Category Archives: numeracy

What’s in a Name? Having Fun with Data

by Sarah Lonberg-Lew

My name is Sarah. (If you don’t believe me, check the byline.) On my first day of high school, I met four other girls named Sarah. It has often been my experience to not be the only Sarah in a group, especially a group of people close to my age. I’ve had to add descriptors to my name to be able to be uniquely identified. I’ve been “short Sarah,” “Sarah short hair,” “Sarah L,” and on more than one occasion I’ve been “math Sarah.” If you also have a common name, you can probably relate.

Just how common is my first name? There is a fantastic internet tool that can answer that question for me with a wealth of interesting data. It is WolframAlpha, the computational search engine. If you are looking for quantitative information of any kind on the internet, this is a great place to look. One exciting thing this site can do is provide current and historical data about the popularity of a name. For example, here are some things I learned about my own name:

  • About 1 out of every 303 people in the United States is named Sarah.
  • There are about 909,700 Sarahs living in the United States right now. That means I may not be one-in-a-million, but I’m close!
  • The most common age of a Sarah in the United States is 30.

WolframAlpha also showed me data that helped shed some light on why I so often have to qualify my name in large groups. This graph shows the popularity of my name over time:

When I was born, my parents didn’t know that my name was getting more popular. Looking at this graph, I wonder if a lot of other parents might have been similarly unaware. Can you guess about when I was born?

The same site also provided a graph showing the age distribution of people with my name in the United States:

What do you notice about the two graphs? What do you wonder?

There are many mathematical avenues to explore within the name results at this website. Beyond looking up your own name or your students’ names, you can also compare multiple names. For example, here’s a graph that compares the popularity of my name with the name Donna (the name of the esteemed director of the Adult Numeracy Center):

What do you notice about this graph? What do you wonder about? One thing I wonder about is whether the Ritchie Valens 1958 hit song “Oh Donna” had anything to do with the name’s popularity. What do you think?

The trends in naming babies can reflect what is going on in the world as well as what is popular in music, literature, movies, or TV shows. Here’s a striking example – this graph shows the historical popularity of the name Barack:

One interesting thing to note on these graphs is that the scale on the y-axis is not always the same. The peaks for the graphs of Sarah and Donna are at over 1.5% of babies born in a given year. The peak for Barack is at about 0.0035% of babies born that year.

A few notes about exploring names with WolframAlpha
  • It’s important when looking at data to know where it comes from. WolframAlpha gets these numbers from the Social Security Administration.
  • If you have an uncommon name, WolframAlpha may not recognize that you are looking for name data. You can fix this by adding the word “name” to your search.
  • WolframAlpha may make assumptions about your search term that affect the results. The site lists the assumptions it made in a blue box at the top of the results. You can modify those assumptions if you need to.
How you can use this resource with your students
  • Have them look up their own names or compare their name to a classmate’s. Are they surprised by what they find? Or does it confirm their lived experience?
  • Do you have students who are thinking about baby names? They may want to look up the names they are thinking about to see how common or unique certain names are.
  • Challenge students to draw connections between the different ways that information is presented in the search results. For example, WolframAlpha told me that the most common age of a Sarah in the US is 30. Where can I find evidence for that in the graphs?
  • Click on the “more” button at the top right of the history graph to see even more graphs. What does each graph show? Why are the shapes similar or different? What is the y-axis measuring on each graph?
  • Ask your students what they see in their search results. They may surprise you!

There’s so much math you can explore with this engaging and personally relevant data. Clicking on links in the results may also bring you to interesting (although possibly not useful) information. For example, I discovered that the combined weight of all the Sarahs living in the U.S. is about 63,682 metric tons! What will you discover about your name? Go have fun!


Sarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency. Sarah’s work with the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center at TERC includes developing and facilitating trainings and assisting programs with curriculum development. She is the treasurer for the Adult Numeracy Network.

Will This Be on the Test? #5

by Sarah Lonberg-Lew

Welcome to the first installment in 2021 of our monthly series, “Will This Be on the Test?” Each month, we’ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually. (You can find links to the earlier installments in the sidebar list on the right.)

I learned a lot of rules in math class, and when I started teaching math, I taught a lot of rules. It was easier to teach students to memorize the rules rather than help them fully understand the concepts (especially when I didn’t always understand the concepts myself). When I taught math to high schoolers, I remember teaching a chapter on the rules of exponents. There was a separate section of the chapter for each rule and I taught them one at a time, so that students could memorize and practice applying each rule before moving on to the next. We spent at least a week on that chapter, covering a new section each day. For some reason, though, everything always fell apart in the last section of the chapter where the problem types were all mixed together and students had to figure out which rule to apply or sometimes whether to apply more than one rule.

There was a time in my teaching career when, if a student had asked me about the problem below, I would have said, “You have to know the rules to be able to solve this.” I don’t say that anymore. How about you? Could you figure out a way to approach this without knowing “the rules”?

Before you read further, take a few minutes to challenge yourself to come up with more than one way to approach this task. How many strategies can you think of? What visuals could you use to help you solve this? Also ask yourself, what skills and understandings do students really need to be able to answer this?

Before we get into some different approaches, let me be clear that if students do not have a grasp on the idea that a letter can stand for a number, then this problem is most likely out of reach and they should not spend time on it. However, if a student has that concept and knows what an exponent means, they do not have to have memorized a set of rules to be able to reason through this. (And while letters representing numbers is an abstract concept, it is an understanding that students can begin building early on.) Here are some possible approaches.  

1.Try it with a number. Just because the problem is written with variables doesn’t mean it has to be solved that way. One important understanding about equivalent expressions is that they will have the same value no matter what the value of the variable is. This means that a student can change x to a friendly number, like 2, and work out a problem like the one in our example:

24 • 25

16 • 32

512

Now the student only needs to figure out which answer choice has a value of 512 when x = 2.

Note: Even with a small number like 2, the value of the expression is going to be pretty big, so students might want to use a calculator. Choosing 1 is not a good idea. Why is that? Could your students see why?

2.Write it out the long way. A student who knows the meaning of an exponent can write equivalent expressions to x4 and x5 like this:

x4 = x • x • x • x

and

x5 = x • x • x • x • x

The student can now multiply the longer expressions together like this:

Now the student can turn their answer back into an exponential expression to match one of the answer choices.

3. Figure out the rule on the fly. One of the most powerful tools in a mathematician’s toolbox is being able to figure out how complicated things work by thinking about simpler things. A student who is empowered to think of themself as a mathematician may realize that they have the power to discover or recreate the rule by looking at simpler cases. A student might do a few quick experiments with expressions similar to the one in the problem to find out what is going on:

  23 • 22 = 2 • 2 • 2 • 2 • 2 = 25 (Check by hand or with a calculator – that works!)

23 • 23 = 2 • 2 • 2 • 2 • 2 • 2 = 26 (Check by hand or with a calculator – that works, too!)

Hmmm… the exponents tell me how many 2s will be in the long expression. It looks like adding the exponents gives me the total number of 2s, so I can add the exponents to get the final exponent.

The rules in math were not made up by some high mathematical council that decided one day to confuse generations of students by making the rule that when you multiply expressions with exponents you add the exponents. You add the exponents because it makes sense to do so when you understand how exponents work. The reason that students jumble up the rules or misapply them is that rules are abstract concepts and abstract concepts must be understood, not memorized, to be useful to students. Teaching rules without reason, especially when those rules are not what you might guess at first, disempowers students because they just have to accept what they are told even if it doesn’t make sense to them. Teaching that rules are formalizations of reasoning and patterns empowers students to discover rules and even solve problems without them — and they are freed from the burden of relying solely on memorization.


Sarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency.

Sarah’s work with the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center at TERC includes developing and facilitating trainings and assisting programs with curriculum development. She is the treasurer for the Adult Numeracy Network.

Will This Be on the Test? #4

by Sarah Lonberg-Lew

Welcome to the fourth installment of our monthly series, “Will This Be on the Test?” Each month, we’ll feature a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually. (You can find links to the earlier installments in the sidebar list on the right.)

This month let’s dive into one of my favorite types of problem to see on a test – an algebra story problem. I love this kind of problem because I think it’s accessible to almost anyone who has the confidence to play with it. The context makes it accessible. I always encourage my students to look for those problems where there is a story they can understand. With that toehold, they often can figure it out. What about you? Can you figure out this month’s challenge without using procedures you memorized in algebra class?

Before you read further, take a few minutes to challenge yourself to come up with more than one way to approach this task. How many strategies can you think of? What visuals could you use to help you solve this? Also ask yourself what skills and understandings do students really need to be able to answer this?

When you’ve given yourself enough time to really explore, read on to see some possible ways students might approach this problem using conceptual understanding and visuals.

1.A student might reason (proportionally!) that if 4 pencils and 4 erasers cost $3.20, half as many pencils and erasers would cost half as much. That means 2 pencils and 2 erasers cost $1.60. Now the student might compare the cost of 2 pencils and 2 erasers to the cost of 3 pencils and 2 erasers (a picture really helps):

If the only difference between the two pictures is a single pencil and the difference in price is 30 cents, the pencil must cost 30 cents! But wait – the question didn’t ask for the cost of a pencil! This is a great start, but it’s always important to make sure you’ve finished the problem, especially on a test. Once they know the cost of a pencil, the student would have several options for figuring out the cost of an eraser.

2. Alternatively, a student might reason that they could figure out what 1 pencil and 1 eraser cost together by dividing the cost of 4 pencils and 4 erasers by 4. The combined cost of a single pencil and eraser is $0.80. Knowing what they cost together means that you can find the cost of one if you know the cost of the other. This paves the way for a pretty efficient guess-and-check.

  • GUESS 1: If the eraser costs $0.10, then the pencil costs $0.70. This means that the cost of Jason’s purchase would be 3($0.70) + 2($0.10) = $2.40. That’s way too much. (Jason only spent $1.60.)
  • GUESS 2: Since the first guess was way off, the student might jump to answer choice (c) or (d). Let’s try c. If an eraser costs $0.30, then the pencil would cost $0.50. This means that the cost of Jason’s purchase would be 3($0.50) + 2($0.30) = $2.10. That’s closer, but still too much.
  • GUESS 3: At this point there are only two possible answer choices. Trying either one will get a student to the correct answer, either directly or by eliminating the other possibility.

3.Speaking of guess-and-check, that’s almost always a workable approach to a situation like this. Even if a student can’t write an equation to find the cost of a pencil if they know the cost of an eraser, they can probably work it out by thinking about the real items. For example, a student might guess that the eraser costs $0.20. Then they might look at Jason’s purchase: 3 pencils and 2 erasers cost $1.60. If the erasers are $0.20 each, they would account for $0.40 of the cost, leaving the remaining $1.20 as the cost of 3 pencils. This means that the pencils must cost $0.40 each. Now the student could check to see if these two costs work out in the larger purchase. 4 pencils at $0.40 each and 4 erasers at $0.20 each would total $2.40. This doesn’t match Irita’s cost, so the guess was wrong.

A student might even keep track of their thinking in a table like this (based on starting with Jason’s purchase and then checking with Irita’s purchase):

That was a lot of work to eliminate just one answer choice, but remember that there will be questions on the test that are not within students’ reach at all. It’s okay to spend some extra time on a problem they have a good chance of getting right. (Also, try not to worry about the fact that students will probably not make a neat table with everything labeled precisely when they are under pressure in a test situation. Tables are an excellent way to keep your thinking organized and if students practice with and feel confident using them in your class, they will be more likely to make use of a quickly sketched table in a test situation. Even a reduced table where students just keep track of their guesses and the cost of 4 pencils and 4 erasers will help them navigate a complicated situation like this.)

As teachers, we are charged with getting our students to pass the test and that is often what’s driving our students as well. But focusing on the test often brings with it the mistaken idea that there is one right way to solve each problem. A student who sees this problem as a prompt to remember and apply a procedure may freeze up and miss the chance to score some points. On the other hand, a student who values their own ability to reason about quantities and relationships, especially in the familiar context of money, may make sense of this in their own creative way. Teaching students to reason flexibly and creatively, to be problem solvers, and to value their own prior knowledge and experience will not only prepare them for the test but also help them grow as numerate citizens and critical thinkers.


Sarah Lonberg-Lew has been teaching and tutoring math in one form or another since college. She has worked with students ranging in age from 7 to 70, but currently focuses on adult basic education and high school equivalency. Sarah’s work with the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center at TERC includes developing and facilitating trainings and assisting programs with curriculum development. She is the treasurer for the Adult Numeracy Network.