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Hot Chocolate
Hot Chocolate
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Content
Content
Expectations:
Expectations:
B2.9 use the ratios of 1 to 2, 1 to 5, and 1 to 10 to scale up numbers and to solve problems
Key Concepts:
Ratios deal with multiplicative relationships in a variety of contexts. For example:
If the ratio of vowels to consonants in a word is 1 to 2, then there are twice as many consonants in the word as there are vowels.
If a set has 1 red object and 5 blue objects, then the ratio of red to blue is 1 to 5.
If there are ten times as many birds as there are kittens in the pet store, then the ratio of birds to kittens is 10 to 1 or the ratio of kittens to birds is 1 to 10.
A ratio of “a to b” can be written symbolically as a:b, for example, 1 to 2 can be written as 1 : 2.
To scale up means to multiply a starting number by a factor. An application of this is scaling a number line. To support this understanding, use a double number line with one number line showing the starting values and the second number line showing the scaling. For example, scaling the numbers 0 to 4 by 5 is illustrated on a double number line below.
Learning Goal
Learning Goal
We want students to understand...
We want students to understand...
use multiplicative thinking (ratios/double number line) to solve real world problems.
Success Criteria
Success Criteria
I can try more than one way to solve a problem
I will stop and think when I get stuck
I will remain positive when I feel stuck
I can use a variety of strategies (such as a double number line) to solve problems
Materials
Materials
Connecting cubes
Cuisenaire rods
Square tiles
Integer chips (or “circles” with one colour on one side and another on the other)
Large paper with markers
Pedagogy
Pedagogy
Number Talk Resources
Number Talk Resources
Fosnot:
Fosnot:
A2: Bag of Apples, Page 12A3: Baskets of Strawberries, Page 13A4: Baker's Trays, Page 14B13: The Open Number LineC1: Cars and Tires, Page 34
How many shapes do you see?
How many [green shapes] do you see?
Count the number of green triangles that could cover the whole construction. How many blue parallelograms could cover the whole area? What’s the ratio of blue pieces to red pieces? Green to blue?
Minds On
(This lesson is adapted from "Tap Into Teen Minds" - Hot Chocolate by Kyle Pearce)Minds On
Show the following visual. Ask students to think of what they notice and wonder in their minds and then share out with their partners through a think, pair, share. When sharing with your neighbours, do not share any numbers that you might have noticed or wondered. Jot down student thinking on chart paper.
Some possible wonderings could include: (remembering that nothing with numbers would not be shared yet):
I notice hot chocolate.
I notice a glass.
I wonder if the hot chocolate is already made in the container?
I wonder if someone is making hot chocolate?
I wonder if that is YOU in the video.
Next, show the students the next video to see if you can build more wonderings.
After watching this clip, students had more noticings and wonderings:
I notice more glasses.
I wonder if the person is going to make more hot chocolate?
I wonder how many scoops they’ll need. I think I know!
I wonder who is going to drink the other hot chocolates?
And many others…
Action!
Action!
At this point, take their wonderings and say:
I think this person is going to be making 3 whole glasses of hot chocolate!
Why don’t we start by thinking in our minds about how many scoops we needed for the first glass and then how many scoops we’ll need for ALL 3 GLASSES?
BUT – we don’t just want to know how many. We want you to convince us of how many in any way you want.
Manipulatives are on your tables. Try to make a plan of how you’re going to convince us of how many scoops were needed in total to make 3 hot chocolates.
The materials on their table should include:
Connecting cubes
Cuisenaire rods
Square tiles
Integer chips (or “circles” with one colour on one side and another on the other)
Large paper with markers
**Distance learning students can use virtual manipulatives to support their explorations.
Extension Questions:
Some you might consider are:
How many scoops do you need for 3 whole glasses and 1 half glass of hot chocolate?
Can you show your thinking using additive and multiplicative thinking?
How many scoops do you need for [number of] glasses?
How many scoops do you need for ANY number of glasses? How would you describe this?
How many spoonfuls would we need for "our" class to each have a glass of hot chocolate?
Consolidation ideas
Consolidation ideas
Bring students together as a group for the consolidation. Ask students to prepare to share their strategies and the possibilities they have found. Have several students share their math thinking with the group. Elicit from the students the ratio of spoonfuls of chocolate to water and uncover how students can use a double number line as a strategy to support solving the problem.
Possible prompts to surface student thinking:
How did you decide to start? What was your first step?
What strategies do you see?
How did you organize your thinking?
What strategies did you use to help solve the problem?
How did the manipulatives help you solve the problem?
After consolidating learning using student generated solution strategies and by extending their thinking intentionally, we can share what really happened with this video:
Notes for consolidating the learning:
Notes for consolidating the learning:
As you can see in the image below, some students might be at a level of abstraction requiring tangible objects (i.e.: the actual glass and scooping hot chocolate mix or sand or similar into glasses). Other students might be able to DRAW the glasses and spoons. A learning objective might be to try and help students make connections between more abstract drawings such as the representation we see when students draw ovals with 3 circles inside to represent the 3 glasses and 3 scoops.
The importance of moving towards this stage when students are ready is to help them see that while drawing glasses and spoons is very helpful for this context, it is not so helpful when the context changes. When we use something more like the ovals with circles inside, that very same visual/drawn representation can be used to model ANY context involving 3 groups with 3 items in each group. This can be helpful prior to moving on to the abstract representation using symbols like 3 x 3 = 9.
Depending on the students in the room, they might be ready to take a set model of say placing 3 concrete objects in 3 groups (i.e.: linking cubes, counters, etc.) and with a bit of a push, might be able to begin creating linear models or “number trains” as Cathy Fosnot calls them in her Minilessons books. In this image, a student is sharing her strategy of building a number train to help her determine how many scoops are in 3 glasses.
What is useful to note is that while she is explicitly counting the number of scoops, the 3 groups of 3 colours used helps her keep track of how many glasses she has as well. This is a huge step in the direction of proportional reasoning where she is now working with two quantities that scale in tandem.
By helping students to make the connection between set models (i.e.: random counters and concrete materials put into “piles”) and linear models like number lines and in this case, a double number line. When students are ready, they can begin to use their concrete number trains to trace a single (and eventually double) number line on paper with the goal of building more sophisticated strategies that will help in the area of multiplicative thinking and algebraic reasoning.
When students are given opportunities to use tools “with legs” that are far stretching for thinking and representing their thinking, this lowers the floor on tasks and also raises the ceiling through the use of extension problems and prompts.
Supporting a Continuum
Supporting a Continuum
If we are to look at a developmental continuum with a focus around the double number line (linear model) we can see that connections can easily be made to continue mathematizing vertically.
Here, we make use of a double number line and a student might be skip counting in order to determine the number of glasses that can be made:
Here, a student makes use of the double number line, but begins to notice that they can cleverly start scaling in tandem by doubling the number of glasses and scoops:
Independent Task / Assessment Opportunities
Independent Task / Assessment Opportunities
Notice the strategies that students are using as they explore the problem.
Possible Independent Tasks:
At the zoo, the ratio of gorillas to monkeys is 1:5. There are 5 gorillas in the zoo. How many monkeys are there?
At the zoo, the ratio of snakes to lizards is 1:10. There are 20 lizards. How many snakes are there?
At the zoo, the ratio of lions to elephants has to be 2:6. One of the lions is released back into the wild. How many elephants should be kept?
Knowledge Hook
Technology
Technology
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