We’ll still be posting our monthly features and our popular “Will This Be on the Test?” series, but the blog will be hosted within the Adult Numeracy Center at TERC website where you can find lots more information about our professional development work and resources. In other words, one-stop shopping for your adult numeracy needs!

Welcome to the latest installment of our monthly series, “Will This Be on the Test?” (You can find links to the earlier installments in the column to the right.) 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.

Let’s dig into some data this month. What skills and understandings do you think of when you think of the data domain? There’s a lot more to it than calculating mean, median, and mode! Making sense of representations of data is important for not only the math section of the test but also for science and social studies. And on top of that, it’s one of the most useful numeracy skills for adults to have in the real world.

Here is this month’s challenge:

Before you read further, allow yourself to bring your full mathematical reasoning power to bear on this challenge. 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 reallyneed to be able to answer this?

Before you read the possible approaches below, be aware that every single one of them relies on the important first step of understanding what the chart is telling you. If a student does not know how to read a circle graph, they will not be able to make sense of this task at all! It is essential to understand that each wedge of the circle graph tells what percentage of total sales came from each product. Even though we are inundated every day with representations of data, it is important not to assume students know how to make sense of them.

Here are some possible approaches:

1.Estimate! Look at the relative sizes of the wedges for T-shirts and hats. Seeing that the wedge for hats is smaller than the wedge for T-shirts won’t help much because that only tells a student that less money was made from hats than from T-shirts, and all the answer choices are less than $1,600. However, if a student fine-tunes their estimate a bit, they may make some progress. The wedge for hats appears to be a little less than half the size of the wedge for T-shirts. (Looking at the numbers confirms this – 15% is a little less than half of 40%.) Which answer choice(s) could reasonably be described as a little less than half of $1,600?

2.Find one percent from another. A student who has built a mental bank of benchmark percents and is comfortable comparing them to each other can figure out how much 15% of sales by starting with what they know about the 40%. There are a lot of different ways to do this, jumping from one benchmark to another. Here are a few possible paths:

(a)40% >> 20% >> 5% >> 15%. Starting with 40%, a student might reason that half of that is 20%. We know 40% is worth $1,600, so half of that is worth $800. Therefore, 20% of the total sales is $800. From there, she could figure out 5% of the sales by dividing 20% by 4. $800 divided by 4 is $200. The last link in the chain is to get to 15% from 5%. How does 15% compare to 5%? What operation is needed?

You can take the illustration from here. (Hint: How many blocks of 5% does it take to make 15%?)

(b)40% >> 20% >> 100% >> 10% >> 15%. This chain starts with the same first step as the last one, but a student comfortable with the benchmark 20% might go from there to figuring out the total sales (the whole). If 20% of the whole is $800 (see reasoning path (a) above), then the total sales must be five times as much as that, or $4,000. From there, a common strategy for finding 15% of a number is to find 10% (or one-tenth) and then add on half of that. In other words, 10% + half of 10% = 15%.

(c)40% >> 10% >> 15%. Combining ideas from the two chains of reasoning above, a student could jump from 40% to 10% by finding one-fourth of 40% by dividing 40% by 4. Then they could use the strategy of adding half of 10% on to 10% to get to 15%.

(d)Quite a few other possible paths! Students who understand the relationships between percents, fractions, and wholes can move around very flexibly in the world of percents. Starting from a benchmark of 50% (or ½ ) and building from there, students can get to a point where they can navigate to almost any percent with confidence.

3.Good old proportional reasoning. Percents are a special kind of ratio and therefore students can bring proportional reasoning tools to bear on them. As in the chains of moving from one percent to another above, there are several paths a student could take setting up proportions to get to the answer.

Use the known percent to find the whole and then use the whole to find 15%. In our example, use one proportion to find the total sales, then use the total sales to find the $ made from hat sales:

(Note: This is not the only way to set up these proportions to find the total sales and then the money earned from hat sales. And once the proportions are set up, there are many ways to reason about them!)

A more direct path is to compare the known percent and the unknown percent in a proportion.

(Again, there is more than one possible way to set up and solve this proportion.)

Did you notice that reasoning about this data task involved a lot of number sense and proportional reasoning? One thing I love about the data domain is that it brings together elements of other domains in a concrete and applicable context. (Geometry is no stranger to the data domain either – representations of data use area and angles to show differences in the sizes of categories.) Teach data deeply and conceptually! We all need to understand it and it makes so many kinds of reasoning real and relevant.

SarahLonberg-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.

In math, seeking patterns is a helpful strategy. Once we think we’ve found a pattern, we can make a rule or generalization, or sometimes even a formula. In some life situations, however, we need to be more cautious about making rules when we think we see a pattern.

Finding patterns helps to make learning easier and faster, but it can also lead to biases. All of us have biases of some kind. Some of them are innocuous (a preference for certain colors, music, food, clothes) and some are even helpful (avoiding certain things or situations that resemble negative experiences we’ve experienced). Biases like these are about our own selves, not about judging others, and they may be implicit or explicit.

Our biases, or prejudices, are part of who we are and they often protect us. Biases are not negative until we act on them to create discrimination. And it’s not just explicit biases that can be harmful. It is often the subconscious, implicit biases that can cause damage if we are not aware of them.

For example, let’s imagine you are a teacher who has just received a class roster with the following last names: Soares, Rivera, Schuler, and Curry. What would be your immediate reaction to the names?

Would you assume that you have at least three new students from cultures other than that of the United States? Which of them might not speak English as a first language?

What would be your gut reaction to a name like Soares? Do you think that maybe Rivera can help you translate for students whose first language is Spanish? And, what do you expect from someone with a name like Schuler? Which of the students are you already thinking might be better at math? Do you assume that ESOL students might be able to do simple math computation, but definitely not word problems?

Do an internal check-in to see whether you have already made some assumptions about who these students are and what their abilities may be based solely on their names. Have you linked these new names with your past experiences (patterns that you’ve noticed over time)? Or have you made assumptions based on media representations or the attitudes of your friends and family? Did you prejudge without even realizing it?

We don’t just make judgments based on names. We also make snap judgments of our peers. How often do you prejudge based on position or education without even realizing it? Do you assume that someone with a formal math education is a good math teacher? Do you assume that the ESOL teacher in your program can’t effectively teach math?

The best way to break some of those patterns, those implicit biases, is to get to know the students and your peers. They are each unique individuals whose identity is not limited to being a student in your class or the teacher in the next room, but also an individual who may or may not affirm some of your initial assumptions. Create opportunities to find out about each individual from a personal perspective. See each person as a unique individual who has something positive to contribute. Appreciate what they can share.

We math teachers might think that we have to focus solely on the content, but this isn’t so. We should focus on the student. Think about teaching students math, not teaching math to students. What you believe is how you will focus your efforts. Get to know Soares, Rivera, Schuler, and Curry. In doing so, you’ll learn that only one of them —Curry— had an immigrant parent and spoke another language fluently as a young child.

Welcome to the latest installment of our monthly series, “Will This Be on the Test?” (You can find links to the earlier installments in the column to the right.) 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.

Last month we looked at a question about a snail crossing a garden. Modeling the question with number lines or Singapore strips made it very accessible. (If you haven’t read that column yet, I recommend you do so before continuing with this one.) What if the fractions involved in the question were a little less friendly? Would the same visuals and conceptual strategies still work? This month I invite you to challenge yourself to solve a slightly thornier variation using visuals and conceptual understanding.

Here is this month’s challenge. The snail is back, but the numbers have changed.

Before you read further, allow yourself to bring your full mathematical reasoning power to bear on this challenge. 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?

Here are some possible approaches:

1.Estimate!The snail has been traveling for 2/3 of an hour (or 40 minutes) already and is only going to travel for another 10 minutes. Drawing a quick sketch can help a student think about how much farther it looks like the snail would get. About how far do you think the snail would get in 10 more minutes?

The snail is already just over halfway across the garden, so answer choice (c) can be eliminated because it is less than half. Answer choice (a) is more than half, but not by much. From the sketch, it looks like the snail will be well past the halfway point in another ten minutes, so answer choice (a) is not likely. Answer choice (b) is more than one, and it doesn’t look like the snail is going to make it past the end of the garden in only more 10 minutes. There are still a couple of possible answers left. Which one would you choose just from eyeballing the picture? What fraction understandings were used here?

2.Use a Singapore strip diagram (also called a bar model or tape diagram). A student might start with a bar representing the 4/7 of the garden the snail has already crossed. Notice that it bears a resemblance to the sketch above; both show 4/7.

This is where it gets a little trickier. The snail traversed those four blocks in 2/3 of an hour. A student who understands that unit fractions (fractions whose numerator is 1) are single pieces and non-unit fractions are groups of pieces may reason that those four blocks are covered in two chunks of time that are each 1/3 of an hour. In other words, the snail covers two blocks every 1/3 of an hour, or every 20 minutes.

If we know how many more blocks the snail will travel in 20 minutes, how many blocks will it cover in another 10 minutes? What total fraction will that make? (The diagram above doesn’t show the final step of adding on the last 10 minutes of travel.)

3.Reason proportionally. This task is about something moving at a constant rate. In other words, the distance the snail covers is proportional to the amount of time it has been traveling. A student might reason that the 10 minutes the snail has yet to travel is one-fourth of the 40 minutes it has traveled so far, so the snail would cross an additional fourth of the distance it had already covered. Thinking about finding 1/4 of 4/7 of a garden might make your head spin, but looking at the picture of 4/7 represented as four blocks out of seven makes it easier to see that 1/4 of 4/7 is one block.

Once again, we found that visual and conceptual approaches carried us through without ever needing to “choose and apply an operation.”

This question of course can also be answered using fraction operations, but it takes a few steps. First, divide the distance covered by the time it took to cover it to get the rate: 4/7 ÷ 2/3 = 6/7 of the garden per hour. Then, multiply the rate per hour by the fraction of an hour the snail adds onto its journey (10 minutes). Ten minutes is 1/6 of an hour so 6/7 x 1/6 = 1/7 more of the garden that would be covered in the extra time. Finally, to determine the total amount of garden covered, add the new distance to the original distance: 4/7 + 1/7 = 5/7, which is answer choice (d). That was three operations and not a very intuitive path through them. (It’s also not the only possible operations approach.)

Even though the operations approach is valid, it tends to elicit the I’m-never-going-to-remember-all-these-steps! response from students. Having to do three different fraction operations to get to the answer can make a student want to run the other way. On the other hand, giving students visual and conceptual tools prepares them to chart their own path through problems in a way that makes sense to them.

SarahLonberg-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.

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!

SarahLonberg-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.

Welcome to the latest installment of our monthly series, “Will This Be on the Test?” (You can find links to the earlier installments at the bottom of this post.) 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.

I have heard an argument made that it’s not worth spending time teaching fractions in adult ed classes. One reason I’ve heard for this is that students are mostly in class to prepare for the test, and they can use a calculator on the test. Isn’t it more sensible and efficient to just spend one class period teaching them how to do fraction operations on a calculator and then moving on to higher level topics? After all, fractions are difficult and frustrating and it seems like you are always reteaching them anyway. Why not take the easy way out?

Unfortunately, taking the easy way out often doesn’t pay off in the long term – and in this case, it doesn’t pay off in the short term either. It doesn’t pay off in the long term because students miss out on real learning and sense-making that can help them make sense of the real-world math they will encounter. It doesn’t pay off in the short term because there’s more to solving test questions about fractions than just doing operations. High school equivalency tests are designed to test students’ mathematical reasoning, not their ability to use calculators.

Here is this month’s challenge. Imagine you are a student who is expert in doing fraction operations on the calculator but that is all you have learned about fractions. Would you be able to figure this out?

Before you read further, allow yourself to bring your full mathematical reasoning power to bear on this challenge. 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?

Here are some possible approaches:

1.Estimate! The snail has been traveling for 3/4 of an hour already and is only going to travel for another 15 minutes (or 1/4 of an hour). Would it get a little further? A lot further? Twice as far? A quick sketch could help a student get a handle on what kind of answer might be reasonable:

The snail is already more than halfway across the garden, so answer choices (a) and (b) don’t make sense because they are less than half. Answer choices (c) and (d) are both more than one, and it doesn’t look like the snail is going to make it past the end of the garden in only more 15 minutes, so those don’t make sense either. That leaves only one possible answer! (What fraction understandings were used here?)

2.Use a Singapore strip diagram. A student might start with a bar representing the 3/5 of the garden the snail has already crossed. Notice that it bears a resemblance to the sketch above: both show 3/5.

A student who understands that 3/4 of an hour means three groups of 1/4 of an hour, will be able to recognize that if the snail traverses three blocks in 3/4 of an hour, it is covering one block every 1/4 of an hour. (This is a non-trivial understanding and is important enough to have its own standard in the College and Career Readiness Standards for Adult Education: Understand a fraction a/b with a > 1 as a sum of fractions 1/b. (4.NF.3))

Filling in the fact that the snail covers one block every 1/4 of an hour will show the student the total fraction of the garden the snail will have covered after 15 more minutes:

3.Reason proportionally.This task is about something moving at a constant rate. In other words, the distance the snail covers is proportional to the amount of time it has been traveling. A student might reason that 15 minutes is one-third of 45 minutes so the snail would cross an additional third of the distance it had already covered. A student who understands that 3/5 is three groups of 1/5 can recognize that 1/5 is one-third of 3/5, so the snail will cover an additional 1/5 of the garden.

Did you notice that none of these approaches involved that old standard, “Choose and apply an operation”? Even after building a deep understanding of what is going on in this task, it may not be obvious which fraction operation, if any, is the ‘right’ one. It turns out that you can get the answer to this task using a fraction operation, so a calculator could be helpful to you if you can figure out which operation makes sense for the situation.

(Psst… want to know which operation it is? You can find the answer by dividing 3/5 by 3/4 because that is dividing distance by time which gives you a rate. Since the rate is in fractions of a garden per hour, the answer to this division gives you the total fraction traversed by the snail in one hour, which is what the question was asking for! This is also a good option, but the approaches explored above are probably more accessible to students.)

SarahLonberg-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.

Imagine that it’s the end of the school year and you’re planning a class celebration with food and drinks. You ask your students to choose which beverage they’d prefer: juice, water, or soda. The students respond as follows:

So, you decide to go with soda for everyone since more students preferred that to juice or water.

During the party, you realize that less than half of the students drank any soda! They either drank from the water fountain or didn’t drink anything. What you didn’t realize is that maybe those few who wanted juice may have preferred water to soda. But how could you have known that? That’s where ranked-choice voting could have come in handy.

There’s a myth that ranked-choice voting means some people’s votes don’t count, but that’s not accurate. In this system, even if a voter’s top choice doesn’t win out, his or her next-favorite choice might. Had this voting method been used in the class party example, all of your students would have had a chance to vote for their favorite beverage as well as their second- and third-favorite (if they had wanted to).

Had ranked-choice voting been used, the results might have looked like this:

As you can see, 5 of the 12 students chose soda as their first choice. But, the majority of students didn’t want soda – 7 of the 12 chose something else instead (water or juice). In ranked choice voting, you have to have a majority to ‘win’, even if it’s only to choose a beverage. You can also see that not everyone voted for a 2^{nd} or 3^{rd} choice. That is absolutely fine! What you cannot do is vote for a choice more than once. So, even if a student absolutely loved soda and nothing else, she could only vote once for soda.

In ranked-choice voting, if no first-choice option wins a majority of votes, the option that received the fewest votes is removed from consideration. In this case, juice was the ‘loser’ because it received only three votes. But, the people who voted for juice still get to have a say in the result if they indicated a second-choice option. Now we can compare the 2^{nd} choice votes with the 1^{st} choice votes.

If you tally the first choices along with the second choices of the ‘loser’, you see that there are now seven votes for water. Since seven is more than half, the majority – people who are OK with drinking water – wins. If you had used this process, you would have discovered that most of your students would have preferred to have water instead of soda and your party would have been more of a hit.

So, to reiterate a few key points about ranked choice voting:

The winner has to be an acceptable option to a majority of those voting. It’s not enough for the ‘winner’ to just have more votes than the second-place option – the winner has to get more than half of the total vote. That means, the winner has to receive at least 50% + 1 vote.

Everyone’s vote counts. Sometimes a person gets to have a result he or she can accept even if it’s not his or her top choice.

People do NOT have to vote for more than one option, but they also cannot vote multiple times for their favorite option.

The beverage example shows that sometimes the option with the most votes is not always what the majority of people really want. There are several times in recent U.S. history when a presidential candidate did not get the majority of votes. For example, in 2016, neither Hillary Clinton nor Donald Trump received the majority of votes in twelve states because a third-party candidate received some of the votes that would have gone to one or the other.

Probably the most significant example of how ranked-choice voting could have had an impact on the presidential election is in 2000 when almost three million voters in Florida chose Ralph Nader over Al Gore and George W. Bush. Had those three million voters had a second option, we may have had a different president. Whenever we have more than two political candidates in the same race, often people have to make the hard choice between the candidate that they personally prefer versus the candidate who has the best chance of getting the most votes. In a ranked choice voting system, people wouldn’t have to make that hard decision.

Most recently, the January 2021 run-off election for two Georgia senate seats could have been a lot simpler – and quicker – had ranked choice voting been used. (As a reminder, because neither candidate received a majority in the November 2020 election, per state law there had to be a run-off.) If those voters whose top choice came in third in the November election had had a chance to indicate their second choice, those votes could have been tallied to help give one of the two leading candidates enough votes to win, eliminating the need for a separate run-off election.

To help your students gain a better idea of how ranked-choice voting works, you might have them participate in an activity similar to the one illustrated in this short, easy-to-understand video How does ranked-choice voting work?

Donna Curry is an educator, curriculum developer and professional development specialist with over 30 years of experience in adult education. For the past 30+ years, she has focused on math standards development at the national level (Equipped for the Future National Standards and Standards-in-Action projects) and at the state level (including states such as Rhode Island, Washington, Oregon, New Jersey, Oklahoma, and Ohio). She has also worked on the National Science Foundation’s EMPower project and served as co-director for the NSF-funded Teachers Investigating Adult Numeracy (TIAN) project. She co-developed and implemented the Adult Numeracy Initiative (ANI) project and Adults Reaching Algebra Readiness (AR)^{2}. Donna currently directs the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center for Massachusetts and the Adult Numeracy Center at TERC.

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 reallyneed 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.

SarahLonberg-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.

I was using a recipe that I had gotten from my cousin in Germany. The recipe called for 200 mL of milk. I wanted to know what that was in cups so I Googled the information. Here’s what I found:

To calculate 200 Milliliters to the corresponding value in cups, multiply the quantity in milliliters by 0.0042267528198649 (conversion factor). In this case we should multiply 200 milliliters by 0.0042267528198649 to get the equivalent result in cups: 200 milliliters x 0.0042267528198649 = 0.84535056397299 cups.

Now, I’m not afraid of decimals (not even ones that are as exceedingly long as the ones above). But, even though I’d found out what the answer was, I realized that I didn’t have a measuring cup that measured in decimals, never mind to the hundred-quadrillionth place! So, the answer itself was useless.

But, I did have enough number sense to know that 0.845… was pretty close to 8/10, or its reduced form 4/5. And, although my measuring cup also doesn’t measure in tenths or fifths, I know that 4/5 is close to 1. So, I knew I needed a little less than a cup of milk for the recipe. I could have also used a slightly different approach to get the same answer. If I had started my calculations by shortening the conversion factor several places to 0.004, I would have come up with 0.8 which would make more sense to me than the answer provided by the online calculator.

This volume conversion experience made me think. Using number sense, I was able to think of these unwieldy numbers in more manageable terms. I was able to connect them to benchmarks that were easier to conceptualize. But what about the many adults who have not been taught to use flexible thinking in math to associate something new with something already understood? The conversion factor procedure may have provided an answer, but how well can the average person understand the concept of a number like 0.84535056397299? What does it mean in everyday terms? How does it relate to the actual problem we’re trying to solve? In short, do we know what to do once we have an answer?

It reminded me of how often we think of math as simply a bunch of procedures to be followed: Don’t worry about making any sense – or being useful in real life – just follow the procedure. When there’s an absence of conceptual understanding, we are prone to:

choosing the wrong calculation method because we don’t understand what is being asked or how the amounts relate to one another;

not being able to recognize whether our answers actually make sense

not being able to “see” our answers in alternative, relatable situations or representations that make sense to us

Is this what we want for our students – to just follow procedures (often being told to memorize them, not even having the luxury of Googling a particular procedure) without any understanding, any estimation, any reasoning about whether the answer makes sense or not?

P.S. After doing some estimating based on the information I found on the internet, I pulled out my measuring cup only to discover that one side had gradations in cups . . . and the other side had gradations in mL. I guess I should pay closer attention to the math tools that I use. At least I was able to check my work another way!

Donna Curry is an educator, curriculum developer and professional development specialist with over 30 years of experience in adult education. For the past 30+ years, she has focused on math standards development at the national level (Equipped for the Future National Standards and Standards-in-Action projects) and at the state level (including states such as Rhode Island, Washington, Oregon, New Jersey, Oklahoma, and Ohio). She has also worked on the National Science Foundation’s EMPower project and served as co-director for the NSF-funded Teachers Investigating Adult Numeracy (TIAN) project. She co-developed and implemented the Adult Numeracy Initiative (ANI) project and Adults Reaching Algebra Readiness (AR)^{2}. Donna currently directs the SABES Mathematics and Adult Numeracy Curriculum & Instruction PD Center for Massachusetts and the Adult Numeracy Center at TERC.

Welcome to the third installment of our monthly series, “Will This Be on the Test?” (You can find the first two installments here and here.) 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.

There are few topics that provoke anxiety and frustration in learners and teachers like fractions. Students are often carrying the weight of having tried to learn fractions operations so many times and never being able to remember the steps, let alone understand them. Teachers are often carrying a similar weight – how many times can you go over those same procedures? Why do you have to keep reteaching this topic? It’s tempting to spend one or two class periods teaching students how to do fraction operations on a calculator and be done with it. After all, students are allowed to use calculators on most HSE tests.

With that in mind, here is this month’s problem:

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 reallyneed to be able to answer this? To what extent, if at all, are memorized procedures required? Would a calculator be helpful here? (For a bonus challenge, try to find the “attractive distractors” in the answer choices. What mistaken strategies might students pursue that would lead them to these incorrect answers?)

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 reallyneed to be able to answer this? To what extent, if at all, are memorized procedures required? Would a calculator be helpful here? (For a bonus challenge, try to find the “attractive distractors” in the answer choices. What mistaken strategies might students pursue that would lead them to these incorrect answers?)

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.Estimate! A student who takes the time to make sense of this problem and think about which answers are reasonable will quickly narrow the possibilities down to two answer choices. It takes a little time and effort to figure out the relationships in this problem, but it’s worth it. A student might do a quick sketch of the fraction in the problem and use it to figure out what they have and what they are looking for:

Three of the answer choices are bigger than 60, but answer choice E is more than twice as big and the sketch shows that the whole circle is only a little bit bigger than the shaded part, so the answer must be C or D.

2.Think about and draw ratios. A student might interpret the fact that 4/5 of the applicants were accepted as meaning that four out of every five applicants were accepted. In other words, for every four that got in, one did not. A student might try drawing all the applicants. Using quick symbols, it doesn’t take as long as you might think. How many Xs and Os are in the picture? A student could count them one at a time, count by fives, or think of this as an array and multiply the dimensions.

It’s also possible that after drawing the first few sets, a student might start to see some patterns and regularity and ask themself, “How many of these will I have to draw?” Since the student already knows that there are five “people” in each row, answering that question will be a quick way to get to the total. (Here is a an example of when a calculator might be helpful. Since there are four accepted people in each row, and 60 accepted people total, the number of rows can be found by dividing 60 by 4. Then the total can be found by multiplying the number of rows by 5. Calculators, used judiciously, can help students use their time well and stay focused in test situations.)

3.Draw a Singapore strip diagram (also called a bar model). Singapore strips are great for making sense of parts and wholes. If students are comfortable drawing and reasoning with them, they provide a way of making sense of the structure of the problem that makes answering the question fairly intuitive.

To solve this problem with the diagram, a student would figure out how many people each box represents. (Four boxes represent 60 people, so how many people are represented by a single box?) Since the total number of people who applied is represented by the whole bar or five boxes, once a student knows how many people each box stands for, they will be able to find the whole in just one more step. (For a similar example using a model like this, see our blog, “How I Learned to Stop Worrying and Love Percents”.)

What about the calculator?

What did you decide about the usefulness of a calculator in addressing this month’s problem? What would your students do? Solve the problem without a calculator? Use a calculator judiciously for some quick computations? Grab the calculator and start doing whatever fraction operations they can think of? A calculator can be a great tool when used strategically, but only teaching students how to do fraction operations procedures instead of teaching how to reason about fractions will not serve them as well on the test or in life.

SarahLonberg-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.