Long Descriptions for Chapter Four
Long descriptions for complex figures and tables in Chapter Four of the Mathematics Framework for California Public Schools, Kindergarten through Grade Twelve.Figure 4.1: The Why, How and What of Learning Mathematics
| Why
Drivers of Investigation |
How
Standards for Mathematical Practice |
What
Content Connections |
|---|---|---|
In order to… DI1. Make Sense of the World (Understand and Explain) DI2. Predict What Could Happen (Predict) DI3. Impact the Future (Affect) |
Students will… SMP1. Make Sense of Problems and Persevere in Solving them SMP2. Reason Abstractly and Quantitatively SMP3. Construct Viable Arguments and Critique the Reasoning of Others SMP4. Model with Mathematics SMP5. Use Appropriate Tools Strategically SMP6. Attend to Precision SMP7. Look for and Make Use of Structure SMP8. Look for and Express Regularity in Repeated Reasoning |
While… CC1. Reasoning with Data CC2. Exploring Changing Quantities CC3. Taking Wholes Apart, Putting Parts Together CC4. Discovering Shape and Space |
Figure 4.5: Shapes Task—How Do You See the Shapes Growing?
Shapes Task—How do you see the shapes growing? The growth patterns for three groups of square shapes (units) are pictured. The first has 4 units in three columns (1-2-1). The second has 9 units in five columns (1-2-3-2-1). The third has 16 units in seven columns (1-2-3-4-3-2-1). Source: From Ruth Parker; a task used in MEC courses (as cited by Mathematics Education Collaborative (2023)
Figure 4.6: Multiple Methods for Describing Growth Patterns
Six solution methods for describing growth patterns for a series of three shapes that grow from left to right. The first shape in the series shows four squares represented in three columns, with one in the first column, two in the second column, and one in the third column. The second shape in the series shows nine squares represented in five columns, with one in the first column, two in the second column, and three in the third column, two in the fourth column, and one in the fifth column. The third shape in the series shows 16 squares represented in seven columns, with one in the first column, two in the second column, and three in the third column, four in the fourth column, three in the fifth column, two in the sixth column, and one in the seventh column.
The “raindrop method” shows growth from the first to the second shape by adding one square to the top of each column, which visually is similar to raindrops dropping from the sky. Similarly, growth from the second to the third shape is shown by adding one additional square to the top of each column.
The “parting of the red sea” method visually looks like the middle column arriving between the columns to the left and right of it in the second and third shapes in the series. For example, in the second shape in the series (where the first two columns are similar to the first two columns of the first shape in the series), the third column of three squares visually drops in to the right of them. This new added third column pushes the second to the last and last columns of squares (which are similar to the second to the last and last column of squares from the previous shape) to the right.
The “bowling alley method,” similar to the raindrop method, shows growth from the first to the second shape by adding one square to the bottom of each column, which visually looks like a new line of arriving pins in a bowling alley, creating a larger triangular shape with each additional row. Similarly, growth from the second to the third shape is shown by adding one additional square to the bottom of each column.
With the ”triangular growth” method, the growth pattern across the three shapes can be seen as increasingly larger triangles. For example, the first shape shows a triangle with a base of three squares and a height of two squares, with one square at each of the three vertices. The second shape shows a triangle with a base of five squares and a height of three squares. The third shape shows a triangle with a base of seven squares and a height of four squares.
In the “volcano method,” the middle column of squares grows high and squares are added to the other columns like lava erupting from a volcano cone and flowing down the sides of the volcano to cover the columns to the left and right. This is similar to the raindrop method, starting the growth from the middle column.
Finally, the “square method” shows how the squares distributed across columns in each shape can be rearranged as a square in each new shape in the series. The first shape in the series can be rearranged to show a 2 x 2 square. The second shape can be rearranged to show a 3 x 3 square. The third shape can be rearranged to show a 4 x 4 square.
Figure 4.8: Euler’s Polyhedron Formula
Demonstrates the formula Vertices – Edges + Faces = 2 with four polyhedrons. The first polyhedron is a tetrahedron, and the features of the tetrahedron are shown beneath it: four vertices, six edges, and four faces. Underneath that is the calculation showing Euler’s formula for the tetrahedron of 4 – 6 + 4 = 2. Three additional polyhedrons are also included in the image, with features and Euler’s formula for each. The next figure is a hexahedron or cube, with eight vertices, 12 edges, and six faces where Euler’s formula is 8 – 12 + 6 = 2. Next is an octahedron, with six vertices, 12 edges, and eight faces where Euler’s formula is 6 – 12 + 8 = 2. The last figure is a dodecahedron with 20 vertices, 30 edges, and 12 faces where Euler’s formula is 20 – 30 + 12 = 2.