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?