Ontario Alignment By Overall Expectation
Grade 4 [Number Sense and Numeration - NS3]
Grade 5 [Number Sense and Numeration - NS3]
Grade 6 [Number Sense and Numeration - NS1]
Grade 7 [Number Sense and Numeration - NS3]
Grade 8 [Number Sense and Numeration - NS3]
MFM1P [Number Sense and Algebra - NA1]
MPM1D [Number Sense and Algebra - NA2]
Sparking Curiosity to Promote Conceptual Understanding of Multiplication
In this real world math task, we’ve attempted to create a task that is accessible by students in Kindergarten and primary grade levels, but could be extended into the later grades by simply asking more complex questions with intentionality.
So let’s get going!
Act 1: Sparking Curiosity
Show students this video.
While I typically have students do a rapid write of what they notice and what they wonder, if we are in a kindergarten through grade 3(ish) class, it’s likely that students are sitting with you on the carpet. In that case, it might make more sense for them to think of what they notice and wonder in their minds and then share out with their partners through a think, pair, share.
While Yvette Lehman and I were leading this task in a grade 3 classroom recently, we intentionally told students that they could notice and wonder ANYTHING and EVERYTHING that comes to mind with a big BUT…
When sharing with your neighbours, do not share any numbers that you might have noticed or wondered. We were lucky that this pre-planned instruction came to mind right before we led the task, because otherwise, some students may have immediately came up with “9” as an answer and then the task as well as the sense making would likely be dead.
Here’s some of the “everything and anything” students noticed and wondered on chart paper (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.
- And many others…
At this point, we hadn’t landed on any particular wonder yet.
Act 2: Reveal More Information
We then said… let’s watch this video to see if we have any more noticings and 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…
At this point, we took their wonders and said:
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, so with your partner head to your stations and 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.
And, they were off to the races.
The materials on their table included:
- Connecting cubes.
- Square tiles.
- Integer chips (or “circles” with one colour on one side and another on the other).
- Large paper.
We would have liked to have relational rods on the table as well, but there were none available in the manipulative kit we were working with. Bummer.
My Challenge to YOU: Fuel Sense Making
What do you think the students might do with this task?
My challenge to you is to leave a comment below anticipating what you think students might do in a grade 3 class.
You can also anticipate what would students in YOUR class do to convince you of the total number of scoops?
How might you modify this task to work in your classroom with your diverse learners?
If you want to be bold, test what you’ve anticipated by doing the task in your classroom and come back to report your thinking in the comments.
If we have enough people taking interest in this task, I’ll update this post with some of the intentional ways we fuelled sense making in this particular situation and we will share student work
DOWNLOAD THE TASK TIP SHEET & RESOURCES
Want to make sure this task goes off without a hitch?
Download the media resources and 2-page Hot Chocolate 3 Act Task Tip Sheet that you can print and have with you close by to ensure that you maximize your chances of Making a Math Moment That Matters for your students!
Act 3: The Big Reveal
After consolidating learning using student generated solution strategies and by extending their thinking intentionally, we can share what really happened with this video.
I hope you enjoy the task!
Don’t forget to take on the CHALLENGES I’ve set out for you.
Looking forward to hearing from you ALL in the comments! Remember, we don’t learn if we don’t reflect.
There are quite a few different directions you could go in after this.
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 27 glasses?
- How many scoops do you need for ANY number of glasses? How would you describe this?
- How many glasses of hot chocolate could you make if there were only 55 scoops of mix total in the container?
Cara had a great question about how we might ask a question that would illicit division, which had me thinking. Check out her comment and my suggestion here.
Can you think of others?
Post in the comments below!
Consolidating The Learning:
As mentioned in many of my tasks, the learning goal you have selected for a task may be different than what I have in mind. This is especially true when we consider that this task could be used in a variety of classrooms with varying levels of student readiness.
In a primary classroom, we might be focusing on helping students build their additive thinking skills with the hope to push their mathematical thinking vertically towards multiplicative thinking through equal groups.
As you can see in the image below, my 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. One of my learning objectives 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 the image below, 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.
By building an understanding of linear models and sophisticated strategies like making use of number lines and double number lines, students will be able to tackle more challenging problems as they move through the grades. Here is a photo of a consolidation in a grade 7 class where students were introduced to the double number line for the first time and could use it to not only solve the question of how many glasses you could make with 55 scoops, but also without resorting to reaching for the calculator:
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.
A Sample 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:
Another student might also see that they can scale both the scoops and glasses, but also leverages rate reasoning to see the rate of scoops per glass is 3 and the rate of gasses per scoop is 1/3:
Over time, the level of abstraction students are able to work with continues to increase and they may no longer require the scale that a double number line provides and instead moves to a ratio table. Note that the same values we can place on a double number line are in the ratio table, however we are more “free” to skip / jump values through ratio reasoning (scaling in tandem) and rate reasoning (constant of proportionality, or the rate):
Interestingly enough, in all of the classrooms that myself and Yvette Lehman have used this task in across grades 1 through 8, students who are promoted to use tools for thinking and representing their thinking like the double number line, are all able to come up with a correct result AND can convince you of their thinking – but we’ve yet to find a student who can actually name the operation that could have taken us to that same answer in one go:
Yes. It’s division. Not one student has been able to identify that we could have come up with the same result by simply dividing the number of scoops by the number of scoops per glass (55 scoops ÷ 3 scoops per glass). Furthermore, when we say it is quotative division, students have never heard that word before and are unaware that there are two types of division.
Over time, you can move from the double number line and ratio table to a proportion.
It should be noted that cross multiplication, the magic circle, y-thingy-thingy, or any other tricks are not necessary by this stage because students are visualizing the double number line in their head and are very flexible with ratio and rate reasoning.
Lastly, we can think algebraically:
y = 3x
Where x represents the number of glasses and y represents the number of scoops.
We could graph and have all kinds of fun from here.
So. What is the learning objective for this task in your class? Let me know in the comments!
New to Using 3 Act Math Tasks?
Download the 2-page printable 3 Act Math Tip Sheet to ensure that you have the best start to your journey using 3 Act math Tasks to spark curiosity and fuel sense making in your math classroom!
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Grade 2 [Measurement - M1, Number Sense and Numeration - NS1, Number Sense and Numeration - NS2, Number Sense and Numeration - NS3]
Grade 3 [Measurement - M1, Number Sense and Numeration - NS1, Number Sense and Numeration - NS3]
Grade 4 [Measurement - M1, Number Sense and Numeration - NS1, Number Sense and Numeration - NS3, Patterning and Algebra - PA2]
Grade 5 [Measurement - M1, Measurement - M2, Number Sense and Numeration - NS1, Number Sense and Numeration - NS3, Patterning and Algebra - PA2]
Grade 6 [Data Management and Probability - DP3, Measurement - M1, Measurement - M2, Number Sense and Numeration - NS1, Number Sense and Numeration - NS2, Number Sense and Numeration - NS3, Patterning and Algebra - PA1, Patterning and Algebra - PA2]
Grade 7 [Data Management and Probability - DP3, Geometry and Spatial Sense - GS1, Measurement - M1, Measurement - M2, Number Sense and Numeration - NS1, Number Sense and Numeration - NS2, Number Sense and Numeration - NS3, Patterning and Algebra - PA1, Patterning and Algebra - PA2]
Grade 8 [Data Management and Probability - DP1, Data Management and Probability - DP3, Geometry and Spatial Sense - GS2, Measurement - M1, Measurement - M2, Number Sense and Numeration - NS1, Number Sense and Numeration - NS2, Number Sense and Numeration - NS3, Patterning and Algebra - PA1, Patterning and Algebra - PA2]
MAP4C [Mathematical Models - MM1, Mathematical Models - MM2, Mathematical Models - MM3]
MAT1LMAT2LMBF3C [Data Management - DM1, Data Management - DM2, Geometry and Trigonometry - GT1, Geometry and Trigonometry - GT2, Mathematical Models - MM1, Mathematical Models - MM2, Mathematical Models - MM3]
MCF3M [Exponential Functions - EF2, Quadratic Functions - QF1, Quadratic Functions - QF2, Quadratic Functions - QF3, Trigonometric Functions - TF1, Trigonometric Functions - TF3]
MCR3U [Characteristics of Functions - CF1, Characteristics of Functions - CF2, Exponential Functions - EF2, Exponential Functions - EF3, Trigonometric Functions - TF3]
MCT4C [Exponential Functions - EF1, Trigonometric Functions - TF3]
MCV4U [Derivatives and Their Applications - DA2]
MDM4U [Counting and Probability - CP2, Organization of Data For Analysis - DA2, Probability Distributions - PD1, Statistical Analysis - SA1, Statistical Analysis - SA2]
MEL4EMFM1P [Linear Relations - LR1, Linear Relations - LR2, Linear Relations - LR3, Linear Relations - LR4, Measurement and Geometry - MG1, Measurement and Geometry - MG2, Measurement and Geometry - MG3, Number Sense and Algebra - NA1, Number Sense and Algebra - NA2]
MFM2P [Measurement and Trigonometry - MT1, Measurement and Trigonometry - MT2, Measurement and Trigonometry - MT3, Modelling Linear Relations - LR1, Modelling Linear Relations - LR2, Modelling Linear Relations - LR3, Quadratic Relations in y = ax^2 + bx + c Form - QR1, Quadratic Relations in y = ax^2 + bx + c Form - QR2, Quadratic Relations in y = ax^2 + bx + c Form - QR3]
MHF4U [Characteristics of Functions - CF3, Exponential and Logarithmic Functions - EL2, Exponential and Logarithmic Functions - EL3]
MPM1D [AG3, Analytic Geometry - AG1, Analytic Geometry - AG2, LR1, LR2, LR3, MG1, MG2, MG3, NA1, Number Sense and Algebra - NA2]
MPM2D [AG1, AG2, AG3, QR2, Quadratic Relations - QR3, Quadratic Relations - QR4, T2, T3]
Functions [F-BF.1, F-BF.3, F-IF.4, F-LE.1, F-LE.2, F-LE.3, F-TF.5]
Geometry [G-C.5, G-C.8, G-C.9, G-GMD.3, G-GMD.4, G-GPE.4, G-GPE.5, G-GPE.7, G-MG.1, G-MG.2, G-SRT.11]
Grade 1 [1.NBT.4, 1.OA.1]
Grade 2 [2.NBT.5, 2.OA.2]
Grade 3 [3.NBT.2, 3.NF.1, 3.NF.3, 3.OA.1, 3.OA.5, 3.OA.9]
Grade 4 [4-MD.3, 4.MD.1, 4.MD.2, 4.NBT.6, 4.NF.3, 4.OA.1]
Grade 5 [5.MD.1, 5.MD.3, 5.MD.4, 5.MD.5, 5.NBT.6, 5.NF.1, 5.OA.1, 5.OA.2, 5.OA.3]
Grade 6 [6.EE.1, 6.EE.2, 6.G.1, 6.G.2, 6.NS.1, 6.RP.2, 6.RP.3]
Grade 7 [7.EE.4, 7.G.3, 7.G.4, 7.G.6, 7.RP.1, 7.RP.3, 7.SP.2, 7.SP.6]
Grade 8 [8.EE.1, 8.EE.5, 8.EE.6, 8.EE.7, 8.EE.8, 8.F.2, 8.F.3, 8.F.4, 8.F.5, 8.G.5, 8.G.7, 8.G.9, 8.SP.1]
Practice [MP.1, MP.2, MP.3, MP.4, MP.6, MP.7]
Statistics & Probability [S-ID.6, S-MD.4]