Archive for the ‘Math Concepts’ Category

The graphing calculator has changed how mathematics is taught and what mathematics is taught in school mathematics (Burrill, Allison, Breauz, Kastberg, Leatham, Sanchez, 2002). Students can experience mathematics in ways that were inaccessible prior to the introduction of the graphing calculator into mathematics education. With the availability of graphing technology as well as other technology, it is important for students to interpret the meaning of computational outputs provided by this technology. It is our contention that many students are not able to provide meaningful interpretations of the computational outputs provided by technology and feel they are engaged in a game of trivial pursuit – a mindless exercise of seeking facts and trivia. To illustrate my point, consider a situation when a student is asked to create a linear regression model and is asked to interpret the meaning of the coefficient of determination ( r^2 ). A typical response would be: “if the value of r^2 is close to 1 then that is “good” and if the value of r^2 is not close to 1 than that is “bad”.  We suggest that this is evidence that the student is engaged in the pursuit of trivia and may respond with a true statement when asked, but, could not explain the meanings involved in the coefficient of determination. Furthermore, our experience tells us that students are often unable to give the details of how the coefficient of determination is computed and as a consequence cannot articulate what this particular statistic is measuring thereby rendering the whole exercise of determining the value pointless.

We believe that an overarching goal of mathematics education ought to be to engage students in the activity of sense making which can lead to a profound understanding of a particular mathematical idea (Ma, 1999). The National Council of Teachers of Mathematics claim that a “high school mathematics program based on reasoning and sense making will prepare students for citizenship, for the workplace, and for further study.” (NCTM, 2009). Furthermore, “a focus on sense making, when developed in the context of important content, will ensure that students can accurately carry out mathematical procedures, understand why those procedures work, and know how they might be used and their results interpreted.” (NCTM, 2009).

In this article, we specifically focus on making sense of the coefficient of determination and suggest that this be an important endeavor for students studying linear functions in an algebra course and not just for students taking statistics. With the wide spread use of the graphing calculator in the algebra classroom, we believe that teachers should be encouraged to help students make sense of all numerical quantities that they are asked to calculate including the coefficient of determination. The importance of statistical literacy is highlighted in a recent article written by NCTM president J. Michael Shaughnessy who says, “statistical literacy has risen to the top of my advocacy list, right alongside numeracy, and perhaps even ahead of “algebra for all”” (Shaughnessy, 2010).

The goal of this article is to present a way that students can be encouraged to make sense of the coefficient of determination while studying linear function models (linear regression models) in an algebra course. We hope that readers might consider the approach taken here and adapt it for students so that they are afforded the opportunity to make sense, and ultimately demonstrate a profound understanding of the coefficient of determination.\

Why Is It Called Regression?

Why is the process of generating a linear function model for a given data set called “linear regression?” During the 1870‟s, Sir Francis Galton studied the heights (he called it stature) of parents and their offspring. He investigated the relationship between the average height of parents that the height of their offspring. What Galton observed and recorded was that the offspring of particularly tall parents were also tall – but not as tall as their parents. The offspring of particularly short parents were also short – but not as short as their parents. That is, the offspring of these parents tended to be less tall or less short – they regressed toward the mean height of the population.

We see this “regression toward the mean” in many real-life situations. If a basketball player scores an extraordinarily high number of points in one game, he most likely will not score as many points in the next game. The number of points will “regress toward the mean” or be closer to the player‟s average number of points per game. The coefficient of determination is computed, and ultimately understood, as we compare data values to the mean (average) of the values in the data set.

Coefficient of Determination

Graphing calculators can be set to output the coefficient of determination, r^2, when computing a linear regression model.


The coefficient of determination is a value that describes the strength of the fit of a linear regression model to a set of data. The stronger the fit, the closer this value, r^2, is to 1. However similar claims can be determined by interpreting the correlation coefficient, r , so being able to understand the coefficient of determination can help one distinguish between these two values. This leads us to ask, how is r^2 computed and what does it mean? We explore a contextual situation in answering these questions.

According to a U.S. Internal Revenue Service review of tax returns and a survey conducted by Indiana University, the total amount of private philanthropy funds increased during the 21st century (Source: Statistical Abstract of the United States, 2006; Table 570). Private philanthropy is the act of donating money, by individuals, corporations, or foundations, to support a charitable cause. For example, many colleges and universities accept private philanthropy to fund scholarships for financially needy students. Another example is the George Carver Academy in San Antonio, Texas that is funded by the private David Robinson foundation. We will compute a linear regression model for private philanthropy data then we explore the computation of the coefficient of determination and discuss its meaning.

The table shows the amount of money donated by U.S. residents, corporations and foundations for philanthropic purposes from 2000 to 2003 (Source: Statistical Abstract of the United States, 2006; Table 570). Algebra teachers often ask students to generate a linear regression model for the data, to interpret the parameters of this model (the vertical intercept and the constant rate of change) in the context of the situation, and to use the model to make a prediction (extrapolate or interpolate).


We use a graphing calculator to compute the linear regression model.


We write this linear regression model F(t) = 4.41t + 226.26 . The model suggests that in 2000 ( t = 0 ), $226.26 billion in private philanthropy funds was given and the amount increases each year at a constant rate of $4.41 billion per year. We can use this model to predict the amount of private philanthropy funds that will be given in 2010. Since 2010 is ten years after 2000, we substitute into this regression model.

F(t) = 4.41t + 226.26
F(10) = 4.41(10) + 226.26 = 270.36

Using the regression model, we predict that $270.36billion in private philanthropy funds will be given. Furthermore, we can see that the coefficient of determination is approximately 0.93. But how is the coefficient of determination value computed and what does it represent? The coefficient of determination is computed by determining the percentage of “error” that is explained by the linear regression model. In this context, error does not mean mistake. Rather, error is a vertical measurement on the graph of the scatterplot with the regression model. More specifically, there are three types of error – explained error, unexplained error, and total error. These error measurements involve the arithmetic mean (average) and are computed by finding the difference between this mean and the actual data values or the values predicted by the linear model. It is important to recognize that the difference between a data value and the mean of the data values may be a positive or negative number. The absolute value of this number
represents the vertical distance (either above or below) between the data value and the mean of the data values. An alternative way to make negative differences into positive values is to square each difference. This is the approach used in calculating the total error. The total error is the sum of the squares of the differences between the actual data value and the mean of the data values.

To visualize the total error, we compute the arithmetic mean and place it on the scatter plot of the data (see the horizontal line below at 232.875).

graph-1To compute the total error, we find the difference between mean and the data point value, as shown in the graph, by subtracting the data value from the mean. We square this difference so that we do not have to consider whether the value is positive or negative. However, we will recognize that differences (prior to squaring) that are negative indicate data values that are below the mean and differences that are positive indicate data values that are above the mean. The total of the squares of these differences is known as the total error.
table-2This total error can be split into two pieces – the explained error and the unexplained error. The explained error is the sum of the squares of the differences between the regression model output values and the mean.

graph-2The sum of the squares of the differences between the actual data values and the regression line output values is known as unexplained error. This is the rest of the total error that is not taken up by, or explained by the model.

graph-3As we examine these graphs, we see that the explained error (total sum of the squares of the differences between the regression line and the mean) seems to be greater than the unexplained error (total sum of the squares of the differences between data values and the regression line). The coefficient of determination is a computation showing the percent of the total error that is explained or “taken up by” the regression line. Recall that the total error is 104.5275. What percentage of this total error is explained? What percentage is unexplained?

table-3table-4The total error is 104.5275. Of this, 97.2405 is explained and 7.287 is unexplained. As a percentage we find that 93.03% of the error is explained. This value is known as the coefficient of determination. This computation confirms the calculator output for r^2.

Making Sense of the Coefficient of Determination

We can go beyond just computing the coefficient of determination to see if it matches with the value that the calculator gives. By examining the computation process, we can see that this value provides useful information about the strength of fit of the regression model to the data set and helps one to further differentiate between r and r^2. We look at two extreme hypothetical situations to make sense of this value. Consider the following hypothetical situation of a perfectly linear data set.

table-5The mean of the y values of the data set is 12.5.

mean-computation-2We see, by looking at a scatter plot of the data, that these data are perfectly linear. Since each data point lies on the regression line, we see that the total error differences (difference between the data value and the mean) and the explained error differences (difference between the regression line and the mean) are the same.

graph-4Since the total error differences (difference between the data point and the mean) and the explained error differences (difference between the regression line and the mean) are the same, the ratio of explained error to total error is 1.
equals-1Also, the unexplained error differences (difference between data point and regression line) total 0. That is, all of the total error is taken up by or explained by regression model. We now consider a situation where two quantities have very little or no relationship.
The mean of the y values of the data set is 2.

mean-is-2In this special situation, the regression model is y = 2; the same as the mean.

ti84-2The total error (differences between the actual data value and the mean) is a relatively large value, as seen in the graph.

graph-5However, the total explained error (difference between the linear regression model and the mean) will be 0 since these two lines coincide.

equals-0By examining these distances, we can estimate the value of r^2 , the coefficient of determination, and do not need to rely solely on the calculator to output the number. Remember, the coefficient of determination measures the strength of fit of the linear regression model to the actual data. The stronger the fit, the closer r^2 will be to 1.

Estimating the Coefficient of Determination

The data in the graph show the number of registered vehicles in the United States for selected years after 1980 (Statistical Abstract of the United States, 2006; Table 1078). The horizontal line in the graph is the mean of the data set. We can estimate whether the coefficient of determination is closer to 0, closer to 0.5, or closer to 1 and explain how we know.

graph-6It appears as though the coefficient of determination would have a value that is close to 1. We see that the total error (difference between the actual data points and the mean) and the explained error (differences between the regression line and the mean) are very similar. The coefficient of determination value, r^2 , is the ratio ratioThis ratio is close to 1 since the values of the numerator and denominator are nearly equal.

The data in the graph show the average winter temperature in New York City for years after 1900 ( The horizontal line in the graph is the mean of the data set. We can estimate whether the coefficient of determination is closer to 0, closer to 0.5, or closer to 1 and explain how we know.


It appears as though the coefficient of determination would have a value that is close to 0. We see that the total error (differences between the actual data points and the mean) will be relatively large due to the fact that the data are so spread out around the mean of 23.38o. However, the explained error (differences between the regression line and the mean) is relatively small. The coefficient of determination value, r^2 , is the ratio ratioThis ratio is close to 0 since the numerator is relatively small and denominator is relatively large.


This article raises several issues. One issue is related to the role that learning about the coefficient of determination ought to play in the algebra classroom. We support the notion that the coefficient of determination ought to be explored fully while students are studying linear function models developed from real-world data. With the widespread use of graphing calculators, this statistic is available to students and therefore should be investigated in terms of its method of computation which should then lead to its interpretation. A second issue is related to the notion that students should be afforded many opportunities to make sense of the mathematics they study. That is, students ought to learn that mathematics is something to be made sense of rather than a litany of trivia that is to be memorized. Students can learn to make sense of mathematical ideas only if they are given many opportunities to practice the mental exercise of doing so. The process of making sense leads to understanding of foundational mathematical ideas. It is our belief that students who strive to understand mathematics will be more successful and progress further in studying mathematics. Hiebert, et al. claim that “understanding breeds confidence and engagement; not understanding leads to disillusionment and disengagement (1999).” We have found that Algebra students will work to make sense of the coefficient of determination in the context of a linear function modeling situation and can demonstrate their understanding of the this statistic by estimating the value of the coefficient of determination given the opportunity and a supporting educational environment and curriculum.


Burrill, G., Allison, J., Breauz,G., Kastberg, S., Leatham, K., Sanchez, W. (2002). Handheld graphing technology in secondary mathematics: Research findings and implications for classroom practice. Texas Instruments

Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Wearne, D., Murray, H., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

National Council of Teachers of Mathematics (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: NCTM

Shaughnessy, J.M. (2010). Statistics for all—The flip side of quantitative reasoning. NCTM Summing Up August 2010 Message from the President. Retrieved August 2, 2010 from /content.aspx?id=26327

“Five percent of the people think;
ten percent of the people think they think;
and the other eighty-five percent would rather die than think.”

-Thomas Edison

I don’t know exactly what Edison meant by this or how he determined the percentages, but the message, to me, is this: thinking is not what people do well!

On a recent Sunday news show, Texas governor Greg Abbott claims that the Common Core State Standards are bad because “it takes a teacher more than a minute to teach a student how to learn that 9+6=15”. He is referencing this video. Check it out.

Let’s think about this. How is this an argument against Common Core?

Is teaching a student how to think about 9+6 = 15 a bad thing? There is a difference between just doing 9+6 and thinking about 9+6. The Common Core Standards clearly have an emphasis on procedural fluency. Students should be able to determine that 9+6=15 quickly. But, how will such fluency happen? It will happen when students have a way of thinking about 9+6. Because it takes a minute to explain this thinking does not mean that it will take a student a minute to compute! Furthermore, the expectation is to push mental math…a good thing! For a student to flexibly be able to see 9+6 as equivalent to 10+5 is very good!

Governor Abbott does not have the opportunity to explain how he believes students should be taught that 9+6=15, but I will make a conjecture that he will argue that students should just memorize it. And in saying this, what would he mean by “just memorize it”? Perhaps something like this (and I embellish):

Teacher: “OK Johnny, this shape right here (9) is called the number 9 and this shape right here (6) which kind of looks like an upside-down 9 is called the number 6. And when you put this symbol in between this (9) and this (6) the answer is two shapes…a 1 which sits right next to a 5. So 9 + 6 = 15”
Johnny: “But why?”
Teacher: “Don’t ask why…just memorize it. Let me get the flashcards.”

We might as well do something like this:

Teacher: “OK Johnny, let me tell you something that I want you to memorize. If you take purple and add dog the answer is banana.
Johnny: “But why?”
Teacher: “Don’t ask why…just memorize it. Let me get the flashcards.”

The teacher in the video is just unpacking the mathematical thinking necessary for a student to make sense of what 9+6=15 means and how this fact might be memorized in a meaningful way. Certainly, we expect students to be able to respond to 9+6 in just a few seconds. This memorized fact will be retained when the mathematical reasoning and thinking, done by the student, allows the student to make meaning and make sense of the computation.

And by the way…this kind of thinking, to allow students to memorize basic math facts, has been around long before the Common Core Standards! This is not a Common Core thing…this is just good teaching.

Diane Briars, president of the National Council of Teachers of Mathematics, says this about the issue: “Teachers have used techniques like splitting a number into parts of 10 for addition–rather than straight memorization–since the 1950s at least, and the research showing its benefits goes back to the 1920s. It has long been best practice for early childhood math.”

I encourage everyone…please read the Common Core Standards and understand their purpose. Then, if you want to debate the issue of the federal government offering federal money tied to the implementation of these standards, great! If you want to debate standardized testing, its cost, its taking instructional time, its impact on the mental health of students, teachers, and parents, great! But please…do not conflate these issues. Standards are standards…teaching is teaching…curriculum is curriculum…testing is testing. My experience is, when people actually consider the Standards, the issue is not the Standards! When they see that students are expected to become effective problem solvers, mathematical thinkers, and fluent in computations, most agree that the Standards are not a problem.

When people try to show that the Standards (not the role of the feds, not standardized testing, etc.) are bad, it is clear that they are not really thinking…



Here is the latest (for me) bashing of the common core standards:

Again, the subject of the bashing is subtraction…I find it interesting that subtraction seems to be taking the biggest hit when it comes to common core bashing!


The blogger says, “Here is the insane method” they came up with to utterly confuse 4th graders about subtraction:”

Insane? Utterly confuse? I think not!

Consider this: Albert Einstein was born in 1879 and died in 1955. How old was he when he died? Think about it…then pay attention to how you thought about it…


Many adults think about this situation like this:

  • Starting with 1879, it takes 21 years to get to 1900.
  • Then, another 55 years takes us to 1955.
  • So, Einstein was 21 + 55 = 76 years old (or would have turned 76) in 1955.

This is the “counting up method” described in the textbook! This method seems SANE and UTTERLY sensible to me!

As I have explained in other posts (Part I, Part II, Part III, Part IV, Part V, Part VI), the main issue is how one thinks about subtraction. If one only thinks about subtraction as “take away” then their understanding is incomplete. We can also think about subtraction as a comparison…which is really what is happening here. We are comparing 1955 to 1879 to see how many years have elapsed. Furthermore, think about the method (yes, it is a method!) described early…first add 21 years…then add 55 years…and compare that method to the standard subtraction algorithm from a “take away” perspective.

CaptureWith the double borrowing, mistakes can be made. In addition, I am a fan of mental math…it can absolutely be a helpful and healthy thing to keep track of quantities mentally!

The goal…and who can really argue this…is for students to become fluent, flexible, and efficient with computations.

  • Fluent in a way similar to becoming fluent in a foreign language…comfort, ease, speed, and confidence are all hallmarks of fluency.
  • Flexible in the sense that a student will use an algorithm if necessary, use a counting up strategy if possible, or other strategies learned in school.
  • Efficient in the sense that a student will choose a strategy that is appropriate for the task at hand. For example, in the Einstein example, I argue that the counting up strategy is more efficient than the traditional algorithm…at least for those who have been trained to think that way!


Subtraction of Mixed Numbers

Posted: April 21, 2013 in Math Concepts

Consider subtracting 5 7/11 – 6 1/2 (five and 7/11 minus 6 and 1/2). Try working through the situation with paper and pencil and then think about how you might represent this problem conceptually.

In a recent exchange with some middle school teachers, this issue arose. How do you represent this situation conceptually? If students understand the meaning of subtraction and the meaning of fraction (mixed number), could they work to develop an algorithmic strategy to subtract mixed numbers? I think so! This video might show how…

Subtracting Mixed Numbers


The standard algorithm for the division of fractions is often one of the procedures that we struggle with finding a conceptual way to make sense of.

Consider this video…when the focus is on the meaning of fraction and the meaning of division, maybe students CAN make sense of the procedure!

Operations with Fractions

Posted: September 22, 2012 in Math Concepts

If you are interested in considering ways to think about operations with fractions so that the operations emerges from powerful ways of thinking about fractions, watch the following:



A teacher asked me the following question. My response follows.

My students are having trouble adding and subtracting signed numbers correctly.   Many had the line “two negatives equals a positive” memorized and applied it universally, which would usually get them the right answers when dividing or multiplying, but clearly doesn’t work with addition and subtraction. How do I get them to remember the rules of basic arithmetic with signed numbers? And are there any resources that might help?

This is a problem that I have experienced frequently! Students struggle since they remember a rule but apply it in the wrong situation. This can be frustrating for students since they might think that they are doing well, remembering the mathematical rules, but then discover that they are applying the rule incorrectly.

The underlying issue in a situation like this is that the students are just remembering rules rather than working to make sense of the rule. Recall that we must focus on developing mathematical practices in our students. These mathematical practices include:

  • Making sense of problems and persevering in solving them.
  • Reasoning abstractly and quantitatively.
  • Constructing viable arguments and critiquing the reasoning of others.
  • Modeling with mathematics.
  • Using appropriate tools strategically.
  • Attending to precision.
  • Looking for and making use of structure.
  • Looking for and expressing regularity in repeated reasoning.

Working with integer operations is a wonderful opportunity for developing these mathematical practices with our students!

I encourage you to take some time to work with your students to help them to make sense of operations with integers using manipulatives. You can use two color chips or any similar objects that come in two different colors, one color to represent positive quantities and one to represent negative quantities. In addition, students will need to think and develop a definition for what it means to add and subtract.

In my experiences, I have found that students often come to the conclusion that addition means to combine quantities and subtraction means to take one quantity away from another. Starting with addition, work with your students to use the manipulatives to model problems like:

  • 5+3
  • 5+(–3)
  • –5+3
  • –5+(–3)

It is critical that students model these situations with the manipulatives so that mental imagery can begin to be formed. Later, when we transition away from the manipulatives, we want students to be able to close their eyes and imagine the situation at hand. This mental imagery will prove to be very powerful in the sense-making process and helping students to become proficient at working with integer operations.

Furthermore, students must realize that a positive quantity and the same negative quantity combine together to make zero. This idea can be developed using the context of money. Spending money can be considered “negative” and depositing money can be considered “positive.” If an equal quantity is deposited and spent, the net effect is a $0 change to the bank account. In my classes, students refer to this idea as “making zero.”

When all of these ideas are in place (meaning of addition, making zero), students are ready to tackle the problems presented. It will be necessary to work through them in sequence as you work to help your students build the mental imagery needed to fully make sense of integer operations. I will share how the last two problems will look as your students build the models with the manipulatives. In this case, the red circles represent negative quantities and the green circles represent positive quantities.


Notice that three of the negative circles (3 negatives) join with three of the positive circles (3 positives) to make zero. In terms of the value of this combinations, we have two negatives (2 red circles) so –5+3=–2. Let’s consider another example.


With the mental imagery formed with activities using manipulatives like these, students are much less likely to make the “two negatives always make a positive” error. It becomes clearer in the minds of our students when they can envision combining (focus on the definition of addition) a negative quantity with another negative quantity and seeing a combination that can be nothing but negative! Clearly, the combination of two negatives with addition cannot be positive!

With subtraction, a little additional work is needed. Recall the definition of subtraction as “taking one quantity away from another quantity.” Also recall that we can “make zero” without changing the overall value of the quantity of items. With this in mind, let’s consider two subtraction problems.

–5 – (–3)

This problem tells us to start with a quantity of negative 5 (5 red circles) and take away negative 3 (3 red circles).

In modeling it this way, it becomes much more clear that –5 – (–3) = –2. Imagine students manipulating objects so that they are physically taking away a quantity of red objects from a larger quantity of red objects. It becomes clear and the mental imagery can be formed.

Let’s look at another situation where we are to take away a quantity that doesn’t seem to be present.

–3 – (–5)

This problem tells us to start with a quantity of negative 3 (3 red circles) and to take away negative 5 (5 red circles). Since we cannot take away more than we have, we take advantage of the idea of “making zero.”

Note that we start with a quantity with a value of negative 3 (3 red circles). By adding two pairs of red/green circles (making zero), the value of the quantity remains negative 3. But now, we can take away the required negative 5 (5 red circles)!

This will help students to see that –3 – (–5) = 2.

If it’s not too late, it is important for students to explore, build mental imagery, make sense, reason, construct arguments, etc. before being handed the traditional rules for operating with integers. In fact, what I have found to be the case is that the students will eventually come to discover the rules for themselves. For example, students may say, “I notice that when I subtract a negative, it has the same effect as adding the opposite of the sign of the second quantity…does that always happen?” Celebrate those moments! Ask the student to share this with the class and ask the class to think about this observation! Discuss it, talk about it, and ask the students to work to confirm it! Once the students all agree to this claim, post it on the wall and give it a name like, “Susie’s Rule.”

As time goes by, students will begin to transition from using the manipulatives to using paper and pencil (they may use ‘+’ and  ‘–’  symbols instead of the color objects). Continue to push students to develop mental math as they imagine the manipulatives or these symbols.

I have a PowerPoint that animates the red and green circle idea that I can send to you if you email I also  have some classroom activities that connect to and extend the ideas presented in this post – again email if you are interested.

In conclusion, focus your classroom activities on the mathematical practices that should be developed in students. Use the context of integer operations to develop these mathematical practices. Make students aware that this is truly what doing mathematics is all about! Avoid any tricks, gimmicks, sayings, or songs that may or may not hold mathematical meaning for students. Rather, build mental imagery and allow students to make sense!

It is a common practice to use patterns involving exponents to show why b^0 = 1 and also why negative exponents such as b^-a give us fractions….

Can we use this same type of thinking to help us understand why  Let’s first take a look at 

To continue the pattern established in Figure 1, 1 times some unknown factor, b, will have to give us a missing number, k, such that when we multiply k by the same factor b we must get 3.

Therefore, since

we have

Reflecting over my years of teaching, I have found that students are challenged by what would seem to be an easy question – “How do we convert from one unit of measure to another?”  When confronted with this type of question, I have come to recognize that many students fall back on relying on a procedure that they try to recall. For example, when asked how many feet 15 yards is equal to many students seem to begin contemplating in their mind “Do I multiply or divide by 3?”. Since they have had many problems of this sort over their mathematical career, are quite familiar with the units of measure, and have enough number sense to realize the answer should be bigger, they generally do see the need to multiply by 3 and arrive at the correct answer of 45 feet.  The obstacle to this problem solving process, however, is that it is not very robust. It will not guarantee them success when they are asked to convert units they are not as familiar with such as in mole conversion in Chemistry or converting area or volume units. I contend that there is a much better option.

As a mathematics educator, we have a powerful way of thinking that can be used in all unit conversions. We need to make it very clear to students that when we convert from one unit of measure to another we are NOT changing the length, area, volume, speed, or quantity of the item under consideration but are simply using another unit of measure to describe it. For example, when we convert 15 yards of rope into feet we are NOT lengthening or shortening the amount of rope.  Therefore, we will be multiplying the measurement by a factor of 1 so that we do not increase or decrease the quantity of rope.  In other words,

We would NOT get the same amount of rope if we multiplied by any other amount other than 1 such as 2.

Therefore when we convert from one unit of measure to another we need to choose the unit of one that will allow us to convert to the appropriate unit we hope to achieve.  The following conversion from yards to feet technically WILL give us the same quantity of rope we had initially because we are multiplying by a factor of 1. However, we do not get the unit of measure we hope for.

We need to choose the unit of one that will do the proper conversion. Let’s try again.

This is incorrect too. With one more try, we see the result in the unit of feet.

The idea of multiplying by a unit of one is very powerful. If we use an example of converting area units we see how the notion can be applied in another context. Assume we are asked to convert 2 square miles (mi2) to square feet (ft2) and we apply this way of thinking. Notice that since there are two mile units multiplied together we have to convert each unit of measure to feet.

Likewise, this can be done with measurements of volume involving cubic units of measure too. Rather than looking at an example of volume however let’s try converting imaginary units of measure to demonstrate how this way of thinking can be used even if we don’t understand what the units of measure are actually quantifying. Given the following equivalent units of measure do the conversion.

For one final example, let’s take a look at a typical Chemistry problem involving unit conversion.

How many moles are in 25 grams of water? We are given that 1 mole of water contains 18 grams of water. We do the calculation.

Returning once more to the problem of converting 15 yards to feet, another equally powerful way of thinking is to envision 1 yard as 3 one foot units (or 1 yard = 3 feet). That is, every yard has 3 copies of 1 foot length segments of rope. So, 1 foot is 1/3 times as large as 1 yard or 1 yard is 3 times as large as 1 foot. Therefore, 15 yards is  times as large as 1 foot or 45 times as large as 1 foot equaling 45 feet.

A visualization for the problem “convert 15 yards to feet”, we might think like this…imagine that the line segment has a length of 15 yards.

We can think of cutting this into 15 pieces of equal length.

Since we cut the length of 15 yards into 15 equal parts, each segment is 1 yard and each of those yards has 3 feet within it.

In this way of thinking, multiplicative reasoning is at the heart of the explanation. Combining this multiplicative/proportional reasoning with a deep understanding of unit conversion creates a more robust understanding that not only serves to help with unit conversion but will help with the big idea of proportional or multiplicative thinking.

In conclusion, as is the case with much of mathematics if we can expose for students powerful ways of thinking then they will have tools they can use in multiple contexts and academic disciplines. If we teach students the concept of the unit of one and the notion that when converting units of measure we are not changing the quantity of an object, students will be well served.



When asked what the average of a set of data means what answer do you think most people (including mathematics teachers) give? Our experience from leading many professional development workshops and in classroom instruction is to hear a response such as “you add up all of the numbers and divide by how many numbers there are”. Well, that is true if the question had been “How do you calculate the average of a data set?” but is not the answer to what the average means. The response typically given describes the process of how to find the average NOT what the numerical value represents in a real world context.

If we use as an example the average G.P.A. of the students in our classroom, most of our students know that the process for finding the average G.P.A. is to total up all of the G.P.A. points of the students in the classroom and then divide that number by the total number of students in the class.  Once we get a result like 3.12 the question becomes what does that number mean? The meaning is closely linked to the calculation process but is not the same thing. When finding the average we find the total number of G.P.A. points (envision all of the points accumulated on a desk in the classroom) and then divide them up equally (think about handing one G.P.A. point to one student, then the next student, etc. until every point has been physically handed out). We can now clearly see what the average G.P.A. of 3.12 means –  if every student in the class had the same number of G.P.A. points they would posess 3.12.

Likewise, the average shoe size of students in class would be the size of shoe every student in the class would have if every student in class had the same size shoe. Note that the average does not have the same meaning as the mode (the most frequent shoe size) or the median (the shoe size directly in the middle).

How does having a thorough understanding of average help one more clearly make meaning of the average rate of change? The average rate of change truly is an average but it is the average of a change!  We calculate the average rate of change by determining the total change in the ouput value and dividing by the total change in the input value. This results in a value that describes how the output values would change for every change of the input value if the output values changed by the same amount for each change in the input

Let’s consider calculating the average rate of change in the population of Austin, Texas. In 2009 the population was 1,705,075 and is anticipated to be 1,730,267 in the year 2015. To calculate the average rate of change we find the total change in population 1,730,267 – 1,705,075 = 25,192 and the total change in time 2015 – 2009 = 6 years. We then divide the total change in population by the total change in time which is 25,192 people/6 years =  4199 people/year. This means that the population of Austin will increase by 4199 people each year IF the population changes by the amount each year.

Understanding the concept of average is fundamental to the proper understanding of average rate of change in Algebra and Precalculus (and subsequently the notion of derivative in the Calculus series).