Positive And Negative Numbers

 

I've hit a unique spot in my math career where I'm having to take children from the idea that positive and negative numbers represent the distance from zero when in reality it might represent going somewhere and coming back.  It could also be going up in elevation or down, depending on things.

My 7th graders just hit the part with 3x - 7x and totally got lost.  They were trying to picture how far 3 was from zero and how far -7 was from zero.  So I had to pull out a couple of number lines and other manipulatives to show that you start and 3 and move seven places to the right.  I know that we'll be doing number lines and such to help them understand that the 3 units is taken from the 7 going left.

Then in Algebra, I introduced Absolute Value.  Students have learned that all answers come out positive but they don't know why so out came the explanations about distance.  Absolute value is like distance in that it is always a positive number but the signs indicate direction. so the distance might be 5 miles away so going there might be -5 because you are going away from your starting point, and + 5 as you come back. It could be the other way round depending on the kids do.  

When you put it as traveling to a near by village, they understand things much better why the answer to an absolute value problem is always positive and when you solve it, why you have to set it equal to two values. They relate so much better to this explanation than any other I've done.

Then in Geometry, my students are doing transformations, specifically translations.  This meant they had to learn to express movement using signed numbers so that when someone read x +6, y-2, they knew the point moved right six spaces and down two.  We have had multiple discussion over the past couple of days on the horizontal movement + is movement to the right, - signifies movement to the left.  For vertical movement, + indicates moving up and - moving down.    I think it is finally sinking in. 

I try to talk about context and how the context helps students decide how to interpret what a sign means.  Let me know what you think, I'd love to hear.  Have a great day.

New 13 Sided Shape!

They discovered a new 13 sided shape that is classified as the first true Einstein shape. It is a shape with 13 sides that can be used to cover a plane without ever repeating the pattern and it has some cool applications in Material Science. 

This 13 sided shape is also referred to as an "aperiodic monotile" which is a specific shape - one shape to that can cover a plane without repeating the pattern. The term Einstein does not refer to the famous physicist. Instead, it refers to the German term for one stone.  Furthermore, all the conditions for this shape are found in geometry and has no additional constraints inflicted via matching conditions. 

This shape also known as a polykite aka The Hat because it vaguely resembles a fedora hat and it  assembles tiles in through substitution. Originally, they had over 20,000 tiles but that has been decreased to 2 back in 1974 but at that point couldn't get it down past the two. Then around 2016, someone became interested in it and found the solutions.  To prove it,  they used computer coding using software that had recently been written. Prior to this software being written, they would have had trouble proving it.  They also discovered that the doesn't lose its aperiodic nature when the sides of the shape are changed.

Furthermore, it stands between order and disorder.  The shapes fit neatly together but the pattern can never repeat itself and that is what makes it an einstein shape. It is actually a two dimensional shape with straight edges so its classified as a polygon.  This particular discover can be used in physics, chemistry, and even in the study of crystals.  

Most mathematicians love the idea because it is so simple, easy to visualize and understand.  Let me know what you think, I'd love to hear.  Have a great day.

Out Sick

 

Got something bad, back on Wednesday!

5 Practices Of A Good Math Lesson Plan

Today we are going to take time to look at the five parts that are considered to make up a good lesson plan in math.  Not all math groups need the same type of lesson plan or your administration might tell you what they want to see.  I pick and choose depending on where my students are and which grade I'm working with.

Overall, most people want to see a lesson plan that has students actively involved with learning. So the first step is determine what should be taught.  This can be done by asking yourself what the students should know and what they need to learn. You should also ask yourself how students are able to show they know something.  You might get a bunch of rich math tasks that allow students an easy entry but are quite interesting.

One thing to do is to think about how students will solve a problem.  For example, so many of my student automatically do the opposite to any numbers on the same side rather than combining them.  I have to watch out for that.  Basically you want to know how the students will do it, how will they do it, and what misconceptions will they have with the concept.

Next, think about monitoring students and the strategies they will choose to use to solve those particular problems.  In other words, spend time identifying the strategies they use and how they solve the problems by going around to various groups and checking in with them.  This opens up the opportunity for a small group lesson or allow the teacher to share a few words to help students get back on the right path to solve the problem.

When students are done, decide which students should be allowed to share their work with the others.  When they share, they are practicing their communicating skills while explaining the why and how.  One should choose the work that meets the goals of what students should know and how they got it done. 

Furthermore, the teacher should take time to determine the sequence things should be taught in.  Sequencing can also refer to going from informal to formal, simple to sophisticated, common to unusual, and decide whether to address misconceptions immediately or later. No matter which sequencing is used, students should be able to see the connections. 

The final step is to have the teacher make connections either directly, or indirectly. This is when the teacher uses student work to see if the students are connecting the ideas or even use it to show what needs to be connected.  So these items are said to make a good lesson plan.  Let me know what you think, I'd love to hear.  Have a great weekend.

5 Common Types Of Lesson Plans.

 

Since I looked at the 5e lesson plan last time, I'm taking time to look at five general categories.  Some schools require a week's worth of daily lesson plans turned in on the Monday.  Others look at unpacking standards to create a unit or semester plan, and then there is the yearly pacing guide.  I think it it important to look at each of the general categories in detail.

The first type of lesson plans are the daily ones. A daily lesson plan usually has the learning objective and some sort of assessment to determine if the students learned the material.  It is also the most detailed lesson plan filled with standards, purpose, activities, timing and are based on unit lesson plans.  This is the one I've been most often required to do.  The best situation was when I had to turn in the plans at the end of the week showing what I actually got done.

Next is the weekly lesson plan.  This lists the learning objective for the whole week.  These can be fairly detailed because they provide the overall view of the whole week from start to finish.  The idea is the teacher uses the first three days to cover the topic and lets the students practice the material for the last two days.  In addition, there is an assessment planned for the final day so the teacher can see how well students learned the material.

The third type of lesson plan is the unit lesson plans which looks at a whole unit.  The whole unit might be a complete chapter in the textbook, or it might be fractions from start to finish.  This type of lesson plan allows the teacher to vary the pace based on how well the students learn the material.  This type of lesson plans looks at the scope and sequence for the unit along with the sequential lessons, that make up that part of the scope and sequence.   These are better for teaching students to meet long term goals.

Next is the subject specific lesson plan. This type of lesson plan is used by a teacher who teaches more than one subject.  This could apply to the math teacher who teaches geometry, algebra I, and algebra II, or the elementary teacher who teaches science, reading, and math.  Such lesson plans can allow more time to differentiate so students are able to make certain goals.

Finally, there is the grade specific lesson plan.  This applies to those who teach social studies across 5th, 6th, and 7th, grades where they have to have a separate lesson plan for each grade.  Each grade requires certain things be covered so that each grade is done separately so the standards are met.  I teach a 6th grade math, 7th grade math, and 8th grade math and I have a lesson plan for each grade because each book covers different topics in a different order.  Let me know what you think, I'd love to hear.  Have a great day.

The 5 E Lesson Plan

 

Every time, I've moved to a new school, I've been told to turn in my lesson plans on Monday morning for the week.  Most of the time, I ask about the format and usually, they didn't bother having a specific request so I've done a bunch of research. Today, I thought I'd look at the 5 E lesson plan.

The 5 E's stand for engage, explore, explain, elaborate, and evaluate so that students are lead to deeper understanding of the material. This method was based on the process used by scientists and it became part of the educational scene back in the late 1980's. Time to look at each step of the way.

Engage, is where the teacher poses a question, an object, or a situation that has students looking on the topic while helping them activate prior knowledge.  In math, this might look like a notice and wonder, a number talk, or the beginning of a 3-act task.  When you do a notice and wonder, the answers do not have to be in mathematical language since you are interested in students to participate.

Explore is the next stage.  In this stage, students and the teacher dig into the topic so they have some common experiences that can be drawn upon when explaining a new concept or topic.  This might be done through  an activity with manipulatives, or a video set up they can watch. 

Explain comes after explore. Although this is where you might think the teacher takes over but this is where the students share their observations or what they learned about the concept or topic. Instead, this is where the teacher supports the student's observations with formal language. This is also where teachers might address misconceptions that arise.

Elaborate is where students have the opportunity to extend their thinking and transfer thinking to new thinking.  This is also the stage where students work through their misconceptions, and work in groups for discussions that help extend their thinking. 

Evaluate is the final step where students and the teacher assess student learning. Formal assessment should occur throughout the whole process but there should be summative assessment at the end.  One way is to have students return to the original engage stage where they look at the original notice and wonder to see if they can apply the mathematical concept and vocabulary to it.

Today, I just touched on the 5E lesson plan.  I've used it before when I did hyperdocs and it made it easy to set up the whole lesson from start to finish.  I don't use it as much right now because of the range of grades I teach.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If you get 225 pounds of sugar from one ton of sugar cane and one acre of land produces 38 tons of sugar cane, how many pounds of sugar are produced per acre?

Warm-up

 If one sugar cube equals 0.08 of an ounce, how many cubes do you need for 8 ounces of sugar?

Are Manipulatives Viable In High School Math?

 

Currently, I am using manipulatives with my middle school and Algebra I math classes.  One big reason for this is that most of my students are below grade level.  The other reason is that the ones who can do the procedure, do not understand the why behind what they are doing.  Using manipulatives seems to help these students tremendously.

If you do a web search, you'll find articles both supporting and stating manipulatives are worthless but I think so much depends on what you are doing.  I've developed a way to use manipulatives that seems to help clarify process for my students while helping others connect things so they understand how to do the process.

For instance.  in Algebra I, I pulled out my base 10 manipulatives to show them how to solve two step equations.  I designated the 10's as x's and the ones as ones so when I had the problem 2x + 3 = 9.  There would be two base 10 blocks with three singles on the right side and nine singles on the other side.  I showed them how you take the 3 away on both sides, then divide the remaining singles into two groups.  

I then had students try a couple of other problems using the blocks.  Once they had the movement down, I began connecting the movements with the process so they could see when they took away 3 blocks on both sides, it was the same as subtracting.  When they divided the remaining singles into two groups, it represented dividing by 2.  I actually took away three with my hands, and wrote -3 on the problem.  When I divided the individual blocks into two groups, I showed that I divided by 2 on the paper.

When I showed them how to solve x/2 + 3 = 5.  I explained the 10's bar represented a half an X, like half a cup of flour with 3 extra blocks.  So then  we removed three from each side and were left with one bar representing half equaling 2.  So we have to double the x's to get one so we double the other.  It took a few times but the kids learned how to do it.  Then we redid the problems showing the movement with the math step by step and they grasped it quickly.

I realize that I should probably let them explore all they want but due to having only 3 students in that class, I don't always have time so I work with them.  I like the way this is going.  Manipulatives can be used but it takes planning.  Let me know what you think, I'd love to hear.  Have a great weekend.

Mastery Learning Versus Spiral Learning

In the past week or two, I have looked at both master learning and spiral learning but today, we'll look at the ways in which they are different and if one if better than the other.  In short, mastery learning is focused on having students master a specific concept before moving on while spiral learning encourages students to review concepts as they move on even if they haven't fully mastered it.  

Mastery learning takes each concept, breaks it down into smaller manageable chunks that the child masters before moving onto the more complex ideas.  In spiral learning, students are exposed to the big idea first, then revisit it multiple times, expanding knowledge and moving on to the more complex ideas.

Ideally, mastery learning allows the students to progress at their own pace so they can master each concept or topic before moving on the the other.  In mastery learning, the students begin with the simplest form such as one step equations with whole numbers.  Once they've mastered these, they will solve one step equations containing fractions and/or decimals.  From there they might solve two step equations only using whole numbers before solving them with fractions and/or decimals.  No matter what, they have to prove they have mastered one before they move on to the next.

Although it lets the student work at their own pace, use critical thinking skills when problem solving, it is more difficult to manage a large group of students.  In addition, since students are working on different lessons, there is less time for collaboration among the students. It also requires the teacher perform more detailed grading so they prove they have mastered each step.

On the other hand, spiral learning emphasizes the idea that learning is cylindrical in nature.  In this, they revisit previous topics, deepening their understanding, expanding their knowledge, and master it.  As they revisit the topic, they become more familiar with it and understand it better. 

In spiral learning, they may introduce students to solving one step equations but the class will move on before it is fully mastered because the teacher knows they will be revisiting the topic multiple times in the future.  When students are introduced to solving one step equations, they may be exposed to whole numbers, fractions, and decimals at the same time so they learn the procedure associated with solving these. The teacher can move on knowing students will be exposed to this again and again throughout the year so they build on the foundation while expanding their understanding.

One major disadvantage is they might get bored with revisiting the material again and again.  In addition, research has found that certain topics tend to get revisited more than others regardless of whether the students need the repeated exposure.

So which is better?  It really depends on the child but for me, I like breaking the material down into bite sized chunks that students work on mastering while providing multiple opportunities to revisit the topics they still need to work on.  Let me know what you think, I'd love to hear.  Have a great day.

What Is The Spiraling Technique In Teaching.

I often see recommendations for including spiraling in the classroom but for me it is hard because I teach grades 6 to 12 in one room and spend two class periods working with each of five math groups.  It means that I don't often get a chance to follow best practices the way I want.  Thus, spiraling is something I can do.  

Spiraling technique is simply teaching the same material across several grades using developmentally appropriate methods. It involves introducing the concept in a simple form and then building on it to lead students to more complex forms.  

Spiraling comes from brain based research in that it reinforces previously taught concepts while helping reinforce them as students move on. Remember it is not repeating but revisiting a concept.  It is a way of incorporating new information with the old that they already know.  This gradual buildup works better with the way our brains function consequently, students are better able to remember and connect concepts.

In addition, there are three principles involved in spiraling. It consists of  cylindrical learning, increasing depth each time through, and learning by building on prior knowledge. Cylindrical learning is defined as returning to the same topic throughout their time in school.  Each time the student returns to the topic, they are exploring it at a deeper level with more complexity.  Finally, when they look at what they already know, they build from their foundations rather than treating it as something new.  

When done properly, the material that is normally taught within a short period of time, is actually broken into smaller pieces, spread out over a longer period of time, and revisited on a regular basis.  Think of it as starting at the surface and moving deeper with each visit.  

One way to introduce spiraling to your classroom is to it by assigning anywhere from one to five problems a day depending on the time you have available.  The time might only take two to four minutes if you are doing one problem per day and this fits in at the beginning to help settle students down or during the last few minutes of class to wrap it up.

The time you put the spiraling in will depend on your students.  For some classes, doing it at the beginning of the period as a bell ringer or warm-up is fantastic but for other classes, it makes the perfect exit ticket. 

Another thing to look at for the spiral review is the form of the problem.  You might post the problem on a whiteboard while you have students do the work on individual whiteboards.  Or you can set it up so all the problems are on one sheet of paper.  They do one problem a day and at the end of the week you collect it and check it for misconceptions.

In addition, it is important to hold them accountable by checking their work.  You can spot check it, looking for students who might need a bit more attention or look at the completed sheet done during the week and give it a couple points grade.  It depends on you and what motivates your students.

One important thing is to go over the work with students.  One way is to make note of various strategies used by students that you want to highlight without having to ask for volunteers.  If your students are willing ask for volunteers to share their work with the rest of the class.  

Finally, it is often difficult to decide what to ask question on.  Some teachers look at the major topics their students should know by the end of the year and provide questions on those.  Others look at the scope and sequence listed for the grade and choose from that.  It is best to look at the topics that give the students the hardest time and return to those again and again.  

No matter what topics you decide to hit, make sure you have the questions planned out so they build on the students knowledge beginning from the easiest forms to the more complex forms.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If you need 2500 plants to make one ounce of rose oil and one acre can hold 5000 plants.  Then you can sell it for $200.00 per ounce, how many acres would you need to make enough oil to earn $25,000 dollars?

Warm-up

 If one plant produces 27 roses on average, how many plants will you need to get 60,000 roses needed to produce one ounce of rose oil?

Using Math To Unlock The Meaning Of Shakespeare.

 

Think about it!  Linguists have begun using numbers and math to help unlock the meanings behind many of the words used by Shakespeare. Some of this is because the meanings of words have changed significantly over time and may not mean the same thing now as it did then.  For instance, intercourse meant having a conversation years ago but has acquired a sexual overtone more recently. 

Think about it!  Back when Shakespeare was writing the word bastard was used to describe certain plants while the word success referred to an outcome.  So bad success was a undesirable outcome to something.

Within linguistics, there is a branch called corpus linguistics which uses computers to help explore words and their meanings within huge collections of words.  These numbers, the counts of how often they appear within certain texts is the key to understanding meaning. 

When I say these numbers, I refer to how often words such as often, associations, odd, or unusual appear and  which can be said to be a soft way to actually count things.  In addition, a dictionary was created from the full works of Shakespeare but this dictionary differed from the others because it also was comparative.  In other words, it took the words from Shakespeare works and compared them with other writings and plays of the time.

Consequently, one finds out that the term bastard referred to hybrid plants because they were not the normal offspring and occurred in a multitude of horticultural texts.  The term also appeared in the plays to mean someone born outside of marriage and thought of as a hybrid.  As mentioned earlier, success refers to outcome so bad success meant an undesired outcome.  You might also see the terms, disatrous, unfortunate, ill, or unhappy, or unlucky linked to bad success.  

Furthermore, the researchers took time to see what other words were frequently associated with that base word.  Another thing is to look at words that appear frequently and are classified as high frequency words.  These words are often excluded from early dictionaries yet have a real meaning.  One such word is by and it is associated with religion because it is found with God in by God.

In addition, certain words within Shakespeare's plays are associated with certain characters or sexes.  For instance, alas, or ah are said more often by his female characters showing they provide the emotional work of lamentation within the play. 

On the other hand, infrequent words also provide some interesting thoughts. One such word, ear-kissing in King Lear means whispering and only appears one time.  Other words such as sweet may appear multiple times but due to the differences of spelling, are often more difficult to determine. Sweet is spelled as sweete, swete, or svveet and if you are using a find function, the nonstandardization of spelling can make it difficult to count the number of times it was found. 

It has only been recently that computer programs have been developed that allow researchers to do this type of in depth search.  So we have a new way to use numbers and math.  Let me know what you think, I'd love to hear.  Have a great day.  

Variation As Part Of Mastering Math.

 The final big idea in mastery math, is variation.  Variation refers to drawing closer attention to a key aspect of a mathematical concept by varying certain elements while keeping the rest constant.  This is similar to what a scientist
s does when they vary one thing and keep the rest the same.  

Variation can be done either in the conceptual region or the procedural element.  In conceptual variation, the variation happens in how it is represented and the representation is varied so students see it from more than one view point.

In the procedural variation, how the student goes through the learning sequence. The variation draws students attention to key features while scaffolding student knowledge so they are better able to see connections.

In addition, procedural variation provides repeated practice in one of three ways.  First, one can extend the numbers used by varying the number, the variable, or the context.  Variations might be 6 + 19, 16 + 19, 26 + 19........96 + 19 and then asking what do you notice?  In multiplication it might look like 7 x 4, 70 x 4, 700 x 4, etc so students have the opportunity to notice the pattern.

Next, is to vary the processes involved in solving a problem.  It might be 8(44) which could be solved that way or look at it as 8(40+4) which gives 8(40) + 8(4) = 320 + 32 = 352.  This is two different ways to solve the problem or the second one might be 8(50) - 8(6) = 400 - 48 = 352.  If you are looking at money, it might be something like Paul bought food for $23.76 and gave the person $30.00 to pay for it.  A student could subtract the numbers so $30.00 - $23.76 = $6.24 or a student can add coins to $23.76 and work their way to $30.00.  They would go with 4 pennies to $23.80, two dimes to $24.00 and $6.00 to $30.00 or $6.24.

Third, to vary the problems by applying the same process to it so students see how its the same and how its different.  For instance, if you use base 10's for multiplying 27 x 32, one can use the same process for (x + 1) (x -2) or even lattice multiplication works for both.  This helps students see the connections between the two. You could also ask what has changed and what has stayed the same.  This allows students to think about the answers and communicate them

This wraps up an in depth look at the five big ideas in mastery math.  Let me know what you think, I'd love to hear.  Have a great day.

Happy Labor Day

 

Warm-up

 

Your tree produced 456 pounds of apples.  If you made it all into pies and each pie uses 3 pounds of apples, how many pies will you make?

Warm-up

If your tree produces 521 apples this year and it takes 3 apples to make a pound, how many pounds will you get this year?

 

Fluency.

 

The fourth overall topic for teaching for mastery is fluency.  Fluency is defined as being able to use what they learn effectively and easily.  The foundation of fluency is conceptual understanding, strategic reasoning, and problem solving to do math automatically.  In other words, students connect conceptual understanding with strategies and methods so the methods make sense to them.

Unfortunately, fluency is often associated with computing basic facts quickly.  Fluency is this and so much more.  It is more than memorization, accuracy, and speed.

Fluency allows students to solve deeper problems that are more meaningful and are likely to encounter in real life. It also helps students do better in the math class and in life.  Fluency is considered to include how efficiently a student solves a problem using the fewest number of steps.  They do not get bogged down trying to use too many different steps to solve the problem, and they are able to immediately use the results.

They also need to be accurate in their record keeping (how they show their work), knowledge of number facts, and other relationships.  They are also double checking any results.  Finally, they need to know more than one way to solve a problem. They need to have a flexibility which allows them to choose the most appropriate method to solve the problem and a different method to check their work.

It has been shown they need to be fluent both in procedure and in concepts.  Conceptual fluency helps them understand why.  When they don't understand why, they often forget or remember incorrectly.  Furthermore, when they memorize everything, they have difficulty bringing meaning to their work later on. 

Students need to have procedural fluency and conceptual fluency to be considered fluent.  It is important to include activities which help the student develop an understanding of concepts and relate them to procedure.  Manipulatives are a good place to start.  Let me know what you think, I'd love to hear.  Have a great day.