Elementary geometry: how much time should you devote to it?

A geometry question from a visitor:

1. How much time should be invested teaching geometry at an elementary level?
2. How much time is actually dedicated towards geometry in a tradicional textbook

Your guidance will be extremely appreciated!

During elementary mathematics, geometry plays more of a sideline role at first. It is intimately tied with measuring topics - and really, the word "geometry" means "measuring the earth", the science to measure the land.

The goal of elementary geometry seems to be that the student be able to find perimeters, areas, and volumes of common two and three dimensional shapes.

I would add to that the goal that the student can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes before entering 10th grade. (I've written about that before in the article Why is high school geometry difficult?.

According to the Curriculum Focal Points report recently released by National Council of Teachers of Mathematics, the following geometry topics play a major role in elementary grades:

Grade	Explanations (from Curriculum Focal Points by NCTM)
Grade 1 Geometry: Composing and decomposing geometric shapes.	Children compose and decompose plane and solid figures (e.g., by putting two congruent isosceles triangles together to make a rhombus), thus building an understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine figures, they recognize them from different perspectives and orientations, describe their geometric attributes and properties, and determine how they are alike and different, in the process developing a background for measurement and initial understandings of such properties as congruence and symmetry.
Grade 3 Geometry: Describing and analyzing properties of two-dimensional shapes.	Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry.
Grade 4: Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes	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 that is 1 unit on a side is the standard unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that involve estimating or measuring area. Students connect area measure to the area model that they have used to represent multiplication, and they use this connection to justify the formula for the area of a rectangle.
Grade 5: Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area	Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems.
Grade 7: Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity	Students also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects.
Grade 7: and Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes	By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. ... Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three dimensional shapes to model real-world situations and solve a variety of problems (including multistep problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders.
Grade 8: Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle	Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean theorem is valid by using a variety of methods—for example, by decomposing a square in two different ways. They apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.

Notice how the focal point below for grade 8 is different: no longer is the focus on area and volume of shapes, but on reasoning with lines and angles.

(Note: The absence of a geometry focal point for grades 2 and 6 does not mean that geometry is not studied on those grades. NCTM's focal points are only three per grade so on those grades there were other three topics that were in the focus.)

In a traditional textbook, how much time is spent on geometry? I checked a few books page counts to get an idea:

3rd 24/336 = 7%
4th 23/340 = 6.7%
4th 18/196 = 9.2%
6th 42/340 = 12.3%
6th 31/224 = 13.8%
7th 56/372 = 15.1%

These did not include measuring topics, but just geometry having to do with shapes, lines, angles, area, perimeter, volume.

Of course on lower grades, measuring topics are another 'slice', usually at least about as large as geometry.

So basically you would spend from 1/12 to 1/7 of the total time on geometry topics, increasing as you proceed to higher grades (while decreasing the amount of time devoted to measuring topics). Obviously various arithmetic topics take the bulk of time in elementary mathematics instruction.

Elementary geometry: how much time should you devote to it?

A geometry question from a visitor:

1. How much time should be invested teaching geometry at an elementary level?
2. How much time is actually dedicated towards geometry in a tradicional textbook

Your guidance will be extremely appreciated!

During elementary mathematics, geometry plays more of a sideline role at first. It is intimately tied with measuring topics - and really, the word "geometry" means "measuring the earth", the science to measure the land.

The goal of elementary geometry seems to be that the student be able to find perimeters, areas, and volumes of common two and three dimensional shapes.

I would add to that the goal that the student can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes before entering 10th grade. (I've written about that before in the article Why is high school geometry difficult?.

According to the Curriculum Focal Points report recently released by National Council of Teachers of Mathematics, the following geometry topics play a major role in elementary grades:

Grade	Explanations (from Curriculum Focal Points by NCTM)
Grade 1 Geometry: Composing and decomposing geometric shapes.	Children compose and decompose plane and solid figures (e.g., by putting two congruent isosceles triangles together to make a rhombus), thus building an understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine figures, they recognize them from different perspectives and orientations, describe their geometric attributes and properties, and determine how they are alike and different, in the process developing a background for measurement and initial understandings of such properties as congruence and symmetry.
Grade 3 Geometry: Describing and analyzing properties of two-dimensional shapes.	Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry.
Grade 4: Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes	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 that is 1 unit on a side is the standard unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that involve estimating or measuring area. Students connect area measure to the area model that they have used to represent multiplication, and they use this connection to justify the formula for the area of a rectangle.
Grade 5: Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area	Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems.
Grade 7: Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity	Students also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects.
Grade 7: and Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes	By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. ... Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three dimensional shapes to model real-world situations and solve a variety of problems (including multistep problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders.
Grade 8: Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle	Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean theorem is valid by using a variety of methods—for example, by decomposing a square in two different ways. They apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.

Notice how the focal point below for grade 8 is different: no longer is the focus on area and volume of shapes, but on reasoning with lines and angles.

(Note: The absence of a geometry focal point for grades 2 and 6 does not mean that geometry is not studied on those grades. NCTM's focal points are only three per grade so on those grades there were other three topics that were in the focus.)

In a traditional textbook, how much time is spent on geometry? I checked a few books page counts to get an idea:

3rd 24/336 = 7%
4th 23/340 = 6.7%
4th 18/196 = 9.2%
6th 42/340 = 12.3%
6th 31/224 = 13.8%
7th 56/372 = 15.1%

These did not include measuring topics, but just geometry having to do with shapes, lines, angles, area, perimeter, volume.

Of course on lower grades, measuring topics are another 'slice', usually at least about as large as geometry.

So basically you would spend from 1/12 to 1/7 of the total time on geometry topics, increasing as you proceed to higher grades (while decreasing the amount of time devoted to measuring topics). Obviously various arithmetic topics take the bulk of time in elementary mathematics instruction.

Word problem question

An interesting math teaching related question again from someone.

Hi Maria,

I have purchased some e-books from you and they are very helpful. My daughter is in 4th grade and I felt that she is weak in understanding the following problems. Would you please suggest which e-book we need to buy from you to help improve her basics.

Problems are as below:

1) Marge spent $4 for a magazine. She spent half of her remaining money on T-shirt. Then she spent $2 on a snack. Marge had $14 remaining. How much money did Marge begin with?

2) Dan is 4 inches taller than Mike. Together they are 8 feet 8 inches tall. How tall, in feet and inches, is each boy?

I have not written an ebook that would concentrate on those kind of problems. I do have some similar ones included in the 6th grade worksheets collection.

However, I wouldn't worry so much about those problems at 4th grade, especially the first one, because it requires quite many steps. She's still young.

You might be interested in Singapore math's word problem books; they might be helpful. They often employ diagrams that enable children to solve these kind of problems without the use of algebra.

But let's try to find the easiest way to solve these.


1) Marge spent $4 for a magazine. She spent half of her remaining money on T-shirt. Then she spent $2 on a snack. Marge had $14 remaining. How much money did Marge begin with?

|-- $4 --|----- half -----|----- half -----|
                          | $2 |----$14 ---|

From the above you can see that $16 was the 'half of what was remaining'. Total is $32 + $4 = $36.


2) Dan is 4 inches taller than Mike. Together they are 8 feet 8 inches tall. How tall, in feet and inches, is each boy?

|-------Dan ---------|----------- Mike -----------|

This one is easiest to solve with this reasoning:
If the two boys were of equal height, they'd both be 4 feet 4 inches. The DIFFERENCE in their heights is 4 inches, so just add half of that-or 2 inches- to the "midpoint" 4 ft 4 in to find the taller boy's height, and subtract that same 2 inches to find the shorter boy's height.

So Dan is 4 ft 6 in and Mike is 4 ft 2 in.

Word problem question

An interesting math teaching related question again from someone.

Hi Maria,

I have purchased some e-books from you and they are very helpful. My daughter is in 4th grade and I felt that she is weak in understanding the following problems. Would you please suggest which e-book we need to buy from you to help improve her basics.

Problems are as below:

1) Marge spent $4 for a magazine. She spent half of her remaining money on T-shirt. Then she spent $2 on a snack. Marge had $14 remaining. How much money did Marge begin with?

2) Dan is 4 inches taller than Mike. Together they are 8 feet 8 inches tall. How tall, in feet and inches, is each boy?

I have not written an ebook that would concentrate on those kind of problems. I do have some similar ones included in the 6th grade worksheets collection.

However, I wouldn't worry so much about those problems at 4th grade, especially the first one, because it requires quite many steps. She's still young.

You might be interested in Singapore math's word problem books; they might be helpful. They often employ diagrams that enable children to solve these kind of problems without the use of algebra.

But let's try to find the easiest way to solve these.


1) Marge spent $4 for a magazine. She spent half of her remaining money on T-shirt. Then she spent $2 on a snack. Marge had $14 remaining. How much money did Marge begin with?

|-- $4 --|----- half -----|----- half -----|
                          | $2 |----$14 ---|

From the above you can see that $16 was the 'half of what was remaining'. Total is $32 + $4 = $36.


2) Dan is 4 inches taller than Mike. Together they are 8 feet 8 inches tall. How tall, in feet and inches, is each boy?

|-------Dan ---------|----------- Mike -----------|

This one is easiest to solve with this reasoning:
If the two boys were of equal height, they'd both be 4 feet 4 inches. The DIFFERENCE in their heights is 4 inches, so just add half of that-or 2 inches- to the "midpoint" 4 ft 4 in to find the taller boy's height, and subtract that same 2 inches to find the shorter boy's height.

So Dan is 4 ft 6 in and Mike is 4 ft 2 in.

Math Mammoth Grade 6 Worksheets now ready

This is what I've been working on behing the scenes so to speak... and now some of it is finally come to the fruition and is available to the public:

(now comes the sales pitch as was made up by my dear husband)

Meet Mrs. Maria Miller's Most Marvellous & Magnificent Math Mammoth Modified Modern Mathematics Meticulous Multiplication Methodology Major Madness - It's Mmm Mighty Majestic!

Ok, back to normal...
I have been making worksheets for SpiderSmart, Inc. tutoring company and the 6th grade ones are now available as two downloadable ebooks at:

Math Mammoth Grade 6 Worksheets

These aren't your run-of-the-mill worksheets, but more like carefully hand-crafted quality problem sheets, with varying problems that both emphasize understanding of concepts and computation.

The worksheets do NOT have explanations and therefore best suit math teachers or others who can explain the mathematical subject matter to the student(s).

I will later on (next year) be making more worktexts (with full explanations) using some of this material, and those will better serve the homeschool community.

Math Mammoth Grade 6 Worksheets now ready

This is what I've been working on behing the scenes so to speak... and now some of it is finally come to the fruition and is available to the public:

(now comes the sales pitch as was made up by my dear husband)

Meet Mrs. Maria Miller's Most Marvellous & Magnificent Math Mammoth Modified Modern Mathematics Meticulous Multiplication Methodology Major Madness - It's Mmm Mighty Majestic!

Ok, back to normal...
I have been making worksheets for SpiderSmart, Inc. tutoring company and the 6th grade ones are now available as two downloadable ebooks at:

Math Mammoth Grade 6 Worksheets

These aren't your run-of-the-mill worksheets, but more like carefully hand-crafted quality problem sheets, with varying problems that both emphasize understanding of concepts and computation.

The worksheets do NOT have explanations and therefore best suit math teachers or others who can explain the mathematical subject matter to the student(s).

I will later on (next year) be making more worktexts (with full explanations) using some of this material, and those will better serve the homeschool community.

Free course materials for college level math

Some of you (if you're a mathematics professor in some college or university) might find this interesting to browse: Massachusetts Institute of Technology offers a LONG list of mathematics undergraduate AND graduate course materials as free downloads...

Some of them have lecture outlines or notes, some have student assignments and tests. I saw even Java applets for calculus with applications, but mostly they seemed to be PDF files.

So here's the link:
http://ocw.mit.edu/OcwWeb/Mathematics/index.htm

Free course materials for college level math

Some of you (if you're a mathematics professor in some college or university) might find this interesting to browse: Massachusetts Institute of Technology offers a LONG list of mathematics undergraduate AND graduate course materials as free downloads...

Some of them have lecture outlines or notes, some have student assignments and tests. I saw even Java applets for calculus with applications, but mostly they seemed to be PDF files.

So here's the link:
http://ocw.mit.edu/OcwWeb/Mathematics/index.htm

Homeschooling Carnival, week 42

I haven't made it to the carnival for a while, but this week I did because Shannon emailed me directly.

The theme of this 42th carnival is "42"! You might recognize that number from the hilarious sci-fi novel "The Hitchhiker’s Guide to the Galaxy". In it, the supercomputer Deep Thought computes the Ultimate Answer to Life, the Universe, and Everything - and spouts out the answer as "42". (Well, then they go to finding out the exact question... if the answer is 42, what is the question?)

Anyways, I enjoyed that book as a teenager. Wikipedia lets us know that the author Douglas Adams just had that there as a joke. (Well, that whole book is just full of jokes, as far as I can remember...)

But, the 'meat' of all this is of course the carnival itself... Is homeschooling the ultimate answer to life, the universe, and everything?

My submission was how teaching math in early grades does not have to cost anything.

Homeschooling Carnival, week 42

I haven't made it to the carnival for a while, but this week I did because Shannon emailed me directly.

The theme of this 42th carnival is "42"! You might recognize that number from the hilarious sci-fi novel "The Hitchhiker’s Guide to the Galaxy". In it, the supercomputer Deep Thought computes the Ultimate Answer to Life, the Universe, and Everything - and spouts out the answer as "42". (Well, then they go to finding out the exact question... if the answer is 42, what is the question?)

Anyways, I enjoyed that book as a teenager. Wikipedia lets us know that the author Douglas Adams just had that there as a joke. (Well, that whole book is just full of jokes, as far as I can remember...)

But, the 'meat' of all this is of course the carnival itself... Is homeschooling the ultimate answer to life, the universe, and everything?

My submission was how teaching math in early grades does not have to cost anything.

Gender differences in math abilities

This is an interesting study about whether there are differences between men's and women's abilities in math.

The researchers first asked the participants (college students) a question, then they took he Vandenberg Mental Rotation Test, a standard test of visual-spatial ability.

One group was first asked whether they lived in a single-sex or co-ed dorm. That subtly triggers a person to think about their gender. Men in this group did 25 percent to 30 percent better than the women.

In the control group, the students were first asked about how it is to live in Northeastern United States. The results of the visual-spatial test were familiar: men performing 15-20 percent better. (That is the typical result whenever this test is given to men and women.)

BUT the surprise came in the group where students were first subtly made to think about their strengths. They were first asked about why they chose to attend a private liberal arts college.

In this group, there were no significant differences between men and women in the Vandenberg Mental Rotation Test!

Very interesting indeed.

The study was conducted by University of Texas psychologist Matthew S. McGlone and Joshua Aronson of New York University.

Read more details of the study here.

Gender differences in math abilities

This is an interesting study about whether there are differences between men's and women's abilities in math.

The researchers first asked the participants (college students) a question, then they took he Vandenberg Mental Rotation Test, a standard test of visual-spatial ability.

One group was first asked whether they lived in a single-sex or co-ed dorm. That subtly triggers a person to think about their gender. Men in this group did 25 percent to 30 percent better than the women.

In the control group, the students were first asked about how it is to live in Northeastern United States. The results of the visual-spatial test were familiar: men performing 15-20 percent better. (That is the typical result whenever this test is given to men and women.)

BUT the surprise came in the group where students were first subtly made to think about their strengths. They were first asked about why they chose to attend a private liberal arts college.

In this group, there were no significant differences between men and women in the Vandenberg Mental Rotation Test!

Very interesting indeed.

The study was conducted by University of Texas psychologist Matthew S. McGlone and Joshua Aronson of New York University.

Read more details of the study here.

Kindergarten math ideas

I recently talked with a friend who was concerned about the cost of math curricula for her soon-5-year-old, doing kindergarten math. Even regarding my ebooks which I've given her free access, she mentioned how even printing costs money and could get costly in the long run.

(And I know some people can print things real cheap, but not everyone. It depends on your printer.)

So I told her, teaching math doesn't have to cost anything in kindergarten or the early grades.

It's not of utmost importance to do worksheet work. You can largely just play games and explore various things. After all, playing is what that age kids do best anyway.

* Playing board games where you roll one die teaches them to recognize the dot patterns on the die.
* Later on, playing board games where you roll two dice can be used for addition practice.
* After learning the dot patterns on a die, use dominoes as "flash cards" for addition. Or better yet, make a simple game out of it: lay them right side down on table, then take turns turning one and if you can add the dots, you get to keep it.
* Try 10 out math card game for learning sums of 10. Adaptable for other sums as well.
* Game worth 1000 worksheets - this is a simple card game. Deal two cards to each player, each person adds those two, and the one with highest sum wins the cards to himself.
* Make cuisenaire rods out of cardboard and play with them.
* Let the child play freely with measuring tape, scales, and measuring cups.
* Make 'ten-bags' or 'ten-bundles' with marbles or sticks. Then you can learn place value with tens and ones with those.

And even with worksheets or written math problems, you can write those yourself in a notebook, and thus not spend a penny. You can write as few problems as you'd like.

I used to draw many empty number charts for my child in her notebook.

I also often made missing addend problems with balls for her. She had to draw more and then write the addition.

You can use colors freely, and let the child color in this notebook. For example, write a few problems on a page, and let the child draw a colored box around the problems when done.

You can draw geometric shapes, or make (colorful) patterns to be continued, such as

square square triangle


All this should be totally free.

---

See also:
My ebooks.

Math Lessons and Teaching Tips - scroll down the page to see addition, subtraction, multiplication, division, place value, fraction, geometry and decimals lessons taken from my ebooks. Most of those aren't for kindergarten but for elementary grades.

Note especially Teaching tens and ones.

Kindergarten math ideas

I recently talked with a friend who was concerned about the cost of math curricula for her soon-5-year-old, doing kindergarten math. Even regarding my ebooks which I've given her free access, she mentioned how even printing costs money and could get costly in the long run.

(And I know some people can print things real cheap, but not everyone. It depends on your printer.)

So I told her, teaching math doesn't have to cost anything in kindergarten or the early grades.

It's not of utmost importance to do worksheet work. You can largely just play games and explore various things. After all, playing is what that age kids do best anyway.

* Playing board games where you roll one die teaches them to recognize the dot patterns on the die.
* Later on, playing board games where you roll two dice can be used for addition practice.
* After learning the dot patterns on a die, use dominoes as "flash cards" for addition. Or better yet, make a simple game out of it: lay them right side down on table, then take turns turning one and if you can add the dots, you get to keep it.
* Try 10 out math card game for learning sums of 10. Adaptable for other sums as well.
* Game worth 1000 worksheets - this is a simple card game. Deal two cards to each player, each person adds those two, and the one with highest sum wins the cards to himself.
* Make cuisenaire rods out of cardboard and play with them.
* Let the child play freely with measuring tape, scales, and measuring cups.
* Make 'ten-bags' or 'ten-bundles' with marbles or sticks. Then you can learn place value with tens and ones with those.

And even with worksheets or written math problems, you can write those yourself in a notebook, and thus not spend a penny. You can write as few problems as you'd like.

I used to draw many empty number charts for my child in her notebook.

I also often made missing addend problems with balls for her. She had to draw more and then write the addition.

You can use colors freely, and let the child color in this notebook. For example, write a few problems on a page, and let the child draw a colored box around the problems when done.

You can draw geometric shapes, or make (colorful) patterns to be continued, such as

square square triangle


All this should be totally free.

---

See also:
My ebooks.

Math Lessons and Teaching Tips - scroll down the page to see addition, subtraction, multiplication, division, place value, fraction, geometry and decimals lessons taken from my ebooks. Most of those aren't for kindergarten but for elementary grades.

Note especially Teaching tens and ones.

Math misconceptions

It's very good to know something about the most common misconceptions students might have.
The website CountOn.org has 22 examples of them at http://www.counton.org/resources/misconceptions/.

Here are some examples:

#2. Multiplication always increases a number... is that really so?

Well, to small kids it appears to be so - but only if you just try whole numbers. Take 10 for example. If you multiply it by 2, 3, 4, 5, etc., it does get bigger.

But all you have to do is multiply it by a fraction less than 1, or by a negative number, and 10 does not get bigger... 1/2 x 10 is 5. 1/4 x 10 is 2.5. -3 x 10 is -30.

We must remember that repeated addition is not the only meaning or definition for multiplication. That's what it is for whole numbers.

For fractions, 1/3 x 12 is better understood as 1/3 of 12.



#3. As 1 x 1 = 1, then 0.1 x 0.1 = 0.1.
This one was new to me. Quite curious. A child might think of 0.1 as some kind of "unit" like 1.

But seriously, 0.1 is one tenth. Taking tenth part of a tenth is a hundredth, or 0.01.



#5. 3 divide 1/4 is same as 3 divide 4.
This must be due to forgetting the second part of the "invert and multiply" rule. If you remember to think of division as "how many times does the divisor fit into the dividend?", then it's easy and clear: 1/4 fits into 3 exactly 12 times.


#7. What is the value that satisfies -4 + ___ = -10 ? Many possible wrong answers, such as 6, 14, or -14.

It's a missing addend problem. We already have -4, and end up with much more negatives, namely -10. So the debt increased, by -6.


#20: 2.3 x 10 = 2.30 ? 20.3 ?
Well, multiply in parts: 10 x 2 and 10 x 0.3. You get 20 and 3, total 23.



These and more are presented at Counton.org/resources/misconceptions/. They also have a one single PDF file that contains them all.

Another site listing popular (or common) math mistakes is MathMistakes.info.

Tags: math, mistakes

Math misconceptions

It's very good to know something about the most common misconceptions students might have.
The website CountOn.org has 22 examples of them at http://www.counton.org/resources/misconceptions/.

Here are some examples:

#2. Multiplication always increases a number... is that really so?

Well, to small kids it appears to be so - but only if you just try whole numbers. Take 10 for example. If you multiply it by 2, 3, 4, 5, etc., it does get bigger.

But all you have to do is multiply it by a fraction less than 1, or by a negative number, and 10 does not get bigger... 1/2 x 10 is 5. 1/4 x 10 is 2.5. -3 x 10 is -30.

We must remember that repeated addition is not the only meaning or definition for multiplication. That's what it is for whole numbers.

For fractions, 1/3 x 12 is better understood as 1/3 of 12.



#3. As 1 x 1 = 1, then 0.1 x 0.1 = 0.1.
This one was new to me. Quite curious. A child might think of 0.1 as some kind of "unit" like 1.

But seriously, 0.1 is one tenth. Taking tenth part of a tenth is a hundredth, or 0.01.



#5. 3 divide 1/4 is same as 3 divide 4.
This must be due to forgetting the second part of the "invert and multiply" rule. If you remember to think of division as "how many times does the divisor fit into the dividend?", then it's easy and clear: 1/4 fits into 3 exactly 12 times.


#7. What is the value that satisfies -4 + ___ = -10 ? Many possible wrong answers, such as 6, 14, or -14.

It's a missing addend problem. We already have -4, and end up with much more negatives, namely -10. So the debt increased, by -6.


#20: 2.3 x 10 = 2.30 ? 20.3 ?
Well, multiply in parts: 10 x 2 and 10 x 0.3. You get 20 and 3, total 23.



These and more are presented at Counton.org/resources/misconceptions/. They also have a one single PDF file that contains them all.

Another site listing popular (or common) math mistakes is MathMistakes.info.

Tags: math, mistakes

Quite curious music box

This music box shows dots of different colors that move around in a spiraling motion.. as a dot hits a horizontal line, it makes a sound. Well I can't explain it, you have to go see and hear it.

It's not really your "easy listen" music that you'd enjoy for hours but just a fascinating way of combining mathematics and music.

A musical realization of the motion graphics of John Whitney

Quite curious music box

This music box shows dots of different colors that move around in a spiraling motion.. as a dot hits a horizontal line, it makes a sound. Well I can't explain it, you have to go see and hear it.

It's not really your "easy listen" music that you'd enjoy for hours but just a fascinating way of combining mathematics and music.

A musical realization of the motion graphics of John Whitney

Addition surprise

With my dd, I've had her study addition and subtraction lately, and fact families, practicing the connection between addition and subtraction.

But she's young. So I know games go over well.

To illustrate the fact families better, I made math "rods" out of cardboard - you know, all different lengths. She really liked them.

But I got quite a surprise at her reaction to my made-on-the-spot Addition Surprise game...

I took three of those rods, so that two of them sum up to the third, and used the two to cover the third. I held them up and said there was a "surprise number" behind.

She saw the numbers on the two rods, added them, and told me the answer. I just said something like, "Ta da ta daa!!!" and uncovered the one on the bottom for her to grab.

She just giggled and giggled, and absolutely loved it.

Such a little thing to us adults - such an impact on a little one.

Amazon actually sells cuisenaire rods, too, so here's a pic if you don't know what they are. Just rods of different lengths. My cardboard ones had numbers printed on them, too.

Addition surprise

With my dd, I've had her study addition and subtraction lately, and fact families, practicing the connection between addition and subtraction.

But she's young. So I know games go over well.

To illustrate the fact families better, I made math "rods" out of cardboard - you know, all different lengths. She really liked them.

But I got quite a surprise at her reaction to my made-on-the-spot Addition Surprise game...

I took three of those rods, so that two of them sum up to the third, and used the two to cover the third. I held them up and said there was a "surprise number" behind.

She saw the numbers on the two rods, added them, and told me the answer. I just said something like, "Ta da ta daa!!!" and uncovered the one on the bottom for her to grab.

She just giggled and giggled, and absolutely loved it.

Such a little thing to us adults - such an impact on a little one.

Amazon actually sells cuisenaire rods, too, so here's a pic if you don't know what they are. Just rods of different lengths. My cardboard ones had numbers printed on them, too.

MathAbacus.com offer

If you're interested, I just got word that MathAbacus.com is having a Package sale: 3 books and 1 abacus for only US$35.90. This includes shipping and handling charges.

MathAbacus.com sells workbooks to go along with the abacus method of learning basic arithmetic.

The offer will only last till October 31, 2006.

MathAbacus.com offer

If you're interested, I just got word that MathAbacus.com is having a Package sale: 3 books and 1 abacus for only US$35.90. This includes shipping and handling charges.

MathAbacus.com sells workbooks to go along with the abacus method of learning basic arithmetic.

The offer will only last till October 31, 2006.

Free scripts to put on your site, plus statistics rant

I got this in the emails today; www.helpingwithmath.com has worksheets and explanations, but above all, you can DOWNLOAD and freely put on your own website (or just on your home computer) all their games, fraction calculator, charts, and worksheets and stuff.

On a personal note, I've been diligently working on 6th grade worksheets for a tutoring company. But, I'm also going to make a book to sell with the material, and fairly soon too.

So lately I've been just burying myself under statistics and playing with Excel, making charts and graphs and pictures for those worksheets. They're made with Virginia state standards as guidelines, and you know, on 6th grade, the students are supposed to study box-and-whisker plots and stem-and-leaf plots, among the usual stuff such as bar graphs and circle graphs.

My husband didn't even know what those were. I first encountered them during my university studies, as I took statistics as a minor. And I really enjoyed studying it, by the way. It was fascinating to see the power of mathematical methods in the statistical analysis.

I still enjoy statistics. But knowing something about it makes me wonder, what are 6th graders able to learn about all these things? Boxplots, stemplots - are they understanding how to interpret them, read the differences between data sets by looking at the boxplots?

Mean, median, mode, range? Can a 6th grader understand much about mean and media, besides being able to calculate them? (Such as, which is better to use when?) It seems to me they are bound to stay as just numbers without much meaning.

As you probably realize, statistics and data analysis used to be taught on much later grades in times past. I realize that in the day and time in which we live, it is essential to be able to read graphs and know how to interpret them. But it just sometimes feels like there's too much of it too early, that they would comprehend it better if it was a little later.

But, it's not up to me to decide; if it's in the standards, I do a worksheet.

Free scripts to put on your site, plus statistics rant

I got this in the emails today; www.helpingwithmath.com has worksheets and explanations, but above all, you can DOWNLOAD and freely put on your own website (or just on your home computer) all their games, fraction calculator, charts, and worksheets and stuff.

On a personal note, I've been diligently working on 6th grade worksheets for a tutoring company. But, I'm also going to make a book to sell with the material, and fairly soon too.

So lately I've been just burying myself under statistics and playing with Excel, making charts and graphs and pictures for those worksheets. They're made with Virginia state standards as guidelines, and you know, on 6th grade, the students are supposed to study box-and-whisker plots and stem-and-leaf plots, among the usual stuff such as bar graphs and circle graphs.

My husband didn't even know what those were. I first encountered them during my university studies, as I took statistics as a minor. And I really enjoyed studying it, by the way. It was fascinating to see the power of mathematical methods in the statistical analysis.

I still enjoy statistics. But knowing something about it makes me wonder, what are 6th graders able to learn about all these things? Boxplots, stemplots - are they understanding how to interpret them, read the differences between data sets by looking at the boxplots?

Mean, median, mode, range? Can a 6th grader understand much about mean and media, besides being able to calculate them? (Such as, which is better to use when?) It seems to me they are bound to stay as just numbers without much meaning.

As you probably realize, statistics and data analysis used to be taught on much later grades in times past. I realize that in the day and time in which we live, it is essential to be able to read graphs and know how to interpret them. But it just sometimes feels like there's too much of it too early, that they would comprehend it better if it was a little later.

But, it's not up to me to decide; if it's in the standards, I do a worksheet.