Jo Boaler: Fluency Without Fear

Jo Boaler is one of my go to people for insight into best practices in math education. Educated in England, she is now a professor at Stanford University and supports a site called You Cubed that has high quality resources for K – 12 educators.  I participated in her online course on Fractions,  an area of mathematics that causes many students and teachers consternation.  What I like about Jo is her focus on conceptural understanding and number sense over memorization.
The following paper is worth a read:
Here is a 15 minute video in which Jo Boaler discusses “Number Talks”, which is a practice that helps students develop number sense.
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Open Middle: Open-Ended Problems for K – 12

Nannette Johnson, Robert Kaplinsky, and Bryan Anderson have joined forces to create the site Open Middle .  Below is their blurb:

 Dan Meyer introduced us to the idea of “open middle” problems during his presentation on “Video Games & Making Math More Like Things Students Like” by explaining what makes them unique:

  • they have a “closed beginning” meaning that they all start with the same initial problem.
  • they have a “closed end” meaning that they all end with the same answer.
  • they have an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.

Open middle problems require a higher depth of knowledge than most problems that assess procedural and conceptual understanding. They support the Common Core State Standards and provide students with opportunities for discussing their thinking.

Some additional characteristics of open middle problems include:

  • They often have multiple ways of solving them as opposed to a problem where you are told to solve it using a specific method. Example
  • They may involve optimization so that while it is easy to get an answer, it is more challenging to get the best or optimal answer. Example
  • They may appear to be simple and procedural in nature but turn out to be more challenging and complex when you start to solve it. Example
  • They are generally not as complex as a performance task which may require significant background context to complete. Example

 

I like that they are all short problems, so they would be easy to do as a warm-up or assign as a homework assignment.  I also like that there is not always one answer.  This can lead to rich discussions.

There are problems for Kindergarten through High School, all aligned to the Common Core. Below are some examples from Grade 3, Grade 8, and High School.  When a student starts the problem, only the problem is shown.  The “Hint” and “Answer” are accessed by clicking.

Grade 3:

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Grade 8:

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High School:

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Teachers can subscribe to the site to receive emails when new problems are added. Also, teachers are encouraged to submit their own problems.

Good stuff! Check it out!

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SBAC Practice Problems Make it Clear that the Bar is Rising

I am a member of a team of math educators in Vermont that is working with the Vermont Agency of Education to help spread the word around the state about math education initiatives.  Vermont is a small, rural state and teachers are often isolated.  The Agency of Education employs 2 (actually 1.5) very dedicated educators to oversee mathematics for the whole state.  Tracy Watterson oversees K – 5 and Lara White oversees 6 – 12, but for only half of her time. Her other time is spent in another .5 Agency of Ed position.  This year, the 6 – 12 team is hosting bi-monthly meetings in 7 regions around the state to bring information directly to the teachers rather than the teachers having to come to a central location.  We’re calling these gatherings Math Morsels … we bring food morsels as an enticement!  We meet from 4 to 6 PM, so teachers do not have to take time off during the school day.  We are in the midst of Round 2. The focus of this round has been to familiarize teachers with the Smarter Balanced (SBAC) assessment that will be administered for the first time this spring to Vermont students.

The SBAC is new to everyone, so this first year will most likely be full of much consternation and headaches for educators and students.  As in any new thing, my advice to teachers  is to go in as informed and prepared as you and your students can be, and accept the inevitable fact that issues will arise that you didn’t even consider.  Don’t beat up on yourself or your students. Learn from the experience and make changes to improve the process over time.  Also, There will be complaints about the sorry state of  education and how awful the Common Core is and there will be an appeal to go back to the basics, whatever they are/were.

The “basics” have changed.  The predominant metric that we have traditionally used to assess students is the ability to get a correct answer quickly. We have many tools that can give us an “answer”. The tools that our students (and we) lack are the habits of mind that are needed to persevere and solve non-routine problems. These habits are encapsulated in the 8 Practice Standards:

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

# 2 Reason abstractly and quantitatively.

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

# 4 Model with mathematics.

# 5 Use appropriate tools strategically.

# 6 Attend to precision.

# 7 Look for and make use of structure.

# 8 Look for and express regularity in repeated reasoning.

The most important change that we need to make as educators  in order to prepare our students for success on the SBAC is not the content that we teach. The most important instructional change we can make is to teach our students how to persevere in solving a problem.

As we looked through the test resources, it became clear that there are some problems that are asked in very different ways than we and the students are used to.  In order to correctly answer the questions, students need to be creative problem solvers.  Here’s an example of  a 6th grade task. Students are shown a square and told that the area of the square is 324 square units.  The question asks for the length of a side.

This is a pretty straightforward question. The concept being assessed is that in a square the area is found by multiplying a side length by itself.  However, while a calculator is provided for many problems, there was not one available for this problem.  (Each problem is worked on a computer with one problem per page.) A teacher that I was sitting next to and I were surprised that there was no calculator, since we normally don’t expect students to have the square root of 324 memorized. In fact, I myself didn’t know it off the top of my head. However, I had some strategies that I have developed over the years that could lead me to the answer pretty quickly.

This type of question has embedded within it several of the Common Core Math Practice Standards:

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

# 2 Reason abstractly and quantitatively.

# 7 Look for and make use of structure.

Students could of course guess and check 1 x 1, 2 x 2, until they found it,  This would be an example of # 1, making sense (knowing that a number times itself needs to have a product of 324).  It would also be an ad nauseum example of perseverance.

However, if the student added the abstract and quantitative reasoning expected by # 2, they might start with larger easy numbers, such as 10 x 10 and 20 x 20 to get a rough idea of the magnitude of the number they were looking for.  Once they looked at 100 and 400, they knew that they number they were looking for was closer to 400 and could guess and check 19 x 19, 18 x 18, and voila, they are there.

If they did the above, but instead of finding 19 x 19, and 18 x 18, they looked at the structure of 19 and realized that the ones place of 19 x 19 would be 1 (since 9 x 9 = 81), they wouldn’t need to do that multiplication. They could move to 18 x 18, see that the ones place would be 4 (since 8 x 8 = 64), then test out 18 x 18, which would prove that a side length of 18 is the correct answer.

Another approach that was mentioned at the meeting that I had not thought of was to factor 324 and group the factors so that there were two sets of the same factors multiplied together.  324 = 2 x 162, which is 2 x 2 x 81, which is 2 x 2 x 9 x 9, s0 324 could be expressed as (2 x 9) x (2 x 9), which is 18 x 18.  This is another example of using Practice Standards # 1, 2 and 7.

The problem was not an extremely difficult problem. It just had a number that a student might not have memorized as a perfect square. A student who had been given plenty of opportunities to make sense of problems and persevere,  who were comfortable with the structure of numbers and operations would have the tools to work through this problem fairly quickly. A student who expected a quick answer or a quick tool to figure it out would most likely skip it and move on.

The Practice Standards provide students with habits of mind that will serve them well as they work through non-routine problems.  Let’s face it. All problems that our students will face in life are non-routine.  Let’s prepare our students for a future in which they will persevere and expect that any problem worth solving may take some thought and can be solved using multiple strategies.  Doing this will mean changing the way we instruct students to approach problems.  It’s not enough to have the 8 Practice Standards posted on the wall.  We must refer to them often, use them in our own practice,  and expect students to do the same.

 

 

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Tracy Zager shares a problem accessible to many levels of learners

Tracy Zager, in her blog Becoming the Math Teacher You’d Wish You Had shares a wonderful problem that looks simple.  However, scratching below the surface a bit, the problem opens up a lot of different approaches.  She aptly names the post A Problem with the Space Inside it to Learn.

Here’s the problem, which she found in a Twitter Post by Justin Lanier:

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Tracy’s post discusses how her family (2 young daughters and husband) approached the problem.  Comments from readers revealed even more approaches to the problem and, after initially seeing and solving the problem fairly superficially, I was drawn in.  Here is my comment:

Love the problem! As you illustrated, It is accessible to people with very different levels of mathematical expertise. The question “What do you think?” turns it into a sort of 3-Act problem. What I thought was “Are they parallel or not?” I simply counted the squares to notice that the top line had a slope of 1/5 and the bottom line had a slope of 1/4, so no, they weren’t parallel.

 I was done…or so I thought until I started reading the comments. Someone else may care about where they will intersect. So I guess I should care, too. Julie went about it in a way that I never would have thought by generalizing the two equations. I would have not been as clever as Julie and just used the equations y = 1/5 x + 5 and y = 1/4 x – 4. Setting 1/5 x + 5 = 1/4 x – 4 and solving for x, and then y, I would get the point of intersection at (180,41), the same answer as Julie.

What if I don’t know algebra yet and so can’t find the point of intersection by setting the y=values of the two equations and solving for x? I graphed it and noticed that at x = 0, the lines were 9 vertical units apart. At x = 20, they were 8 vertical units apart. Hmmm….could the 20 come from 4 x 5? I continued the graph to x = 40 and they were 7 vertical units apart!

Screen Shot 2015-02-02 at 1.58.01 PM

Practice Standard # 8 tells me to “look for and express regularity in repeated reasoning” and # 7 tells me to “look for and makes sense of structure.” I’m pretty convinced that when x = 60 that the lines will be 6 vertical units apart. If I want to graph this, I can check out my hypotheses. At some point I will be convinced that for every increase of 20 units for x, the lines are 1 unit closer together. I fill out the following table. For the y-value of the upper line, I notice that for every increase of 20 units for x, there in an increase of 4. For the y-value of the lower line, I notice that for every increase of 20 units for x, there is an increase of 5! Wow! Another pattern! For every increase of 20 units, the y-value of the line with a slope of 1/4 increases by 5 and the y-value of the line with slope 1/5 increases by 4!

 x-value of both lines             distance apart             y-value of upper line     y-value of lower line

             0                                      9                                       5                                   -4

             20                                     8                                       9                                   1

             40                                     7                                     13                                   6

             60                                     6                                     17                                   11

            80                                     5                                     21                                   16

           100                                     4                                     25                                   21

           120                                     3                                     29                                   26

           140                                     2                                     33                                   31

           160                                     1                                     37                                    36

           180                                     0                                     41                                   41

 So, the two lines intersect at (180, 41).

In the CCSSM, the coordinate plane is introduced in Grade 5. Standard 5.G.2 states “Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.”

 I love problems that are accessible to a wide range of problem solvers. Thanks for sharing! I’m going to re-post this on my own blog www.watsonmath.com.

So, now I’m posting this on my own blog, which has lain fallow for a while.  I’m just starting to get my mojo back and have another post almost ready, which focuses on the Multiplication Machine at MoMath in NYC. Stay tuned!

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A real life geometry problem proposed by Howard Phillips

Howard Phillips, on his blog  “Saving School Math” proposed a real life geometry problem that proved harder to solve than it first appeared. I love to banter about mathematics education with Howard.  We agree on some things about math education and disagree on other things, most likely because his background is in higher ed and mine is in K-12 ed. Nevertheless, the back and forth is always interesting.

Howard hails from from England and was a senior lecturer in mathematics and more at the University of Huddersfield.  He retired from teaching in 2004 and moved to Puerto Rico with his wife, who is the director of the Western Ballet Theater.

I suspect that this real life geometry problem was somehow related to his building a stage prop for the next ballet performance.

Here’s Howard’s problem:

While designing a system for connecting “educational” cubes together I figured that the holes in the faces had to be positioned very carefully. To achieve what I wanted the holes had to be positioned with  length a equal to length b, and length c had to be twice the length a.

So what is length a, as a fraction of the side of the cube ?

geometry from real life
There will be eight holes altogether, and all cubes are the same size.

Howard wrote back and challenged me: “And now without trigonometry…”

I took the challenge. After several messy hand-drawn diagrams, I created the shape on GeoGebra and messed around for a long time, getting nowhere.  I concluded that the length of the side of the square consisted of 6 copies of length “a” plus 2 copies of a length shorter than “a”.  I couldn’t figure out the value of the shorter length nor how it was related to “a” .  I was wondering if somehow phi was involved, but this was late at night, so visions of sugarplums and phi were dancing in my head. It would not make sense for phi to be involved with something so static and “non-growthy”.  Phi is often involved in nature, but rarely in stage props.

To get an idea of how far flung my messing around took me, check out the image below of the GeoGebra mess. This is where I noticed the  6 copies of length “a” plus 2 copies of a length shorter than “a”.  If you look at the horizontal lines cutting the vertical sides of the square, you will see along the vertical sides the 6 equal lengths and the 2 smaller equal lengths. You will also notice the octagon that informed a later solution.

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Howard wrote me back with a BIG HINT.  He used the fact that an angle bisector in a triangle divides the opposite side into lengths proportional to the two sides.  He wrote down the first few lines.  I would like to say that I immediately jumped on his idea and came up with my own solution. However, if I said that I would be lying!  I labored over hand-drawn diagrams.  Then I decided that I would again use GeoGebra to create a a simpler diagram than the one you see above, but one that I could print out and mess around with.

Creating the clean GeoGebra diagram was helpful in allowing me to really wrap myself around the problem.  It made it easier to follow Howard’s approach and finally get the same answer that I got with trig.  Rather than let the side length of the square be x, I set the length of the side of the square to be 1 unit (whatever unit that may be).  This allowed me to not worry about yet another variable.

Here’s my work using Howard’s approach of the angle bisector:

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I had not forgotten about the octagon formed from the 8 holes, so decided to try out a solution using it. I got the same answer as I got using trig and using Howard’s angle bisector: If the side of the square is 1 unit, then “a” = 0.14644 units.

While solving it this way, I was reminded of my discovery the night before.  I had noticed that the side of the square had a length that consisted of 6 copies of length “a” plus two shorter lengths, but I had no idea what the shorter lengths were or how they were related to a.

The relationship became clear when I broke up the top of the square into different lengths. It’s easy to see on the top edge that there are 4 copies of length “a” plus 2 copies of length “(sq rt 2)a”.  Square root 2 is approximately 1.414.  So the 2 copies could be expressed as having length “1.414 a” units, which could also be written as “(1 + .414) a”, which is the same a “1a + .414a”.  We have two of these lengths of “1a + .414a”, so now we have the other 2 copies of “a” to make “6a” and we have the 2 copies of the shorter length, which we now know each have a length of .414 units.

This is a great example of showing how dead-ends like I thought I had during my late night playing around with GeoGebra actually informed my final geometric solution.  Not only that, but my finally geometric solution verified my observation that the side consisted of 6 copies of length “a” plus two shorter lengths.  I figured out the shorter lengths!

Now I am inspired to pull off the cobwebs of a post I started writing in 2013 about a “multiplication machine” at MoMath in NYC.  Thank you, Howard, for helping me get my blog post mojo back!

 

 

Posted in 6 - 8 Math, 9 - 12 Math, Common Core, GeoGebra, Geometry, Ideas from other math blogs, K - 12 Math Tasks, Uncategorized | 1 Comment

At last…a balanced critique of CCSSM

Geoff, in his emergent math blog, has posted a balanced critique of the Common Core Math Standards.

Worth reading!

 

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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.

Posted in 3 - 5 Math, 6 - 8 Math, Fractions | 1 Comment

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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