The Learning that Led to Today

“You aren’t these kids’ first teacher, and you won’t be the last.”

-Jennifer N.

When I was an early career teacher, teaching 4th grade, I wanted to teach my students everything. I convinced myself that this was a drive born out of my ambition. I wanted to teach them to think critically about texts! I wanted to teach them to write stories with poetic details that catch in your throat, and how to add fractions with unlike denominators, and why we have seasons, and how immigrants have shaped our nation and yet still face discrimination, and… and…

One of the fifth grade teachers sat me down. “Jenn, you aren’t these kids’ first teacher. And you won’t be their last.”

Now, as a K-8 math specialist, I get to see learning progressions in action every day. It’s beautiful to see how learning builds over time, and I wish all of my colleagues that teach one grade level had the opportunity to see how different stories build over the course of elementary and middle school. We are a team.

Today, in 7th grade, we were working on determining the volume of complicated prisms.

Desmos 6-8 Math: Grade 7, Unit 7, Lesson 11

How did we get here?

Kindergarten: Shapes Take Up Space

Two months ago in Kindergarten, students were deep into their investigation of three-dimensional figures. Through play, students discovered that shapes are defined by their properties, and so their name doesn’t change based on their orientation (CCSS K.G.A.2). This is critical in 7th grade as students are determining a base on a prism.

Kindergartners also compare and contrast shapes (CCSS K.G.B.4). With this, they start to realize that some shapes are flat, and others take up space.

What It Looks Like

Students in our kindergarten classroom examined 3D blocks. They looked at how we can identify faces, many with familiar shapes like rectangles and triangles.

Then, they tried to build the blocks out of multilink cubes. (From TERC Investigations: Grade K, Unit 5, Lesson 1.7)

Students recognized which shapes were easiest to build out of multilink cubes (rectangular prisms, shapes with layers) and which were trickier (prisms, shapes with triangles). We can examine why a shape is easier to build using a middle school geometry lens, e.g. the angles of the shapes are congruent.

Grade 3: We Can Measure Flat Space

There is learning in grades 1 and 2 that contributes critical ideas to a 7th graders layered understanding of volume, including the fact that we can decompose and compose 3D objects, and how we measure length. However, for the sake of brevity, let’s focus how third graders learn about measuring flat space.

In third grade, students recognize that flat shapes (plane figures) have area, and that we can measure that using square units that are tiled across the space. (3.MD.C.5, 5a, and 5b). Students may calculate the area by counting (3.MD.C.6) or by using addition, skip counting, and multiplication (3.MD.C.7). Students apply learning they did in primary grades about decomposing shapes to calculate the area of rectilinear figures (which are composed entirely of right angles, and thus break down nicely into rectangles).

What It Looks Like

From Illustrative Mathematics: Grade 3, Unit 2, Lesson 3

To launch a third grade lesson about area, we showed these four images to students and asked them “which one doesn’t belong?”

  • “A doesn’t belong, because it’s the only one that where you can see diagonals. All of the other ones have tiles that are stuck together.”
  • “B doesn’t belong because it’s the only one with a big block of white. The other ones have lots of blue squares.
  • “C doesn’t belong because it’s the only one that looks perfect and even. All of the other ones have missing parts of gaps.”
  • “D doesn’t belong because it’s the only one that looks wrinkled and all over the places. There are wiggled boxes overlapping. All of the other ones are flat.”

From there, students discover how important it is to have square units tiling a space in order to measure area. This relates to the ‘perfectly packed’ units we will see in volume later.

Above, third graders are determining how many square tiles it will take to cover a rectangle. (This lesson is from TERC Investigations: Grade 3, Unit 1, Lesson 3.3.) The student on the left tiled the rectangle, and then counted by ones. The student in the middle discovered he could count faster if he made equal rows: 5, 10, 15, 20, 25, 30, 35. The third student realized that he could make rows of 5, and determine how many rows of 5. To synthesize this lesson, I shared the work of these three students, and other students started to make the connection between the last student’s work and multiplication: 7 rows of 5 is 7 x 5, or 35. This connection to multiplication would become especially critical.

Grade 5: We Can Measure The Space Inside

Students continue making connections to multiplication and area in fourth grade. Then, in fifth grade, they’re ready to make connections between multiplication and measuring the space inside of a solid figure. (5.MD.C.3). Instead of measuring with perfectly tiled unit squares, which we first learned about in third grade, we are now measuring with perfectly packed unit cubes.

Most critically, fifth graders are tasked with understanding and applying two formulas for volume: l \cdot w \cdot h and also b \cdot h. Students recognize that they can multiply the three defining attributes (length, width, and height) of a rectangular prism to calculate the volume, and also that they can find the area of the base and then multiply it by the number of ‘layers’ (the height), and why these are equivalent.

What It Looks Like

Fifth graders are wrestling with solid figures, sometimes tangibly (blocks, or built out of multilink cubes), and sometimes in 2D representations. Here, in an activity from Illustrative Mathematics Grade 5, Unit 1, Lesson 1, a student is wrestling with how to compare figures. The ambiguity here — “which is bigger?” — is intentional, to draw out student ideas about size and measurement.

This student speaks Mandarin natively. She did not know the English word for “layer,” so she recorded it here in Mandarin (层). “[these blocks are the] same, because the first is two layers (层), every layer (层) have 4 blocks, so 4 x 2 = 8 blocks. The second has two layers (层), the top has 3 blocks and the bottom has 5 blocks, 3 + 5 = 8.”

With this one problem, the student shows how we can think flexibly about volume as both multiplicative (4 x 2) and also additive (3 + 5). This understanding will be especially important in seventh grade as students wrestle with the volume of more complicated composite figures.

The various equivalent expressions to represent the volume of these figures represent some beautiful ideas about the properties of multiplication. For example, 3 \cdot 7 \cdot 4 = (7 \cdot 4) \cdot 3, because, per the associative property, we can group numbers in any way and the product remains the same.

Grade 7: Moving Beyond “Layers”

As an early career teacher, teaching fourth and fifth grade, I emphasized that volume could be calculated with the formula l \cdot w \cdot h. Rectangles are two dimensional, so we use two factors, and prisms are three dimensional, so we use three factors. It’s a lovely connection. (Minds were blown.)

Then, when the common core came out, I gained new appreciation for use of area of the base multiplied by the height. It is not only equivalent but connects better to work with prisms without rectangular bases that happens in middle school — something I couldn’t see as easily when I was immersed fully in the world of 4th.

Previously, students have calculated the volume of right rectangular prisms and composite figures made of rectangular prisms. The formula l \cdot w \cdot h works well for this. However, one look at a triangular prism shows us why l \cdot w \cdot h has its limitations. That l \cdot w represents the area of the rectangular base, but the triangle base would be calculated with \frac{1}{2}(l \cdot w) \cdot h or maybe even \frac{l \cdot w}{2} \cdot h.

But that only works for triangular prisms. What about prisms with other bases?

The area of the base multiplied by the height will consistently work for prisms.

And how do we know if two shapes are prisms? In kindergarten, students talked about how some shapes come to a point. In fifth grade, students may conceptualize the right rectangular prisms we worked with as starting with a base and stretching up and up and up, adding layers along the way.

Seventh graders identify and describe cross-sections, the two-dimensional figures that result from slicing three-dimensional figures. (7.G.A.3) High schoolers build on this work when they explore conic sections. However, this learning also serves a critical purpose in 7th grade. Students here learn that we can add nuance to the idea that prisms are layers of the base: indeed, we can now define prisms by the fact that they have parallel bases — a word familiar from a study of plane figures in third, fourth, and fifth grade — and identical parallel cross-sections.

The images above show a triangular prism built out of magnatiles. (I have no financial stake in magnatiles, but I will say that they work brilliantly for 7th grade geometry!) Turning the figure to the side reveals that we can compose it out of stacked layers of congruent equilateral triangles — indeed, each cross-section parallel to the base will yield a figure congruent to the base! It doesn’t matter where you slice. This is also an application of the concept of infinite density, introduced in the study of decimals (6.NS.C.6).

What This Looks Like

In Desmos 6-8 Math (Grade 7, Unit 7, Lesson 9: Slicing Solids), students play around with describing the cross sections of different prisms and pyramids. Here, in activity 2 (screen 6), students compare and contrast the cross-sections of figures with the same base.

Students might describe that both sections are hexagons, or that they have congruent bases. Students might also draw on their understanding of scale copies (7.G.A.1) to say that all of the cross sections produce scale copies of the bases, however, the pyramid produces smaller and smaller copies. This understanding can be built on in eighth grade, when students study similarity (8.G.A.4).

Here are three responses from seventh grade students at my school:

STUDENT 1
Similarities both can have hexagon cross sections.
Differences the pyramid's cross section get smaller.
STUDENT 2
Similarities: they each have six sides, the cut is a hexagon.
Differences: The hexagon for the pyramid changes size as you move it, and the prism doesn't.

STUDENT 3
Similarities: they hold the same shape.
2. Differences: they have different quantity of area.

Which Brings Us Back to Today…

Now that students have discovered more about the structure of prisms, they see why we can use the formula area of the base \cdot height to calculate the volume of various prisms. Several lessons later (Desmos 6-8 Math: Grade 7, Unit 7, Lesson 11: More Complicated Prisms), students are sketching the base and using strategies from developed in sixth grade to determine the area of composite figures, e.g. by decomposing the shape into familiar polygons or enclosing the shape and subtracting away the negative area.

Here are a few work samples from our seventh graders. Check out all four samples using the arrows to navigate.

Students then calculated the volume.

In Praise of Coherence

I am not the first teacher my students have, and I am not the last.

As an elementary classroom teacher, I found it surprisingly easy to develop a laser focus for the experiences happening with my room. Teaching my students to add and subtract fractions with unlike denominators seemed like a natural extension of the work we were doing with equivalence — so I taught it. Never mind that this was taught in fifth grade, and adding it to the fourth grade curriculum meant that we had to cut something else. I was using my professional judgment about what was important.

Now multiply that by 250,000 educators teaching fourth grade in the US. Our fragile educational ecosystem starts to break down — fast.

This does not mean that I never make a connection for students to future learning, giving them what is essentially a “sneak preview” of what is to come. It also does not mean that every single student comes into 7th grade fully and completely ready, with all of the learning described in this post. Sometimes, students need to work with concepts from previous grade levels — whether it’s a quick reactivation of ideas or a just-in-time intervention — in order to access the grade level material. Even a coherent system is dealing with the nature of humans, and all of our beautiful complexities. We don’t learn in perfect lines.

Understanding the learning progression — the prior learning that contributes to today’s — helps me identify access points for students. What learning do we need to leverage? When a student is struggling, what concepts would help?

I have tremendous appreciation for the learning progressions built into our standards. As Bill McCallum, the lead writer of the Common Core State Standards in Mathematics, recently wrote in an Illustrative Mathematics blog post, “Coherent standards led to coherence of agreement among states about the progression of concepts and procedures across grade levels.” These standards are a beginning, and from there we can start to develop other coherent systems: curriculum, assessment, professional learning, etc.

It’s a vision of educators working together. We build mathematical ideas together. The learning the seventh graders did today was not without the help of educators who have taught them for the last 8+ years, and it is hopefully in service of work they will do at the high school. K-12 public education should be a glorious collaboration between all of the stakeholders.

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