The internet is full of attention grabbing headlines about this including:

**Chinese space station carrying TOXIC chemicals to CRASH into Europe or New York NEXT WEEK**

**Chinese space station: Tiangong-1 could crash into Earth and hit Europe next week**

And this one that sounds like a movie ad:

**China’s Out of Control Space Station Coming to a City Near You!**

Don’t Panic! Despite the dire warnings, you are not at risk of being obliterated by a space lab crashing through your roof. Most of the lab will burn up as it re-enters the Earth’s atmosphere with remaining debris scattering over a large area.

But scientists around the world are tracking Tiangong-1 and making models to predict when and where it will land. Here’s one article that explains how they are doing it:

https://www.livescience.com/62077-tiangong-re-entry-date.html

While the models being developed to pinpoint the re-entry and landing sites must consider an incredible number of variables and depend on sophisticated modeling tools, the basics of understanding the path of this or any other satellite or projectile only require an understanding of the polynomial functions that we’ve been studying in Algebra this month.

And this one that sounds like a movie ad:

Don’t Panic! Despite the dire warnings, you are not at risk of being obliterated by a space lab crashing through your roof. Most of the lab will burn up as it re-enters the Earth’s atmosphere with remaining debris scattering over a large area.

But scientists around the world are tracking Tiangong-1 and making models to predict when and where it will land. Here’s one article that explains how they are doing it:

https://www.livescience.com/62077-tiangong-re-entry-date.html

While the models being developed to pinpoint the re-entry and landing sites must consider an incredible number of variables and depend on sophisticated modeling tools, the basics of understanding the path of this or any other satellite or projectile only require an understanding of the polynomial functions that we’ve been studying in Algebra this month.

While in orbit the Tiangong-1 Space Lab, like all other satellites, travels in an ellipse. We won’t work with ellipses until Pre-Calculus, but the equation is just a two-variable version of the equations we’ve been working with this week:

Where

Although we haven’t studied this in class yet, you can try to graph an ellipse using the same technique of picking values for one variable and solving for the other then plotting the points. Try finding points. Then, graph these ellipses:

Look at your graphs and use the analysis tools we learned in class to think about what happens when you change the value of any of the constants in the equation.

Many things will make a satellite, such as the Tiangong-1 Space Lab, change course or slow down. Normally, ground control corrects the path by activating one or more engines on the satellite. In 2016, China’s space agency lost its connection to the Tiangong-1 Space Lab, so it hasn’t been able to correct its path for two years.

As a satellite slows down, its path becomes less elliptical and more circular. Mathematically, the values of

Notice that the first equation gave you a long, stretched out ellipse. Each subsequent equation gives you a rounder ellipse until you end up with a circle in the last equation. (Mathematicians call this degree of stretchiness “eccentricity” and is usually calculated using the variable *e*.)

Notice that if*A = B*, the equation for the ellipse become the same as the equation for a circle with : *C = r^2*

Notice that if

If a satellite traveling in a circle slows down further, its path become a spiral and eventually follows the path of a projectile.

This week we worked on solving equations in the form

Substituting these variables gives us:

Since the altitude of any object on the ground is zero, we can find the time it takes for a projectile to land by solving the following equation:

This is the standard equation used to figure out how long a baseball or rocket will be airborne before it lands on the ground. Then the time can be used to find the distance traveled by putting it into a simple equation that normally looks like this:

Where:

We can use this equation for debris from Tiangong-1 Space Lab after it re-enters the Earth’s atmosphere.

Last year, in Pre-Algebra we played with the equation for the Force Due to Gravity:

Last year, in Pre-Algebra we played with the equation for the Force Due to Gravity:

We used this to confirm the acceleration due to gravity close to the surface of the Earth and to understand why there is very little gravitational force in the space station. When we get to Calculus we’ll be able to work with cases where *g* is a variable this is constantly changing over time.

Using the quadratic equations that we learned in class, it seems like the path of the Tiangong-1 Space Lab has three distinct parts. But we can see that the circle is just a special type of ellipse.

To see how projectile motion and orbiting are related think about this:

- Imagine that you throw a baseball. It will travel some distance following the path of a parabola and land somewhere on the ground.
- Imagine that you threw the same baseball with more force. It will travel further before landing on the ground.
- Now imagine that you are superman. You throw the ball so hard that the distance it travels before landing on the ground is greater than the radius of the Earth. There is no ground for it to land on. It will keep falling toward the Earth but never land. It goes into orbit.

In “Hidden Figures” Katherine Goble wrestled with the problem of how the path of a rocket would change from an elliptical orbit to a parabolic fall to Earth. Rewatch the movie and to find her “aha” moment when she realized that these two paths could be described with one equation. That equation involves trigonometry and polar coordinates. You can see Ms. Goble work out the mathematics of this in the movie. The unifying equation is:

I know what you’re going to ask me and the answers are:

“No, we can’t spend our entire next class meeting playing with these equations. You don’t yet have that math background to derive all of them. BUT we can spend part of the class period playing with projectile motion since it ties into the quadratic functions that we worked on.”

“Yes, some of these equations are the ones that Dylan derived and wrote on the window last year. As sad as it was to erase his work, I’ve very proud that you were able to add the derivation of the quadratic equation to the window last week.”

“Yes,I recommend that you spend some of your time off of school doing your own research on this.” I suggest the following:

http://www.physicsclassroom.com/class/circles/Lesson-4/Circular-Motion-Principles-for-Satellites

Rewatch the movie Hidden Figures.

Logic, often translated as thought or reason, builds the foundation for critical thinking skills. It is essential for success in higher level math, computer science, philosophy, science, engineering problem solving and decision-making. Logic demands that each statement be supported by reason. In logic puzzles, each conclusion can be supported by a “because” statement. At higher levels, students begin to use ‘if…then…” statements to support their conclusions. Logic puzzles require students to practice using both deductive and inductive reasoning skills. Deductive reasoning is the practice of using information from a larger set of information to understand other sets of related information. Inductive reasoning is the practice of using specific data to draw a much larger conclusion. These skills will prove indispensable as children emerge in their professional and academic lives. |

In the United States, formal training in logic is often limited to writing formal proofs in high school geometry classes. Yet, logic can easily be introduced to younger students through fun games and puzzles.

The key to teaching students logic through games and puzzles is to train them to consistently explain “why” they are able to make each of their moves. I train them to precede each move with a statement in the form of:

*“I know ________________ because ______________________.”*

Students are not allowed to make moves if they “think” something is true or if something “could” be true.

I always start with Binary Puzzles because the rules are very simple. Each puzzle is a square array with some cells containing 1s or 0s. The goal is to fill in the empty cells using the following rules:

Students are not allowed to make moves if they “think” something is true or if something “could” be true.

I always start with Binary Puzzles because the rules are very simple. Each puzzle is a square array with some cells containing 1s or 0s. The goal is to fill in the empty cells using the following rules:

In order to discuss our example puzzle, we will label each column and row so we can easily identify each cell. So in the example below, there is a 1 in cell Aa and a 0 in cell Bd.

Notice how this statement is supported by one of the rules.

Look at the updated puzzle. What other cells can you find based on the rules of the puzzle?

You should be able to make the following statements:

Look at the updated puzzle. What other cells can you find based on the rules of the puzzle?

You should be able to make the following statements:

“I know that cell Ab is a 0 BECAUSE rule #3 says that we can’t have three 1s next to each other in any row." “I know that cells Af and Df are 0s BECAUSE rule #3 says that we can’t have three 1s next to each other in any column.” “I know that cells Bb, Be, Ca, Cd, Ec, and Fe are 1s BECAUSE rule #4 says we can’t have three 0s next to each other in any row." |

Update the puzzle for each of these statements then use the additional information to try to fill in more spaces.

Based on this updated puzzle you should be able to make the following statements:

Based on this updated puzzle you should be able to make the following statements:

“I know that cell Ae contains a 1 BECAUSE rule #1 says that each row must have the same numbers of 1s as 0s and there are already three 0s in that row.’“I know that cells Ba and Ce contain 0s BECAUSE rule #1 says that each row must have the same numbers of 1s as 0s and there are already three 1s in those rows.’ |

After filling in these cells, you should be able to easily finish the puzzle.

Review each rule to make sure that your puzzle is correct:

- Confirm that each row contains three 0s and three 1s.
- Confirm that each column contains three 0s and three 1s.
- Scan each row and column to make sure that there are no instances where three 0s or three 1s are next to each other.

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 20 puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 20 puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 9 puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 9 "8x8" puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 5 "10x10" puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

$1.00

Binary Puzzles are the perfect introduction to logic. Students only need to know how to count to three to complete these logic puzzles! Each puzzle is a square array with cells containing 1s and 0s. Fill in the empty cells using the following rules:

**Each puzzle pack contains 5 "10x10" puzzles and step-by-step directions. **

- Each row must have the same number of 1s and 0s
- Each column must have the same number of 1s and 0s
- Three 1s can't be next to each other in any row or column.\
- Three 0s can't be next to each other in any row or column

The first time I saw him, he reminded me so much of Q. The way he rolled his head back when he was thinking. His lack of eye contact when speaking. His abnormally precise language, his obsession with numbers, especially very large numbers resulting from doubling, irrational numbers and those resulting from functions unknown to any mathematician other than himself. I played with him the way I’ve played with countless other children I’ve met like this. I created patterns and let him expand on the series. I invented functions using made up notation to see what he would do with them. His mother, anxious that he would make a good impression, encouraged him to look at me and answer my questions. “Wait” I hush her, almost in a whisper. “He’s thinking.”

I will never tire of watching these children think. They go into their own worlds. They don’t make eye contact. It’s not like they are avoiding eye contact; it’s that they just don’t see me or anything in this world for that matter. I can almost hear the wheels turning in their heads. I see them suspended between two worlds – the physical one that I occupy, and the sacred realm accessible only through the portal controlled by their own minds.

Their solutions to my challenges are never based on standard algorithms. I don’t expect them to be. In fact, I present problems that I’m sure they’ve never seen. It’s how their minds work that I’m studying. Can they explain their answer to me? I ask deeper questions to help them more clearly define their logic.

I’m no longer asking him questions to determine whether or not I’ll take him on as a student. I’ve already decided that. I want to see how he reacts to me and the problems I give him. I want to see if his mother trusts me enough to let us play our game without interference. He can talk math with me forever, but I know that I don’t really exist to him. I am just the source of the fascinating puzzles. His mother is also just a sounding board for the product of his mental exercises. At this point, I’m sure the endless monologues just exhaust her but soon they will also bewilder her when he crosses over into topics beyond her ability to comprehend. In time I will teach her to take her cues from his tone of voice and facial expressions to know when to smile and nod like she’s trying to converse with someone who’s speaking a foreign language.

I flip through my mental Rolodex as I think about where to place him. D is a perfect personality match but he is at least 3 years ahead. C would be perfect but she is 2 years behind. E would be perfect but his parents already push him so much that I’m considering dropping him. I will help any child explore their passions but I won’t be party to pushing a child just for bragging rights.

I could tutor him. We would have fun. But I know that there is something he needs more than math instruction. His mother doesn’t know what I know yet; if I don’t intervene, in four years, I will be holding her hands and listening to her cry about a crisis that she can’t yet imagine.

Some people would say he has Asperger’s or is ‘on the spectrum’. In my day, we would have called him ‘quirky.’ I’m not a psychologist so I don’t label him. To me, he is just a little boy who loves math. I know I can use his love of math to help him develop social skills that he doesn’t realize he needs. I also know that I will have to prepare him to deal with the college classroom before he is old enough to get a driver’s permit.

There was a time when I thought that this quirky social awkwardness must be tied to the same gene that determined mathematical genius. I believed that their social limitations had to be accepted as an immutable part of their personalities. I remember the joy of finding other boys like this for my son.

I no longer accept that brilliant children are destined to be hampered by their social skills. I no longer believe that they are predisposed to spectrum disorders. I’m not a mental health professional. I’m just a mother who has worked with hundreds of these children over the past decade. I’ve participated in dozens of meetings like the one above.

Their solutions to my challenges are never based on standard algorithms. I don’t expect them to be. In fact, I present problems that I’m sure they’ve never seen. It’s how their minds work that I’m studying. Can they explain their answer to me? I ask deeper questions to help them more clearly define their logic.

I’m no longer asking him questions to determine whether or not I’ll take him on as a student. I’ve already decided that. I want to see how he reacts to me and the problems I give him. I want to see if his mother trusts me enough to let us play our game without interference. He can talk math with me forever, but I know that I don’t really exist to him. I am just the source of the fascinating puzzles. His mother is also just a sounding board for the product of his mental exercises. At this point, I’m sure the endless monologues just exhaust her but soon they will also bewilder her when he crosses over into topics beyond her ability to comprehend. In time I will teach her to take her cues from his tone of voice and facial expressions to know when to smile and nod like she’s trying to converse with someone who’s speaking a foreign language.

I flip through my mental Rolodex as I think about where to place him. D is a perfect personality match but he is at least 3 years ahead. C would be perfect but she is 2 years behind. E would be perfect but his parents already push him so much that I’m considering dropping him. I will help any child explore their passions but I won’t be party to pushing a child just for bragging rights.

I could tutor him. We would have fun. But I know that there is something he needs more than math instruction. His mother doesn’t know what I know yet; if I don’t intervene, in four years, I will be holding her hands and listening to her cry about a crisis that she can’t yet imagine.

Some people would say he has Asperger’s or is ‘on the spectrum’. In my day, we would have called him ‘quirky.’ I’m not a psychologist so I don’t label him. To me, he is just a little boy who loves math. I know I can use his love of math to help him develop social skills that he doesn’t realize he needs. I also know that I will have to prepare him to deal with the college classroom before he is old enough to get a driver’s permit.

There was a time when I thought that this quirky social awkwardness must be tied to the same gene that determined mathematical genius. I believed that their social limitations had to be accepted as an immutable part of their personalities. I remember the joy of finding other boys like this for my son.

I no longer accept that brilliant children are destined to be hampered by their social skills. I no longer believe that they are predisposed to spectrum disorders. I’m not a mental health professional. I’m just a mother who has worked with hundreds of these children over the past decade. I’ve participated in dozens of meetings like the one above.

I no longer accept that brilliant children are destined to be hampered by their social skills. I no longer believe that they are predisposed to spectrum disorders. I’m not a mental health professional. I’m just a mother who has worked with hundreds of these children over the past decade.

I started HEROES as the mother of a child like this. After reading all the research and speaking to all the living experts I still wanted to meet another mother of a child like mine. I wanted to look at her in the eye and ask her if her child eventually grew up to be a productive, happy adult. I wanted to see that her grown child was OK when he grew up. I also wanted to be the mentor to other mothers who shared the same anxiety over their children.

Over the past decade, I’ve met thousands of mothers of brilliant children. I’ve leant my shoulder to hundreds of them as they cried; at times shedding tears of joy; at other times tears of despair. I’ve been enchanted by children making new friends at conferences. I’ve mourned the decision to commit a child to a psychiatric facility. I’ve celebrated with pre-teens who won a math completion against college students. I’ve grieved for a child hospitalized for eating disorders. I’ve congratulated teens who’ve earned patents. I’ve counseled teens locked up in juvenile detention.

Why do some become rising stars and others end up on suicide watch? There is no simple answer to that question but I’ve observed some critical variables.

Balance is probably the trickiest part of being a parent. In the movie Gifted, Mary enjoys the perfect balance between academic stimulation and social integration. Unfortunately, life is not as simple as a movie. Nevertheless, there is an important lesson to learn from Frank Adler’s battle with his mother over Mary’s education. Geniuses need academic stimulation. They need guidance. Mary didn’t learn differential equations on her own. Someone introduced her to the concepts of limits and infinity and the meaning of delta and epsilon. Geniuses also need love and companionship. They need friends who connect with them not just for what they know but for who they are. As Mary says, ‘Frank loved me before I was a genius.’

If your child does not find school adequately challenging then you may need to look for outside resources that will reinvigorate the thirst for learning. But it is important to not go overboard. I’ve met many children who used to love math until they burned out over too many tutors, competitions, and classes. Remember, it is more important to nurture the love of learning than to accelerate the rate of learning.

Over the past decade, I’ve met thousands of mothers of brilliant children. I’ve leant my shoulder to hundreds of them as they cried; at times shedding tears of joy; at other times tears of despair. I’ve been enchanted by children making new friends at conferences. I’ve mourned the decision to commit a child to a psychiatric facility. I’ve celebrated with pre-teens who won a math completion against college students. I’ve grieved for a child hospitalized for eating disorders. I’ve congratulated teens who’ve earned patents. I’ve counseled teens locked up in juvenile detention.

Why do some become rising stars and others end up on suicide watch? There is no simple answer to that question but I’ve observed some critical variables.

Balance is probably the trickiest part of being a parent. In the movie Gifted, Mary enjoys the perfect balance between academic stimulation and social integration. Unfortunately, life is not as simple as a movie. Nevertheless, there is an important lesson to learn from Frank Adler’s battle with his mother over Mary’s education. Geniuses need academic stimulation. They need guidance. Mary didn’t learn differential equations on her own. Someone introduced her to the concepts of limits and infinity and the meaning of delta and epsilon. Geniuses also need love and companionship. They need friends who connect with them not just for what they know but for who they are. As Mary says, ‘Frank loved me before I was a genius.’

If your child does not find school adequately challenging then you may need to look for outside resources that will reinvigorate the thirst for learning. But it is important to not go overboard. I’ve met many children who used to love math until they burned out over too many tutors, competitions, and classes. Remember, it is more important to nurture the love of learning than to accelerate the rate of learning.

It is more important to nurture the love of learning than to accelerate the rate of learning.

If your child is not connecting socially at school then you may also need to look into additional resources. Contrary to popular belief, brilliance does not equate to social awkwardness or isolation. The most amazing children I’ve met were not only nationally or internationally recognized as brilliant but also popular and possessed sophisticated senses of humor. Most of the children at HEROES are happy and well rounded. They also play soccer, compete in fencing, enjoy drama, play musical instruments and often rush from class to friends’ birthday parties.

Other children need help to find other children close to their age with similar abilities and interests. I’ve worked with dozens of children who were diagnosed with ADHD or Asperger’s Spectrum Disorders who were able to fully engage in programs with other children with similar abilities and interests. When I look at my class of students, I don’t see the ADHD or the ASD. I see children who are enjoying learning and the camaraderie of friends mutually working to solve problems and meet challenges. I also note that the monologues have given way to genuine conversation and debate and that eye contact naturally developed as classmates became friends.

]]>

Other children need help to find other children close to their age with similar abilities and interests. I’ve worked with dozens of children who were diagnosed with ADHD or Asperger’s Spectrum Disorders who were able to fully engage in programs with other children with similar abilities and interests. When I look at my class of students, I don’t see the ADHD or the ASD. I see children who are enjoying learning and the camaraderie of friends mutually working to solve problems and meet challenges. I also note that the monologues have given way to genuine conversation and debate and that eye contact naturally developed as classmates became friends.

Before each math class, I check and restock all the supplies my students will need: wide ruled paper, colored pencils, crayons, and pens. Pens? Not Pencils? Yes, pens not pencils – black and purple pens to be precise.

I know this is unconventional but there’s a method to my madness. Think about it: Why do we use pencils for math? Normally this question is met with a blank stare since the use of pencils for math is so universal that most have never considered questioning this convention. The obvious answer, of course, is to erase our mistakes but, why do we want to erase our mistakes when we know that we learn from our mistakes?

I know this is unconventional but there’s a method to my madness. Think about it: Why do we use pencils for math? Normally this question is met with a blank stare since the use of pencils for math is so universal that most have never considered questioning this convention. The obvious answer, of course, is to erase our mistakes but, why do we want to erase our mistakes when we know that we learn from our mistakes?

I teach my students to embrace mistakes rather than erase their mistakes. My students self-grade with purple pens. I teach them to not just mark a problem right or wrong but to annotate their work with notes on their mistakes. Was it a concept that they did not understand? Was it an arithmetic mistake? My students must write a note about every mistake they find on their paper. After a few pages of work, they look back at their purple notes find a pattern. Most days, most students are making only one or two types of mistakes over and over. By highlighting their mistakes, each student is able to focus on specific individual learning goals.

Students are usually surprised by what they find out about themselves by highlighting their mistakes. They learn that:

Neatness matters – The number #1 cause for getting the wrong answer is not being able to read their own handwriting.

Basic arithmetic matters – Students often dismiss ‘careless’ arithmetic mistakes but they quickly realize that these errors matter when they actually have to calculate an answer rather than simply select from a set of multiple choice options.

Once students realize the importance of neatness they understand some of my classroom rules which include:

Telling the children is one thing, getting them to adopt these practices is quite another. In my classroom these rules are enforced by The Crumple Monster. Any page that does not meet my standards is removed, crumpled, placed in the recycling bin and replaced by a fresh sheet of paper. The Crumple Monster destroys any and all messy work. It is rare for a student to not have at least one page destroyed by The Crumple Monster on the first day. In fact I normally fill two commercial sized recycling bins on the first weekend of class.

How do the kids react to these tactics? I thought they might think I’m mean but they almost universally think I’m funny. They love writing with the purple pens and joke with each other that The Crumple Monster is going to get them.

Students are usually surprised by what they find out about themselves by highlighting their mistakes. They learn that:

Neatness matters – The number #1 cause for getting the wrong answer is not being able to read their own handwriting.

Basic arithmetic matters – Students often dismiss ‘careless’ arithmetic mistakes but they quickly realize that these errors matter when they actually have to calculate an answer rather than simply select from a set of multiple choice options.

Once students realize the importance of neatness they understand some of my classroom rules which include:

- All work must be written between, not across, the lines on the paper. Most new students seem to have never noticed that notebook paper is covered by a series of parallel horizontal lines. They certainly don’t demonstrate any understanding of the purpose of these lines.
- Work must begin at the top left corner and continue down the paper with writing going from left to right. I am no longer shocked to find that students literally do not know which way is up when it comes to paper. I flip and turn paper over on the students’ desks so that it faces the correct way. I demonstrate on my classroom poster by randomly placing the words “If I write all over the paper like this you can’t understand what I wrote.” This usually gets a few laughs while making the point.

Telling the children is one thing, getting them to adopt these practices is quite another. In my classroom these rules are enforced by The Crumple Monster. Any page that does not meet my standards is removed, crumpled, placed in the recycling bin and replaced by a fresh sheet of paper. The Crumple Monster destroys any and all messy work. It is rare for a student to not have at least one page destroyed by The Crumple Monster on the first day. In fact I normally fill two commercial sized recycling bins on the first weekend of class.

How do the kids react to these tactics? I thought they might think I’m mean but they almost universally think I’m funny. They love writing with the purple pens and joke with each other that The Crumple Monster is going to get them.

They eventually stop getting problems wrong because they can’t read their own handwriting. In doing so, they are taking their first steps to learning how to develop study skills that they will need for higher level math.

The bottom line is that they do learn to write to my standards. They eventually stop getting problems wrong because they can’t read their own handwriting. In doing so, they are taking their first steps to learning how to develop study skills that they will need for higher level math.

After students learn to complete their work so that it can be read, they can learn to add notes so that their math practice transforms from mindless, repetitive tasks that must be completed simply to earn a grade into meaningful exercises that they can use to learn from their mistakes. Only after their work is legible does it make sense to teach them how to take useful notes and keep an organized notebook.

Students do not magically acquire organizational skills as they get older. They learn organizational skills when they realize that they are beneficial. Whether it is writing legibly, writing notes or keeping a notebook, students must see the benefit of these skills before we can expect them to consistently adopt them.

]]>After students learn to complete their work so that it can be read, they can learn to add notes so that their math practice transforms from mindless, repetitive tasks that must be completed simply to earn a grade into meaningful exercises that they can use to learn from their mistakes. Only after their work is legible does it make sense to teach them how to take useful notes and keep an organized notebook.

Students do not magically acquire organizational skills as they get older. They learn organizational skills when they realize that they are beneficial. Whether it is writing legibly, writing notes or keeping a notebook, students must see the benefit of these skills before we can expect them to consistently adopt them.

Learn(v.) - [lurn] gain or acquire knowledge of or skill in (something) by study, experience, or being taught. *'they'd started learning French'**[with infinitive] 'she is learning to play the piano.**[no object] 'we learn from experience.'*
| Every child should be learning in every class every day. This means that each child should walk out of each class session having acquired some new knowledge or skill. Simply walking through a lesson plan and completing a checklist of activities does not ensure that every student is learning in every class. |

In order to assure that each child is learning, we must must determine what a child knows both before and after the lesson. Nationally, we try to assess learning by standardized test scores. The large scale of most school systems requires assessments that can be recorded in databases to be tracked and analyzed. Parents often feel that their child already knows something because they were exposed to material and children often claim that they ‘already know that’ because they’ve seen something similar before. None of these methods can provide a comprehensive answer to the question: "Did the individual child gain or acquire knowledge of or skill in something?"

Just because a child has been exposed to material, does not mean that the child learned it. Scoring well on a multiple choice standardized test does not guarantee mastery. Children can parrot back information or memorized steps without full comprehension. What many parents fail to understand is that there are degrees to understanding each concept. In my classroom I strive to make sure that each child walks away with a greater degree of understanding than when they entered my classroom. This often requires me to adjust my learning objective as the lesson progresses.

Just because a child has been exposed to material, does not mean that the child learned it. Scoring well on a multiple choice standardized test does not guarantee mastery. Children can parrot back information or memorized steps without full comprehension. What many parents fail to understand is that there are degrees to understanding each concept. In my classroom I strive to make sure that each child walks away with a greater degree of understanding than when they entered my classroom. This often requires me to adjust my learning objective as the lesson progresses.

I strive to make sure that each child walks away with a greater degree of understanding than when they entered my classroom.

For example, this past weekend my third grade classes began a unit on fractions. My first goal was to make the connection between division and fractions. I also wanted my students to understand what a fraction is, how the denominator relates to the parts of a whole, that fractions can be added or ‘counted’ like any other object and to relate whole numbers to equivalent fractions

I started with a review of division:

*Miss Danielle bakes 48 cookies for her class. If there are 9 students in her class, how many cookies can she give to each student? How many cookies will she have left over?*

Then, I transitioned to an almost identical problem to introduce fractions:

*Our class is going to have a pizza party. If I order two pizzas for the class, how much pizza should I give each student?*

Then, I transitioned to an almost identical problem to introduce fractions:

We discussed the difference between these two problems until I was convinced that every student understood the difference between division problems with remainders and ones with fractions. While we CAN turn every remainder into a fraction, sometime it simply does not make sense. Each child provided an example of something that they could divide into fractions and something that could not be divided into fractional parts. Following my belief that laughter helps children remember, I celebrated their often silly examples such as:

*“We couldn't each have 1 1/4 dogs. Someone would have an extra puppy to take home.” *

It wasn’t until the children started to play with the manipulatives that I realized that they were going to learn much more than I had originally anticipated.

First, we played with magnetic fraction bars. My original intent for this project was to allow the students to see the connection between the denominator and the number of pieces required to make a whole. I also wanted them to see that larger denominators meant smaller pieces. But then the students started moving the pieces around and reported on their own discoveries. Two quarters were the same size as one half. One half was the same size as three sixths. You could replace 1/5 with 2/10 but if you tried with 1/12ths it was always too short or too long.

They were discovering equivalent fractions on their own, not through formal study or by being taught a formal lesson but through experience. The fractions weren’t just numbers on a worksheet that needed to be completed but an abstract concept that was now real and comprehensible. The greatest learning moments are when children experience the joy and excitement of making their own discoveries. It's moments like these, when their faces light up with each new discovery, that I live for.

It wasn’t until the children started to play with the manipulatives that I realized that they were going to learn much more than I had originally anticipated.

First, we played with magnetic fraction bars. My original intent for this project was to allow the students to see the connection between the denominator and the number of pieces required to make a whole. I also wanted them to see that larger denominators meant smaller pieces. But then the students started moving the pieces around and reported on their own discoveries. Two quarters were the same size as one half. One half was the same size as three sixths. You could replace 1/5 with 2/10 but if you tried with 1/12ths it was always too short or too long.

They were discovering equivalent fractions on their own, not through formal study or by being taught a formal lesson but through experience. The fractions weren’t just numbers on a worksheet that needed to be completed but an abstract concept that was now real and comprehensible. The greatest learning moments are when children experience the joy and excitement of making their own discoveries. It's moments like these, when their faces light up with each new discovery, that I live for.

My students presented me with another surprise when we played with fraction circles. Not only were they able to use them to perform division with fractional remainders, but they again were drawn back to equivalent fractions. By the end of the period each child had created an addition problem that required at least one conversion to common denominators. One boy even predicted, and then proved, that he needed a 1/20th to complete the circle that he started with ½ + ¼ + 1/5.

“The use of equivalent fractions as a strategy to add and subtract fractions” is a Common Core Standard for 5th grade. Yet all of my 3rd Graders were happily discovering this strategy on their own through guided play with manipulatives.

While children may have different attitudes about school, I have never met a child who did not love to learn. While children may have different attitudes about school, I have never met a child who did not love to learn. Children restricted to a rigid curriculum designed to ensure that everyone meets a minimum standard are often robbed of the joy of learning. Classes focused on completing standardized lesson plans to prepare for standardized tests often deprive students of the excitement that new discoveries stimulate. The most important job of a teacher is to nurture the love of learning, whether it is through study, being taught, or by experience. When working with advanced learners, that often requires providing students with opportunities to discover relationships that are beyond the standard curriculum. | Use equivalent fractions as a strategy to add and subtract fractions.CCSS.MATH.CONTENT.5.NF.A.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) |

Math is not about crunching numbers; true math is about pattern recognition and logic. This does not mean that mastering basic calculations is not important. Learning math facts is like learning the alphabet. A child who must sound out words, syllable by syllable is not able to comprehend a reading passage. Likewise, a child who struggles to complete basic computations is not able to recognize new mathematical relationships. |

While the ability to memorize math facts is not related to mathematical reasoning skills required for higher level math it greatly influences the perception of a child’s math ability. Compare two children learning to add fractions. They are given the task of calculating 1/14 + 1/21. Both equally understand the concept and procedure to add fractions.

Sophia knows her times tables but uses skip counting for a few of the harder math facts such as 6 X 8 and 7 X 7. She remembers that a common denominator may be found by multiplying the two denominators. After performing long multiplication to find that 14 X 21 = 294, Sophia converts the fractions to 21/294 + 14/294 = 35/294. She then factors both 35 and 294, probably using long division twice to find that 294 may be factored to 2 X 3 X 7 X 7. After recognizing that 7 is a common factor, Sophia converts 35/294 to 5/42. While Sophia is able to eventually get the correct answer, this process required her to perform at least one three-step double-digit multiplication and probably three long division calculations.

Emma has instant recall of all of her math facts and quickly recognizes that both 21 and 14 are multiples of 7. Emma quickly converts the problem to 3/42 + 2/42 = 5/42. By avoiding the need to perform double-digit multiplication and long division Emma is able to finish at least five times as many problems as the Sophia.

Although Sophia’s mathematical reasoning skills matches Emma’s, Emma is perceived as the “better” at math. Since Sophia is the last to finish the math worksheet and has several errors due to simple calculation mistakes, she believes that she’s “not good at math.” These perceptions are reinforced by classmates, teachers and parents who also equate Emma’s speed and Sophia’s slowness to their math abilities.

Over time these perceptions become reality. Emma is able to complete more math problems because she is able to work faster. She is more likely to complete challenge problems, be selected for pull-out programs and eventually accelerated math. Sophia, on the other hand, does not have time to complete challenge problems or participate in enrichment math since it takes her so long to complete her regular assignments.

In reality, the difference between the two girls is the degree to which they learned their multiplication facts. Both understand the concept of multiplication. Both understand how to calculate a product through repeated addition of skip counting. But Emma recognizes multiples like sight words while Sophia needs to calculate a few of them like a child who must still sound out a few words.

While Emma and Sophia are fictional, I’ve worked with hundreds of students from algebra to calculus whose self-perceptions and actual ability to master higher level math have been limited not by their innate mathematical reasoning skills but their mastery (or lack thereof) of basic math facts.

One of my 4th graders, Lucas*, recently said to me, “I don’t need to learn my multiplication facts anymore because we’re done with that in school. Now I’m doing division instead.” In his mind, he was done with the task of multiplication and didn’t need it anymore.

Sophia knows her times tables but uses skip counting for a few of the harder math facts such as 6 X 8 and 7 X 7. She remembers that a common denominator may be found by multiplying the two denominators. After performing long multiplication to find that 14 X 21 = 294, Sophia converts the fractions to 21/294 + 14/294 = 35/294. She then factors both 35 and 294, probably using long division twice to find that 294 may be factored to 2 X 3 X 7 X 7. After recognizing that 7 is a common factor, Sophia converts 35/294 to 5/42. While Sophia is able to eventually get the correct answer, this process required her to perform at least one three-step double-digit multiplication and probably three long division calculations.

Emma has instant recall of all of her math facts and quickly recognizes that both 21 and 14 are multiples of 7. Emma quickly converts the problem to 3/42 + 2/42 = 5/42. By avoiding the need to perform double-digit multiplication and long division Emma is able to finish at least five times as many problems as the Sophia.

Although Sophia’s mathematical reasoning skills matches Emma’s, Emma is perceived as the “better” at math. Since Sophia is the last to finish the math worksheet and has several errors due to simple calculation mistakes, she believes that she’s “not good at math.” These perceptions are reinforced by classmates, teachers and parents who also equate Emma’s speed and Sophia’s slowness to their math abilities.

Over time these perceptions become reality. Emma is able to complete more math problems because she is able to work faster. She is more likely to complete challenge problems, be selected for pull-out programs and eventually accelerated math. Sophia, on the other hand, does not have time to complete challenge problems or participate in enrichment math since it takes her so long to complete her regular assignments.

In reality, the difference between the two girls is the degree to which they learned their multiplication facts. Both understand the concept of multiplication. Both understand how to calculate a product through repeated addition of skip counting. But Emma recognizes multiples like sight words while Sophia needs to calculate a few of them like a child who must still sound out a few words.

While Emma and Sophia are fictional, I’ve worked with hundreds of students from algebra to calculus whose self-perceptions and actual ability to master higher level math have been limited not by their innate mathematical reasoning skills but their mastery (or lack thereof) of basic math facts.

One of my 4th graders, Lucas*, recently said to me, “I don’t need to learn my multiplication facts anymore because we’re done with that in school. Now I’m doing division instead.” In his mind, he was done with the task of multiplication and didn’t need it anymore.

“I don’t need to learn my multiplication facts anymore because we’re done with that in school. Now I’m doing division instead."

Not only will Lucas need to know his multiplication facts for long division and working with fractions, but he won’t recognize the patterns formed by exponents, factorials, calculations of permutations and combinations, factoring polynomials or infinite series just to name a few. I often find it difficult to convince students and parents to invest the time and effort to memorize their math facts until they inevitably encounter the frustration of no longer being able to succeed in class.

Last month my 4th graders learned long division. All the students picked up the concept quickly and could easily dictate the next step required to complete a problem when we worked on group exercises. Completing problems independently was not as easy. One of the students, Jackson*, was visibly upset when he tried to use multiplication to check his answers and found that he didn’t get any of the correct. In each case, he found a multiplication error. Jackson’s an exceptionally bright boy who clearly understood the concept but didn’t know his multiplication facts well enough. He had been ignoring my admonishments to complete his math facts homework each week. His parents had also viewed learning math facts as less important than completing the other homework assignments.

Finally Jackson could see how not knowing his math facts was limiting his ability to advance. The next week, he brought in about 50 pages of math facts practice and was excited for math minutes. He finished his 50 multiplication facts up to 10 X 10 in only 40 seconds and was giddy waiting for it to be graded. He was also able to complete the long division exercises correctly.

Last week, Aiden*, one of my prealgebra students stayed late to talk to me after class. He’s another very bright student who doesn’t want to put in the work to memorize his math facts. His mother believes that Aiden isn’t good at memorizing and shouldn’t be excused from this requirement. As we talked, my fourth graders started to arrive. Jackson overheard the conversation. He told Aiden, “I was the same way for a long time but then I decided I was going to just DO IT. I did about 10 practice sheets every day for a week. It was hard, but now I know them. You can do it too.”

Last month my 4th graders learned long division. All the students picked up the concept quickly and could easily dictate the next step required to complete a problem when we worked on group exercises. Completing problems independently was not as easy. One of the students, Jackson*, was visibly upset when he tried to use multiplication to check his answers and found that he didn’t get any of the correct. In each case, he found a multiplication error. Jackson’s an exceptionally bright boy who clearly understood the concept but didn’t know his multiplication facts well enough. He had been ignoring my admonishments to complete his math facts homework each week. His parents had also viewed learning math facts as less important than completing the other homework assignments.

Finally Jackson could see how not knowing his math facts was limiting his ability to advance. The next week, he brought in about 50 pages of math facts practice and was excited for math minutes. He finished his 50 multiplication facts up to 10 X 10 in only 40 seconds and was giddy waiting for it to be graded. He was also able to complete the long division exercises correctly.

Last week, Aiden*, one of my prealgebra students stayed late to talk to me after class. He’s another very bright student who doesn’t want to put in the work to memorize his math facts. His mother believes that Aiden isn’t good at memorizing and shouldn’t be excused from this requirement. As we talked, my fourth graders started to arrive. Jackson overheard the conversation. He told Aiden, “I was the same way for a long time but then I decided I was going to just DO IT. I did about 10 practice sheets every day for a week. It was hard, but now I know them. You can do it too.”

“I was the same way for a long time but then I decided I was going to just DO IT. I did about 10 practice sheets every day for a week. It was hard, but now I know them. You can do it too.”

**WHAT PARENTS CAN DO**

Parents should accept the responsibility of making sure that their children are fluent in their math facts.

There is nothing complicated about teaching these to your children. It’s just a matter of time on task. While there it is impossible to completely avoid drilling of some sort, I recommend combining drills with fun games and activities.

Children must first understand the meaning of multiplication as repeated addition, skip counting and area before working on memorizing the math facts. After your child understands the concept of multiplication start working with them to learn small groups of simple multiplication facts, such as the two times tables. Gradually add more math facts as your child masters each set.

After your child is able to recall 50-60 math facts without assistance begin working on fluency or automatic recall. Your goal is to help the child transition from rapid skip counting to an automatic association. Provide math minutes worksheets starting with 30 multiplication facts up to 6 X 10. Instruct the child to complete as many of the problems as possible in 60 seconds. Specifically instruct the child to skip any “hard” problems. Pick three of the problems that the child skipped and focus on mastering those three math facts for the week. The child should write these three facts at least three times a day for a week. I tell my children to say the math fact while they write it. Continue working on mastering three math facts each week until the child is able to consistently complete the entire set in 60 seconds. Since my classes meet weekly, I require each child to correctly complete each level for three weeks before progressing to the next level. Parents who are working with their children daily may want to require their child to correctly complete a set every day for five consecutive days before progressing to the next level.

Gradually add more math facts and more problems per page until you child can consistently correctly answer at least 60 math facts at least up to 12 X 12.

**Games and other Activities**

While it is critical that students learn to study, it is equally important for them to love the learning process. Make sure that you balance drills with fun games and activities. Here are few that my students enjoy:

Parents should accept the responsibility of making sure that their children are fluent in their math facts.

There is nothing complicated about teaching these to your children. It’s just a matter of time on task. While there it is impossible to completely avoid drilling of some sort, I recommend combining drills with fun games and activities.

Children must first understand the meaning of multiplication as repeated addition, skip counting and area before working on memorizing the math facts. After your child understands the concept of multiplication start working with them to learn small groups of simple multiplication facts, such as the two times tables. Gradually add more math facts as your child masters each set.

After your child is able to recall 50-60 math facts without assistance begin working on fluency or automatic recall. Your goal is to help the child transition from rapid skip counting to an automatic association. Provide math minutes worksheets starting with 30 multiplication facts up to 6 X 10. Instruct the child to complete as many of the problems as possible in 60 seconds. Specifically instruct the child to skip any “hard” problems. Pick three of the problems that the child skipped and focus on mastering those three math facts for the week. The child should write these three facts at least three times a day for a week. I tell my children to say the math fact while they write it. Continue working on mastering three math facts each week until the child is able to consistently complete the entire set in 60 seconds. Since my classes meet weekly, I require each child to correctly complete each level for three weeks before progressing to the next level. Parents who are working with their children daily may want to require their child to correctly complete a set every day for five consecutive days before progressing to the next level.

Gradually add more math facts and more problems per page until you child can consistently correctly answer at least 60 math facts at least up to 12 X 12.

While it is critical that students learn to study, it is equally important for them to love the learning process. Make sure that you balance drills with fun games and activities. Here are few that my students enjoy:

Prime Climb – Without a doubt the most popular board game with my students from 3rd grade through algebra. Students add, subtract, multiply and divide to be the first to reach 101. I remind students that they are not allowed to “count up” but need to calculate their next position. I often “think out loud” to introduce new concepts. For example, after rolling a 2 I may say “I can add 2 to 25 (my current position) to move to 27 OR I could multiply 25 by 2 and get to….humm, let’s see 25 X 2 is…” and let someone help me out. Great for learning math facts, the understanding order of operations, logic, and strategy. Will not promote math fluency because rapid recall is not important. |

*Jackson, Aiden and Lucas’ names have been changed to protect their privacy.