Calculating on the edge of the newspaper

Recently, 7 million flower bulbs were planted in Keukenhof. In many places, three bulbs are planted right next to each other in the ground. These bulbs bloom at different times, so the flower bed is always full and the garden is beautiful. Spring-flowering bulbs are dug up after they have grown (around June), dried, and stored. They are replanted in October or November. If you want to store 7 million bulbs, how much space do you need: a box, a classroom, the school building, a barn the size of a soccer field?

Stacking boxes is easier than stacking bulbs. Some of the students will probably start stacking small boxes. They then have to think about what happens when you increase the number of boxes in length, width, and height, say ten times as many boxes in length, width, and height. The number of boxes then increases 1,000-fold.

Seven spaces measuring 100 by 100 by 100 bulbs is already 7 million flower bulbs. If an average bulb has a diameter of 5 cm, you will need seven spaces measuring 5 by 5 by 5 meters. A classroom is often 7 by 7 meters and say 2.5 meters high. That is about the same volume as a space measuring 5 by 5 by 5. So 7 full classrooms is about enough. However, you will need to fill each classroom to the brim.

Other groups of students will probably take a different approach. They may have seen boxes of 10 or 100 bulbs at a garden center and decide to use those. That's no problem. They will also determine the number of boxes in a classroom or calculate how large the space needs to be to store a large number of boxes.

When harvesting, bulbs are often sifted (sorted by size). What happens if the bulb size doubles? The question of magnification will now come back to all students. Do you now need twice as many classrooms, or do you need many more classrooms? How many exactly? With a magnification factor of 2, the content will grow by a factor of 8 (2³). Don't tell them the answer, but give the students enough time to figure it out for themselves.

The enlargement question also comes up with the copier. I want to enlarge an A4 to an A3, what settings do I need to use? Or I want to reduce an A3 to an A4. Where exactly do those numbers come from?

There are, of course, many more spherical objects that are sorted by size. Think of onions and potatoes, for example, but oysters are also sorted and named by size.

Webkwestie has developed an assignment about flower bulbs. Below, students investigate how a bulb is structured, how to plant a bulb, and how to care for a flower bulb.

A dike around a reservoir of waste water is about to break. Pumps are being used to try to empty the reservoir in time. In its report, NOS news states that the reservoir contains more than 2 billion liters of water. Thousands of liters of water are being pumped out every minute and it will take 10 to 12 days to empty the reservoir.

Moving two billion liters of water in 10 days. How many liters do you need to pump out per minute?

With a number of pumps, you can move a lot of water very quickly. A small pump can move 60,000 liters per hour; larger pumps can move 3,000 m³ or more per hour. How many pumps do you need to do this in 10 days?

The New York Times provides more information. The reservoir contains approximately 400 million gallons of wastewater. Between 2 and 3 million gallons of water leak from the reservoir every day. On March 26, the reservoir contained 480 million liters, and on April 3, it contained 390 million liters. Currently, 22,000 gallons per minute are being pumped out.

Are the figures quoted by the NOS in its report correct?

This problem requires some clever math. You can do this with a calculator, of course, but you can also use approximate calculations. If you're going to use approximate calculations, you have to choose your numbers carefully. A day is 24 hours, but it's easier to calculate with 25 hours. A day is 24 × 60 minutes, but it is easier to calculate if you approximate the number of minutes in a day as 25 × 60 minutes, or approximately 1,500 minutes.

When estimating, this problem is similar to the question of who in the drawing below is a billion seconds old.

Birthdays are always fun, but you can also celebrate special birthdays and special lengths in class. We are big fans of the 1-meter day. Let students keep track of when they are 1 meter tall. That day is difficult to predict. But the 500 or 1000 week party, the 3,000 day party, or the 1 billion second party can be determined precisely. Let students make their own list of special days in their lives and create a class list together. That way, they can check each other's calculations. The students' 1,000-week celebrations cannot be too far apart, and the difference in age (birthday) says something about the difference between the moments when the 1,000-week celebration can be celebrated. Can they come up with a nice party for family members?

"The police want 35 percent of officers to have a migrant background in five years' time," according to various media reports. The NOS news article also reports that at least 17,000 of the 63,000 officers will retire in the next five years. Can the police actually achieve this 35% standard in five years? It's a good question for the children to puzzle over.

There are different ways of looking at this. Thirty-five percent of 63,000 is just over 22,000. That would only be possible if there were already 5,000 officers with a non-Western background working for the police and all newly recruited officers (17,000) had a non-Western background. If 50% of the new recruits have a non-Western background (8,500 officers), then at least 13,500 officers must currently have a non-Western background (and not leave). So it partly comes down to making some realistic assumptions. Because we don't know exactly who will leave and who will be hired. Students will probably also wonder how many officers with a non-Western background are already in service. Asking good questions and clarifying your assumptions is very important.

They may be able to find the answer to that last question on the internet. On June 12, 2020, the Volkskrant newspaper reported that 7 percent have a non-Western background: "Of the police officers—from cleaners to chiefs of police—7% have a non-Western migration background, meaning people 'with a migration background from one of the countries in Africa, Latin America, and Asia (excluding Indonesia and Japan) or Turkey.'" Let's assume that the Volkskrant's figures are correct.

7% of 63,000 employees is 4,410 employees. If all newly recruited officers have a non-Western background, then there will be a total of 21,410, or just under 34%, officers with a non-Western background.

That will never work. It seems impossible to me to hire only officers with a non-Western background over the next five years. First of all, it is questionable whether there are that many candidates, but such strong discrimination in the selection process is also likely to provoke considerable resistance. If they want to reach 35%, they would have to drastically increase the number of officers.

You can count on that too. To bring the percentage from 7% to 35%, you will in any case have to recruit more than 35% of officers with a non-Western background. Would the students make this argument and explain it to each other? Say 50% of all new officers have a non-Western background. How big would the police force have to be to reach that 35%?

If you calculate this, you will see that the number of police officers would have to almost double to achieve that 35% percentage. So it turns out to be a nice but unachievable statement.

Just give the students the data and ask them if the police will be able to achieve this. Let the students decide for themselves how they will determine this and how they will convince their classmates.

The NOS news headlines: "7684 new infections, increase levels off slightly." What do they mean? Has the number of infections now increased or decreased? Newspapers regularly use headlines like this. In fact, the newspapers are writing about the first derivative, but most readers do not understand that at all. Ask the students to make a graph that could go with this and have them explain why that graph reflects what the newspaper is saying. Also have them explain what is increasing and what is decreasing. At some point in the discussion, the students will start talking about the increase per day and will be able to show that this increase is becoming smaller. The number is rising, but the increase in the number is falling. That is interesting. Could they draw a graph of the increase?

Of course, there are other reports with similar headlines: Rise in house prices slows, Rise in number of ambulances needed slows, Rise in consumer prices falls from 2.7 to 1.8 percent, Rise in number of unemployment benefits slows. Here are two similar reports on meat production and the number of incidents involving running red lights.

Deforestation is a serious problem that threatens species diversity, the climate, and the ecosystem worldwide. Deforestation increased again in 2020. The NOS reports that in 2020, an area of 42,000 square kilometers (the size of the Netherlands) of primeval forest was cut down. That is quite striking. How many trees would fit in the Netherlands if we demolished all the houses, roads, squares, and schools and replaced them with trees? How many trees in primeval forests were cut down last year?

The number of trees that can fit on 42,000 square kilometers depends on tree density, which is not the same everywhere. This is also evident from figures c and d (below) from the article by Hoang, N.T., and Kanemoto, K. (Mapping the deforestation footprint of nations reveals growing threat to tropical forests. Nat Ecol Evol (2021).) The number of trees lost per capita is highest in Sweden, but the number of square meters of forest lost is highest in Canada. I would only give the two graphs to the students and ask them to explain exactly how this works. A comparison can be made between the density of forests in Sweden and Canada. Do you need to know the population and size of the country to do this, or can you calculate it without knowing those figures?

A large ship has run aground in the Suez Canal. "To pull it free, you first have to do some calculations," says a director at Boskalis. This shows once again how important mathematics is. That kind of math isn't for us, though. It's about buoyancy, the weight of the oil, the weight of the water on board, the weight of the containers and the cargo in the containers...

But the reports about this ship and the photos online give us plenty of reason to be amazed and do some calculations. The ship contains about 20,000 containers, is about 400 meters long, and is lying 15 meters deep in the water. We cannot see all the containers, because some of them are in the hold. How many containers are stacked on top of each other? What kind of terrain do you need if you bring those containers ashore and stack a maximum of three containers on top of each other? How long is a train that transports all those containers?

Do you add or multiply when dealing with ratios and percentages?

The United States has already fully vaccinated 44 million people, according to some newspapers. In the Netherlands, 500,000 people have already had two shots. Where is it going faster? In Great Britain, approximately 900,000 people were vaccinated on a Saturday. In the Netherlands, an average of 29,000 people were vaccinated per day around that time. With all this kind of data, it's always about the difference between absolute and relative numbers. In other words, we're bombarded with ratios again. To help readers get a quicker overview, many newspapers publish the coronavirus figures as the number of infections per 100,000 inhabitants. Luckily for us, that's not always the case, and there are plenty of articles that give us a reason to work on ratios.

Tine Degrande (a researcher from Belgium) wondered why students sometimes add and sometimes multiply when dealing with ratio problems. She showed that students have a personal preference for one operation or the other. Among other things, she gave students the following problem: What can be placed in the question mark?

Some students filled in 8 (6+2), others filled in 12 (2 x 6). In this case, both answers can be explained. However, this preference for addition or multiplication also recurred in questions where only one of the operations is correct. To verify this, Tine presented the students with questions such as:

• Two packs of pencils cost 6 euros together. How much do four packs of pencils cost?
• Ellen and Kim are running laps on a track. They run at the same speed, but Ellen started later. When Ellen has run 3 laps, Kim has run 6. When Ellen has run 12 laps, how many laps has Kim run?

Here, the students again showed their preference for addition or multiplication. We also see this preference for an additive or multiplicative approach with percentages. Tine concluded that this preference is persistent.

We must therefore constantly address this in class. Provide students with sufficient situations in which one approach or the other is correct and create a conflict with an accompanying discussion among the students about the question "why can you add here and not multiply?" or "Why can you multiply here and not add?"

There are more possibilities

One more comment. The problem "What can go in place of the question mark?" is not interesting at all from a mathematical point of view:

Mathematically speaking, any number is correct. For example, we can say that the point (2,4) and the point (6,?) must lie on a line. Whatever you put in place of the question mark, there is always a straight line that passes through those two points.

Even if we expand the problem further, mathematically speaking there are many more possible solutions. Look at:

It seems as if we have to add 2 everywhere. That certainly gives a good solution; the question mark is then 12. Multiplying certainly does not give a good answer. However, if we also consider a parabola, for example, then more is possible. For example, the points (2.4), (6.8), and (10.44) lie on the parabola y= x²-7x + 14. So I could also have entered 44. And so, mathematically speaking, there are an infinite number of possible answers.

Apple is becoming increasingly active in the health sector. It is now possible to use an Apple app to check whether you have hearing problems. Most people with hearing problems would probably like to try this out. They usually want to know how serious the problem is.

Apple recently reported that 1 in 5 participants has some degree of hearing problems. What does this mean? Does 20% of the world's population have hearing problems?

It is always good to discuss articles like this with children. The question is always: what is the whole picture? This is always an important question, and children often do not pay enough attention to it. It is important when dealing with percentages, fractions, everything. In this case, it is hidden in the word "participants." People who experience hearing problems are more likely to take this test than people who do not experience hearing problems. The group is therefore not a good reflection of society as a whole. This is also the case with Covid-19 tests, for example. If 11% of the people tested are positive for Covid-19, you cannot conclude that 11% of the population will test positive.

The question "what is the whole" also arises in famous questions such as:

• I want to buy a drill. I know that tomorrow the price will increase by 10%. However, the day after tomorrow the price will decrease by 10% again. When is the best time to buy the drill?
• I want to buy a drill. The store "The Golden Drill" is advertising a 20% discount. "The Handy Driller" is selling drills at a 50% discount. Where is the best place to buy the drill?
• My sister bought some candy. She put it in a jar, counted the number of pieces, and ate half of them. The next day, I looked in the jar and counted the number of pieces. I also ate half of them. Did my sister and I eat the same number of pieces?

"What is the whole" is a great idea. You can never pay enough attention to it in class. The above questions are not the best questions to raise this issue in class. It is better to look for bigger problems in which this issue keeps coming up.

On the same day, NOS Teletekst showed how the question "What is the whole?" is important in a more complex setting. Minister De Jonge says that two-thirds of 80 to 85 percent of adults will be fully protected against Covid-19 in July. The article describes this in a few steps.

• 80 to 85 percent of adults want to be vaccinated
• Two-thirds can be fully protected in July with two doses
• The rest have had one shot.

What percentage of the Dutch population will be fully protected in July? That requires quite a bit of thought. Two-thirds of 80-85 percent of adults. How many Dutch people is that? Or what percentage of the 17 million Dutch people is that? To understand this situation properly, students need to realize that there are three different groups involved: all Dutch people (17 million), all adults (14 million), 80-85 percent who want to be vaccinated (between 11 and 12 million). Can you say that this is 2/3 of 80 percent? Can you multiply here? Why exactly?