Making a nice number line in PowerPoint is quick and easy – it should only take about 10-15 minutes. These instructions are really just a frame for learning a few neat formatting tricks that will make using PowerPoint for graphical interfaces like posters (or even presentations!) a snap.The first thing is to “Insert” a “Shape” (in this case a line) and choose the color, thickness, and other effects using “Format” in “Drawing Tools”. To make sure the line is straight, hold the “Shift” key down as you drag it out.Make a smaller line for the tick marks, and copy paste however many times you’d like. The position doesn’t matter – we’ll straighten those out in a minute.Left-click and drag the mouse to highlight all of the ticks fully. When you release the mouse button, all of the ticks will be selected. PowerPoint will only select shapes (or images) if it is entirely within the highlighted area, so you can be very selective by leaving out even a tiny particle of a shape.If you select objects that you didn’t want to select, just hold the <CTRL> key down and click on them to deselect.Once you’ve selected everything, go to the “Format” window and look to the far right for the “Align” menu. This menu gives you many options and, if you haven’t yet, you should definitely experiment and figure out what they all do and think about how they can be useful.These tickmarks need to be aligned horizontally, so choose “Align Middle” (although really top or bottom would also work fine in this case.)Now, click anywhere else on the slide to deselect all of the ticks. Choose one and drag it all the way to one side so that the tick on the left and the tick on the right are in the desired positions. (You can drag it around without ruining alignment by holding the <Shift> button down.Once they are in place, go back to the “Align” menu and choose “Distribute Horizontally”. Congratulations! You have a number line (albeit blank).Adding the numbers is easy too – just use text boxes and the align menu to get them straight and in place.

# Monthly Archives: July 2012

# Definitions of Volume from Math Ed Textbooks

In a side project of *Strengthening Tomorrow’s Education in Measurement* data, I analyzed definitions from elementary education mathematics content textbooks. I began with the list of content textbooks from McCrory (2006) and found updated versions when possible – some textbooks were out of print and others were not accessible to me. (see references section for list of textbooks – I was able to find copies of the textbooks with a number in brackets on the left side, and others are greyed out.) Definitions across textbooks varied, especially the level and focus.

**Bassarear, T. (2012). Mathematics for elementary school teachers (5th ed). Belmont, CA: Brooks/Cole-Cengage.**

- p. 608 – Over time, two related systems of measure for what we call volume evolved, depending on whether the thing being measured was dry or wet.
- p. 612 – volume (dry) and capacity (liquid);
- p. 637- the amount of space contained within that object
- p. 641 – volume reflects how much it will take to fill an object

**Beckmann, S. (2011). Mathematics for elementary school teachers (3rd ed). Boston, MA: Pearson/Addison Wesley.**

- p. 153 – The volume, in cubic units, of a box or box shape is just the number of 1-unit-by-1-unit-by-1-unit cubes that it would take to fill the box or make the box shape (without gaps or overlaps).
- p. 483 – A volume describes the size of an object (or a part of an object) that is three-dimensional; the volume of that three-dimensional object is how many of a chosen unit of volume (such as cubic inches, cubic centimeters, etc.) it takes to fill the object without gaps or overlaps, where it is understood that we may use parts of a unit, too. Roughly speaking, an object is three-dimensional if at each location there are three independent directions along which to move within the object.

**Bennett, A.B., Burton, L. J. & Nelson, L. T. (2012). Math for elementary teachers: A conceptual approach (9th ed). New York, NY: McGraw-Hill Science/Engineering/Math. **

- p. 701 – To measure the amount of space in tanks, buildings, refrigerators, cars and other three-dimensional figures, we need units of measure that are also three-dimensional figures. The number of such units needed to fill a figure is its volume. Cubes are convenient because they pack together without gaps or overlapping.

**Billstein, R., Libeskind, S., & Lott, J. W. (2010). A problem solving approach to mathematics for elementary school teachers (10th ed). Boston, MA: Pearson/Addison-Wesley.**

- p. 906 – volume describes how much space a three-dimensional figure contains. The unit of measure for volume must be a shape that tessellates space. Cubes tessellates space; that is, they can be stacked so that they leave no gaps and fill space. Standard units of volume are based on cubes and are cubic units. A cubic unit is the amount of space enclosed within a cue that measures 1 unit on a side.

**Sowder, J., Sowder, L., & Nickerson, S. (2010). Reconceptualizing Mathematics for Elementary School Teachers: Instructors Edition. New York: W.H. Freeman and Company.**

- p. 558 – Any real object takes up space. Even a scrap of paper, a speck of dust, or a strand of cobweb takes up some space. The quantity of space occupied is called the volume. As with the terms length and area, the term volume is sometimes used for the attribute as well as the measurement of the attribute.
- p. G-16 – the number of 3D units that could fit inside the region, or that could be used to make an exact model of the region. The units are usually cubic regions (e.g., cm^3).

**Long, C. T., & DeTemple, D. W. (2012). Mathematical reasoning for elementary teachers (6th ed.). Boston, MA: Pearson/Addison Wesley.**

- p. 530 – Volume is the measure of space taken up by a solid three-dimensional space. The unit… is the volume of a cube whose side length is one of the standard units of length.

**Musser, G. L., Burger, W. F., & Peterson, B. E. (2011). Mathematics for elementary school teachers: A contemporary approach (9th ed.). New York: John Wiley & Sons, Inc.**

- p. 680 – The volume of the vase would be the amount of material comprising the vase itself.

**Parker, T. H., & Baldridge, S. J. (2008). Elementary geometry for teachers (Volume 2). Okemos, MI: Sefton-Ash Publishing.**

- p. 193 – volume of a solid is the number of unit cubes needed to fill the solid
- p. 193 – Volume measurements come in three different forms: liquid, solid, and air/space. Each has unique features that make it easy or hard to measure.
- The volume of a
**Liquid**is easily measured – use a measuring cup. - The volume of a
**Solid**is easily measured for rectangular solids, but volume is not intuitive or easy to measure for irregular solids. - The volume of
**Air/Space**is neither intuitive nor easily measured.

p. 198 – [image provided illustrating the 5 types of measurement that is usually studied in elementary school and their relationships – I just thought the Volume/Capacity is nicely shown]

- The volume of a

**Sonnabend, T. (2010). Mathematics for Elementary Teachers: An Interactive Approach for Grades K-8 (4th ed.). Belmont, CA: Brooks/Cole-Cengage.**

- p. 571 – Volume is the amount of space occupied by a three-dimensional figure. Volume is usually measured in cubic units. Whereas surface area is the total area of the faces of a solid, volume is the capacity of a solid.

**Van de Walle, John A., Karp, Karen S. and Bay-Williams, Jennifer M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally (8th ed.). New Jersey: Pearson Education, Inc.**

- p. 391 –
*Volume*and*capacity*are both terms for measures of the “size” of three-dimensional regions…The term*volume*can be used to refer to the capacity of a container but is also used for the size of solid objects. Standard units of volume are expressed in terms of length units, such as cubic inches or cubic centimeters.

**References**

*Notices of the AMS*,

*51*(1), 20–29.

**List of Content Textbooks**

[1] Bassarear, T. (2012). Mathematics for elementary school teachers (5th ed). Belmont, CA: Brooks/Cole-Cengage.

[2] Beckmann, S. (2011). Mathematics for elementary school teachers (3rd ed). Boston, MA: Pearson/Addison Wesley.

[3] Bennett, A.B., Burton, L. J. & Nelson, L. T. (2012). Math for elementary teachers: A conceptual approach (9th ed). New York, NY: McGraw-Hill Science/Engineering/Math.

[4] Billstein, R., Libeskind, S., & Lott, J. W. (2010). A problem solving approach to mathematics for elementary school teachers (10th ed). Boston, MA: Pearson/Addison-Wesley.

[5] Sowder, J., Sowder, L., & Nickerson, S. (2010). Reconceptualizing Mathematics for Elementary School Teachers: Instructors Edition. New York: W.H. Freeman and Company.

Darken, B. (2003). Fundamental mathematics for elementary and middle school teachers: Kendall/Hunt.

Devine, D. F., Olson, J., & Olson, M. (1991). Elementary mathematics for teachers. New York, NY: Wiley. (NO LONGER IN PRINT)

Jensen, G. R. (2003). Arithmetic for teachers: With applications and topics from geometry: American Mathematical Society.

Jones, P., Lopez, K. D., & Price, L. E. (1998). A mathematical foundation for elementary teachers. New York: Addison Wesley Higher Education.

Krause, E. F. (1991). Mathematics for elementary teachers: A balanced approach. Lexington, MA: D.C. Heath and Company. (NO LONGER IN PRINT)

[6] Long, C. T., & DeTemple, D. W. (2012). Mathematical reasoning for elementary teachers (6th ed.). Boston, MA: Pearson/Addison Wesley.

Masingila, J. O., Lester, F. K., & Raymond, A. M. (2002). Mathematics for elementary teachers via problem solving: Prentice Hall.

[7] Musser, G. L., Burger, W. F., & Peterson, B. E. (2011). Mathematics for elementary school teachers: A contemporary approach (9th ed.). New York: John Wiley & Sons, Inc.

O’Daffer, P. G. (1998). Mathematics for elementary school teachers. Reading, Mass.: Addison-Wesley.

[8] Parker, T. H., & Baldridge, S. J. (2004). Elementary mathematics for teachers (Volume 1). Okemos, MI: Sefton-Ash Publishing. [Note: FIRST VOLUME, PRIMARILY NUMBER AND OPERATIONS.]

[8a] Parker, T. H., & Baldridge, S. J. (2008). Elementary geometry for teachers (Volume 2). Okemos, MI: Sefton-Ash Publishing.

Sgroi, R. J., & Sgroi, L. S. (1993). Mathematics for elementary school teachers: Problem-solving investigations. Boston: PWS Publishing Company. (NO LONGER IN PRINT)

[9] Sonnabend, T. (2010). Mathematics for Elementary Teachers: An Interactive Approach for Grades K-8 (4th ed.). Belmont, CA: Brooks/Cole-Cengage.

Wheeler, E., & Brawner, J. (2005). Discrete mathematics for teachers (Preliminary ed.): Houghton Mifflin. (THIS BOOK FOCUSES ON DIFFERENT CONTENT THAN THE OTHERS.)

Wheeler, R. E., & Wheeler, E. R. (2002). Modern mathematics (Eleventh ed.): Kendall/Hunt Publishing Company. (THIS BOOK IS USED FOR GENERAL MATHEMATICS CLASSES AS WELL AS CLASSES FOR ELEMENTARY TEACHERS.)

[10] Van de Walle, John A., Karp, Karen S. and Bay-Williams, Jennifer M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally (8th ed.). New Jersey: Pearson Education,

# Definitions of Length from Math Ed Textbooks

In a side project of *Strengthening Tomorrow’s Education in Measurement* data, I analyzed definitions from elementary education mathematics content textbooks. I began with the list of content textbooks from McCrory (2006) and found updated versions when possible – some textbooks were out of print and others were not accessible to me. (see references section for list of textbooks – I was able to find copies of the textbooks with a number in brackets on the left side, and others are greyed out.) Definitions across textbooks varied, especially the level and focus.

**Bassarear, T. (2012). Mathematics for elementary school teachers (5th ed). Belmont, CA: Brooks/Cole-Cengage.**

- p. 607 – refers to length as a “linear measure”

**Beckmann, S. (2011). Mathematics for elementary school teachers (3rd ed). Boston, MA: Pearson/Addison Wesley.**

- 481 – A length describes the size of something (or a part of something) that is one-dimensional; the length of that one-dimensional object is how many of a chosen unit of length (such as inches, centimeters, etc.) it takes to cover the object without gaps or overlaps, where it is understood that we may use parts of a unit, too. Roughly speaking, an object is one-dimensional if at each location, there is only one independent directions along which to move within the object.

**Bennett, A.B., Burton, L. J. & Nelson, L. T. (2012). Math for elementary teachers: A conceptual approach (9th ed). New York, NY: McGraw-Hill Science/Engineering/Math. **

- p. 655 – 1 length (or linear) unit

**Billstein, R., Libeskind, S., & Lott, J. W. (2010). A problem solving approach to mathematics for elementary school teachers (10th ed). Boston, MA: Pearson/Addison-Wesley.**

- p. 838 – linear measure

**Sowder, J., Sowder, L., & Nickerson, S. (2010). Reconceptualizing Mathematics for Elementary School Teachers: Instructors Edition. New York: W.H. Freeman and Company.**

- 528 – We speak of the length of a piece of wire or a rectangle in two ways. The term length might refer to the quality or attribute we are focusing on, or it might refer to the measurement of that quality. The context usually makes clear which reference is intended.
- p. G-7 – the characteristic of one-dimensional shapes that is measured with a ruler; for example, width, height, depth, thickness, perimeter, and circumference all refer to the same characteristic

**Long, C. T., & DeTemple, D. W. (2012). Mathematical reasoning for elementary teachers (6th ed.). Boston, MA: Pearson/Addison Wesley.**

- [none]

**Musser, G. L., Burger, W. F., & Peterson, B. E. (2011). Mathematics for elementary school teachers: A contemporary approach (9th ed.). New York: John Wiley & Sons, Inc.**

- p. 680 – Regardless, in every case, we can select some appropriate unit and determine how many units are needed to span the object. This is an informal measurement method of measuring length, since it involves naturally occurring units and is done in a relatively imprecise way.

**Parker, T. H., & Baldridge, S. J. (2008). Elementary geometry for teachers (Volume 2). Okemos, MI: Sefton-Ash Publishing.**

- 5 – In geometry one also has a notion of length and distance. The distance between two points
*A*and*B*is the length of the segment ;*in this book we will denote this length by AB*(some textbooks use a different notation for length). Note that*AB*is a number whereas is a segment. - p. 6 – Lengths are not numbers because any measurement of length involves a two-step process:
- Choose a unit length. [picture omitted – shows example of 1 unit]
- Express other lengths as multiples of that unit. [picture omitted – shows multiple of unit “
*The bar is 4 units long.”*]]

The resulting length is then a

*quantity*; a number times a unit. [picture omitted – shows that in 48 inches, 48 is the number and inches is the unit]

**Sonnabend, T. (2010). Mathematics for Elementary Teachers: An Interactive Approach for Grades K-8 (4th ed.). Belmont, CA: Brooks/Cole-Cengage.**

- [none]

**Van de Walle, John A., Karp, Karen S. and Bay-Williams, Jennifer M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally (8th ed.). New Jersey: Pearson Education, Inc.**

- p. 382 – There are four important principles of iterating units of length, whether they are nonstandard or standard (Dietiker, Gonulates, Figueras, & Smith, 2010, p. 2): *All units must have equal length – if not, you cannot accumulate units by counting. *All units must be placed on the path being measured – otherwise, a different quantity is being measured. *The units must be without gaps – if not, part of the quantity is not being measured. *The units must not overlap – otherwise, part of the quantity is measured more than once.

**Wu, H. H. (2011). Understanding Numbers in Elementary School Mathematics. AMS.**

- [none]

**References**

*Notices of the AMS*,

*51*(1), 20–29.

**List of Content Textbooks**

[1] Bassarear, T. (2012). Mathematics for elementary school teachers (5th ed). Belmont, CA: Brooks/Cole-Cengage.

[2] Beckmann, S. (2011). Mathematics for elementary school teachers (3rd ed). Boston, MA: Pearson/Addison Wesley.

[3] Bennett, A.B., Burton, L. J. & Nelson, L. T. (2012). Math for elementary teachers: A conceptual approach (9th ed). New York, NY: McGraw-Hill Science/Engineering/Math.

[4] Billstein, R., Libeskind, S., & Lott, J. W. (2010). A problem solving approach to mathematics for elementary school teachers (10th ed). Boston, MA: Pearson/Addison-Wesley.

[5] Sowder, J., Sowder, L., & Nickerson, S. (2010). Reconceptualizing Mathematics for Elementary School Teachers: Instructors Edition. New York: W.H. Freeman and Company.

Darken, B. (2003). Fundamental mathematics for elementary and middle school teachers: Kendall/Hunt.

Devine, D. F., Olson, J., & Olson, M. (1991). Elementary mathematics for teachers. New York, NY: Wiley. (NO LONGER IN PRINT)

Jensen, G. R. (2003). Arithmetic for teachers: With applications and topics from geometry: American Mathematical Society.

Jones, P., Lopez, K. D., & Price, L. E. (1998). A mathematical foundation for elementary teachers. New York: Addison Wesley Higher Education.

Krause, E. F. (1991). Mathematics for elementary teachers: A balanced approach. Lexington, MA: D.C. Heath and Company. (NO LONGER IN PRINT)

[6] Long, C. T., & DeTemple, D. W. (2012). Mathematical reasoning for elementary teachers (6th ed.). Boston, MA: Pearson/Addison Wesley.

Masingila, J. O., Lester, F. K., & Raymond, A. M. (2002). Mathematics for elementary teachers via problem solving: Prentice Hall.

[7] Musser, G. L., Burger, W. F., & Peterson, B. E. (2011). Mathematics for elementary school teachers: A contemporary approach (9th ed.). New York: John Wiley & Sons, Inc.

O’Daffer, P. G. (1998). Mathematics for elementary school teachers. Reading, Mass.: Addison-Wesley.

[8] Parker, T. H., & Baldridge, S. J. (2004). Elementary mathematics for teachers (Volume 1). Okemos, MI: Sefton-Ash Publishing. [Note: FIRST VOLUME, PRIMARILY NUMBER AND OPERATIONS.]

[8a] Parker, T. H., & Baldridge, S. J. (2008). Elementary geometry for teachers (Volume 2). Okemos, MI: Sefton-Ash Publishing.

Sgroi, R. J., & Sgroi, L. S. (1993). Mathematics for elementary school teachers: Problem-solving investigations. Boston: PWS Publishing Company. (NO LONGER IN PRINT)

[9] Sonnabend, T. (2010). Mathematics for Elementary Teachers: An Interactive Approach for Grades K-8 (4th ed.). Belmont, CA: Brooks/Cole-Cengage.

Wheeler, E., & Brawner, J. (2005). Discrete mathematics for teachers (Preliminary ed.): Houghton Mifflin. (THIS BOOK FOCUSES ON DIFFERENT CONTENT THAN THE OTHERS.)

Wheeler, R. E., & Wheeler, E. R. (2002). Modern mathematics (Eleventh ed.): Kendall/Hunt Publishing Company. (THIS BOOK IS USED FOR GENERAL MATHEMATICS CLASSES AS WELL AS CLASSES FOR ELEMENTARY TEACHERS.)

[10] Van de Walle, John A., Karp, Karen S. and Bay-Williams, Jennifer M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally (8th ed.). New Jersey: Pearson Education, Inc.

[11] Wu, H. H. (2011).* Understanding Numbers in Elementary School Mathematics.* AMS.

# Definitions of Area from Math Ed Textbooks

In a side project of *Strengthening Tomorrow’s Education in Measurement* data, I analyzed definitions from elementary education mathematics content textbooks. I began with the list of content textbooks from McCrory (2006) and found updated versions when possible – some textbooks were out of print and others were not accessible to me. (see references section for list of textbooks – I was able to find copies of the textbooks with a number in brackets on the left side, and others are greyed out.) Definitions across textbooks varied, especially the level and focus.

**Bassarear, T. (2012). Mathematics for elementary school teachers (5th ed). Belmont, CA: Brooks/Cole-Cengage.**

- p. 621 – Questions about area generally deal with “how much” it takes to cover an object – for example, how much fertilizer to cover a lawn, how much material to cover a bed. In order to answer area questions, we have to select an appropriate unit, and thus the answer takes the form of how many of those units.;
- p. 637 – surface area is the area needed to cover all the faces of a three-dimensional object

**Beckmann, S. (2011). Mathematics for elementary school teachers (3rd ed). Boston, MA: Pearson/Addison Wesley.**

- p. 151 – In general, for any unit of length, one square unit is the area of of a square that is 1 unit wide and 1 unit long. The area of a region, in square units, is the number of 1-unit-by-1-unit squares it takes to cover the region without gaps or overlaps (where square may be cut apart if necessary)
- p. 482 – An area or a surface area describes the size of an object (or a part of an object) that is two-dimensional; the area of that two-dimensional object is how many of a chosen unit of area (such as square inches, square centimeters, etc.) it takes to cover the object without gaps or overlaps, where it is understood that we may use parts of a unit, too. The surface area of a solid shape is the total area of its outside surface. Roughly speaking, an object is two-dimensional if, at each location, there are two independent directions along which to move within the object.

**Bennett, A.B., Burton, L. J. & Nelson, L. T. (2012). Math for elementary teachers: A conceptual approach (9th ed). New York, NY: McGraw-Hill Science/Engineering/Math. **

- p. 676 – To measure the sizes of plots of land, panes of glass, floors, walls, and other such surfaces, we need a new type of unit, one that can be used to cover a surface. The number of units it takes to cover a surface is called its area.Squares have been found to be the most convenient shape for measuring area.
- p. 704 – Another important measure associated with objects in space is their amount of surface. Surface area is expressed as the number of unit squares needed to cover the surface of a three-dimensional figure.

**Billstein, R., Libeskind, S., & Lott, J. W. (2010). A problem solving approach to mathematics for elementary school teachers (10th ed). Boston, MA: Pearson/Addison-Wesley.**

- p. 854 – quote from “Focal Points”: “Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps. They understand that a square unit that is 1 unit on a side is the standard unit for measuring area. “
- p. 854 – Area is measured using square units and the area of a region is the number of nonoverlapping square units that covers the region.

**Sowder, J., Sowder, L., & Nickerson, S. (2010). Reconceptualizing Mathematics for Elementary School Teachers: Instructors Edition. New York: W.H. Freeman and Company.**

- p. 547 – We speak of the area of a field, a lake, a country, a geometric shape, a wall, or your body, all of which are examples of surfaces or regions – area is a characteristic of surfaces or regions. As with length, the term area is used to refer both to the attribute (“The wolf wandered over a wide area”) and to the measurement (“The area of the field is 15 acres”).
- p. G-2 – the number of square units that would be required to cover the region. the region could be a 2D region, or it could refer to all the surfaces of a 3D figure, in which case it is called the surface area.

**Long, C. T., & DeTemple, D. W. (2012). Mathematical reasoning for elementary teachers (6th ed.). Boston, MA: Pearson/Addison Wesley.**

- p. 530 – Area is a measure of the region bounded by a closed plane curve. Any shape could be chosen as a unit, but the square is the most common. The size of the square is arbitrary, but it is natural to choose the length of a side to correspond to a unit measure of length. Areas are therefore usually measured in square inches, square feet, and so on.

**Musser, G. L., Burger, W. F., & Peterson, B. E. (2011). Mathematics for elementary school teachers: A contemporary approach (9th ed.). New York: John Wiley & Sons, Inc.**

- p. 680 – To measure the area of a region informally, we select a convenient two-dimensional shape as our unit and determine how many such units are needed to cover the region.

**Parker, T. H., & Baldridge, S. J. (2008). Elementary geometry for teachers (Volume 2). Okemos, MI: Sefton-Ash Publishing.**

- p. 107 – 108 – A portion of the plane is called a
**region.**A triangular region is a triangle together with its interior, and a polygonal region is a polygon together with its interior. A circular region is a*disk*. A region formed from several pieces is a*composite region.* - [image omitted – examples of triangle, triangular region, rectangular region, disk or circular region, composite region]
**Area**is a way of associating to each regiona quantity Area(*R**R*) that reflects our intuitive sense of “how big” the region is without reference to the shape of the region. Area is defined by the same two-step scheme used to define length, weight, and capacity:- Choose a “unit region” and declare its area to be 1
*unit of area* - Express the areas of other regions as multiples of this unit area.

- Choose a “unit region” and declare its area to be 1
- p. 110 –
**Definition 1.2**(School Definition).*The area of a region tiled by unit squares is the number of squares it contains.* - p. 193 – area is the number of unit squares needed to cover a region

**Sonnabend, T. (2010). Mathematics for Elementary Teachers: An Interactive Approach for Grades K-8 (4th ed.). Belmont, CA: Brooks/Cole-Cengage.**

- p. 531 – If you want to know the size of the interior of a field, or which package of gift wrap is a better buy, you measure the area. Area is the measure of a closed, two-dimensional region.
- p. 564: The total surface area of a closed space figure is the sum of the areas of all its surfaces. A surface area in square units indicates how many squares it would take to cover the outside of a space figure.

**Van de Walle, John A., Karp, Karen S. and Bay-Williams, Jennifer M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally (8th ed.). New Jersey: Pearson Education, Inc.**

- p. 384 –
*Area*is the two dimensional space inside a region.

**Wu, H. H. (2011). Understanding Numbers in Elementary School Mathematics. AMS.**

- p. 45 – We begin with a definition of area that is adequate for the present need. A square with length 1on each side is called a unit square. The area of the unit square is by definition equal to 1. We say a collection or rectangles {R_j} tile or pave a given rectangle R, if, by combining the R_j’s together we get the whole rectangle R, and if the R_j’s intersect at most along their boundaries. With all this terminology in place, the area of a general rectangle is by definition the number of unit squares required to pave that rectangle. (Remember that we are dealing only with whole numbers at this point and therefore the lengths of all rectangles are whole numbers.)
- p. 191 – Because we will have to use the concept of area in a more elaborate fashion, let us first give a more detailed discussion of the basic properties of area. The basic facts about area that we need are rather mundane and are summarized below.
- The area of a planar region is always a number.
- The area of the unit square is by definition the number 1.
- If two regions are congruent, then their areas are equal.
- If two regions have at most (part of) their boundaries in common, then the area of the region obtained by combining the two is the sum of their individual areas.

**References**

*Notices of the AMS*,

*51*(1), 20–29.

**List of Content Textbooks**

[1] Bassarear, T. (2012). Mathematics for elementary school teachers (5th ed). Belmont, CA: Brooks/Cole-Cengage.

[2] Beckmann, S. (2011). Mathematics for elementary school teachers (3rd ed). Boston, MA: Pearson/Addison Wesley.

[3] Bennett, A.B., Burton, L. J. & Nelson, L. T. (2012). Math for elementary teachers: A conceptual approach (9th ed). New York, NY: McGraw-Hill Science/Engineering/Math.

[4] Billstein, R., Libeskind, S., & Lott, J. W. (2010). A problem solving approach to mathematics for elementary school teachers (10th ed). Boston, MA: Pearson/Addison-Wesley.

[5] Sowder, J., Sowder, L., & Nickerson, S. (2010). Reconceptualizing Mathematics for Elementary School Teachers: Instructors Edition. New York: W.H. Freeman and Company.

Darken, B. (2003). Fundamental mathematics for elementary and middle school teachers: Kendall/Hunt.

Devine, D. F., Olson, J., & Olson, M. (1991). Elementary mathematics for teachers. New York, NY: Wiley. (NO LONGER IN PRINT)

Jensen, G. R. (2003). Arithmetic for teachers: With applications and topics from geometry: American Mathematical Society.

Jones, P., Lopez, K. D., & Price, L. E. (1998). A mathematical foundation for elementary teachers. New York: Addison Wesley Higher Education.

Krause, E. F. (1991). Mathematics for elementary teachers: A balanced approach. Lexington, MA: D.C. Heath and Company. (NO LONGER IN PRINT)

[6] Long, C. T., & DeTemple, D. W. (2012). Mathematical reasoning for elementary teachers (6th ed.). Boston, MA: Pearson/Addison Wesley.

Masingila, J. O., Lester, F. K., & Raymond, A. M. (2002). Mathematics for elementary teachers via problem solving: Prentice Hall.

[7] Musser, G. L., Burger, W. F., & Peterson, B. E. (2011). Mathematics for elementary school teachers: A contemporary approach (9th ed.). New York: John Wiley & Sons, Inc.

O’Daffer, P. G. (1998). Mathematics for elementary school teachers. Reading, Mass.: Addison-Wesley.

[8] Parker, T. H., & Baldridge, S. J. (2004). Elementary mathematics for teachers (Volume 1). Okemos, MI: Sefton-Ash Publishing. [Note: FIRST VOLUME, PRIMARILY NUMBER AND OPERATIONS.]

[8a] Parker, T. H., & Baldridge, S. J. (2008). Elementary geometry for teachers (Volume 2). Okemos, MI: Sefton-Ash Publishing.

Sgroi, R. J., & Sgroi, L. S. (1993). Mathematics for elementary school teachers: Problem-solving investigations. Boston: PWS Publishing Company. (NO LONGER IN PRINT)

[9] Sonnabend, T. (2010). Mathematics for Elementary Teachers: An Interactive Approach for Grades K-8 (4th ed.). Belmont, CA: Brooks/Cole-Cengage.

Wheeler, E., & Brawner, J. (2005). Discrete mathematics for teachers (Preliminary ed.): Houghton Mifflin. (THIS BOOK FOCUSES ON DIFFERENT CONTENT THAN THE OTHERS.)

Wheeler, R. E., & Wheeler, E. R. (2002). Modern mathematics (Eleventh ed.): Kendall/Hunt Publishing Company. (THIS BOOK IS USED FOR GENERAL MATHEMATICS CLASSES AS WELL AS CLASSES FOR ELEMENTARY TEACHERS.)

[10] Van de Walle, John A., Karp, Karen S. and Bay-Williams, Jennifer M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally (8th ed.). New Jersey: Pearson Education, Inc.

[11] Wu, H. H. (2011).* Understanding Numbers in Elementary School Mathematics.* AMS.