It's Not a Window Pane... It's an Area Model

By Cassandra Hatfield, RME Assessment Coordinator

As students develop an understanding of multiplication, instruction often moves from an equal sets into an area model. In my experience as a math specialist, sometimes an area model was called a “window pane.” In this blog we will focus on the conceptual understanding of an area model and the need to shy away from calling it a “window pane.”

Just last month, I was visiting my nephew who was very proud that he had memorized several multiplication combinations and he asked me to quiz him. He was giving me answers at a rapid pace until I got to 8 • 7. His response, “I don’t know that one yet!” He had just told me what 8 • 5 was and what 8 • 2 was. However, my nephew was simply memorizing combinations and did not have a strategy to compose the combination with other combinations he knows. The area model is a powerful way to assist students in composing combinations.

When students know and understand combinations through 10 • 10 then they can decompose numbers to find any other combination.

I’ve been in classrooms and seen anchor charts titled “Window Pane Strategy” with an example of finding the product of a 2 digit number by a 2 digit number. However, notice that prior to this area models did not look like window panes. The models shown are proportional. The units are squares, so in the problem 12 • 8 the factor 12 is longer than the factor 8. When teaching students to multiply 2 digit by 2 digit numbers before transitioning to an open area model (without the grid lines) it is extremely important for students to make the connection to their prior understanding by using a model with grid lines. Using an area model is not a procedure for finding an answer; it’s a conceptual understanding of the distributive property.

Even after moving to an open area model, while students do not need to measure to make the parts perfectly proportional, it is important that students still draw the open area models to show that each of the parts is a different area. The open area model can continue to be used to support students in multiplying larger numbers.

In another post next month, I’ll share how the area model connects to the standard algorithm and can be a powerful model to support student understanding of the standard algorithm beyond a procedure.

After High School: What's Next?

By Elizabeth Howell, RME Research Assistant

High school graduation is certainly cause for celebration! All of the hard work, studying, and preparation have paid off. Students have taken mathematics courses and learned advanced algebra skills, geometry, and maybe even more. So what’s next?

In Texas, students pursuing a postsecondary degree require a stamp of college readiness. There are many ways that a student can demonstrate college readiness: SAT/ACT scores, TAKS scores, math dual credit courses, to name a few.

But what if a student's SAT/ACT or TAKS score in math wasn’t so hot, and the student never took a dual credit math class in high school? Without one of the many approved exemptions, a student will be required to take a placement test upon enrolling in college. This test will be used to determine if a student is indeed college ready in reading, writing, and mathematics. And the results will determine which classes a student can enroll in for their first semester.

Most placement tests will be administered via a computer, and there is typically a fee associated with taking the test. Common placement tests are Accuplacer, Compass, and THEA. Each of these tests will have a reading, writing, and mathematics component in order to assess a student's skill level before enrolling in classes. Each test has a pre-set cut off score. Any student that does not meet the cut off score will be required by state mandate to enroll in remedial or developmental classes.

Approximately forty-one percent of students in Texas higher education require remediation upon entering college (THECB, 2013). Remedial classes have homework, tests, and grades, just like any other class. They cost tuition dollars, just like any other class. BUT…they do not count toward any degree! Remedial or developmental classes are designed to reteach the material that high school mathematics courses should have taught, and they are required if a student's math placement score is not passing. Completion of the remediation specified by a student's test score is required to enroll in credit mathematics courses.

Sadly, students that require developmental education are far less likely to graduate from a college or university (Morales-Vale, 2012). Developmental courses delay degree completion, cost tuition dollars, and can be a major roadblock to a student’s academic goals.

So how can student's avoid developmental courses?

1. Take high school courses seriously. The reading, writing, and mathematics skills a student learn sin high school is critical to college success.
2. Take the placement test seriously. If a student is required to take a placement test for college, emphasize  that reviewing notes and looking at practice questions is critical. The importance of the test cannot be overstated. The placement test will determine the academic path a student will start on, and being on the right path is crucial.
3. If remediation is needed, take the remedial classes seriously. These classes are designed to improve academic skills, but sadly many students do not realize the importance of these classes because they think that they do not count. In a sense that is true, these classes do not transfer or count toward a degree. But, remedial classes can be the gatekeeper between a student and the degree they want -  because not completing them successfully means that a student cannot move on to the classes that DO count toward their desired degree.

In Texas colleges and universities, far too many students end up in developmental coursework. College readiness is a demonstrated skill, and students' need to take the initiative to brush up on skills before taking a placement test. Practice versions of many tests are available online for free. Make sure students talk to a high school counselor or a college advisor if they have concerns. In addition, have them visit websites dedicated to college readiness such as http://gentx.org for resources, hints, and checklists to help transition successfully from high school to college.


Morales-Vale, S. (2012). TSI and developmental education updates. Presented at CRLA/CASP Convention, November 8, 2012, Austin TX. 

Texas Higher Education Coordinating Board (THECB). (2013). Developmental education/Texas success initiatives. Retrieved May 10, 2013 from http://www.thecb.state.tx.us/index.cfm?objectid=233A17D9-F3D3-BFAD-D5A76CDD8AADD1E3.

The Pythagorean Relationship

By Saler Axel, RME Research Assistant 

Math has a reputation of being dull. Luckily, there are some fun math holidays that exist throughout the year. Two popular ones are Pi Day (3/14) and Mole Day (10/23). Last week was 5-12-13 Triangle Day! What makes the 5-12-13 right triangle worth celebrating?

Let’s spend time considering special right triangles, which are some of geometry’s extraordinary shapes. A right triangle contains sides lengths that can be calculated using the Pythagorean Theory, a² + b²= c². We will spend time discussing right triangles like the 5-12-13 right triangle.

Side-based special triangles, such as a 5-12-13 right triangle, contain proportionate side lengths that make computing easier. Called Pythagorean Triples, these triangles contain angles with degrees that are never rational numbers. If students understand the relationships of a special right triangle’s side lengths, they can calculate other side lengths in geometric problems without having to employ difficult strategies.

An easy way to calculate Pythagorean Triples: a = m²– n², b = 2mn, c = m²+ n². where m and n are relatively prime positive integers and m>n.

Below are some things that you can do in your classroom to celebrate this extraordinary shape.

• Challenge students to calculate scaled examples of 5-12-13 triangles.

• Draw a 5-12-13 right triangle on grid paper. (An example of a 3-4-5 triangle is below.) Have your students make a square from each side. The diagram should have a 5•5 square on the left, a 12•12 square on the bottom, and a 13•13 square off of the hypotenuse. Encourage your students to measure the number of square units. They will discover that 5²+ 12²= 13². Then ask your students to try the same activity with an isosceles triangle (or any other type of triangle except a right triangle). This will help them understand that if they measure the squares, the sides will not make a right triangle.
• Here, the two squares together are a "proof without words." Here we see that:
  
  a² + 2ab+b²= c²+ 2ab

a²+ b²= c²

Other common Pythagorean Triples include those with side length ratios of: 3-4-5, 8-15-13, 7-24-25, and 9-40-41, though the possibilities are endless using the formula (3n)²+ (4n)²= (5n)². For an extensive list of Pythagorean Triples, visit www.mathisfun.com/numbers/pythagorean-triples.html.

How can you tailor these and other classroom lessons to expand your students’ thinking about special right triangles and their importance in geometric calculations?

Teaching Fractions Beyond Pizza and Pie Charts

By Sharri Zachary, Mathematics Research Coordinator

Research shows that “algebra is the [career] gateway to success for many students” (Williams, 2011, pg. 1). For students to understand the concepts and symbolic representations for everyday situations in algebra, students must master certain foundational skills and concepts at the elementary and middle school levels. According to the U.S. Department of Education (2006), a solid foundation in math in early grades will assist in the development of critical thinking skills necessary to pass algebra. The National Mathematics Advisory Panel (2008) identifies the skills that demonstrate algebra readiness as: (1) whole number computation, (2) fraction and decimal proficiency, (3) number concepts (including percents), (4) general concepts, proportions, and geometry, (5) problem-solving, and (6) basic understanding of integers, variables, and simple equations.

Fraction and decimal proficiency is often developed with the use of pizza and pie charts as the introductory piece. Visually, this allows students to see one “whole” partitioned into equal parts, which in turn, helps students understand the part-whole idea of fractions. However, this leaves out the idea that fractions are numbers with magnitude that can be compared (Siegler et al, 2010).

If students are to be truly algebra-ready, visual representations must extend to model fraction concepts that teach the part-whole relationship, as well as, fractions as a distance and magnitude. The IES practice guide recommends the use of number lines as the “central representational tool” to help students recognize fractions as numbers and expand student thinking beyond whole numbers (Siegler et al, 2010, p. 19).

The use of number lines can help students visualize and understand the magnitude of fractions, the relationship between fractions and whole numbers, and the relationship between fractions, decimals, and percents (Siegler et al, 2010). This conceptual understanding is foundational to understanding algebra (Siegler et al, 2010). Based on the recommendations put forth in the IES practice guide, teachers should center fraction instruction around the number line model and support that instruction with other models that include (but are not limited to) fraction circles and strip diagrams. If number lines are recommended as the “central representational tool” for fraction and decimal proficiency, then teachers must consider moving beyond pizza and pie charts in their instruction and assessments to prepare students for mastery of this component of algebra readiness.

National Mathematics Advisory Panel (Spring 2008). Final report, Washington, D.C. 

Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade (NCEE 2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/practiceguide.aspx?sid=15 

U.S. Department of Education. (2006). Math now: Advancing math education in elementary and middle school. February 2006. Retrieved from: http://www.ed.gov/about/inits/ed/competitiveness/math-now.html 

Williams, T. G. (2011). Reaching algebra readiness (RAR): Preparing middle school students to succeed in algebra – the gateway to career success. Rotterdam, The Netherlands: Sense Publishers.

PEMDAS and the Number-Gobbling Dragon

By Erica Simon, RME Project Specialist

In our digital world, creativity and a little storytelling can be a powerful mnemonic to understanding a basic principal of mathematics that all students need to be successful. Many students love to watch YouTube videos, and now, the new TED Ed site has organized these educational videos to allow teachers to find them and "flip" the videos into lessons. These videos are displayed on lesson pages already created with multiple-choice quizzes, open ended questions, and links to more information about the topic. A teacher can create her own lesson with any YouTube video by simply "flipping" it!

So how does this work? Let's look at a mathematics content example. PEDMAS – the well-known acronym for parenthesis, exponents, division, multiplication, addition and subtraction, is taught as students are introduced to problem solving strategies for equations and word problems. Although many a “trick” has been used to help students understand the order of operations, the clever folks at TED Ed(ucation) have used the power of animation and storytelling to deliver a swashbuckling tale of the Land of Pi where the numbers run wild. But only with the sequential attack of the mathematical symbols for the operations, are the numbers tamed and “order” is restored. Puff, the number-gobbling dragon is squashed by the symbolic musketeers who attack with precision and order to save the Land of Pi.



After students watch the video, there is a lesson already created if a teacher doesn't create her own. There are five multiple-choice questions and three open ended questions, including one where students must work to simplify an expression using PEMDAS. Last there are three resources where students can "dig deeper."

The IES Practice Guide for Problem Solving in Grades 4-8 provides recommendations that support procedural knowledge and flexibility as students develop skills to efficiently and correctly solve math problems. By tapping students’ prior knowledge of mathematical operations and supporting a solid understanding of the order in which operations are used, teachers help students become comfortable with symbols and correctly use them in their problem solving.

By sharing this whimsical approach to the order of operations with your class as a support for honing procedural knowledge, conceptual understanding and procedural flexibility, students will see mathematical rules come to life in a way that exemplifies why the order of operations matters when solving math problems.

Let us know what you think and how you could use TED Ed videos in your classroom.

Help Your Students Experience Fractions Conceptually

By Dawn Woods, RME Elementary Mathematics Coordinator

Many students find fraction concepts difficult to understand yet the understanding of fractions is essential for learning algebra and advanced mathematics (National Mathematics Advisory Panel, 2008). As an elementary mathematics educator, I noticed that many of my students struggled with fraction concepts across the curriculum. I wondered how I could help my students experience and understand fraction concepts conceptually so they could succeed not only in my classroom but also in advanced mathematics.

My search for answers began with research. I discovered that to understand fractions means to recognize the multiple meanings and interpretations of fractions. Furthermore, I needed to explicitly present these different constructs in a contextual way to build understanding. Mathematics educators generally agree that there are five main fraction constructs and that they are developmental in nature.

The first construct presents fractions as parts of wholes or parts of sets. Research suggests that this construct is an effective starting point for building fractions (Cramer & Whitney, 2010). However, it is important to realize that the part-whole relationship goes way beyond the shading of a region. For example, it could be part of a group of animals such as (¼ of the animals are dogs), or be part of a length, (we ran 1 ½ miles) (Van De Walle, Karp, Bay-Williams, 2013).

Researchers such as Cramer, Wyberg, and Leavitt suggest that the fraction circle manipulative is a powerful concrete representation since it helps to build understanding of the part-whole relationship as wells as the meaning of the relative size of fractions (2008). Here, they use fraction circle models to help build mental images that aid in the ability to judge relative sizes of fractions. It is also important to remember that the fractional parts do not need to be identical in shape and size, but must be equivalent in some other attribute such as area, volume, or number (Chapin & Johnson, 2006). However, it is important to teach beyond this first construct to include other fraction representations and models.

The second construct presents fractions as measures. Measurement (Van De Walle, Karp, Bay- Williams, 2013) involves identifying a length and then uses that length to determine a length of an object. The number line plays an important role in this construct by partitioning units into as many subunits that one is willing to create (Chapin & Johnson, 2006). For example, in the fraction ¾, you can use the unit fraction ¼ as the selected length and then measure to show that it takes three of those to reach ¾ (Van De Walle, Karp, Bay-Williams, 2013). Research suggests that students who develop an initial understanding of rational numbers as measures, develop ideas of unit, partitioning, order, addition and subtraction (Cramer & Whitney, 2010) while using the number line as a model. Essentially, this powerful construct illustrates that there are an infinite number of rational numbers on the number line as it focuses on how much rather than parts of a whole.

Fractions can also result from dividing two numbers. This construct is often called the quotient meaning, since the quotient is the answer to a division problem (Chapin & Johnson, 2006). Think about the number of cookies each person receives when 15 cookies are shared between 3 people. This problem is not a part-whole scenario (Van De Walle, Karp, Bay-Williams, 2013) but it still means that each person will receive one-third of the cookies expressed as ¹⁵⁄₃, ⁵⁄₁, or 5. Connecting division to fractions enables students to feel comfortable with seeing division expressed in multiple ways such as 16 ÷ 3, ¹⁶⁄₃, and 5¹⁄₃ and is important for continued success in advanced mathematics.

The fourth construct presents fractions as operators. In this construct, a fraction is a number that acts (or operates) on another number to stretch or shrink the magnitude of the number (Chapin & Johnson, 2006). For an example, a model of a car may be 1/16 the size of the original or a cell maybe magnified under a microscope to 400 times the actual size demonstrating a multiplicative relationship between the quantities. This construct takes fractions beyond representation to a place where students know how to use fractions to solve problems across the curriculum.

The fifth and final construct characterizes fractions as the ratio or comparison of two quantities. A ratio such as 1/3 can mean that the probably of an event is one in three (Van De Walle, Karp, Bay-Williams, 2013). Or a ratio can also represent part-whole relationships such as 11 children at the park compared to the total number of 18 people. We could write this part-to-whole relationship as the fraction ¹¹⁄₁₈. However, it is important that realize that all fractions are ratios but not all ratios are fractions (Chapin & Johnson, 2006). Part-to-part comparisons such as the number of children to the number of people at the park, 11:18, is not a fraction because this comparison does not name a rational number but presents a comparison of two numbers.

Fraction understanding, although a challenge to students, is a critical mathematics concept. For students to really understand fractions, they need to experience fractions across all five constructs in meaningful ways that build conceptual understanding. This conceptual understanding, in turn, provides students with mental representations that enable students to connect meaning to fractions across a variety of contexts.

Chapin, S.H., & Johnson, A. (2006). Math matters, 2nd edition. Sausalito: Math Solutions Publications.

Cramer, K., & Whitney, S. (2010). Learning rational number concepts and skills in elementary school classrooms. In D.V. Lambdin & F.K. Lester, Jr. (Eds.), Teaching and learning mathematics: Translating research for elementary school teachers (pp. 11-22). Reston, VA: NCTM

Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School 13(8), 490-496.

National Mathematics Advisory Panel (2008). The final report of the national mathematics advisory panel. Jessup, MD: Education Publications Center. Retrieved from http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013). Elementary and middle school mathematics: Teaching developmentally, 8th edition. Boston: Pearson.