Review of IXL - online math practice system

I've written a review of IXL.com (from "I excel"). It is an online math practice system - would be perfect for those of you who want your kids to have something for the summer months.

IXL has unlimited questions on hundreds of math topics and a comprehensive reporting system.

Their pricing is $9.95 per month for 1st child and only $2 for the next child.

Read my review here.

Review of IXL - online math practice system

I've written a review of IXL.com (from "I excel"). It is an online math practice system - would be perfect for those of you who want your kids to have something for the summer months.

IXL has unlimited questions on hundreds of math topics and a comprehensive reporting system.

Their pricing is $9.95 per month for 1st child and only $2 for the next child.

Read my review here.

Math card game for all four operations (elementary)

I'm sure youngsters will like this card "game"! I'm going to try it with mine.

Actually it's not really a game, but a fun way for kids to write some of their own math problems using a deck of cards. She even provides a free printout:

Little Blue School: Math Card Game for Addition, Subtraction, Multiplication, Division

HT: This week's Carnival of Homeschooling

Math card game for all four operations (elementary)

I'm sure youngsters will like this card "game"! I'm going to try it with mine.

Actually it's not really a game, but a fun way for kids to write some of their own math problems using a deck of cards. She even provides a free printout:

Little Blue School: Math Card Game for Addition, Subtraction, Multiplication, Division

HT: This week's Carnival of Homeschooling

Does the child need to add completing the ten?

Question:

...have an other question about using your worksheets for my daughter. She is six years old - home schooled.
Currently, we are using Addition & Subtraction 2

She is able to add and subtract any numbers from 1 to 100 but when I try to explain "complete the ten" concept, she doesn't like thinking about it. She would rather solve 9+6 by counting on the finger 6 after 9 ... . She complains that I have to first subtract then add to make 10.

My question is: Should I let her complete the exercises without making her think in this manner?

It is important that she understands the IDEA in completing the ten. I gather that she does indeed understand the idea, but doesn't want to do it, since you'd have to both subtract and add, right? She just likes to do one operation, not two?

The adding though, is really easy, because you add 10 + 5 or 10 + 7 or something to ten. If she's counting with fingers, she's not yet seeing the easiness of this adding. It is also faster to do 9 + 6 by first subtracting one from 6 (getting 5), and then adding 10 + 5, because you can do 10 + 5 really easy: it is one ten and five, or we call that fifteen.

Of course she must understand place value to realize how easy that addition is.

Now, the goal is ALSO to eventually memorize the answers. You might let her know that she can't always count with fingers to find 9 + 6 but a quicker method is needed.

You can let her do some exercises "her way" for now as long as she also is able to understand and do the "complete the ten".

But come back to this idea how completing the ten makes the whole calculation easier and quicker. She WILL need to understand the idea when studying CARRYING in addition, (I mean to totally understand how it works. I realize kids can learn carrying mechanically as well.)

So you can say 1-3 months later reprint some pages and try again.

Does the child need to add completing the ten?

Question:

...have an other question about using your worksheets for my daughter. She is six years old - home schooled.
Currently, we are using Addition & Subtraction 2

She is able to add and subtract any numbers from 1 to 100 but when I try to explain "complete the ten" concept, she doesn't like thinking about it. She would rather solve 9+6 by counting on the finger 6 after 9 ... . She complains that I have to first subtract then add to make 10.

My question is: Should I let her complete the exercises without making her think in this manner?

It is important that she understands the IDEA in completing the ten. I gather that she does indeed understand the idea, but doesn't want to do it, since you'd have to both subtract and add, right? She just likes to do one operation, not two?

The adding though, is really easy, because you add 10 + 5 or 10 + 7 or something to ten. If she's counting with fingers, she's not yet seeing the easiness of this adding. It is also faster to do 9 + 6 by first subtracting one from 6 (getting 5), and then adding 10 + 5, because you can do 10 + 5 really easy: it is one ten and five, or we call that fifteen.

Of course she must understand place value to realize how easy that addition is.

Now, the goal is ALSO to eventually memorize the answers. You might let her know that she can't always count with fingers to find 9 + 6 but a quicker method is needed.

You can let her do some exercises "her way" for now as long as she also is able to understand and do the "complete the ten".

But come back to this idea how completing the ten makes the whole calculation easier and quicker. She WILL need to understand the idea when studying CARRYING in addition, (I mean to totally understand how it works. I realize kids can learn carrying mechanically as well.)

So you can say 1-3 months later reprint some pages and try again.

Teaching elementary mathematics - don't skip stages

I received recently a book called "Arithmetic for Parents. A book for Grownups about Children's Mathematics" by Ron Aharoni.

I want to tell you a little bit about this book, because I think it is exceptionally good, and a very worthwhile book to read if you're a teacher OR a parent.

Ron Aharoni was like many: he thought he could easily teach elementary school math because he knew college level math and beyond (he was teaching math in a university).

But... he was in for a big surprise when he entered the fourth, fifth, and first grade classes in a backward town in northern Israel, in 2000.

One surprise he had was that...

• ...he did not know how to teach elementary mathematics.

For example, during his first lessons he took children outside to measure shadows of trees, buildings, and the children themselves. The idea was to use the ratio of a child's shadow to his height to find the height of a tree or a building.

He took children outside to draw circles on the pavement and measure diameters, and compare them.

But as he admits in the introduction to his book, his first lessons were just a mess and confusion, with little meaningful teaching going on.

He wanted to do "hands-on experimentations" with children. But he didn't know how to integrate the hands-on experiences into the learning process, how to be systematic.

He lacked the idea of building concepts upon many small elements, one upon another, and the idea that you cannot skip any of these steps.

But he found out all that, and wrote an excellent book about how to teach elementary math.

...and the other big surprise was that

• During his teaching career in the elementary school he actually learned a LOT of mathematics... not new facts, but subtleties: how that concepts we adults think are easy, are actually built upon other simpler concepts and notions, and how children need to be explained ALL those little steps.

For example, Aharoni learned that he couldn't just present first graders a question such as

"Donna has 4 pencils more than Joseph. How many pencils does Donna have, if Joseph has 5 pencils?"

He had skipped stages. Children couldn't do it. They needed first to learn the concept of there being a certain number more than another number.

You can read how he handled the lesson from that on, and many other things.

The last part of the book goes through all important concepts in elementary mathematics from grade 1 to grade 6, and explains these subtleties and little steps that you should teach.

I highly recommend this book. It costs $19.95, you can find it at www.sumizdat.org.

Please note, I'm not affiliated with this; I'm not getting a penny for promoting the book. I just truly feel that reading it is a MUST for each and every teacher and parent who's teaching elementary math.

Teaching elementary mathematics - don't skip stages

I received recently a book called "Arithmetic for Parents. A book for Grownups about Children's Mathematics" by Ron Aharoni.

I want to tell you a little bit about this book, because I think it is exceptionally good, and a very worthwhile book to read if you're a teacher OR a parent.

Ron Aharoni was like many: he thought he could easily teach elementary school math because he knew college level math and beyond (he was teaching math in a university).

But... he was in for a big surprise when he entered the fourth, fifth, and first grade classes in a backward town in northern Israel, in 2000.

One surprise he had was that...

• ...he did not know how to teach elementary mathematics.

For example, during his first lessons he took children outside to measure shadows of trees, buildings, and the children themselves. The idea was to use the ratio of a child's shadow to his height to find the height of a tree or a building.

He took children outside to draw circles on the pavement and measure diameters, and compare them.

But as he admits in the introduction to his book, his first lessons were just a mess and confusion, with little meaningful teaching going on.

He wanted to do "hands-on experimentations" with children. But he didn't know how to integrate the hands-on experiences into the learning process, how to be systematic.

He lacked the idea of building concepts upon many small elements, one upon another, and the idea that you cannot skip any of these steps.

But he found out all that, and wrote an excellent book about how to teach elementary math.

...and the other big surprise was that

• During his teaching career in the elementary school he actually learned a LOT of mathematics... not new facts, but subtleties: how that concepts we adults think are easy, are actually built upon other simpler concepts and notions, and how children need to be explained ALL those little steps.

For example, Aharoni learned that he couldn't just present first graders a question such as

"Donna has 4 pencils more than Joseph. How many pencils does Donna have, if Joseph has 5 pencils?"

He had skipped stages. Children couldn't do it. They needed first to learn the concept of there being a certain number more than another number.

You can read how he handled the lesson from that on, and many other things.

The last part of the book goes through all important concepts in elementary mathematics from grade 1 to grade 6, and explains these subtleties and little steps that you should teach.

I highly recommend this book. It costs $19.95, you can find it at www.sumizdat.org.

Please note, I'm not affiliated with this; I'm not getting a penny for promoting the book. I just truly feel that reading it is a MUST for each and every teacher and parent who's teaching elementary math.

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.

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.

Online math resources

Resources

These are some of the links I've added to my site recently. Maybe there's some that interest you.

Mathopenref.com
Free online textbook for high school geometry; not finished.

GapMinder
Visualizing human development trends (such as poverty, health, gaps, income on a global scale) via stunning, interactive statistical graphs. This is an interactive, dynamic tool and not just static graphs. Download the software or the reports for free.

How to write proofs
A 12-part tutorial on proof writing. Includes direct proof, proof by contradiction, proof by contrapositive, mathematical induction, if and only if, and proof strategies.

Money Math
Crystal clear tutorial on interest.

Graph Mole
A fun game about plotting points in coordinate plane. Plot points before the mole eats the vegetables.

---

All sorts of sites to explore! But if those didn't fit your bill, if you're in need of a game or tutorial about specific math topic, check my link lists of online math resources; they're organized by level and topic:

Math Help & Tutoring

Math help, Homework help, tutoring

General Math Resources

Math history, Problem solving, Hands-on, For Gifted, Mental Math, Test prep

Elementary Math 1

Basic operations, Times tables, Place value

Elementary Math 2

Time, Money, Measuring, Geometry

Middle School Math 1

Fractions, Decimals, Percent, Integers

Middle School Math 2

Measuring, Coordinate Plane, Geometry

High School 1

Algebra, Graphing, Calculus

High School 2

Geometry,  Trigonometry, Statistics, Logic

Fun & Games Math Sites

Math games, quizzes, or interactive tutorials websites

Online math resources

Resources

These are some of the links I've added to my site recently. Maybe there's some that interest you.

Mathopenref.com
Free online textbook for high school geometry; not finished.

GapMinder
Visualizing human development trends (such as poverty, health, gaps, income on a global scale) via stunning, interactive statistical graphs. This is an interactive, dynamic tool and not just static graphs. Download the software or the reports for free.

How to write proofs
A 12-part tutorial on proof writing. Includes direct proof, proof by contradiction, proof by contrapositive, mathematical induction, if and only if, and proof strategies.

Money Math
Crystal clear tutorial on interest.

Graph Mole
A fun game about plotting points in coordinate plane. Plot points before the mole eats the vegetables.

---

All sorts of sites to explore! But if those didn't fit your bill, if you're in need of a game or tutorial about specific math topic, check my link lists of online math resources; they're organized by level and topic:

Math Help & Tutoring

Math help, Homework help, tutoring

General Math Resources

Math history, Problem solving, Hands-on, For Gifted, Mental Math, Test prep

Elementary Math 1

Basic operations, Times tables, Place value

Elementary Math 2

Time, Money, Measuring, Geometry

Middle School Math 1

Fractions, Decimals, Percent, Integers

Middle School Math 2

Measuring, Coordinate Plane, Geometry

High School 1

Algebra, Graphing, Calculus

High School 2

Geometry,  Trigonometry, Statistics, Logic

Fun & Games Math Sites

Math games, quizzes, or interactive tutorials websites

Division of fractions

This topic is often not understood real well by teachers or students. But we want them to learn, not only the rule, but also the meaning.

These ideas can help you to explain and understand division of fractions:


1) The rule of "invert and multiply" applies to division in general - not just to division of fractions. It is a general principle. For example:

20 ÷ 4

I can invert and multiply:

20 × 1/4 = 5.

With whole numbers, division can be thought of as making equal parts. When you divide something by 7, you're dividing it into 7 parts, so might as well just take 1/7 part - multiply by 1/7.

You can always change division into multiplication with this principle: 18 ÷ 2.51 = 18 × 1/2.51





2) Think of fraction division this way: how many times does the divisor fit into the dividend? You can use this to judge the reasonableness of your answer.

For example consider 1 3/5 ÷ 2/3. Clearly 2/3 can fit into 1 3/5 more than two times.

1 3/5 ÷ 2/3 = 8/5 × 2/3 = 16/15 = 1 1/15 - just a tad over 1 whole. CAN YOU SPOT THE ERROR?????


Or, 3/8 ÷ 11/12. Here the divisor is greater than dividend. Well, that means that it won't fit even once into 3/8; it only "fits" into 3/8 about half ways, so the answer should be near half.

And indeed, using the rule, 3/8 ÷ 11/12 = 3/8 × 12/11 = 3/2 × 3/11 = 9/22.



3) One alternative method for fraction division is to first change both the dividend and divisor into equivalent fractions, and then simply divide the numerators.

 5/6 ÷ 1/8 =
20/24 ÷ 3/24
= 20 ÷ 3 = 6 2/3.

The answer makes sense since 1/8 can "fit" more than six times into 5/6.

I think this is neat, because it helps make sense of the procedure: how many times can 3/24 fit into 20/24? It's the same as asking how many times can 3 fit into 20.

Another:
2 2/11 ÷ 2/5 = 24/11 ÷ 2/5
= 120/55 ÷ 22/55
= 120 ÷ 22 = 5 10/22 = 5 5/11.


4) Back to the rule of "invert and multiply". Let's think about number 1 as the dividend first.

How many times does 1/2 fit into one? Two times. 1 ÷ 1/2 = 2.

How many times does 3/4 fit into one? It fits there once, and there's 1/4 left, and into 1/4 we can fit 1/3 of 3/4. So total 4/3 times.
1 ÷ 3/4 = 4/3.

How many times does 1 2/5 fit into one? Not even once, clearly. But if you think of 1 has 5/5, you can see that five of the 7 fifths can fit... so 5/7 times. You might have make a picture of this in your mind or paper. Draw one whole as 5/5, then draw 7/5 next to it. Exactly five of the 7 parts of 7/5 fit into one.

1 ÷ 7/5 = 5/7.

You can follow this thinking with any fraction m/n: 1 ÷ m/n = n/m.

OK, so if 5/6 goes to one exactly 6/5 times, then how many times can 5/6 fit into 3 13/15 ?

Exactly 6/5 × 3 13/15 times.

Or, 3 13/5 × 6/5, if you like. Invert and multiply!


Hope this helps!!

Division of fractions

This topic is often not understood real well by teachers or students. But we want them to learn, not only the rule, but also the meaning.

These ideas can help you to explain and understand division of fractions:


1) The rule of "invert and multiply" applies to division in general - not just to division of fractions. It is a general principle. For example:

20 ÷ 4

I can invert and multiply:

20 × 1/4 = 5.

With whole numbers, division can be thought of as making equal parts. When you divide something by 7, you're dividing it into 7 parts, so might as well just take 1/7 part - multiply by 1/7.

You can always change division into multiplication with this principle: 18 ÷ 2.51 = 18 × 1/2.51





2) Think of fraction division this way: how many times does the divisor fit into the dividend? You can use this to judge the reasonableness of your answer.

For example consider 1 3/5 ÷ 2/3. Clearly 2/3 can fit into 1 3/5 more than two times.

1 3/5 ÷ 2/3 = 8/5 × 2/3 = 16/15 = 1 1/15 - just a tad over 1 whole. CAN YOU SPOT THE ERROR?????


Or, 3/8 ÷ 11/12. Here the divisor is greater than dividend. Well, that means that it won't fit even once into 3/8; it only "fits" into 3/8 about half ways, so the answer should be near half.

And indeed, using the rule, 3/8 ÷ 11/12 = 3/8 × 12/11 = 3/2 × 3/11 = 9/22.



3) One alternative method for fraction division is to first change both the dividend and divisor into equivalent fractions, and then simply divide the numerators.

 5/6 ÷ 1/8 =
20/24 ÷ 3/24
= 20 ÷ 3 = 6 2/3.

The answer makes sense since 1/8 can "fit" more than six times into 5/6.

I think this is neat, because it helps make sense of the procedure: how many times can 3/24 fit into 20/24? It's the same as asking how many times can 3 fit into 20.

Another:
2 2/11 ÷ 2/5 = 24/11 ÷ 2/5
= 120/55 ÷ 22/55
= 120 ÷ 22 = 5 10/22 = 5 5/11.


4) Back to the rule of "invert and multiply". Let's think about number 1 as the dividend first.

How many times does 1/2 fit into one? Two times. 1 ÷ 1/2 = 2.

How many times does 3/4 fit into one? It fits there once, and there's 1/4 left, and into 1/4 we can fit 1/3 of 3/4. So total 4/3 times.
1 ÷ 3/4 = 4/3.

How many times does 1 2/5 fit into one? Not even once, clearly. But if you think of 1 has 5/5, you can see that five of the 7 fifths can fit... so 5/7 times. You might have make a picture of this in your mind or paper. Draw one whole as 5/5, then draw 7/5 next to it. Exactly five of the 7 parts of 7/5 fit into one.

1 ÷ 7/5 = 5/7.

You can follow this thinking with any fraction m/n: 1 ÷ m/n = n/m.

OK, so if 5/6 goes to one exactly 6/5 times, then how many times can 5/6 fit into 3 13/15 ?

Exactly 6/5 × 3 13/15 times.

Or, 3 13/5 × 6/5, if you like. Invert and multiply!


Hope this helps!!

Adding 2-digit numbers - followup

So today I experimented, and gave my daughter a problem 15 + 25 for starters, to see what she'd do.

1 5
+ 2 5

3 10

She promptly wrote down 3 and 10. Then she looked at the numbers, kind of wondering, saying "It's ten three." I asked her how would we say the number, pointing with my finger and making a sweeping motion from 3 towards 10, and then she said, "Thirty-ten."

I said yes, it's thirty-ten, but we say it differently normally. Then I asked her to find it out using abacus. She easily found it was 40. Then we erased her numbers and wrote in 40.

She did two more similar problems on her own, first adding tens, then ones (as her custom seems to be), and then erasing the numbers and changing them to the right result. She seemed very happy for being able to understand the gimmick.

So we did all that without discussing 'carrying' or putting little number 1 up on top, or anything.

I figure next time I'll try a problem where the ones add up to 11 or 12.

Adding 2-digit numbers - followup

So today I experimented, and gave my daughter a problem 15 + 25 for starters, to see what she'd do.

1 5
+ 2 5

3 10

She promptly wrote down 3 and 10. Then she looked at the numbers, kind of wondering, saying "It's ten three." I asked her how would we say the number, pointing with my finger and making a sweeping motion from 3 towards 10, and then she said, "Thirty-ten."

I said yes, it's thirty-ten, but we say it differently normally. Then I asked her to find it out using abacus. She easily found it was 40. Then we erased her numbers and wrote in 40.

She did two more similar problems on her own, first adding tens, then ones (as her custom seems to be), and then erasing the numbers and changing them to the right result. She seemed very happy for being able to understand the gimmick.

So we did all that without discussing 'carrying' or putting little number 1 up on top, or anything.

I figure next time I'll try a problem where the ones add up to 11 or 12.

Adding 2-digit numbers

Just today I made my dd some more math problems with adding 2-digit numbers - the easy type where you don't have to carry. She's getting it now pretty good.

I always talk about adding so many tens to so many tens so she won't forget it's about adding tens and adding ones. Somehow she's doing the tens first, then the ones, and she tends to say, after adding tens:

52
+ 37

8  

"Oh, it's going to be eighty-something."

You know, that's the way many people add numbers mentally: 52 + 37, we add tens first and get 80. Then add ones.

Even with 46 + 36, you probably add tens first and get 70, then add ones and go over to 82.

So a thought hit my mind: one day I will present to her a problem where the ones make a ten, such as

26
+ 34

Is she going to get an answer fifty-ten?

Well, that's what it is, basically. We just need to recognize that's not the normal way of saying it but that we need to add that ten to the fifty so it becomes sixty.

Anyway, I really feel I'm going to experiment with something like that soon. And I wonder if doing it this way would make it more obvious why we need to 'carry'.

You see, if you first add the ones, you don't really yet know what you're coming to and so the whole addition is more 'uncertain'.

26
+ 58

14

Maybe the child adds ones and gets 14, and feels perfectly happy about that.

But if you add tens first and are expecting a single-digit number, would it make it easier to grasp that the tens number needs to change??

Or, here's one more 'variation':

26
+ 58

14

Then add tens... get seventy... and erase the one of the 14 and replace it with 8. This, of course, would just be an alternate way of 'carrying'.

26
+ 58

84

What do you think? Or maybe it's a matter of, well it won't matter too much whichever way, as long as the end result is the same and we learn our math.

Tags: math, elementary

Adding 2-digit numbers

Just today I made my dd some more math problems with adding 2-digit numbers - the easy type where you don't have to carry. She's getting it now pretty good.

I always talk about adding so many tens to so many tens so she won't forget it's about adding tens and adding ones. Somehow she's doing the tens first, then the ones, and she tends to say, after adding tens:

52
+ 37

8  

"Oh, it's going to be eighty-something."

You know, that's the way many people add numbers mentally: 52 + 37, we add tens first and get 80. Then add ones.

Even with 46 + 36, you probably add tens first and get 70, then add ones and go over to 82.

So a thought hit my mind: one day I will present to her a problem where the ones make a ten, such as

26
+ 34

Is she going to get an answer fifty-ten?

Well, that's what it is, basically. We just need to recognize that's not the normal way of saying it but that we need to add that ten to the fifty so it becomes sixty.

Anyway, I really feel I'm going to experiment with something like that soon. And I wonder if doing it this way would make it more obvious why we need to 'carry'.

You see, if you first add the ones, you don't really yet know what you're coming to and so the whole addition is more 'uncertain'.

26
+ 58

14

Maybe the child adds ones and gets 14, and feels perfectly happy about that.

But if you add tens first and are expecting a single-digit number, would it make it easier to grasp that the tens number needs to change??

Or, here's one more 'variation':

26
+ 58

14

Then add tens... get seventy... and erase the one of the 14 and replace it with 8. This, of course, would just be an alternate way of 'carrying'.

26
+ 58

84

What do you think? Or maybe it's a matter of, well it won't matter too much whichever way, as long as the end result is the same and we learn our math.

Tags: math, elementary