Fractions: The Gatekeeper to Algebraic Thinking

In my work as a mathematics consultant for the past six years, I have become convinced that a clear conceptual understanding of fractions is the biggest deficit that I run into with both teachers and students.  Fractions are the gateway to proportional reasoning and algebraic thinking. Any student who doesn’t understand the structure of fractions and their operations will struggle in middle and high school mathematics.

This is not news to middle and high school teachers.  When I was a high school math teacher, it frustrated me that my students didn’t understand fractions.  However, at that time I didn’t have the tools myself to break down fraction ideas conceptually.  I, too, had learned to manipulate fractions via algorithms and was good at memorizing procedures. I’ve spent years after my stint as a high school teacher working with elementary and middle math teachers. I’ve used many resources to help reinforce my foundational understanding of how fraction concepts unfold over the grades so that I could help teachers make sense of how to work with fractions.  Now all teachers can access  an excellent free resource online:  the Progressions Document for Numbers and Operations: Fractions

The Progressions Documents for those of you that are not familiar with them are the original documents, each written by a group of talented educators, that were first step in developing the Common Core (CCSSM). The progressions documents were used to guide the three lead writers of the CCSSM in writing the individual standards.  The Common Core approach to fractions in Grades 3 – 5  focuses on conceptual understanding of fractions and their operations. There is a clear connection between fraction concepts and later algebraic concepts. The Progressions Document for Numbers and Operations: Fractions should be required reading for all math teachers grade 3 – 12.

In their article Fraction proficiency and success in algebra: What does research say?, George Brown and Robert J. Quinn assert the following:

There are at least three critical achievements in the mathematical life of a student: mastering the idea of ten as a unit, understanding fractions, and grasping the concept of the unknown. Consequently, when attempting to learn algebra without the aid of understanding fractions, “it is no wonder that many students’ seeming mastery of fractions begins to fall apart” (Driscoll, 1982, p. 107) Australian Mathematics Teacher, v63 n3 p23-30 2007

Robert Siegler of Carnegie Mellon, in his article Early Predictors of High School Mathematics Achievement  points out that understanding of fractions and long division by 10-year-old students is a predictor of later success in Algebra. He refers to research by Case and Okamoto (1996) who support the idea of introducing fractions to students as rational numbers on a number line at the beginning of their study of fractions: “During mathematics learning, the central conceptual structure for whole numbers, a mental number line, is eventually extended to rational numbers.”

Siegler is also the lead author on a 90 page Practice Guide “Developing Effective Fractions Instruction for Kindergarten Through 8th Grade” which includes the following Five Recommendations for Teaching Fractions:

 Recommendation 1 is to build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts. Learning is often most effective when it builds on existing knowledge, and fractions are no exception. By the time children begin school, most have developed a basic understanding of sharing that allows them to divide a region or set of objects equally among two or more people. These sharing activities can be used to illustrate concepts such as halves, thirds, and fourths, as well as more general concepts relevant to fractions, such as that increasing the number of people among whom an object is divided results in a smaller fraction of the object for each person. Similarly, early understanding of proportions can help kindergartners compare, for example, how one-third of the areas of a square, rectangle, and circle differ.

­Recommendation 2 is to ensure that students know that fractions are numbers that expand the number system beyond whole numbers, and to use number lines as a key representational tool to convey this and other fraction concepts from the early grades onward. Although it seems obvious to most adults that fractions are numbers, many students in middle school and beyond cannot identify which of two fractions is greater, indicating that they have cursory knowledge at best. Number lines are particularly advantageous for assessing knowledge of fractions and for teaching students about them. They provide a common tool for representing the sizes of common fractions, decimals, and percents; positive and negative fractions; fractions that are less than one and greater than one; and equivalent and nonequivalent fractions. Number lines also are a natural way of introducing students to the idea of fractions as measures of quantity, an important idea that needs to be given greater emphasis in many U.S. classrooms.

Recommendation 3 is to help students understand why procedures for computations with fractions make sense. Many U.S. students, and even teachers, cannot explain why common denominators are necessary to add and subtract fractions but not to multiply and divide them. Few can explain the “invert and multiply rule,” or why dividing by a fraction can result in a quotient larger than the number being divided. Students sometimes learn computational procedures by rote, but they also often quickly forget or become confused by these routines; this is what tends to happen with fractions algorithms. Forgetting and confusing algorithms occur less often when students understand how and why computational procedures yield correct answers.

Recommendation 4 involves focusing on problems involving ratios, rates, and proportions. These applications of fraction concepts often prove dif­ficult for students. Illustrating how diagrams and other visual representations can be used to solve ratio, rate, and proportion problems and teaching students to use them are important for learning algebra. Also useful is providing instruction on how to translate statements in word problems into mathematical expressions involving ratio, rate, and proportion. These topics include ways in which students are likely to use fractions throughout their lives; it is important for them to understand the connection between these applied uses of fractions and the concepts and procedures involving fractions that they learn in the classroom.

Recommendation 5 urges teacher education and professional development programs to emphasize how to improve students’ understanding of fractions and to ensure that teachers have suf­ficient understanding of fractions to achieve this goal. Far too many teachers have diffi­culty explaining interpretations of fractions other than the part-whole interpretation, which is useful in some contexts but not others. Although many teachers can describe conventional algorithms for solving fractions problems, few can justify them, explain why they yield correct answers, or explain why some nonstandard procedures that students generate yield correct answers despite not looking like a conventional algorithm. Greater understanding of fractions, knowledge of students’ conceptions and misconceptions about fractions, and effective practices for teaching fractions are critically important for improving classroom instruction.

The CCSSM writers, using the approach recommended by the research of Case and Okamoto (1996),  introduces fractions as units on the number line starting in Grade 3. This is a new approach that is at present not understood and fully embraced by elementary teachers who don’t have a deep understanding of fractions on the number line, or of fractions as single numbers with a single value (magnitude). Fractions are often understood only as a comparison of part to whole. While the part to whole interpretation of fractions is an important model to understand, it breaks downs when operations such as addition, subtraction, multiplication, and division are applied to two or more fractions.  One of the misconceptions that students have is that operations (addition, subtraction, multiplication, and division) on fractions have the same effect as operations on whole numbers. Note: In the following examples, we will only take into account positive rational numbers, since introducing negative values adds another layer of complexity to operations, which is not introduced until Grade 6 and also presents many misconceptions for students.

When two whole numbers are added, the resulting sum is larger than either addend. Adding two positive fractions will also result in a sum that is larger than either addend. Multiplying two whole numbers will result in a larger whole number.  However, multiplication does not have the same effect on fractions, which can be very confusing to students unless the misconception is recognized and explained with concrete examples.

When the two factors in a multiplication problem are fractions, each factor acts as a “scalar” on the other fraction.  In the case of 1/2 x 3/4, the product will be less than 1/2 (since we are finding 3/4 of 1/2) and the product will be less than 3/4 (since we are finding 1/2 of 3/4).

The operation of division also presents a different story than division with whole numbers.  When asked to divide a larger whole number by a smaller whole number, say 6 divided by 3, the answer is 2, which is less than 6.  When asked to divide 6 by 1/2, the answer is 12, which is more than 6.  This presents a lot of misunderstanding and rather than delve into the reasons that this happens (1/2 is smaller half as big as 1, so twice the number of 1/2s will fit into 6), most students simply memorize the “invert the second fraction and multiply” rule, but often forget whether that rule is for division or for multiplication, or which one they invert, or whether they have to get a common denominator when multiplying and dividing.

All of these misconceptions can be circumvented, but only if teachers anticipate the misunderstanding and provide guidance to students to understand how operations on numbers can affect the numbers differently, depending upon the value of the numbers being operated on.  Are both numbers less than one? Are both numbers more than one? Is one number more and one and another less than one?  And, of course, the introduction of integers in Grade 6 adds another layer of complexity.

Elementary students can grasp these ideas when they are introduced in a concrete way, using manipulatives, the number line and the area model. However, students are often given the rules and, since the rules are counter-intuitive to their understanding of how whole numbers work, they are often confused.

Unit Fractions:  The Building Blocks of Fractions

CCSSM bridges fractions and algebra by introducing fractions as units.  For example, 1/3 is introduced as the quantity formed by 1 part when a whole is partitioned into 3 equal parts.  (3.NF.A.1)

We count things using units. When using 1 as our unit, we count 1, 2, 3, 4, 5…By the same token, when we use 1/3 as our unit, we count 1/3, 2/3, 3/3, 4/3, 5/3…When we use apple as our unit, we count 1 apple, 2 apples, 3 apples, 4 apples, 5 apples. When we use x as our unit, we count 1x, 2x, 3x, 4x, 5x…The  introduction of fractions as units should help students understand why it is important to have a common denominator when adding and subtracting. The denominator expresses the size of the unit. The numerator tells how many units of the denominator exist.

Linking the understanding of units fractions to the number line, CCSSM first defines a fraction such as 3/5 as being the quantity formed by 3 parts, each with size 1/5. (3.NF.A.1). This idea is extended to the number line in 3.NF.A.2a.  The whole on the number line is first defined as the interval between 0 and 1. Then, if the interval is divided into, say, 4 equal parts, each part has size 1/4. The fraction 1/4 is then defined as the endpoint of the part based at 0. Finally, 3.NF.A.2b completes the number line story by explaining that the placement of 3/4 on the number line is achieved by starting at 0 and marking off 3 intervals each of length 1/4 to make an interval of size 3/4 and that the endpoint of that interval represents the number 3/4 on the number line.

These ideas are complex, not only for students, but for teachers as well.  The language in the standards (seen below) is not easy to comprehend. A significant amount of professional development for teachers needs to focus on how the concept of fraction unfolds between Grades 3 and 6.

  • CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
  • CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
  • CCSS.Math.Content.3.NF.A.2a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
  • CCSS.Math.Content.3.NF.A.2b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Students who intuitively know that you can’t add apples and oranges often add 2/3 + 4/5 and get 6/8.  If they understand 2/3 as 2 copies of 1/3 of a whole and 4/5 as 4 copies of 1/5 of a whole, it will be clear why they can’t be added in the form that they are presently in.

When adding fractions with unlike denominators in Grade 5, the CCSSM does not require that a lowest common denominator must be found, only that a common denominator must be found.

5.NF.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.)

The most efficient common denominator is found by multiplying the denominators of the addends. This is how fractions are multiplied in algebraic expressions, when the actual numbers are not known.

a/b +c/d

–> multiply a/b by 1 in the form of d/d –>  a/b * d/d = ad/bd

–> multiply c/d by 1 in the form of b/b –> c/d * b/b = cb/bd

–> ad/bd + cb/bd = (ad + bc)/bd

Conclusion:

The Grade 3 CCSSM standards and those in Grades 4 through 6 will hopefully result in students and elementary teachers having a conceptual understanding of fractions that will transfer to algebraic understanding in later years.  However, due to the pipeline effect, it will be several years before many students reach high school having learned about fractions using this approach. Even then, the understanding will most likely be fuzzy.  I highly recommend that middle and high school teachers reinforce the conceptual understanding of fractions outlined in the Progressions Document for Numbers and Operations: Fractions in their classes.  It will pay off.

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Visual Patterns Seen Through the Eye of the Beholder

Yesterday, in my post Fawn Nguyen’s New “Math Talks” Site Enhances her “Visual Patterns” Site, I discussed how Fawn and I saw Pattern # 115 in her Visual Patterns site differently.  I also sent Fawn a pattern that fit my first erroneous way of interpreting Pattern # 115.  Shed added the pattern I sent to the site. It is Pattern # 116.  Under Pattern 116, were instructions to go to the link to get the equations.  The link, however, was to yesterday’s Watsonmath post, which only included the equation to Pattern # 115, not # 116.  I’m creating this post so that if Fawn wants to change the link to #116, it will help students to see how two different people interpreted how the shape was constructed.

Below I will explain how I envision Pattern # 116 and how I guessed that Fawn might envision the same pattern, based on the different ways she and I interpreted Pattern # 115.

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The fact that both formulas simplify to y = 8x – 1 does not mean that the simplified formula provides information about the shape.  The most efficient way to show the 10th pattern, which would have 8(10)-1 = 79 circles, would be to draw a large rectangular array of circles that has a vertical dimension of 8 and a horizontal dimension of 10, with one circle taken out somewhere.  The total number of circles is the same, but the pattern looks nothing like the Patterns 1, 2 and 3.

Using my formula y = 3(2x – 1) + 2 (x + 1), I can easily create the 10th pattern that has the same general “shape” as the original 3, but would have 79 circles.  I know that there would be a large rectangle that has a vertical dimension of 3 circles and a horizontal dimension of  2(10) – 1 = 19 circles.  There would be 10 + 1 = 11 towers that are 2 circles tall that are evenly spaced along the top of the large rectangle. The outer 2 towers would be placed alongside the top row of the large rectangle.  The remaining 9 would be placed on top of the large rectangle and there will be a space that is one circle diameter wide between each tower.

Although these pattern problems aren’t “real world” problems,  they provide students with a way to practice   problem solving.  They elicit student use of  the following CCSSM Practice Standards:

1. Make sense of problem and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics. (Although they are not real world situations, an abstract model is being developed.)

5. Use appropriate tools strategically. (The tool of the x/y table helps students to keep track of their thinking.)

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

If you missed my previous post, make sure to check it out to see how Fawn uses Patterns in her classroom on a regular basis.

 

 

 

 

 

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Fawn Nguyen’s New “Math Talks” Site Enhances her “Visual Patterns” Site

For my first post after a very long hiatus…due to a mixture of professional and personal reasons…I want to highlight my favorite math educator, Fawn Nguyen, who I suspect doesn’t sleep.  She somehow finds the time to produce the work of two or three of us mortals.  She is a full time middle school math teacher, which tends to keep the normal person fairly busy.  On top of that, she has created and regularly maintains three excellent websites:

Finding Ways to Nguyen Students Over is Fawn’s blog where you can get the gist of her educational outlook and where her unique personality reveals itself.  She manages to be both irreverent and passionate as she shares her day to day adventures as a middle school math teacher in California.  One doesn’t have to teach middle school  to mine gems from her blog.  Her insights and issues resonate with upper elementary as well as high school math teachers.

Visual Patterns Site

Fawn must have been bored over Christmas/New Years vacation last year.  On December 27, she posted in her blog her idea for a new site Visual Patterns, created for the  “purpose of helping students develop algebraic thinking through visual patterns.” The patterns on the site are created by Fawn, other math teachers, and students.   The site is simple.  There are tabs on the top: 1 – 20, 21 – 40, …, 101 – 120. Each tab accesses 20 different patterns.  There are only 15 patterns on the 101 – 120 tab, with space for 5 more before she adds another tab.  There is a tab called Gallery, in which there are 11 patterns created by students.  Another tab, called Teachers, is one that I hadn’t looked at yet, thinking that it was the patterns created by teachers. When I checked it out, I found a wonderful resource in which Fawn explains how she assigns patterns to students.

With 115 good patterns to choose from, it is hard to decide which ones to assign.  Also, perhaps she has run across the problem that I have when I set loose a classroom of students to figure out one pattern.  I find that if students are all working on the same pattern, especially if they are working in groups, some students allow others do the thinking and simply copy the groupthink with no understanding.  (On the Math Talks site, discussed below, Fawn does have all students work on the same pattern, but they do it independently for 5 minutes,  then share their work with their neighbors, and then the whole class discusses the pattern.)

Fawn developed a system to randomly assign different patterns to students. For independent work that can be given for homework or classwork, click on the Teachers tab to see how Fawn assigns the patterns to her students.  She wanted students to be able to choose from 3 patterns and wanted each student to have their own 3 patterns to choose from.  To accomplish this, she used the free online Research Randomizer. When you go to this site, a blank form appears.  Fawn explained how she filled out the form for 39 students. At the time she posted this, there were 111 patterns.  Now there are 115.

In addition, Fawn created a downloadable Word document template form which has room for students to show their work on 2 patterns for each 8.5 x 11 page.  On the form, the student fills in  their name, date, Pattern #, and the following:

1. Draw the next step

2. Draw a quick sketch of step 27

3. Complete this table (2 columns: Step #, # of ________)  Step #s 1 – 5, 10, 27, then a blank

4. Write the equation

Wanting to try out the Random Number generator and the form, I went to the site and followed Fawn’s instructions.  Student #1 was assigned 64, 115, 32, Student # 2 was assigned 47,105, 53. I followed Student # 1 and worked through Patterns # 64 and #115 on the template.  There was plenty of room for my messy calculations that I did in the margin.

On each Pattern in the Visual Pattern site, the correct value of units for Pattern # 43 is given. This is a good way for students to check their answer after they create an equation on the Student Sheet. It doesn’t give the answer away, since they still have to come up with the value for # 27 in addition to an equation.  What I love about the patterns is that there are many ways to envision the same pattern.  In fact, I have to admit that I envisioned a pattern incorrectly in my haste to write this post.  My Step # 43 did not agree with Fawn’s.  I checked it over a few times, and then figured that it had to be a mistake on the site!  (Fawn nicely asks people to comment if they find a mistake.)

Here’s the pattern:

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I emailed her and included my filled in template.  Within a couple of hours I received a reply from her and she very gently explained to me that I had counted the number of squares on my Step 2. I had visualized it as the large 3 x n rectangles on the bottom and the small 2 x 1 rectangles on the top.  My mistake was that after the first step, the 3 x n rectangles were overlapped by the 2 x 1 rectangles that were in the “middle” (not on the left end and right end). As a result, I said that there were 15 squares in # 2 when there were really only 14.  On Step 3, I had 2 too many squares and my error increased by 1 for each step.

Here’s how I saw it once Fawn set me straight on my error:

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Here’s how Fawn saw it:

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She showed how she had visualized the pattern, which was totally different from how I had visualized it.  This is magic in these patterns.  For any given pattern, there are multiple ways that the equation can be created.  Her equation was Squares = 4(2n+1) – n – 4.

Here’s how some teachers might want the equation to be simplified.

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Once I see this simplified equation, I can go back to the figure and “see” it.  But I would never see y = 7x on my own.  In my opinion,the pattern just doesn’t visually elicit that equation without some mental somersaults being turned.

I’m not sure how often Fawn gives students the 3 random patterns to choose from.  I can envision giving students 3 to choose from each week with the expectation that they hand in a sheet with 1 or 2 patterns filled in by the end of the week.

Math Talks Site

Not wanting to rest on her laurels with only two sites, Fawn recently added a new site early this month, Math Talks, that meshes well with the Patterns Site. If you click on the link, Fawn has clearly outlined how she uses Math Talks in the classroom.  Several screen shots from her site are below:

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In her Visual Patterns Talks, she reinforces the idea that people see the patterns very differently. That is the beauty of working with patterns.  Students can be creative.

Wondering about how the scribing happens, I wrote a comment on Fawn’s Math Talks Site in early November.  Here is my comment and her reply:

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Yes, Fawn is my hero.  She not only is an excellent teacher, but takes the time to share her strategies with the world (or at least those of us interested in math education.) By the way, MTBoS stands for Math Twitter Blogosphere for those of you who haven’t heard of it.  I attended the 2nd annual MTBoS Twitter Camp this summer in Philadelphia, where I met Fawn and many other talented, generous math teachers. It was a whirlwind of activity presented by and for math teachers from all over the country…and some from overseas.  I plan to attend the 3rd annual MTBoS Twitter Camp next summer, wherever it is held.

I highly recommend subscribing to Fawn’s three sites…as well as the next site she is sure to dream up soon!

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Smarter Balanced Item Preview Makes it Apparent That Students Need to Be Able to THINK!

I’ve finally gotten a chance to check out the recently released SBAC item preview. I am impressed with the rigor of the problems that they have created.  The tasks make it clear that students will no longer be successful simply by memorizing procedures. Rather, they will need to have a deep understanding of concepts.

Below is a list of the of sample problems for the different grade levels.

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So far, I’ve only looked at the Problem Solving tasks. For some of the problems, students are asked to choose the correct answers.  Yes, answers is plural. There may be one correct answer, but there may be more.   Some have students choose from a list of answer options and move them to correctly fill in the blank.   After each of these types of problems, the participant can check to see if the answer is correct and receive a score.

Other tasks are free response items and are not scored, but a rubric is provided. In this post, I will look at two related HS problem solving tasks: Circle 1 and Circle 2.

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My approach to coordinate geometry has always been to have students apply the Pythagorean Theorem when finding the distance between two points and when deriving the equation of a circle with a given center and a given radius.  Students who understand that the Pythagorean Theorem is the foundation for finding a distance between two points and who understand that the distance between two points is the foundation for finding the equation of a circle (set of all points that are a given distance from the center) can easily approach this problem.  Although the problem description suggests that trig can be used, I did not approach it in this way.

In my approach, the essence of the problem is creating congruent right triangles, in which the hypotenuse is the radius of each circle.  There is not even a need to find the length of the hypotenuse in order to answer the question. In fact, applying the distance formula would, in my opinion, cloud the issue. Once the vertical and horizontal legs of the initial right triangle are known, the student can simply copy those vertical and horizontal distances starting at the point on the initial circle.  The ending point will be the center of the second circle. However a student needs to understand that there is a unique right triangle that connects the center of the circle to every point on the circle. This requires a conceptual understanding of two important ideas: (1) the definition of a circle as the set of all points (x,y) that are equidistant from a given center and (2) the distance between two points is the hypotenuse of the right triangle whose legs are vertical and horizontal line segments on the coordinate plane.

Each problem has an explanation as well as a rubric.  Below is what is provided for the Circle 1 problem. Note that the student only gets 1 out of 2 points for the correct answer.  Also, the student can get an incorrect answer due to a slight error, but if the reasoning shows that they knew what they were doing, they will score 1 point. The reasoning is given as much weight as the correct answer! Hallelujah!

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Circle 2 uses the same original circle, but throws in the fact that the area of the second circle is 1/4 the area of the original circle.  For this problem, the student needs to understand that a radius that is 1/2 the radius of the original circle will be needed in order to result in an area that is 1/4 the area of the original circle.

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The Practice Standards that students needs to use to solve this problem include:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments. (explaining their approach)

5. Use appropriate tools strategically. (using the coordinate plane as a tool)

6. Attend to precision.

7. Look for and make use of structure. (use the fact that the structure of circles being made up of an infinite number of right triangles.)

Both of the problems are the type of conceptually-based problem solving that I espouse, so I am excited that the SBAC folks are are the same wavelength.  If this is what will be expected of students, it is clear that many high school teachers will have to change their approach from a procedural-based approach to a conceptual approach. I’m glad to finally have some rich examples to show teachers in the SBAC states. I’m anxious to peruse the other types of tasks.  I’ll save that for another post.

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Fawn Nguyen Rocks with Fraction Division

Ms. Win does it again! I’ve decided that Fawn Nguyen is my most favorite math teacher that I’ve never met! I look forward to meeting her and many other admired math bloggers at Math Twitter Camp 2013 in Philly at the end of July.

In this blog post, she presents a very conceptual way for students to visualize division of fractions.  I’ve tried different approaches using the area model, but none have touched this one for it’s efficiency.  Here’s why it’s better than what I’ve done in the past:

  • She uses graph paper, so the area models are neat.
  • She has students sketch the area model for both the dividend and the divisor.
  • She uses color to differentiate between the dividend and the divisor.
  • She gently leads students to discover that the dimensions of each whole are the most efficient when students use the two numbers that are  (1) the denominator of the dividend and (2) the denominator of the divisor.
  • She starts off simply and then gradually shows all cases such as mixed numbers, and finally a larger fraction dividing into a smaller fraction, resulting in a quotient that is a fraction less than one (this example is added in the comments).
  • She encourages students to check their answer using an online calculator with a fraction mode.

Here’s a peak at her first example, which asks students to divide 3/4 by 2/3:

 

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Although this lesson doesn’t show it, her comments indicate that she eventually introduces students to short cut numerical methods, first using common denominator and even perhaps (egads!) common numerator a la Chrisopher Danielson’s post, and finally showing the traditional invert and multiply algorithm that is easy and efficient, but leads to zero conceptual understanding.  However, after students have seen the concept unfold visually, at least the students have a chance of remembering the conceptual reasoning of fraction division.

If you were to poll students, or adults that are not math teachers, and ask them WHY 8 divided by 4 is 2, I bet that they would be able to explain it satisfactorily.  However, if you were to ask the same group (and even some adults that ARE math teachers)  and asked them WHY 3/4 divided by 2/3 is 1 and 1/8, I can almost guarantee a pained, puzzled expression.

Thanks, Fawn and Christopher, for adding to the conversation!

Both Fawn’s blog and Christopher’s blog can be subscribed to via email.  I highly recommend that you follow them for some great mathematical instructional ideas.

 

 

 

 

 

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Fawn Nguyen Molds Middle School Mathematical Thinkers

In her blog Finding Ways to Nguyen Students Over, Fawn Nguyen, aka Ms. Win, shows how she worked through a Dan Meyer 3-Act, Pyramid of Pennies,  with her Grade 8 students. I’ve been a Fawn fan for a while and this post illustrates every reason why I am a fan.

First, she makes the mathematics engaging and fun for the students.  The 3-Act task itself could be approached from a variety of ways.  However, once students got the answer, she realized that the problem itself was pretty straightforward;  no students “broke a sweat” figuring the answer out once they developed a plan. She decided to focus on the sequel.  Why does the summation formula work?  Students are provided with manipulatives to help them make  make sense of and model the problem.

Screen Shot 2013-05-19 at 12.32.05 PM

 

Fawn approaches the activity with the expectation that her students can think and figure things out.  She is the “guide on the side” rather than the “sage on the stage”, to quote a tired cliche’. The “letting go” required by a teacher can be scary. Any teacher who uses this open-ended approach must have a great deal of confidence in their ability to guide and also to in their willingness to say, “I don’t know, but I think we can figure this out together.”

In a previous post on this blog, I shared a chart in which I had shown how the Pyramid of Pennies  3-Act elicited all of the 8 Standards for Mathematical Practices.  This chart was created after I was in a class in which Dan Meyer presented the Pyramid of Pennies.  Recently, Dan Meyer revisited the 3-Act process in his blog, which encouraged Fawn to do this activity with her class.

In my work as a consultant with math teachers, I often hear some version of the following statement: “Yes, those 3-Act tasks and other problem solving ideas are all well and good, BUT… We can’t afford to take the time that it takes to work those problems.  We have too much to cover.” As the grade taught goes up, the tendency to hear this statement becomes more predictable.

High School teachers tend to be the most reticent to want to take the time it takes to help their students to learn to be independent, creative problem solvers. Yet the math thinking that Fawn’s students are undertaking will add more to their ability to “do math” than any memorized procedure will ever do.  The procedural memory will be eventually lost, but the habits of mind that are built when making sense of non-routine problems will become, as the phrase Phil Daro used to describe the CCSSM Practice Standards, “the content of a student’s mathematical character”. In another Phil Daro video, Against “Answer Getting”, he asserts that “a high percentage of what we teach kids is not mathematics, but answer-getting techniques.” (8:20) The video compares the different approaches to math problems by American and Japanese teachers.  American teachers first focus on teaching students the mathematical procedures they need to get the answer to the problem and then give them a problem to apply the skills and get the correct answer. Japanese teachers, on the other hand, choose problems based on the mathematics that can be learned by students as they work through the the problem. For American teachers, the focus is on the answer. For Japanese teachers, the focus is on the problem solving process.  My opinion is that teaching students to be independent critical thinkers trumps teaching them streamlined procedures that will lead to the correct answer as quickly as possible.

In the comments to Fawn’s post, she was asked how long it took her to complete this investigation.  She replied that it took her 3.5 days:  2 days for the original 3-Act and 1.5 days for the sequel.  My opinion is that these 3.5 days of making sense of the problem and determining WHY the summation equation works left the students with more tools in their math toolbox than 3.5 days of learning “answer-getting techniques.”

Thank you, Fawn, for sharing your students’ journey.

 

 

 

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Geoff Krall Makes it Easier to Mine Math Online Resources

My love affair with the mathtwitterblogosphere has gone up a notch.  After reading Dan Meyer’s post Geoff Krall Combs The Internet for Lesson Plans So You Don’t Have To,  I went to Geoff’s site Emergent Math to check out his curriculum maps.

Geoff has developed CCSSM-aligned Problem-Based Curriculum Maps for Grade 8, Algebra I, Geometry, and Algebra II.  In the works are maps for Integrated Math 9, Integrated Math 10, Integrated Math 11, and Grade 6/7.   Kudos to Geoff for taking the time to create these lists and share them freely!

BTW: In his post Math Blogging Retrospectus 2012, Geoff wrote a wonderful summary of posts to math blogs that inspired him in 2012.  It’s worth reading to get an idea of the richness of the discourse and resources available via the online math community.

 

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Star Polygons Take 2

The MathTwitterBlogosphere (MTB): Professional Development at its Best

I love to go to conferences and take workshops, but it involves packing, shlepping myself and my stuff to the airport, as well as money and time.  I will always continue to seek out “live” learning opportunities, but I have a PD source right at my fingertips 24-7 in the form of the MTB.  (Full disclosure: I don’t do twitter, but I more than make up with it by following many blogs.) Every once in a while, a blog post lights a fire under me, guiding me into to a line of inquiry that I can’t leave alone until I play around with the idea and write my own post to make sense of the idea.

3.5 gons? What the heck?

This post continues my pursuit of understanding of 3.5-gons and their ilk.  It all started with Dan Meyer’s March 27, 2013 post “Discrete Functions Gone Wild!” His post focused on what a regular polygon would look like when when the number of sides was not a whole number.  He used his understanding of how regular polygons with whole number sides behaved to determine the angle between sides, using the formula (n-2)180/n.  He let n = 3.5 and determined that the measure of each angle in a 3.5-gon is 77.1 degrees.  Then he constructed the 3.5-gon. and created a star shape. Then he left his readers with the following BTWs and went to bed, across the pond in Nottingham:

BTW. One of you enterprising programmers should create the animation that runs through continuous values of n and shows the regular polygon with that many sides. That’d blow my mind. I can only do the discrete values.

BTW. Malcolm Swan demonstrated the 3.5-gon on the back of some scratch paper in the middle of a design session here in Nottingham. That kind of throwaway moment (often before tea, of course) has been a lot of fun these last two months.

BTW. But where is the 3.5 in that shape? Maybe you see how the number 3.5 turned into the number 77.1 and how the number 77.1 turned into that star shape. But where is the 3.5 in the star? I’ll hint at it in the comments but I’ll encourage you to think this through. (It may be helpful to see 3.5 as the rational number 7/2.)

When he awoke, he found that several enterprising programmers had taken his challenge can created the goods.

2013 Mar 28. I love you guys. I fall asleep for a few hours and wake to find out it’s Christmas. Some interesting visualizations of rational regular polygons from Marc Garneau, Eric Berger, Stuart Price, and Josh Giesbrecht.

When I stumbled upon the post, I went to work on BTW # 3, trying to make sense to myself about why a 3.5-gon looks like this:

3.5-gon to 7/2-gon: Different representations of the same value open up new possibilities!

Looking at functions via multiple representations (graphical, numeric, algebraic, verbal) has always served me well. Some representations shine a different light on the function. Putting 3.5 into the form 7/2, a hint that Dan gave, opened up a new door for making sense of the shape.

I played with the different applets that were provided overnight by four readers of Dan’s post…another positive aspect of the MTB…great minds just waiting to use their talent for the betterment of mankind!  The 7 was pretty easy to see since there were 7 line segments.  The 2 took awhile for me to see a pattern.  What I finally noticed by trying out different fractions was that the denominator indicated how many “rotations” around the shape it took to “close up” the polygon.  These terms are used loosely, since a full rotation would take one back to the “starting point”, which would “close up” the polygon.

3 – gon, 3/1 – gon, or you may know it as a triangle

For example, in a whole number-gon, such as a 3-gon (triangle), which, by the way, could also be called a 3/1 – gon, follow this procedure:

Start at A. Heading counter-clockwise and draw segment AA’. Construct a 60 degree angle degrees at A’. Draw A’B the same length as AA’.  Construct a 60 degree angle at B.  Draw BB’ and B’ will coincide with A.  You have closed the 3/1 -gon after 1 rotation.

m/n – gons and partial sweeps or revolutions

When dealing with a m/n-gon, when n is not 1, it will take more than one sweep to get exactly back to the starting point  I am calling each “sweep around” a “partial revolution”, meaning that it gets close to the starting point, but doesn’t quite make it that time around.

Although my explanation was not mathematical precise…it was more like hand-waving and a “kinda” explanation…  I posted my comment to Dan’s post, putting that interpretation out there.

Here is part of my comment to Dan’s post:

Wow! I love math! I was blown away by this post and the idea of thinking of a 3.5 gon as a 7/2 “gon” that means that it has seven “sides” and takes two rotations to complete.

Out of the denizens of the MTB, a comment appeared from none other than Michael Serra, author of one of my favorite Geometry books, Discovering Geometry: An Investigative Approach. Of course, he has a better and more mathematically succinct way to describe how the “n” behaves. Before I explain his way, which is skipping vertices, and much less wishy-washy than my explanation, I will share the following comment that he made:

BTW: Expressing the n as an improper fraction opens the door to two ways of expression each star polygon. The star polygon 12/5 is equivalent to the star polygon 12/7. The numerator expressing the number of vertex points and the denominator expressing how many points to count from one vertex to the next vertex.

It is cool that Elaine Watson noticed that the denominator is also the number of cycles to complete the star polygon. I hadn’t seen that before.

This comment by Michael Serra served two purposes:

(1) it made me curious to find out more about 12/5-gons and 12/7-gons, and

(2) it fed my ego to have a mathematics educator that I highly respected comment on my post that I had noticed something that he didn’t!

So, in honor of Michael Serra, I am going try to make sense of my explanation of the role of the denominator and define more clearly what I mean by “it takes two rotations to complete.” While doing that, I am also going to refer to Michael’s explanation of the role of the denominator, which expresses “how many points to count from one vertex to the next vertex.”

Closer look at 7/n – gons

7/1 – gon

My explanation of the role of the denominator was influenced by how Dan created the original 3.5-gon.  He started with a line segment and the angle measure and added the sides one at a time, until the last segment met up with the starting vertex. On the other hand, Michael Serra’s more precise explanation required that the vertices needed to be placed first, followed by drawing in the sides using the rule for skipping the required number of vertices.

GeoGebra can conveniently create a regular polygon.  To do this, construct two points and a line segment connecting the points to represent one side.  Using the “polygon” tool, choose “regular polygon” and create a 7-gon. Below is a 7/1-gon.  The 7 means there are 7-sides. Using my revolution interpretation, the 1 means that it takes one revolution to complete the polygon. Using Michael’s vertex interpretation, the 1 mean that you go directly to the next vertex, without any skipping over vertices.  This forms the regular 7-gon, we all know and love.


7/2-gon

In a 7/2-gon, use the vertices of the 7/1-gon as a skeleton.  The 7/2-gon will be represented by the black segments. Choose a starting vertex and a direction (I chose counter-clockwise) around the polygon to the 2nd vertex and draw the line segment between the two vertices. Repeat this method until you return to the starting vertex. Since you are skipping vertices, you will have to go around more than once.  In this case of a 7/2-gon, you will go around “about” twice to pick up the missing vertices before you get back to the starting one.

There is still one side to draw to close the 7/2-gon.  This is not a full sweep. The other two sweeps each created 3 sides. The final sweep will create 1 side, so we’ll call it Sweep 2 1/3.

7/3-gon

What about a 7/3-gon? Start with the skeleton of the 7/1-gon.  Choose a starting vertex and count counter-clockwise to the 3rd vertex and draw the segment from the starting vertex to the 3rd vertex.  Repeat that process until you close up the polygon by returning to the starting vertex.  Each “partial  revolution” or “sweep” is illustrated below:

As in the 7/2-gon, we’re not quite there yet.  There is one more segment to draw. Each sweep filled in 2 sides.  This last sweep will fill in 1 side, so we’ll call it sweep 3 1/2.  Although the denominator doesn’t exactly define how many revolutions, it hovers around the ballpark.

Here is the final 7/3 – gon.

Both of the final sweeps here have required only one more line to be drawn. Questions to consider:

  • Will there always be one leftover side to complete the shape?
  • If not, what are the possibilities for leftover sides?
  • Explain and illustrate your conjectures.

Comparing and Contrasting 12/7-gons and 12/5-gons

My next project was to figure out what Michael Serra meant when he said “Expressing the n as an improper fractions opens the door to two ways of expression each star polygon. The star polygon 12/5 is equivalent to the star polygon 12/7.” What did he mean by equivalent?

As I worked through my ideas, I simultaneously began creating an investigation for students, Comparing and Contrasting Congruent Star Polygons.

This is a very rough first draft, so I’m not ready to post it.  I may have scaffolded it too much for it to be considered an investigation.  If anyone has any comments or suggestions, please pass them on.

Conclusion: Fan Letter to the MTB and All of the Math Nerds out there who Keep the Flame Lit

To conclude, all of this started with one post by Dan Meyer that intrigued me.  Because Dan has such a following, there were many more contributors who created applets, comments, and their own ideas.  I don’t think that professional development gets much better than this.  Being  actively involved in keeping up with math blogs and then continuing on with my own investigations sparked by the posts, has made me much more reflective about my work and better at what I do.  Books and articles have their place in my life; I’ll never be willing to burn my large math library.  However, reading is a solitary pursuit.  The MTB is a community pursuit.   I need both.

 

 

 

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What does a regular 3.5-gon look like?

Just when I thought I had no more to learn and discover about regular polygons, Dan Meyer draws me back in and wows me.  Once I know that the sum of the angle measures is (n-2)*180 and, if it’s regular, each angle has measure {[(n-2)*180]/n}, how much more is there to know? So the sum of the interior angles of a triangle is 180° with each angle 60° , a quadrilateral has 360° with each angle 90°, a pentagon has 540° with each angle 108°, etc. And, of course n must be a positive integer…right? Been there, done that.  Let’ move on!

Dan is currently in Nottingham soaking up the knowledge imparted by Malcolm Swan and others at the University of Nottingham Centre for Research in Mathematics Education.  There are a lot of names and organizations in the math education world that are all interrelated: University of Nottingham, Shell Centre, MARS, MAPS, UC Berkeley, The Charles A. Dana Center.  Before I continue here, I want to take some time to figure out a family tree of sorts.  Here’s what I came up with.  I may be wrong, so if anyone cares to comment and set me straight, please do.

No matter who begot whom, all of them have excellent resources for the mathematics educator. If you are not familiar with them, please visit the following:  MARS, MAPS, Shell Centre, CRME, Charles A. Dana Center

Back to 3.5-gons: Dan told the story of Malcolm Swan illustrating a 3.5-gon on the back of a scrap piece of paper one afternoon at a design course, which was probably a throwaway aside for Dr. Swan. However it provided the spark that generated an intriguing post by Dan. That spark, in turn traveled to  several programmers, Marc Garneau, Eric Berger, Stuart Price, and Josh Giesbrecht, to work their magic showing graphs of the n-gons. The same spark traveled to Vermont and I couldn’t let it go until I wrote a post about it.

Check out Dan’s post to see how he introduced the idea starting with a regular 3-gon (equilateral triangle), regular 4-gon (square), etc. Then Dan illustrated using graphing software what Malcolm Swan had shown him on the scrap paper for a 3.5-gon.  He challenged the reader to make sense of it by giving the hint to think of 3.5 being represented as 7/2.

With that hint, I watched his animation of a regular 3.5-gon being built one side at a time. Where is 7/2?  Oh…I see!  Then I got to thinking about how I could make this into an investigative unit for Geometry students.

Have the students look through the different ways that programmers showed the polygons.  Which was their favorite tool for visualizing the n-gons and why?

Have them play with their favorite visualization tool to answer the following questions:

  • What conjectures could students make about regular n-gons?
  • If given a regular 3.2 gon, what would the angles be?
  • How many sides would it have?
  • How many rotations would it take for the last side to meet the first side?
  • How are a 3-gon, a 3.5-gon and a 3.2-gon the same, different?
  • Add a 4.7-gon into the picture.
  • Why do some have many rotations and some have only 1 or 2?
  • What type of regular n-gon will have many rotations before the last side meets with the first side?
  • What type will have few rotations?
  • How do the angle measures change as n-increases? Why?
  • How does the graph of y = [(x - 2)*180/x] relate to the shapes of the polygons?
  • What is the domain of y = [(x - 2)*180/x]?
  • Can we build a regular ∏-gon? If so, what would it look like?  If not, why not?

This is the type of problem that is not a modeling problem, but one that really forces students to think outside the box and look through a different lens of perception.  All of the CCSSM Practice Standards except the Modeling standard would be exhibited by students as they work through this problem.

So, Dan Meyer, thank you once again for sending that spark this direction.  And thank you Marc Garneau, Eric Berger, Stuart Price, and Josh Giesbrecht for sharing your gift for programming to give us tools so that we can analyze regular n-gons visually and discover some fascinating things.  And a final thank you to the Math Twitterblogosphere (MTBS) that provides so much free knowledge and expertise to so many people.

 

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North Country UHS 3-18-13 Inservice

Powerpoint

Part 1 of 2 North Country CCSSM HS March Inservice

Part 2 of 2 North Country CCSSM HS March Inservice

The Lowell bowling ball images are not included in the powerpoint because they were too large.

Continue reading

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