MEASURING THE MARIGOLDS

MEASURING THE MARIGOLDS

https://www.youtube.com/watch?v=fXi3bjKowJU

Measurement is a multidimensional topic.  It includes telling time, determining elapsed time, perimeter, area and the degrees of angles.  While some of these topics seem to fall into other categories, each is a measure of something.

TIME IS A MEASUREMENT

Most of us do not consider time to be a measurement.  However, without the separation of the day into hours, hours into minutes and minutes into seconds, we would have no idea where we were at any given point during the day.  This is a measurement of our activities.  We determine the amount of time or more aptly we measure the amount of time it should or does take to complete certain tasks.  Remember that old Algebra problem: “Two trains leave the station at 9:00 a.m., one traveling south at 55 mph and the other traveling north at 65 mph.  How long will it take for the trains to meet?”  This problem measures the elapsed time of each train to a meeting point at an imaginary location.  

In  the Third grade and into the Fourth grade, students are given word problems to solve involving elapsed time.  The problems are not as advanced as the train problem.  There are three possible types of problems that students could be asked to solve.  They could be required to determine how long a flight took, when a flight took off, or when a flight landed.  In the first type of problem, the problem would provide the take off time and landing time.  The second type would contain the landing time and the length of the flight.  Lastly, the third problem type would provide the take off time, along with the length of the flight.    As with earlier word problems, the students would be doing addition and subtraction to solve these problems.

Determining elapsed time is just one of the measurements related to time.  The other is the division of our lives.  This division is the way we measure our days.  The calendar divides our year into months, weeks and days.  Our clocks divide our days into hours, minutes and seconds.  As adults, most of us are constantly looking at our watches or “watching the clock”.  These are the measurements that drive our lives.  The calendar and the clock are our first introduction into measurements.  As we grow older, we begin to apply these divisions to other things.  We begin to look at how long, wide, high, or heavy something is.  These divisions allow us to make sense of our world.

THE INCHWORM AND UNIT BLOCKS

When students begin to look at length, height, and width, they are required to begin with measuring the length of objects based upon a specified unit.  In Kindergarten, a child might be asked to compare a paper circle to the number of pennies it is across.  This is the beginning of measurement.  The students are tasked with measuring objects by the number of a specified “item” it is long/wide/high.  Once this type of measuring is understood, students progress to using rulers and measuring to the nearest inch, which begins in about second grade.  

By third grade, students are again measuring area and perimeters with a unit block or item.  The beginning concept of area is how much of an item is “covered” by a shape.  In other words, if I place a piece of paper on my desk I will have covered an area of 93,5 square inches.  Students would use a unit cube, which is 1" by 1" and count the number of squares contained in that piece of paper.  In the 4th and 5th grades, students are expected to use the formulas for the area of most regular polygons.  (Rectangle–A= l * w. Triangle–A= (b * h)/2, and so on.)  Also, in the 4th and 5th grades, students begin to experiment with volume and learn the formulas for those measurements, as well.  The most important thing for students to remember when calculating these figures is to appropriately label their answers.  93.5 is the correct calculation for the area of a piece of 8.5" x 11" paper.  However, without the appropriate labeling, the area could be square feet, inches or yards.  It is important for students to label their work with the appropriate measurement unit.

MEASURING ANGLES

In the 4th grade, most students begin to measure angles.  I remember buying “pre-filled” school boxes and measuring tools, as a child.  These always had a protractor included.  I never knew what a protractor was for.  Today, students use the protractor in the 4th grade to not only measure angles to the nearest degree, but to also draw these angles as accurately as possible.  I was intrigued, while observing a 4th grade math class this semester, watching the teacher and the students using a protractor to draw and measure angles.  It was great.  I always had to have a protractor in my supplies, but I can never remember using it.  To get the opportunity now was fantastic.  The understanding of the students on how to create the two rays and draw the angles was very impressive.  I learned so much that day.

POSSIBLE “WEEDS” TO PULL

        As with any math topic, there may be some “weeds” to be pulled, during measurement lessons.   The first possible weed is a confusion between area and perimeter.  These topics are introduced in 3rd grade and honed in 4th and 5th grades.  The perimeter is the distance around a shape.  Another way to think of the perimeter is to imagine putting a fence around your yard.  The perimeter is how much fencing you would need to buy to fence in your yard.  It can also be thought of the distance you walk around your residential block.  Explaining perimeter to our students in this manner will give them a real world “picture”.   In explaining area, it is also important to give students a real world frame of reference, so they are able to visualize it.  A teacher could use the yard analogy for area, as well.  However, rather than putting a fence around it, you are concerned with amount of grass needed to cover the inside of the fence.  A better analogy would be the surface of a table.  If I want to cover my desk with a cloth, but have no over hang.  The amount of space on my desk covered by the cloth is the area.  A table top of any kind will give them a frame of reference for area, as fencing the yard does for perimeter.

Each new topic in math has its own little “weeds” to pull.  Being able to provide real world examples to our students can help bridge the gap between our reality and theory we are teaching.

TEACHERS BEING “SEEDS” PART TWO

TEACHERS BEING “SEEDS” PART TWO

On February 20, 2020, I attended another Twitter chat with the Ohio Council of Teachers of Mathematics.  The title of the chat was “Diving into Discourse.”  As with the other chats, this chat was a half hour and consisted of five questions.

The first question dealt with defining Math Discourse.  There were a great number of responses to the question.  The answers ranged from simplistic to very detailed.  However, each one boiled down to discourse being the language of Math and being able to determine whether or not your students are learning the materials you are teaching.  I believe that true discourse is the teacher and the students being able to talk about numbers, their mathematical concepts and how each applies to real life situations.

The second question’s topic was on the key factors needed for honest and open discourse.  The responses for this question were very similar.  In order for discourse to be effective, the teachers need to be sure they provide a safe zone for their students to discuss mathematical principals without judgment or fear of ridicule.  Only when students feel free to make mistakes will students feel free to share their strategies and approaches to problem solving.  Students also need to know we care about them.

Question 3 dealt with the necessity of teachers completing the tasks, prior to the “lesson”.  It is important for teachers to be prepared for the lessons they are teaching.  In some ways, it is most important with a Math lesson.  For some reason, Math is feared by students.  For a teacher to complete the task prior to giving it to students, allows the teachers to consider the possible strategies their students might choose to complete the task.  I also think it is important for students to know that we can actually solve the problems we ask our students to solve.  It is comparable to a lawyer not asking a witness a question that you do not know the answers to.  Lawyers don’t like surprise answers during trials.  It is the same with teachers.  If we don’t plan ahead and prepare, we are like a lawyer that didn’t prepare for court.

The last question asked which online resources we preferred to obtain the tasks we use.  In the class that I am taking this semester, our teacher begins each class with a “math routine.”  I have discovered some great online resources with these routines.  Splat, Esti-mystery, “What doesn’t belong”–it is a book with four pictures on each page and the students determine  what doesn’t belong.  I have really enjoyed these routines and can’t wait for others.

I really enjoyed reading the responses from everyone who participated in the chat.  I also enjoyed the topic.  However, I had more difficulty with this chat than the last.  I don’t remember having to place the # in my responses on the first chat, but lost my responses that did not contain the # on the second one.  While I enjoy participating in the chats, I personally find Facebook chats more user friendly.  However, I will continue to participate in these chats.  I have learned much from all who participate.  I can’t wait to learn more.  I am really looking forward to the chat with COSI in April. 

WHAT “SHAPES” YOUR GARDEN?

WHAT “SHAPES” YOUR GARDEN?

This week, we turn our thoughts towards geometry.  Geometry is the study of points, lines, rays, segments, shapes.  The study of the geometry is divided into two distinct types of geometry, plane geometry and solid geometry.  Plane geometry contains everything that is not a three-dimensional shape.  Solid geometry contains only three-dimensional shapes.

It is important for a student’s first introduction to geometry be within the context of his or her environment.  The ability to observe the world around us is important in being able to categorize and classify the different types of shapes.  Most children are able to look at a window and recognize that it is shaped like a rectangle or square.  These initial categorizations allow students to build the necessary observation skills that allow students to piece together definitions of the shapes.  These early definitions are refined with each passing year, until the student’s definition of a square is regular rectangle.  The understanding of this definition will not come together without constant observation and refinement.  

In order to refine a definition, a student must be able to observe the shapes in different settings and sort the shapes based on his or her own interpretations of the defining attributes of the shape.  A squares defining attributes are many.  All squares are polygons, quadrilaterals, parallelograms, rhombuses, and rectangles.  However, as we all know, not all polygons, quadrilaterals, parallelograms, rhombuses and rectangles are squares.  Squares are a special 4 sided polygon that has alternate sides being parallel, all line segments form a 90° angle and are all sides are equal. Only by observation will a student be able to understand these relationships between the various classifications to which a square belongs and its special designation as a square.

It is sometimes difficult for children to make the connection between the shapes, as shown above.  They have been taught from an early age that there is a difference between squares and rectangles, that in fact, the two are distinctly different shapes. Only to discover later that the square is a special type of rectangle and not a distinctly different shape after all.  To further their confusion, we begin to introduce other types of shapes that contain the same attributes that squares and rectangles have.   These newer classifications of shapes are not exactly distinct, as their old friends are actually special subsets of the “new” shapes.

I am not advocating that polygon, quadrilateral and parallelogram be introduced prior to the introduction of square and rectangle.  However, I am advocating that these shapes be introduced sparingly in the earlier grades.  Being exposed to these shapes and their names will help assuage some of the confusion later.

The best way to discover the relationships between the shapes and refine the working definitions of the shapes is to sort the shapes.  It is important that the sorts be open with some guidance from the teacher.  These sorts allow the students to determine the rules and make observations of the similarities between the different shapes.  The teacher’s involvement is to direct each sort’s parameters.  The parameters contained on page 126 of Making Sense of Mathematics for Teaching Grades 3-5, by Juli K. Dixon, Edward C. Nolan, Thomasena M. Tobias, and Guy Barmoha are a great example of the teacher’s involvement in the sorting process.   These parameters allow the students to begin with a sort of curved and non curved shapes, so that each sorting loop has distinct and separate shapes contained within.  One of  the instructions asks the students to create 2 new rules, so that there are some shapes outside the loops, some inside one of the two loops and some that are in both loops. In order to succeed, a student will need to look at what makes each shape the same and what makes them different.  Finally, the students are tasked with creating another set of two new rules.  These rules are to sort the shapes  so that some shapes are outside of both loops, some shapes inside the larger loop and some shapes inside embedded the smaller loop. The rules must be such that the student is able to find shapes that while a bit different have the same attributes as another.  These sorts will allow the students to make the connections outlined above that squares, while the shape will always remain a square, it is also a polygon, quadrilateral, parallelogram, and rectangle, as well.

You might be surprised by the additional shapes you can find, while you wander around.  There are shapes everywhere, in tile floors, patio paving stones.  Each of these can make beautiful pictures.  I use many items such as these as inspiration for my quilts.

The Inspiration

The Quilt

Inspiration for future quilts.

Let's Add Multi-Digit Numbers

I really enjoyed making this video.  It took 3 takes and each time I felt more confident.  Hope you enjoy it.


I decided to demonstrate addition of three digit numbers.  I used the based 10 block manipulatives to demonstrate adding 367 + 256 .


It is important that children understand the nature of the numbers they work with.  They also need to have numbers remain within the place value that they begin.  I hope this video helps.

https://photos.app.goo.gl/YMZYrUszEhQQeyC5A

FRACTIONS: MATHEMATICAL “TOWERS OF TERROR” TAKE TWO

FRACTIONS: MATHEMATICAL “TOWERS OF TERROR”

TAKE TWO

In our last episode, we discussed recognizing fractions and the terror created by “stacking numbers” into numerators and denominators.  Most students understand “sharing” items to be shared evenly.   This sharing is simply dividing.  In the progression of mathematical learning, students learn to count, add, subtract, multiply, divide, and finally fractions.  This progression is natural.  Sharing with your friends is an idea in fairness and equity.  The difficulty becomes the actual numbers themselves.  

After teaching our students to look a “part of the whole”, we expect them to begin manipulating those strange numbers and perform our favorite operations upon them.  Students are now expected to add and subtract fractions, along with multiply and divide them.  While in the natural progression of numbers, multiplication and division come after addition and subtraction, these operations are the most difficult operations to perform with fractions.  The reason is three very scary words----“Least Common Denomination.”   Telling students that you can not add halves and quarters, because they don’t have the same denominator sends another wave of terror through the classroom.

  The students need a three-step combination to dispel the terror and crack the LCD code.  The first part of the combination is ensuring that we are clear in our instruction and all parties in the classroom are involved in the learning.  It is important for all parties involved in the classroom to work together to create a key to break the code–the code of adding and subtracting fractions.  The classroom should have a class-created and student-contributed anchor chart, which is in plain sight of all the students.  It is also a great idea for each student to have his or her own smaller version of this anchor chart close by to use as a reference.

The creation of the class-contributed anchor chart will allow the students to assist each other in their respective learning.  It gives the students more practice in supporting and critiquing each other’s work.  Only when the entire class works together, can the true learning begin for all.  Each student needs to know that they are contributing to their own education, as well as each others.  

The second number in the combination is ensuring that each student is knowledgeable in fraction equivalents.  Knowing that ½ = 2/4 = 3/6 = 4/8 = 5/10, along with some equivalents for 1/3, 1/5, 1/4.  These equivalents will give the students some beginnings for the addition and subtraction of fractions with different denominators.

The last number in the combination would be a strong grasp of their multiplication factions.  Factoring the denominators is an important step in computing the LCD, which makes knowing their basic factors a necessity.

Once the students have gathered all of the necessary numbers to enter the combination, they are ready to begin.  However, it is the process to the combination that is the major contributor to their learning.  Students need to see that they can do it, before they will begin they can.  This three numbered combination is key to it.

FRACTIONS: MATH’S “TOWERS OF TERROR”

FRACTIONS: MATH’S “TOWERS OF TERROR”

Standard of Mathematical Practice No. 4 states that students need to be able to model with mathematics.  Modeling is practiced early in a child’s math career.  In kindergarten and first grade, students are taught to “draw” a model of their problems.  For example, Brody and his class went to the apple orchard.  Brody picked 3 apples and his friends, Harry and Monty, each picked 5 apples.  How many apples did they pick all together?  

In order to solve this addition problem, the students will make a model of the problem, by drawing a set of 3 apples and 2 sets of 5 apples.  These beginnings allow students to visualize their problems and how to solve them.  Being able to visualize is the first step to understanding fractions.  It has always amazed me how the creating parts of a whole can confuse otherwise mathematically savvy students and their adult homework help.  I have a friend that is able to calculate relatively well, but just freezes when fractions are involved.  I believe she would benefit from creating a “picture” when working with fractions.

Apple Problem Representation

 Once a new mathematician has his or her model, the calculation to 13 apples becomes more manageable, as the "calculator" can count the number of apples to reach the answer.

Some Practical Manipulative Ideas

Being able to create a model that represents the whole and its equal parts is as necessary to beginning fractions, as it was to begin adding and subtracting.  The models allow a student to see the interactions of the fractions with the whole.  It also allows the students to manipulate the fractions and gain an understanding in a concrete way.  This will also allow their understanding to move from the concrete to abstract.


As the saying goes, “A picture is worth a 1000 words.”  I believe a model representation of fractions would save a lot of heartache.

GROWING MEANS KNOWING WHY!!!!

GROWING MEANS KNOWING WHY!!!!

There are many issues facing mathematics classrooms.  One major issue is to move away from how and help students move toward why.  Why does a certain mathematical formula work?  It is becoming necessary for students to justify their work and not just reach the correct answer.

In their article, Moving Students to “the Why”, written for Mathematics Teaching in the Middle Grades, April, 2015, Volume 20, No. 8, pages 484-491,  Michael Cioe, Sherryl King, Deborah Ostien, Nancy Pansa, and Megan Staples conducted the two year JAGUAR (Justification and Argumentation: Growing Understanding of Algebraic Reasoning) project

As students prepare for more higher mathematics, they need to be able to explain the “why.”  Standard for Mathematical Practice No. 3 require the creation and construction of viable arguments and critique of others.  This standard leads to the idea that each student is to have the ability to support their methodology and critique their peers.  These two practices have a great impact on later mathematics. 

The impact that justifying an answer has on a student’s higher mathematical success is shown as they advance to high school geometry.  Geometry is an entire class dedicated to the justification of why two triangles are congruent.  The logic behind these proofs is in actuality proving “the why of shapes”. 

Secondly, critiquing another student’s thinking allows the students to take charge of their own learning and make connections through their peers’ work, as well as their own.  It allows them to work within their comfort zone, rather than outside of it.  It also allows them to see where they may have erred in their calculation and could also lead to a much more streamlined process later.

These reasoning skills are skills we use everyday.  There is belief that algebraic thinking and “math” will never be needed after we finish school.  However, we use math and algebraic processes everyday.  We need to reason through our daily decisions and determine our budgets, when shopping.  These are skill learned in math class and will be carried through our remaining learning and life.