Section 12.1 Small Number and the Three Berries - Classroom Guide
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Suggested Grades: 2 – 5
Subsection 12.1.1 Mathematics
- Sequencing - following the order of steps in cooking vs. following the order of operations
- Measuring - the amounts of the necessary ingredients; time
- Mathematics - proportions and double counting
- Solving a puzzle - mathematical thinking and understanding of the question and the given conditions
Subsection 12.1.2 Mathematical Vocabulary
- small, number, a lot, one, large, gradually adding, measure, how much, into, everything, two times, follow the recipe, how many handfuls, how many pinches, two, five, ten, fifty, five pieces of, four handfuls of, half, part of, moved, last summer, your age, divided, six, flattening, great, between, first, a few, placed in, faster, took from, each, both sides, equally
Subsection 12.1.3 Cultural Components
- Indigenous: Most Indigenous nations have some version of and ; kinship; love; ; , ; .
- General: Preparing your own meals vs. purchasing already made meals; ; ; .
Subsection 12.1.4 Mathematical Observations (Video)
Opening scene: Reflection of the moon in the ocean. The shape of the moon; shapes of the hats; patterns on the capes
0:13 - Digits of the building number are first five Fibonacci numbers. Lines of symmetry and parallel lines. Other shapes? Who is taller? Who is younger? How many apartments are there in the building?
0:25 - Describe the floor pattern and the tablecloth pattern. Can we be sure how many chairs are there in the kitchen?
0:39 - How many ingredients did Grandma use? Do you notice any symmetrical objects? Any geometrical shapes?
0:54 - “Grandma, how come that you didn’t measure how much flour, salt, and baking powder you put into the bowl? When I help my mom with cooking, she asks me to measure everything two times so that she is sure that we follow the recipe.”
1:19 - “I know how many handfuls of flour and how many pinches of salt I need if I am making bannock for two, five, ten, or fifty people.”
1:37 - “She taught me that for five pieces of bannock, I would need four handfuls of flour.”
2:08 - How would you describe the shape of dough? The shape of the bowl?
2:24 - How would you describe the shape of the canoe? Can you estimate the number of the rocks by the beach? What is the shape of the fireplace? How would you describe the mutual position of the two big logs? How many different kinds of berries did grandma mention?
2:44 - Grandma divided the dough into six balls and started flattening the first ball with the palm of her hand.
3:06 - “Both my Grandpa and my Grandma were great storytellers. I remember sitting between them on the beach, staring at the stars and listening their stories about the ways of our people.”
3:24 - What is the time on the clock? What do you think how much time did Grandma need to fry bannock?
3:52 - How would you describe the shape of each piece of bannock?
4:04 - Grandma's riddle
All - How did you like the sound of drumming in the film? Can you repeat the pattern that the drummer was making?
Subsection 12.1.5 Answer: How many handfuls of flour did Grandma need to make six pieces of bannock? Also, how could two father bears and two son bears share three berries equally?
Small Number to Grandma:
“Grandma, you said that for five pieces of bannock you would need four handfuls of flour. This means that you need less than a handful of flower for one bannock. So, because for five pieces of bannock you need four handfuls of flour and for the sixth piece you need less than a handful of flower, I think that altogether you will need just little bit less than five handfuls of flour.”
Grandma:
“That’s very good thinking, Small Number. But can you be more precise?”
Small Number looks at Grandma, thinking. After a few moment, he says:
“I know that you can calculate this in your head Grandma, but I'll use the calculator in my phone, OK?”
Small Number takes his phone and starts punching numbers, thinking out loud:
“If I divide \(5\) by \(4\) I get \(0.8\text{.}\) This means that for a piece of bannock Grandma needs \(0.8\) handfuls of flour. Now, because Grandma wants to make six pieces, I multiply \(6\times 0.8=4.8\text{.}\) Grandma, you will need exactly \(4.8\) handfuls of flour. You see that I was right, you will need just little bit less than five handfuls of flour.”
Grandma, proudly:
“Very good, Small Number? But what about my riddle? Do you know the answer?”
Small Number scratches his head:
“I don't know. When I divide \(3\) by \(4\text{,}\) my calculator shows \(0.75\) but that cannot be the answer.”
Grandma, smiling:
“Let me give you a hint: When your grandpa, your dad, and you go to see a game together, how many dads and how many sons are there? How many tickets do you need to buy?”
Subsection 12.1.6 Discussion/Activities
Exercise 12.1.1. What is a handful?
Teacher to the class: “Do you remember how Grandma measured the amount of flour she needed to make six pieces of bannock?”
Alice: “Grandma said that for five pieces of bannock, she would need four handfuls of flour.”
Teacher to Alice: “Thank you, Alice!” And then to the class: “What do you think that a ‘handful of flour’ is?”
Bob: “I think that a ‘handful’ means how much flour Grandma can grab from the bag with one hand.”
Teacher: “That is a very good explanation, Bob. Thank you!” To the class: “Let me ask you a question. Say that Small Number grabs a handful of flour from the bag. Do you think that he grabbed more or less flour than Grandma? Or, maybe the same amount? Can we be sure?” [Teacher moderates the follow up discussion accordingly.]
Teacher to the class: “It seems that we cannot agree. Some of you think that Small Number would grab more flour than Grandma, some of you think that he would grab less.”
(After a pause) “I'd like us to do an experiment. I need four volunteers!”
Teacher to the class, after a couple of minutes: “OK, Cory, Dženana, Emma, and Felix, please come forward.”
(While students are approaching the front of the classroom, teacher explains) “This is what we are going to do. Here I have a scale, a bag of flour and an empty container.”
Teacher to Cory and Dženana, while drawing a table on the whiteboard: “Cory, you will do all weighing and each time you will read the weight out loud. Dženana, you will enter Cory's measurements in this table.”
Teacher to Cory: “Cory, please measure the weight of the empty container and leave the container on the scale.”
Cory (putting the empty container on the scale and reading out loud): “The weight of the empty container is 330 grams.” Dženana writes 330g in the corresponding cell in the table.
Teacher to Cory and Dženana: “Good job!” Turning to Emma: “Emma, please take a handful of flour from the bag and put it into the container.”
After Emma completes the task, Teacher to Cory: “Please read the weight that the scale is displaying now.”
Cory (reads out loud): “The scale is displaying 360 grams.” Dženana writes 360g in the corresponding cell in the table.
Teacher, turning to Felix: “Felix, please take a handful of flour from the bag and put it into the container.”
After Felix completes the task, Cory reads: “395 grams.”
Teacher to the class: “Now, I will take a handful of flour from the bag and put it into the container.” A couple of minutes later: “Cory, what is the weight of the container and flour in it?”
Cory: “The weight is 445g.”
Teacher to Cory, Dženana, Emma, and Felix: “Thank you so much, guys!” To the class: “Look at the table on the board:”
“Can you tell me what is the weight of Emma's handful of flour? What about the weight of Felix's handful of flour? What about the weight of my handful of flour?”
Grace: “The weight of Emma's handful of flour is 30 grams and Felix's is 35 grams!”
Teacher: “Very good, Grace! How did you calculate this so quickly?”
Grace: “The empty container weighs 330 grams. When Emma emptied her hand into the container, the weight was 360 grams. Then I calculated in my head: \(360-330=30\) grams. If add 35 to 360 I get 395. So, Felix's handful of flour weighs 35 grams.”
Teacher: “Wow, Grace! That is excellent thinking!” To the class: “Who will tell me the weight of my handful of flour?”
Harpreet: “50 grams! I calculated \(445-395=50\text{.}\)”
Teacher to the class: “You really make me feel proud! But, are you surprised that my handful of flour weighs more than Emma's and Felix's?”.
Irha: “Your hand is bigger, so you can grab more flour from the bag.”
Teacher: “Exactly, Irha! A person with a bigger hand can grab more flour that a person with a smaller hand. So when we say a handful of flour, do we really know how much flour this is?”
Jaspreet: “No, we can't. I am the youngest in my family and all my siblings have bigger hands than I do. Then my mom's hand is bigger that theirs. And my dad's hand is bigger than my mom's.”
Katarina: “I agree with Jaspreet. I think that this is the reason that we really do not use handful as a unit of measurement.”
Teacher: “I would not be able to say it better, Katarina! Thank you! Next class we will talk more about the best ways of measuring substances like flour or sugar.” After a pause: “But, how come that Grandma uses handful as a measure to make her delicious bannock for so many years? What do you think?”
Teacher encouragingly: “C'mon, guys! Just tell me what you think!”
Lauren: “I think that because she uses her own hands all the time, she always grabs the same amount of flour. So that is the amount Grandma needs, and it is not important to her what the weight of the flour is.”
Maria: “And because Grandma's Grandma had a hand of about the same size, the old recipe still works!”
Exercise 12.1.4. Weight vs. Volume Measurements.
Teacher to the class: “As we have seen in the Small Number film, bannock is one of the traditional dishes of the Indigenous Peoples of Canada.” Pointing to the slide on the board:“Here are the ingredients to make eight pieces of bannock from a recipe that the members of a Métis group from Alberta have been using:”
3 cups | 390 g flour
3 tsp | 15 g baking powder
3 tbsp | 45 ml oil
1 cup | 240 ml cold water
“Observe that there are two numbers associated to each ingredient. What do you think that means?”
Nickolas: “When my Grandma is cooking, she always uses her measuring cup. So, I know that a cup is some kind of measure.”
Olga: “My Grandma does the same, but my dad likes to use a small kitchen scale too. He says that the scale is much more precise than the measuring cup.”
Teacher: “Thank you for sharing, Nickolas and Olga! And yes, cup is a unit of measurement. But things, are a bit complicated here.”
Teacher after a short pause: “You see how in the first line of the recipe it says that 3 cups of flour is the same as 360 g of flour. In the last line, the recipe asks for 1 cup of water, which is the same as 240 ml of water.” After another short pause, “What is the meaning of the letter ‘g’ in the recipe? What is the meaning of ‘ml’ ?”
Priya: “I am quite sure that ‘g’ stands for ‘grams’, but I do know what ‘ml’ means.”
Teacher: “You are right, Priya: ‘g’ stands for ‘grams’. We read ‘ml’ as ‘millilitre’. Do you know what one millilitre is?”
Teacher continues after a few moments: “If you see a letter ‘m’ before some unit of measurement that means one thousandth of that unit. So one millilitre means one thousandth of one litre. Imagine that you divide one litre of something into thousand equal amounts. Then the measure of each of those amounts is one millilitre.”
Teacher to Priya, while writing on the board: “Priya, what do you think that ‘one mg’ means and how would you read it?”
Priya: “I think it means one thousand of one gram. Wow, that sounds like something very small. We read it one milligram, correct?”
Teacher: “Very good, Priya!” To the class: “But we need to go back to the bannock recipe. What do we measure in grams? What does 360 grams of flour mean? What do we measure in grams?”
Quinn: “We use grams to measure the weight of something.”
Teacher: “Thanks, Quinn. Yes, the gram is a unit of mass. So, the required weight of flour is 360 grams.” After a short pause: “Is this OK? Do you have any questions?”
Rohan: “So the cup is also a unit of mass?”
Teacher: “That is an excellent question, Rohan! But, please be patient a bit more. Things will be much clearer in a few minutes.” To the class: “What does 240 millilitres of water mean? What do we measure in millilitres? Or, maybe you are more familiar with litres. What do we measure in litres?”
Shifaa: “We use the litre to measure something that is liquid, like water, or oil, or milk.”
Teacher: “You are absolutely right, Shifaa. But maybe you do not know that the litre is is a metric unit of volume.”
Teacher: “To understand what this means, we will do a demonstration.” Placing on the top of the desk a bottle of liquid labeled ‘1L’ and a plastic cube shaped box with open top. “Here I have a bottle with 1 litre of limonade and an open top box. This is important: the length, the width, and the height of this box are all measured exactly 10 centimetres.”
Teacher: “Now, I am going to pour the llimonade from the bottle into the box. Observe what will happen.” Starts pouring the limonade while students are watching. When she places the empty bottle on the desk: “What do you observe?”
Tyra: “The bottle is empty, but the box is full with limonade.”
Teacher: “That is exactly that I wanted you to see! So the litre is the volume of the cube with all sides being exactly 10 centimetres.”
Tyra: “Is this true for any liquid? I mean, if I have one litter of olive oil, would this still work? I ask because oil looks thicker than lemonade.”
Teacher: “Another great observation, Tyra. And the answer is a yes! When ever it says on the bottle ‘1 litre’ that means that you can empty the bottle into the 10cm \(\times\) 10cm \(\times\)10cm cube and that the cube be full of the liquid.”
Rohan: “Is the cup a unit of volume?”
Teacher: “Yes, it is, Rohan.” Then, teasingly to the class: “Can you guess what the volume measured by one cup is?”
Uwe laughingly: “240 millilitres - that is what the recipe is saying!”
Nickolas: “I remember that my Grandma said that one cup is about half a litter. That makes sense now!”
Teacher: “This is what I'd like you to remember, the litre and the cup are units of measure for volume of something. So when the recipe asks for three cups of flour, that means that it asks for the amount of flour with the volume that measures 3 cups, i.e., the amount of flour that will filled up. The recipe says that the weight of this amount of flour is 390 grams.”
Teacher: “What is even more interesting is that originally, long time ago, the kilogram was defined as the mass of one litter of water. So, let us do some mathematical thinking here: what is the mass of 1 cup of water?”
Vicky: “This is easy. By the recipe, the volume of one cup is the same as the volume of 240 millilitres. Since the mass of one litre of water is one kilogram, and since one kilogram has 1000 grams, the mass of one cup of water is 240 grams.”
Teacher: “Wow, Vicky! That was great” Teacher starts clapping and the class joins.
Teacher, after a pause: “Do you remember when Olga said that her dad uses a scale because it is more precise than the measuring cup? I'd like to show you, why he was right.”
Teacher: “This is an experiment that I did in my kitchen yesterday when I was preparing for our class.”
“First I weighed the empty measuring cup. As you can see, the cup weighted 200g.”
“Next, I poured 1 cup of flour into the measuring cup. You can see clearly that the level of flour is marked by 1. The mass of the cup and the flour is 330. So the mass of one cup of flour is 130 grams.”
“Then I took a ladle and started pressing down flour in the measuring cup.”
“As you can see on the photo, the volume of flour in the measuring cup is now less than one cup, but the mass is the same, 130 grams.”
Teacher: “So, what kind of conclusion can you make based on my experiment?”
Olga: “Now I understand why my dad trusts the scale more than the measuring: It is possible that two amounts of some material, like flour in your photos, have the same mass but different volumes.”
Exercise 12.1.5. Fibonacci Numbers.
Remind students about the number of Grandma's building: 11235. Explain to the class that \(1, 1, 2,3 ,5\) is the beginning of the most famous mathematical sequence that is based on a pattern that we can find in the nature. Mention that the sequence is called the Fibonacci sequence and that its terms are called Fibonacci numbers.
Ask your students:
To find out the pattern that determines the Fibonacci sequence. [Starting with the third term each term in the sequence is the sum of the previous two terms.]
To evaluate five terms of the sequence that follow the number 5.
To find in the library, or to ask their siblings, parents/guardians to help them find on the web where Fibonacci sequence appears in the nature.
To find out who was Fibonacci and when he lived.
Exercise 12.1.6. General.
Assign to your students the following task:
Talk to your parents or guardians or grandparents and offer to help them with the preparation of a meal.
Suggest that you measure and/or count all ingredients necessary to prepare the meal.
Take a selfi of you and your parent/guardian/grandparent in the kitchen.
Write a detailed report about your experience. The report must include: the name of the meal; all ingredients together with their quantities/amounts in the appropriate metric units; time needed to prepare the meal; your selfi; and anything else that you find important.
Subsection 12.1.7 Challenge: The Land of If–Then
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If \(4\) were \(6\text{,}\) then \(10\) would be …
If \(5\) times \(4\) were \(100\text{,}\) then one quarter of \(20\) would be …
If \(4\) times \(6\) were \(40\text{,}\) then one third of \(10\) would be …
If \(3\) times \(5\) were \(20\text{,}\) then the square of \(6\) would be …
If two thirds of a number are added to \(24\) the number will be doubled. What is the number?
Answers: 1. If \(4\) were \(6\text{,}\) then \(2\) were \(3\) and \(10\) would be \(15\text{.}\) 2. If one quarter of \(100\) were \(5\text{,}\) then one quarter of \(20\) would be \(1\text{.}\) 3.\(2\text{.}\) 4.\(48\text{.}\) 5.\(18\text{.}\)