engineering. If you have time answer the questions about this contest too:
Whenever you could use extra resources, just let me know! I am here to help you meet your goals.
Goals
We talked some about this, but I wanted to add a little bit more context to our discussion. Long-term goals are inspirational. They are big, grand and represent the best version of yourself. That said, it’s had to know if you are meeting or falling behind meeting your long-term goals. That is why you want to write 1 or 2 long-term goals, then write a few short-term “SMART” goals that work toward your long-term goal.
So, your way to stay inspired and motivated is:
Write 1-2 long-term goals.
Write a few SMART goals (Maybe one for each subject or a couple that focus on study processes you know work for you.)
Proportions
I worked with the people at Khan Academy and I know they value proportional relationships as much as any topic in mathematics. Proportions show relationships that exist continuously (e.g., when you drive 55 miles per hour, your distance traveled changes proportionally). The fundamental idea behind proportions are ratios. Ratios are in recipes (2 cups of water for each 1 cup of rice; there are 3 girls for every 2 boys, etc.).
If you feel like some practice on ratios would be helpful, Khan Academy has a series of videos. (Make sure you actively listen to the videos by having pencil and paper out and working through the examples with him!)
This video connects ratios and proportions with examples.
This video might be the most important because it helps unlock math word problems you will see. If you have any questions, you can send them to me in the comments section below. Thanks!
I am excited to see your progress and work with you again! I think we will have some fun and some good times working hard to make sure you can solve lots of math problems and understand what you read. Practice almost everyday will make these topics easier, and I will try to give you practice that helps you learn about things you care about. I am really looking forward to this!
The start of this page has some introductory information, then there is a reading and math assignment, and finally I listed a couple of the things we used before in case you want to use them again. As a reminder, there are lots of ways to share information. Here are the four best ways you can let me know if you are having issues or need help with something in particular.
you can text me a picture of your work or an assignment at 571.641.7611,
or you can write a comment below.
Reading & Math Assignments to Get Started
These two paragraphs describe a virtual field trip you can take to the Inventors Hall of Fame. Read these paragraphs twice because they have lots of information. Then, answer the questions that will ask you to use words from these paragraphs in a sentence you make up. This activity will take about 30 minutes because you may have to think hard to write your own sentence using these words. Afterwards, you can take a virtual field trip!
The museum’s key attraction is the inspiring Gallery of Icons™. Here, you’ll find icons commemorating each of the more than 500 National Inventors Hall of Fame® (NIHF) Inductees we’ve welcomed since our founding in 1973. Hung in a tessellating hexagon pattern inspired by honeycombs and organized by patent number, this is a stunning visual representation of the history of American innovation.
Familiar names include the prolific Thomas Edison, who filed more than 1,000 patents, including the electric light bulb; the high-flying Wright Brothers, who successfully piloted the first powered aircraft; the brilliant Stephanie Kwolek, who invented the life-saving Kevlar fiber worn by police officers and military personnel across the world; and George Washington Carver, who famously developed crop-rotation methods for conserving nutrients in soil and discovered hundreds of new uses for crops such as the peanut and sweet potato.
It will be great to see you again! I really look forward to talking with you about math, reading, and other things you want to learn. I hope you continue to work hard at school. I know if we talk and email a lot and work hard together, then you will be ready for middle school in the Fall!
The start of this page has some introductory information, then there is a reading and math assignment, and finally I listed a couple of the things we used before in case you want to use them again. As a reminder, there are lots of ways to share information. Here are the four best ways you can let me know if you are having issues or need help with something in particular.
How did Pennsylvania get its name? Its founder, English reformer William Penn, born on October 14, 1644, in London, England, named it in honor of his father. Persecuted in England for his Quaker faith, Penn came to America in 1682 and established Pennsylvania as a place where people could enjoy freedom of religion. The colony became a haven for minority religious sects from Germany, Holland, Scandinavia, and Great Britain. Penn obtained the land from King Charles II as payment for a debt owed to his deceased father.
Born the privileged son of a land-owning gentleman, young William Penn was greatly affected by the preaching of Quaker minister Thomas Loe. Expelled from Oxford University in England in 1662 for refusing to conform to the Anglican Church, Penn joined the Quakers. He was locked up in the Tower of London four times for stating his beliefs in public and in print. After his father died in 1670, Penn inherited the family estates and began to frequent the court of King Charles II, campaigning for religious freedom.
Seeing no prospects for religious tolerance or political reform in England, Penn looked to America, which he had visited briefly in 1677. In a 1682 document, Penn guaranteed absolute freedom of worship in Pennsylvania. Rich in fertile lands as well as religious freedom, the colony attracted settlers and grew rapidly.
Penn is also remembered for peaceful interaction with the Lenni Lenape Indians and his draft of the Plan of Union, a forerunner of the U.S. Constitution. Thanks to William Penn, Pennsylvania, which guaranteed religious freedom for its citizens, was established in the New World.
We personalize a bible by focusing on 20 passages that you choose and we can bring the reading level down to your child’s level if you are interested. You can check out by adding one or both products below to your cart and then check out above.
Keysight Technologies demonstrated critical technology at the 2019 Association of Old Crows conference in Washington, D.C.
You may have seen remote control cars making circles on the
floor. You may even own one yourself. But, do you know how they work?
Batteries provide engine to the car. These electrons flow
from the battery into the motor. The motor rotates the tires so that the car
can move forward and backwards. In addition, the motor controls arms which can
turn the wheels. These two systems allow the car to go forward and backwards as
well as turn.
The handset also has a battery which transfers information
from the buttons to radio waves. These radio waves are sent to the car and
received by a signal receiver. The transmitters send signals to the signal
receiver. These radio waves change frequency based on the information it sends.
The frequency is the code that tells the car what to do.
Keysight Technologies has a great demonstration showing a 21st-century demonstration of this process. They add a signal recorder and a signal transmitter to their set up. Altogether, this demonstration uses:
Remote control handset
Remote control car with signal receiver
Signal recorder
Signal transmitter
Students first drive the remote-control car normally. They use the remote control handset to send waves to the car’s signal receiver. The car interprets those radio waves and perform the driving maneuvers sent by the remote control handset.
Keysight Technologies’ remote control car, remote controller, signal receiver and signal transmitter (bottom).
At the same time, the signal recorder updates the display
showing the frequency of the radio waves. Not only does this device show the
frequency of the radio waves, it records them! These recorded frequencies can
then be transmitted again at any time.
The second part of the demonstration shows how signals can
be recorded to “spoof” a system. “Spoof” means sending controls to a device in
a way that makes it think it is connected to its controlling system. In this
demonstration, the signal transmitter spoofs the remote control handset. Even
when the handset is turned off, the remote-control car can be moved!
The signal transmitter uses the data saved by the signal
recorder. It transmits these signals in the same order at the same frequency.
This signal instructs the car to move in the exact same ways. In the second
part of the demonstration, the remote control handset is turned off and the
signal transmitter sends its signal. The remote-control car exactly duplicates
the motions the student just did.
The purpose of this demonstration is to show that radio
signals used to control remote vehicles can be manipulated by technology. This
technology can take control of another system. For example, a drone that is
operating inappropriately could be controlled by our military using a signal
transmitter that overwhelms the drone’s operation signal.
Your turn! Think about another example of how someone could use this technology. Explain what parts they would need, how they would use them, and what the result would be.
