Steps for Solving Equations

In CCSS, the description of Standard 8 for Mathematical Practice says, "Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts."

This week, I was tutoring an 8th grade student who was very bogged down when solving equations. Her teacher, like many others, was requiring students to "show all of the work." Here is an example of work that the student was showing to solve the equation 7x – 17 = 11. Notice that her first step was to write +17 under each side and draw lines to add. My concern is that this step detracts from key idea of equation solving, which is to write equivalent equations that become simpler and simpler until the solution is clear.

In most textbooks, the second line above would be shown as an equation:
7x – 17 + 17 = 11 + 17
This is OK since it is an equivalent equation, but the goal should be for students to do more steps mentally, as follows (in black text):

When I encourage students to use shortcuts and show only key steps, they usually respond positively. They are glad to try to do more of the work mentally, and it is easier for them to see the progression to simpler and simpler equivalent equations. As a result, students become more confident and their homework papers look neater!

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.

How Dot Arrays on Grids Cause Confusion

A few days ago during math tutoring, a 6th grader was having trouble counting the length and width of a rectangle on a grid. When shown a rectangle 4 units wide and 2 units long, she said the area was 15 square units! The error was caused by counting the intersection points instead of the spaces. Consider the following figures.

Some curriculum materials include practice with making dot arrays on a grid for multiplication. Students who make these arrays may then be confused about finding the length of a segment on the grid. In Common Core, students are expected to draw and count arrays of squares. This is a better approach because students are less confused and it leads perfectly into the concept of area.

Simplified Metric Conversions with Common Core

Recently, one of my 7th grade students needed help with metric conversions. The teacher had given her a "staircase" diagram. From upper left to lower right, the steps were marked kilo, hecto, deca, (base unit m/L/g), deci, centi, and milli. The student was told that to convert by counting steps and then moving the decimal point. For example, meters to kilometers is three steps to the left, so move the decimal point three places to the left.

Even though using this type of diagram can lead to correct answers on homework, there are two reasons why this method troubles me.

• Students are dependent on the diagram. Most students will not remember how to draw and label it if they need to convert metric units a week, month, or year later.
• Many of the metric prefixes on the diagram are rarely used! In fact, the only prefixes needed for high school courses and tests such as ACT and SAT are kilo, centi, and milli.

I'm very glad that for CCSS, students are expected to convert measurements using two-column tables. In Grade 4, students convert from larger units to smaller units. In Grade 5, students convert in either direction. The units mentioned in the standards for Grade 4 are kilometer, meter, centimeter, kilogram, gram, liter, and milliliter.

To convert a measurement, first set up a table with headings for the two units. Under the correct headings, record the relationship and also the measurement to be converted. Here are two examples. The table at the left is set up for converting 12 liters to milliliters (grade 4). The table at the right is set up for converting 15 millimeters to centimeters (grade 5).

From tables like these, it is easy for students to see whether to multiply or divide to convert the measurements! Writing relationships in tables is also excellent preparation for variables and functions in higher grades.