Inventory of “Big Ideas”
(Vital Understandings in Mathematics

II. Geometry and Measurement

Foundation (geometry): Geometry describes and provides understanding of the physical world. This description includes size, shape, position, and dimension.

1.

Can my students recognize, draw, construct, visualize, compare and analyze real and abstract geometric shapes that have one, two, and/or three-dimensional features?

Example Grade K-3:

Take a sheet of paper. Roll it up. What shape have you made? What other things can you think of that are this shape?

II-1.1

Example Grade 3-4:

Build a structure with three or four blocks. Draw the side view, the front view, the top view. See if someone else can build your structure by looking at your drawings.

II-1.1

Example Grade 5-6:

Draw what an ice cream cone would look like if viewed from the side, the top, sliced parallel to the base, sliced at an angle.

II-1.1

Example Grade 7-8:

What shapes can be formed by “slicing a cube”? Can you figure out how to slice a cube so that you get a triangle, a rectangle, a pentagon, and a hexagon?

II-1.1

2.

Can my students identify the characteristics that are necessary to classify and name geometric shapes and can they differentiate one shape from another?

Example Grade K-3:

II-1.1, 1.3

The same shape can be included in many categories:

Find all the four-sided shapes.

All the shapes with all sides the same length

All the shapes with four square corners and opposite sides parallel.

In which categories would the square fit? Do any other shapes fit in more than one category?

Example Grade 3-4:

What information is enough to identify a figure?

II-1.1, 1.2

Are all 3-sided figures triangles? Are all 4-sided figures rectangles? If a shape has 4 right corners, is it a rectangle?

Example Grade 5-6:

After which piece of the information that follows can you say for sure what figure is being described?

II-1.1, 1.3

A figure has four sides. (What could it be?) Not all of the sides are the same length. (What have you eliminated?) Two sides are parallel. It has at least two square corners. (Could it be anything besides a rectangle? What else do you need to know to be sure?)

Can you draw:

a rectangle with 4 sides of equal length?

a parallelogram with 4 right angles?

a parallelogram with 4 sides of equal length and no right angles?

a parallelogram with 4 sides of equal length and 4 right angles?

II-1.1

II-1.3

Describe a variety of polyhedra in terms of numbers of shapes and faces and number of edges and vertices. What pattern can you find for regular solids when you examine the number of faces, vertices, and edges.

3.

Can my students describe geometric shape in terms of their relationships with other shapes? The relationships include relative size, position, orientation, congruence, similarity, and symmetry.

Example Grade K-3:

Match the lids with the containers. How many different ways can you put the lid on a square jewelry box? A margarine tub? A shoe box?

Example Grade 3-4:

Make a border pattern by rotating, flipping, and sliding a single shape or figure. Analyze the border patterns made by other students and tell what transformations they used to create their patterns.

II-2.3

Example Grade 5-6:

Which of these capital letters of the alphabet can be rotated 90 or 180 degrees and still look the same? Which can be flipped (top to bottom or side to side) and still look the same?

II-2.3

O B T L A M U I

Example Grade 7-8:

Which of the following sets of shapes are always similar? Why?

Cubes, spheres, cones, rhombuses, parallelograms, quadrilaterals, regular hexagons, equilateral triangles, isosceles triangles.

II-2.2

Look at a globe. Are the lines of longitude parallel at the equator? Why do they then intersect at the poles? Why do latitudes lines not intersect?

4.

Can my students compose or decompose geometric shapes into other geometric shapes?

Example Grade K-3:

Explore the following shapes. Which ones fit together and fill space with no spaces in between? Which ones don’t?

II-1.5

Example Grade 3-4:

Can you make a square using only triangles? A triangle, using only squares? A triangle, using squares and triangles?

II-1.5

Example Grade 5-6:

Put these shapes in order by area. Which shapes have the same area? How do you know?

II-1.5

Example Grade 7-8:

Here are some figures on geoboards. Can you use some familiar shapes to help you find the area?

II-1.5

5.

Can my students discover and express constant relationships within and between geometric shapes through measuring and looking for patterns? Can my students connect and relate these constant relationships to formulas?

Example Grade K-3:

If we had only five books, how could we tell how many books it would take to cover the table?

Example Grade 3-4:

Measure the circumferences and the diameters of all the circular things you can find. Is there a relationship?

II-1.7

Example Grade 5-6:

Here is a piece of paper that is 10 inches long and 5 inches wide. How many inch square pieces of paper does it take to cover it? What multiplication problem does this represent? How can we express that as a general rule? What is the relationship between the length and width of a rectangle and its area?

II-1.7

Example Grade 7-8:

What is the relationship between a 10 inch pizza and a 14 inch pizza? How much more pizza do you get when you buy a 14 inch pizza than when you buy a 10 inch pizza? If the 10 inch pizza costs $5.75 what would be a fair price for the 14 inch pizza?

II-1.7

Foundation (measurement): Measurement is a process that assigns a number to the magnitude of a quantity.

1. 

Can my students choose appropriate units of measure based on the properties of the quantity to be measured?

Example Grade K-3:

Which is bigger, the top of the table or the desk top?  How can we find out?  Can we use pieces of paper, books, circles, squares, triangles?

II-3.1

Example Grade 3-4: 

Put six books in order by size.  What are the different ways you could do that?  By height?  By weight?  By thickness?  By numbers of pages?  By surface area of the covers?   What different units would you use when measuring these different properties?  Which would you measure with units of length such as centimeters?  With units of mass such as grams?  What kind of units would you use to measure surface area or volume?

II-3.1

Example Grade 5-6:

Put these boxes in order by size.  Estimate first and then measure to find out for sure.  Is there more than one way to do that?  Could you put them in order by the amount of wrapping paper it would take to cover them?  By how much they can hold?  How will you measure how much they hold?  Would small wooden cubes help?  Is there another way?

II-3.1

Example Grade 7-8:

a.

What do we need to consider to measure speed?  What about the speed of a rocket, a bicycle, an earthworm, a car, the speed of light?

II-3.1

b.

Compare the density of yogurt with the density of cottage cheese.

2.

Can my students choose appropriate measuring tools in situations where the size of the object and the use of the measurement is considered?

Example Grade K-3:

Measure the length of the floor, a crayon, the VCR, and a paintbrush using two-inch cubes, paper clips, toothpicks, and straws. Would it be easier to measure some things if we strung some straws or paper clips together?   Why?  Why don’t we have inch rulers?  Why do we put 12 inches together on one ruler?  Why do some rulers have 36 inches on them?

II-3.2

Example Grade 3-4:

What would you use to measure the following items?  Why?

II-3.2

The length and width of a piece of paper

The length of a fingernail

The length of the first base line on the baseball diamond

The area enclosed inside the baselines

Example Grade 5-6:

When is it appropriate to use strides or footsteps to measure a room and when is it better to use standard measures?  Which standard measures?  What if you were trying to determine how to arrange the furniture?  What if you were ordering carpeting? Would you need to know how carpet is purchased (e.g., square feet or square yards)?  Why?

II-3.2

Example Grade 7-8:

How can you measure the following items?

II-3.2

The angle of a regular polygon

The direction the classroom is facing

The height of the building

The thickness of a piece of paper

The diameter of a straw

3.

Do my students understand that all measurement is approximate because the ability to read a measuring instrument and the precision of the measuring instrument is limited?  (More accuracy requires a smaller unit of measurement.)

Example Grade K-3:

When you measure around the wastebasket with this string, how do you know if the string is too short, too long, or just right?  How close is just right?

II-3.3

Example Grade 3-4:

The class is planning a Junior Olympics.  The students need to make a decision about the standards for measuring for the broad jump.  Does measuring to the nearest eighth of an inch make sense in this case?  Why or why not?

II-3.2

Example Grade 5-6:

What is the area of your handprint in square centimeters?  How do you count the number of square centimeters when your handprint doesn’t fit exactly on the squares?  What about the extra spaces?

II-3.3

Example Grade 7-8:

When do you need to be accurate to the nearest gram?  Milligram?  Second?  Millisecond?  Can you name something that we measure more accurately than we need to?

II-3.2

4.

Can my students make a scaled model or drawing of real objects?

Example Grade K-3:

Here is a three-dimensional model of our classroom made from a cardboard box.  The windows are here; the doors are here.  Can you make the tables and the shelves to fit?

II-3.5, 3.6

Example Grade 3-4:

Using this graph paper, draw a model of our classroom.  What length do you want the side of each square to represent?  Put in the windows, the doors, the shelves, and the tables.

II-3.5, 3.6

Example Grade 5-6:

Build a scale model of a building.  Include the lot on which it is situated.  What do you have to consider in deciding what scale to use?

II-3.5, 3.6

Example Grade 7-8:

Which probably shows more detail?  A map with a scale of 1 inch = 1 mile or a map with a scale of 1 inch = 10 miles?  In order for both maps to show the same amount of detail, what would the physical characteristics of the two maps have to be?

II-3.5, 3.6