Why Use Visual Models before Algorithms?

In Common Core, many standards refer to using visual models such as arrays or area models. For example, students in Grade 4 are expected to use place value, arrays, or area models to multiply 2-digit by 2-digit numbers. In Grade 4, students are not required to know the standard multiplication algorithm. This algorithm is delayed until Grade 5. Here is a visual model for multiplying 17 by 14. Notice how the area grid has four parts. By finding the area of each part and then adding, students can find the product without learning a new algorithm.

I feel that requiring students to understand visual models before using algorithms is extremely important for three reasons.

• Many students don't slow down to think about problems. Once they know a shortcut, they just try to get answers quickly. Visual models help students learn that answers need to make sense!
• The visual model is a bridge to understanding the algorithm. With 2-digit by 2-digit multiplication, the model helps students see that both tens and ones need to be multiplied and then added.
• Students often have difficulty with word problems. Drawing visual models for problems helps students make the connection from the word problem to the related operation.

Although drawing and using models requires time and effort, the result is that students will develop a deeper understanding of important concepts.

Generalizing Area as "Average" Length x Width

Why are there so many different formulas for area? Many students become dependent on applying formulas, but they forget the formulas soon after the chapter test.

In tutoring, I emphasize that area is ALWAYS related to length by width. I encourage students to think about the middle or average length of a shape. Since the area is a measurement of square units, segments representing length and width must always be perpendicular to each other. For consistency, I have students consider the length as the horizontal measurement, and the width as the vertical measurement (which can also be called the height).

Here are some examples.

Consider parallelograms and trapezoids, as shown by the blue shapes above. If you draw a line segment across a parallogram, parallel to a base, that segment will always be the same length as the base. So, it can be considered the "average" length. For a trapezoid, the middle segment is the same as half of the sum of the two bases. For triangles, as shown in purple above, the middle length is half of the base.

I often have students identify the length and width of a rectangle by counting the squares on graph paper. Then I  extend this activity to identifying and labeling the middle segment of a parallelogram, trapezoid, or triangle. In this way, students are better able to understand the concept of area of these types of shapes and may not need to memorize individual formulas.

In Common Core, students in Grade 2 are expected to partition rectangles into square units (2.G.2). In Grade 3, students count squares to find the area, and relate area to multiplication (3.MD.6 and 3.MD.7). In Grade 4, students calculate areas of rectangles (4.MD.3). In Grade 6, students find area of other polygons (6.G.1 and 6.G.4).

Please write a comment if you have your own special methods for teaching area.

Another Round of Pi

Some of you may have discussed pi with students on March 14, often called Pi Day. Are you ready for another look at that special number that is very close to 3.14?

Many students are confused by the formulas for circumference and area of a circle. In CCSS, circle relationships and formulas are to be taught in Grade 7, standard 7.G.4. To help my students with circumference and area, I've developed a visual approach.

First consider the circumference or distance around a circle. I ask leading questions to help students estimate the circumference as a multiple of the length of the diameter, based on these diagrams.

For area, I help students relate the area of a circle to the square of the radius.


I help students see that when the diameter of a circle is 1, the circumference is pi! And, if the radius is 1, the area is pi! If students visualize the relationships between parts of circle, they are more likely to develop an understanding of the formulas.