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Section 8.1 Small Number and the Old Arrowhead - Classroom Guide

View the full story: Small Number and the Old Arrowhead

Suggested Grades: 4 – 7

Subsection 8.1.1 Mathematics

  1. Locating - in space and time
  2. Measuring - time
  3. Mathematics - mathematical modelling and radioactive dating

Subsection 8.1.2 Mathematical Vocabulary

  • small, number, a lot, Perfect Number, surroundings, a few days ago, Big Circle, the biggest, ever, how many more, towards, just in time, has just arrived, the next few weeks, straight line, a hundred, two hundred years, measure, calculate, to all of, hide in, above, too thick, squeezed between, tripped over, rolling down, stopped spinning, opened, into, stuck in, above, a long time ago, large, very, once, pointed to, arrow, started rotating, very few, thousands years ago, spinning arrow head, wide open, tightly.

Subsection 8.1.3 Cultural Components

  1. Indigenous: ; Indigenous food: , , , and ; kinship; love; ; ; , ; .
  2. General: What is ? Studying archeology at ԰AV.

Subsection 8.1.4 Mathematical Observations (Video)

Opening scene: Notice various shapes, such as: the mountain peaks, the forrest, the shore, the hills. Notice the perspective and reflection symmetry.

0:40 - Shapes, perspective and reflection symmetry.

1:09 - “I'll be joining a group of my professors and friends from school who are digging on Straight Line Beach. We will look for artifacts that will help us better understand how our culture has changed through time.”

1:43 - “How can you tell if something is a hundred or two hundred years old?”

1:50 - “We measure and calculate.”

2:03 - The next morning, Small Number rode to Big Circle's house. “There are people digging on Straight Line Beach. Let's hide in the bushes on the hill above the beach and watch what they are doing!”

2:13 - “These bushes are too thick,” whispered Small Number. “Let me try to get a bit closer.”

2:21 - He squeezed between the two bushes, tripped over a stone, and suddenly found himself rolling down the hill!

2:28 - When he finally stopped spinning, Small Number opened his eyes and yelped. He was looking into the eyes of a man?s face that appeared stuck in the ground!

3:00 - Describe various shapes. Count people and objects.

3:30 - “They are such large animals and it must have been very hard to catch them.” “Yes, but when you catch one you get a lot of meat.”

3:33 - “This is a ground slate point. It was probably used as the head of an arrow.”

3:45 - She took the stone and very gently started rotating it in her hand. “Very few people can say that they have held an object that was used by our ancestors thousands of years ago.”

4:00 - Shapes, perspective

Subsection 8.1.5 Answer: How can an artifact reveal its age?

Small Number asks Perfect Number to help him answer the question.

Subsubsection 8.1.5.1 Mathematical Model

Perfect Number to Small Number:

“Scientists use mathematics to describe the change of the matter over the time. Since this change is a very complicated process, there are different mathematical ways to describe it. We call each of those ways a mathematical model.”

Small Number:

“I thought that in math everything must be the same all the time. How come that there are different math things to describe the same … whatever?”

Perfect Number looks at Small Number, thinking. After a few moment, she says:

“That is a good observation, Small Number. For many, many centuries everyone who was doing mathematics thought that things in mathematics do not change. But several centuries ago some very smart and very brave people wanted to use mathematics to describe how changing one thing affects the change of another, related thing. Maybe you remember that I had to take a Calculus course when I was a first year student? Well, Calculus is part of mathematics that studies that kind of stuff.”

Small Number, puzzled:

“I have no idea what you are talking about, Perfect Number. What kind of stuff?”

Perfect Number, patiently:

“OK, Small Number. Let me give you an example. Do you remember the photo of you, me, Mom, and Dad that Grandma keeps in her room? You remember how little you and I are in that photo? And look at us now. I am almost as tall as Mom. It is our Mom's tradition that on our birthdays she marks on the doorframe how tall we are on that day. When she did that on your last birthday, she said that you were growing so quickly that in a few years you would be taller than Dad.”

Small Number, still puzzled:

“Yes, I remember, but what does that have to do with math?”

Perfect Number, smiling:

“Those marks on the doorframe can be used to create a mathematical model that would estimate in how many years you will become taller than Dad.”

Small Number, scratching his head:

“I still don't understand what a mathematical model is.”

Perfect Number, patiently:

“You can think about a mathematical model as a formula that works like this. Since you are 11 years old now, please pick a number between 1 and 11.”

Small Number, hesitating:

ٱ𱹱.”

Perfect Number, convincingly:

“If you put the number seven in the formula, the formula would give the number that is super close to your height in centimetres that Mom marked on your seventh birthday. If you decide to put the number 10, the formula would give the number that is again very close to your height that Mom marked when you were 10 years old. OK?”

Small Number, stubbornly:

“But what if I put the number 12, or the number 18, or the number 25 in your formula?”

Perfect Number, nodding her head:

“That is an excellent question, Small Number. And that is the real use of a mathematical model: I can use the formula to predict how tall you will be on your 12th, 18th, or 25th birthday. But getting the formula is a tricky business. This is because your growth depends on many things, and your age is only one of them. For example, we stop growing around our 20th birthday. So if you put 25 in the formula, the outcome should be something that is very very close to the number that formula gives you when you put 20 in it.” After a pause: “This is why it is so important to think if numbers that we get by calculating something make sense.”

Perfect Number, looking at her brother:

“From the formula, by using Calculus, we can estimate by how much your height changes every day. You see, when I look at you, to me it seems that you are of the same height as yesterday. But we know that your height is changing every day by a bit that our eye cannot notice. But mathematics can describe that change. That is what Calculus is about, a small change in time, like a day in your life, causes a small change in your height.”

Perfect Number, seriously:

“Calculus is not easy, but learning it is lots of fun.”

Small Number, impatiently:

“I believe you, Perfect Number. But I need to answer the question: ‘How can an artifact reveal its age?’”

Subsubsection 8.1.5.2 Radioactive Decay and Half-life

Perfect Number picks up her tablet and continues:

“Before we answer the question, Small Number, I'd like to tell you about something that I learned in my science class in high school.”

Perfect Number continues:

“More than 100 years ago, scientists discovered that there are substances that, over a very long period of time shrink in mass. One of such substances is called Carbon-14.”

Perfect Number writes and draws on her tablet:

Figure 8.1.1. Carbon -14: Half-Life

Perfect Number:

“It is known that whatever amount of Carbon-14 you have today, in 5,730 years only one-half of the amount will remain. This is why we say that 5,730 years is the half-life of Carbon-14.”

Small Number, existingly:

“5,730 years is a long time! But, are you saying that the shrinking happens on its own, without someone doing something?”

Perfect Number, nodding:

“Yes, the change happens because of substance's chemical and physical properties and the loss of mass is due to radiation. This process is called radioactive decay. Here the word decay means that the initial amount of the substance gradually decreases.”

Small Number, existingly:

“Wow! I am going to be a scientist!”

Subsubsection 8.1.5.3 Radiocarbon Dating

Perfect Number, excitingly:

“Now, we can talk how to find out the age of an old bone, or the remains of a piece of wood that was buried for long time, or anything else that was alive at some point.” After a pause, Perfect Number continues: “You remember that I mentioned Carbon-14 before? Well, everything that's alive, plants and animals, absorbs Carbon-14 into their tissue. When they die, Carbon-14 that is contained in the tissue starts to change and its amount starts to slowly shrink over time.”

Small Number, sighting:

“Finally, Perfect Number. You've been talking about other stuff forever.”

Perfect Number, smiling:

“Thanks for your patience, Small Number! You will see, how everything comes together.” Perfect Number kisses her brother's forehead and continues: “You remember the stone arrowhead that I showed you the other day on Straight Line Beach? The arrowhead was buried deep in the dirt. When we kept digging around, we found a remain of a piece of wood positioned in the way that we had no doubt that the piece was part of arrow's shaft.”

Small Number, inpetently:

“Please continue, Perfect Number. What happened next.”

Perfect Number, looking at her brother:

“You remember when we talked about a mathematical model as a formula?”

Small Number, confidently:

“Yes, you put my age in the formula and you get how tall I was then.”

Perfect Number, approvingly:

“Exactly, Small Number! We will do something very similar when figuring out the age of the old arrow that we found.”

Small Number, shaking his head:

“I still do not know what you will do.”

Perfect Number, smiling:

“There we go, Small Number! We know how much Carbon-14 there is in a living tree. This, together with the half-life of Carbon-14, determines a mathematical model that connects the amount of Carbon-14 in a dead tree and the time that has passed since the tree died.”

Perfect Number, seriously:

“When we get back to the university, in our lab we will measure the amount of Carbon-14 in the old shaft, plug that number in our formula, and get the number that will tell us how many years have past since that piece of wood was cut from the tree.”

Small Number, disapointingly:

“What, that's all? You just plug a number into a formula?”

Perfect Number, lovingly:

“Plugging the number into the formula was just one step in this process of finding the age of the old arrow. First, before you start digging, to even have a chance to find what you are looking for, you need to know where to dig. Then you need to dig very carefully and very patiently so that you do not damage the objects that may be buried in the dirt for long time. When you find something, you need to understand what that is. Then you need to record everything that you observe before collecting the object. You have to handle each collected artifact very carefully because often those objects are very fragile. Then in the lab, you have to use quite advanced technology to get the information you need. Creating mathematical models is a challenge itself. And even when you evaluate a number, you need to think if that number makes sense. Often, I need to read many pages of the related research, before I am confident to say anything about the object I found.”

Small Number, proudly:

“That is why I always say to Big Circle that you are the smartest person I know!”

Subsection 8.1.6 Challenge: Do You Homework

A teacher assigned homework to their class and told the students that on each day after the first they must do twice the number of problems they had done so far. If at the end of five days, a student had completed one third of the problems, how long will it take the student to do all of the homework problems?

Answer: Six days.