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, Homework help, tutoring
Math history, Problem solving, Hands-on, For Gifted, Mental Math, Test prep
Basic operations, Times tables, Place value
Time, Money, Measuring, Geometry
Fractions, Decimals, Percent, Integers
Measuring, Coordinate Plane, Geometry
Algebra, Graphing, Calculus
Geometry, Trigonometry, Statistics, Logic
Math games, quizzes, or interactive tutorials websites
Thursday, September 28, 2006
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, Homework help, tutoring
Math history, Problem solving, Hands-on, For Gifted, Mental Math, Test prep
Basic operations, Times tables, Place value
Time, Money, Measuring, Geometry
Fractions, Decimals, Percent, Integers
Measuring, Coordinate Plane, Geometry
Algebra, Graphing, Calculus
Geometry, Trigonometry, Statistics, Logic
Math games, quizzes, or interactive tutorials websites
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, Homework help, tutoring
Math history, Problem solving, Hands-on, For Gifted, Mental Math, Test prep
Basic operations, Times tables, Place value
Time, Money, Measuring, Geometry
Fractions, Decimals, Percent, Integers
Measuring, Coordinate Plane, Geometry
Algebra, Graphing, Calculus
Geometry, Trigonometry, Statistics, Logic
Math games, quizzes, or interactive tutorials websites
Tuesday, September 26, 2006
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!!
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!!
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!!
Monday, September 25, 2006
Curriculum focal points
I'm going to talk again about the new report released by NCTM, Curriculum Focal Points.
For each grade, it describes THREE focal points, and also explains to which other mathematical topics these focal points connect to.
By familiarizing yourself with these few points, you can see the basics of school mathematics unfold before your eyes. Knowing the basic goals is essential for being a good teacher.
By the way, these aren't the only things kids might study on a given grade. They are the focus areas. Many of the other topics would connect with these.
Go look at the pics at textsavvy.blogspot.com about the way textbooks and programs typically present the topics, versus a "VISION OF CURRICULUM" laid out by the Focal Points document.
On grade 1, the focus is on
Measuring and simple bar and picture graphs are presented as applications of these.
On Grade 2:
Skip-counting is also included, and prepares children for 3rd grade and multiplication.
On grade 3:
Measuring using fractional parts ties nicely in with the second point.
So we get started on multiplication and division. Notice on next grade, one of the focal points is "developing quick recall", or basically memorizing multiplication/division facts.
On grade 4:
Well, reading these makes teaching school mathematics sound almost easy to teach - compared to reading a big laundry list of objectives and thinking, "Oh well, I'm supposed to cover all that".
You can find the other grades at Curriculum Focal Points by Grade.
For each grade, it describes THREE focal points, and also explains to which other mathematical topics these focal points connect to.
By familiarizing yourself with these few points, you can see the basics of school mathematics unfold before your eyes. Knowing the basic goals is essential for being a good teacher.
By the way, these aren't the only things kids might study on a given grade. They are the focus areas. Many of the other topics would connect with these.
Go look at the pics at textsavvy.blogspot.com about the way textbooks and programs typically present the topics, versus a "VISION OF CURRICULUM" laid out by the Focal Points document.
On grade 1, the focus is on
- addition/subtraction: basic basic addition facts and related subtraction facts. Learning to add and subtract two-digit numbers.
- Understanding whole numbers in terms of tens and ones
- Composing and decomposing geometric shapes
Measuring and simple bar and picture graphs are presented as applications of these.
On Grade 2:
- Understanding the base-ten numeration system and place-value concepts at least till 1000.
- Quick recall of addition facts and related subtraction facts and fluency with multidigit addition and subtraction.
- Measurement: Developing an understanding of linear measurement and facility in measuring lengths.
Skip-counting is also included, and prepares children for 3rd grade and multiplication.
On grade 3:
- Understanding multiplication and division, and strategies for basic multiplication facts and related division facts.
- Understanding fractions and fraction equivalence
- Geometry: Describing and analyzing properties of two-dimensional shapes.
Measuring using fractional parts ties nicely in with the second point.
So we get started on multiplication and division. Notice on next grade, one of the focal points is "developing quick recall", or basically memorizing multiplication/division facts.
On grade 4:
- Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication.
- Understanding decimals, including the connections between fractions and decimals.
- Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes.
Well, reading these makes teaching school mathematics sound almost easy to teach - compared to reading a big laundry list of objectives and thinking, "Oh well, I'm supposed to cover all that".
You can find the other grades at Curriculum Focal Points by Grade.
Curriculum focal points
I'm going to talk again about the new report released by NCTM, Curriculum Focal Points.
For each grade, it describes THREE focal points, and also explains to which other mathematical topics these focal points connect to.
By familiarizing yourself with these few points, you can see the basics of school mathematics unfold before your eyes. Knowing the basic goals is essential for being a good teacher.
By the way, these aren't the only things kids might study on a given grade. They are the focus areas. Many of the other topics would connect with these.
Go look at the pics at textsavvy.blogspot.com about the way textbooks and programs typically present the topics, versus a "VISION OF CURRICULUM" laid out by the Focal Points document.
On grade 1, the focus is on
Measuring and simple bar and picture graphs are presented as applications of these.
On Grade 2:
Skip-counting is also included, and prepares children for 3rd grade and multiplication.
On grade 3:
Measuring using fractional parts ties nicely in with the second point.
So we get started on multiplication and division. Notice on next grade, one of the focal points is "developing quick recall", or basically memorizing multiplication/division facts.
On grade 4:
Well, reading these makes teaching school mathematics sound almost easy to teach - compared to reading a big laundry list of objectives and thinking, "Oh well, I'm supposed to cover all that".
You can find the other grades at Curriculum Focal Points by Grade.
For each grade, it describes THREE focal points, and also explains to which other mathematical topics these focal points connect to.
By familiarizing yourself with these few points, you can see the basics of school mathematics unfold before your eyes. Knowing the basic goals is essential for being a good teacher.
By the way, these aren't the only things kids might study on a given grade. They are the focus areas. Many of the other topics would connect with these.
Go look at the pics at textsavvy.blogspot.com about the way textbooks and programs typically present the topics, versus a "VISION OF CURRICULUM" laid out by the Focal Points document.
On grade 1, the focus is on
- addition/subtraction: basic basic addition facts and related subtraction facts. Learning to add and subtract two-digit numbers.
- Understanding whole numbers in terms of tens and ones
- Composing and decomposing geometric shapes
Measuring and simple bar and picture graphs are presented as applications of these.
On Grade 2:
- Understanding the base-ten numeration system and place-value concepts at least till 1000.
- Quick recall of addition facts and related subtraction facts and fluency with multidigit addition and subtraction.
- Measurement: Developing an understanding of linear measurement and facility in measuring lengths.
Skip-counting is also included, and prepares children for 3rd grade and multiplication.
On grade 3:
- Understanding multiplication and division, and strategies for basic multiplication facts and related division facts.
- Understanding fractions and fraction equivalence
- Geometry: Describing and analyzing properties of two-dimensional shapes.
Measuring using fractional parts ties nicely in with the second point.
So we get started on multiplication and division. Notice on next grade, one of the focal points is "developing quick recall", or basically memorizing multiplication/division facts.
On grade 4:
- Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication.
- Understanding decimals, including the connections between fractions and decimals.
- Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes.
Well, reading these makes teaching school mathematics sound almost easy to teach - compared to reading a big laundry list of objectives and thinking, "Oh well, I'm supposed to cover all that".
You can find the other grades at Curriculum Focal Points by Grade.
Tuesday, September 19, 2006
Rational or not?
is 9/56 rational? when converted to a decimal it seems to be never ending and it seems like there's no pattern (at least as far as the calculator shows)
Well, relying on a calculator is leading this person astray.
Obviously the number is rational - it's a fraction (!); it fits the definition of a rational number.
The calculator gives 0.160714286 (to nine decimal digits), but if you use long division and continue it till you get just a few digits more, you get 0.16071428571428..., or 0.160714285.
Rational or not?
is 9/56 rational? when converted to a decimal it seems to be never ending and it seems like there's no pattern (at least as far as the calculator shows)
Well, relying on a calculator is leading this person astray.
Obviously the number is rational - it's a fraction (!); it fits the definition of a rational number.
The calculator gives 0.160714286 (to nine decimal digits), but if you use long division and continue it till you get just a few digits more, you get 0.16071428571428..., or 0.160714285.
Monday, September 18, 2006
Curriculum Focal Points
National Council of Teachers of Mathematics (NCTM) has released a new report entitled "The Curriculum Focal Points".
It summarizes THREE focal points of math education for each grade, and also explains the major connections between those and other areas of math.
I feel this document can be of enormous help to homeschoolers. If you've ever felt "lost" in the jungle of math standards, objectives, goals, and such, then this document is especially good for you.
The teacher needs to know what the major goals are. Then, he/she can plan how to teach those topics, what tools to use, and so on. Without the goals clearly in one's mind, math instruction can become just mindless wandering from topic to topic.
I'll probably write more about this report later, but for now, just head on over the NCTM site and read:
The Curriculum Focal Points - the main page
or
Curriculum Focal Points by Grade.
It summarizes THREE focal points of math education for each grade, and also explains the major connections between those and other areas of math.
I feel this document can be of enormous help to homeschoolers. If you've ever felt "lost" in the jungle of math standards, objectives, goals, and such, then this document is especially good for you.
The teacher needs to know what the major goals are. Then, he/she can plan how to teach those topics, what tools to use, and so on. Without the goals clearly in one's mind, math instruction can become just mindless wandering from topic to topic.
I'll probably write more about this report later, but for now, just head on over the NCTM site and read:
The Curriculum Focal Points - the main page
or
Curriculum Focal Points by Grade.
Curriculum Focal Points
National Council of Teachers of Mathematics (NCTM) has released a new report entitled "The Curriculum Focal Points".
It summarizes THREE focal points of math education for each grade, and also explains the major connections between those and other areas of math.
I feel this document can be of enormous help to homeschoolers. If you've ever felt "lost" in the jungle of math standards, objectives, goals, and such, then this document is especially good for you.
The teacher needs to know what the major goals are. Then, he/she can plan how to teach those topics, what tools to use, and so on. Without the goals clearly in one's mind, math instruction can become just mindless wandering from topic to topic.
I'll probably write more about this report later, but for now, just head on over the NCTM site and read:
The Curriculum Focal Points - the main page
or
Curriculum Focal Points by Grade.
It summarizes THREE focal points of math education for each grade, and also explains the major connections between those and other areas of math.
I feel this document can be of enormous help to homeschoolers. If you've ever felt "lost" in the jungle of math standards, objectives, goals, and such, then this document is especially good for you.
The teacher needs to know what the major goals are. Then, he/she can plan how to teach those topics, what tools to use, and so on. Without the goals clearly in one's mind, math instruction can become just mindless wandering from topic to topic.
I'll probably write more about this report later, but for now, just head on over the NCTM site and read:
The Curriculum Focal Points - the main page
or
Curriculum Focal Points by Grade.
Thursday, September 14, 2006
Calculating percent with mental math
Would you say that students' understanding of percent is sometimes - or often - hazy?
Solving that way, the unknown number would be Y, "of" would be multiplication, and "is" corresponds to '='. We'd get:
Y × 79.5% = 101.
0.795 Y = 101
Y = 101/0.795 = 127.044025157
I'm a bit leery of this method, as it's so mechanical. What if a question comes that is not worded exactly as the ones in the book, and the student just gets stuck? Or it is worded so that the student gets misled and calculates it wrong?
So while this idea is great and works, it is also necessary for students to understand the concept of percent well.
In the above problem, we are to find a number so that 79.5% of that number is 101. (Obviously, then, the number itself is more than 101.) If you understand the problem, and say the problem that way, it is pretty obvious how to write the equation:
"79.5% of that number is 101"... so 0.795 Y = 101.
Ideas for using MENTAL math for calculating percent problems
I have also made a video of this topic. It shows you how to use mental math for calculating simple percentages.
I recently got this sort of homework question sent to me:
You simply first find 10% of $50, then use that to find 30% of $50, and lastly subtract. Easy as a pie!
(10% of $50 is $5. 30% of $50 is three times as much, or $15. Lastly subtract $50 - $15 = $35. So the discounted price is #$35.)
Find the number of which 79.5% is 101.Often, solving these kinds of problems is taught with the idea that you "translate" certain words in the problem into certain symbols, and thus build an equation.
Solving that way, the unknown number would be Y, "of" would be multiplication, and "is" corresponds to '='. We'd get:
Y × 79.5% = 101.
0.795 Y = 101
Y = 101/0.795 = 127.044025157
I'm a bit leery of this method, as it's so mechanical. What if a question comes that is not worded exactly as the ones in the book, and the student just gets stuck? Or it is worded so that the student gets misled and calculates it wrong?
So while this idea is great and works, it is also necessary for students to understand the concept of percent well.
In the above problem, we are to find a number so that 79.5% of that number is 101. (Obviously, then, the number itself is more than 101.) If you understand the problem, and say the problem that way, it is pretty obvious how to write the equation:
"79.5% of that number is 101"... so 0.795 Y = 101.
Ideas for using MENTAL math for calculating percent problems
I have also made a video of this topic. It shows you how to use mental math for calculating simple percentages.
- Find 10% of some example numbers (by dividing by 10).
- Find 1% of some example numbers (by dividing by 100).
- Find 20%, 30%, 40% etc. of these numbers.
FIRST find 10% of the number, then multiply by 2, 3, 4, etc.
For example, find 20% of 18. Find 40% of $44. Find 80% of 120.
I know you can teach the student to go 0.2 × 18, 0.4 × 0.44, and 0.8 × 120 - however when using mental math, the above method seems to me to be more natural. - Find 3%, 4%, 6% etc. of these numbers.
FIRST find 1% of the number, then multiply. - Find 15% of some numbers.
First find 10%, halve that to find 5%, and add the two results. - Calculate some simple discounts. If an item is discounted 20%, 15%, etc., then find the new price.
- "40% of a number is 56. What is the number?" - types of problems.
You can do this mentally, too: First FIND 10% and then multiply that result by 10, to find 100% of the number (which is the number itself).
If 40% is 56, then 10% is 14. So 100% of the number is 140. This result is reasonable, because 40% of this number was 56, so the actual number (140) needs to be more than double that. - "34% of a number is 129. What is the number?" (Now you need a calculator.)
You don't need to write an equation. You could also first find 1% of this number, and then find 100% of the number.
If 34% of a number is 129, then 1% of that number is 129/34. Find that, and multiply the result by 100.
I recently got this sort of homework question sent to me:
I have a problem and I don't know how to solve it so here is the problem: At a popular clothing store clothes are on sale when they have hung on the rack too long. When an item is first put on sale, the store marks the prices down 30% off. If some shoes are regular-priced at $50.00, how much will they cost after the discount?
You simply first find 10% of $50, then use that to find 30% of $50, and lastly subtract. Easy as a pie!
(10% of $50 is $5. 30% of $50 is three times as much, or $15. Lastly subtract $50 - $15 = $35. So the discounted price is #$35.)
Calculating percent with mental math
Would you say that students' understanding of percent is sometimes - or often - hazy?
Solving that way, the unknown number would be Y, "of" would be multiplication, and "is" corresponds to '='. We'd get:
Y × 79.5% = 101.
0.795 Y = 101
Y = 101/0.795 = 127.044025157
I'm a bit leery of this method, as it's so mechanical. What if a question comes that is not worded exactly as the ones in the book, and the student just gets stuck? Or it is worded so that the student gets misled and calculates it wrong?
So while this idea is great and works, it is also necessary for students to understand the concept of percent well.
In the above problem, we are to find a number so that 79.5% of that number is 101. (Obviously, then, the number itself is more than 101.) If you understand the problem, and say the problem that way, it is pretty obvious how to write the equation:
"79.5% of that number is 101"... so 0.795 Y = 101.
Ideas for using MENTAL math for calculating percent problems
I have also made a video of this topic. It shows you how to use mental math for calculating simple percentages.
I recently got this sort of homework question sent to me:
You simply first find 10% of $50, then use that to find 30% of $50, and lastly subtract. Easy as a pie!
(10% of $50 is $5. 30% of $50 is three times as much, or $15. Lastly subtract $50 - $15 = $35. So the discounted price is #$35.)
Find the number of which 79.5% is 101.Often, solving these kinds of problems is taught with the idea that you "translate" certain words in the problem into certain symbols, and thus build an equation.
Solving that way, the unknown number would be Y, "of" would be multiplication, and "is" corresponds to '='. We'd get:
Y × 79.5% = 101.
0.795 Y = 101
Y = 101/0.795 = 127.044025157
I'm a bit leery of this method, as it's so mechanical. What if a question comes that is not worded exactly as the ones in the book, and the student just gets stuck? Or it is worded so that the student gets misled and calculates it wrong?
So while this idea is great and works, it is also necessary for students to understand the concept of percent well.
In the above problem, we are to find a number so that 79.5% of that number is 101. (Obviously, then, the number itself is more than 101.) If you understand the problem, and say the problem that way, it is pretty obvious how to write the equation:
"79.5% of that number is 101"... so 0.795 Y = 101.
Ideas for using MENTAL math for calculating percent problems
I have also made a video of this topic. It shows you how to use mental math for calculating simple percentages.
- Find 10% of some example numbers (by dividing by 10).
- Find 1% of some example numbers (by dividing by 100).
- Find 20%, 30%, 40% etc. of these numbers.
FIRST find 10% of the number, then multiply by 2, 3, 4, etc.
For example, find 20% of 18. Find 40% of $44. Find 80% of 120.
I know you can teach the student to go 0.2 × 18, 0.4 × 0.44, and 0.8 × 120 - however when using mental math, the above method seems to me to be more natural. - Find 3%, 4%, 6% etc. of these numbers.
FIRST find 1% of the number, then multiply. - Find 15% of some numbers.
First find 10%, halve that to find 5%, and add the two results. - Calculate some simple discounts. If an item is discounted 20%, 15%, etc., then find the new price.
- "40% of a number is 56. What is the number?" - types of problems.
You can do this mentally, too: First FIND 10% and then multiply that result by 10, to find 100% of the number (which is the number itself).
If 40% is 56, then 10% is 14. So 100% of the number is 140. This result is reasonable, because 40% of this number was 56, so the actual number (140) needs to be more than double that. - "34% of a number is 129. What is the number?" (Now you need a calculator.)
You don't need to write an equation. You could also first find 1% of this number, and then find 100% of the number.
If 34% of a number is 129, then 1% of that number is 129/34. Find that, and multiply the result by 100.
I recently got this sort of homework question sent to me:
I have a problem and I don't know how to solve it so here is the problem: At a popular clothing store clothes are on sale when they have hung on the rack too long. When an item is first put on sale, the store marks the prices down 30% off. If some shoes are regular-priced at $50.00, how much will they cost after the discount?
You simply first find 10% of $50, then use that to find 30% of $50, and lastly subtract. Easy as a pie!
(10% of $50 is $5. 30% of $50 is three times as much, or $15. Lastly subtract $50 - $15 = $35. So the discounted price is #$35.)
Tuesday, September 12, 2006
Carnival and more
First of all, carnival of homeschooling is online at Principled Discovery.
Organized under a theme of homeschooling journey, it is shock full of good stuff.
Some picks of mine (though, I didn't really have time to read many of the posts, unfortunately):
One homeschooling parent has chosen to focus on music instruction, hoping it will translate to book learning. Well I don't know how it will go, but I do believe music is powerful, and learning music will help a child's brain grow and develop. (I myself studied piano for 13 years of my life). There are scientific studies that show how music instruction can greatly improve other academic areas.
Also from the carnival: Thinking Blocks has an interactive program that helps your child model or visualize word problems. The models use blocks but you could as well call them diagrams. The idea for the program has come from Singapore Math's usage of diagrams.
My entry was Let It Make Sense.
Organized under a theme of homeschooling journey, it is shock full of good stuff.
Some picks of mine (though, I didn't really have time to read many of the posts, unfortunately):
One homeschooling parent has chosen to focus on music instruction, hoping it will translate to book learning. Well I don't know how it will go, but I do believe music is powerful, and learning music will help a child's brain grow and develop. (I myself studied piano for 13 years of my life). There are scientific studies that show how music instruction can greatly improve other academic areas.
Also from the carnival: Thinking Blocks has an interactive program that helps your child model or visualize word problems. The models use blocks but you could as well call them diagrams. The idea for the program has come from Singapore Math's usage of diagrams.
My entry was Let It Make Sense.
Carnival and more
First of all, carnival of homeschooling is online at Principled Discovery.
Organized under a theme of homeschooling journey, it is shock full of good stuff.
Some picks of mine (though, I didn't really have time to read many of the posts, unfortunately):
One homeschooling parent has chosen to focus on music instruction, hoping it will translate to book learning. Well I don't know how it will go, but I do believe music is powerful, and learning music will help a child's brain grow and develop. (I myself studied piano for 13 years of my life). There are scientific studies that show how music instruction can greatly improve other academic areas.
Also from the carnival: Thinking Blocks has an interactive program that helps your child model or visualize word problems. The models use blocks but you could as well call them diagrams. The idea for the program has come from Singapore Math's usage of diagrams.
My entry was Let It Make Sense.
Organized under a theme of homeschooling journey, it is shock full of good stuff.
Some picks of mine (though, I didn't really have time to read many of the posts, unfortunately):
One homeschooling parent has chosen to focus on music instruction, hoping it will translate to book learning. Well I don't know how it will go, but I do believe music is powerful, and learning music will help a child's brain grow and develop. (I myself studied piano for 13 years of my life). There are scientific studies that show how music instruction can greatly improve other academic areas.
Also from the carnival: Thinking Blocks has an interactive program that helps your child model or visualize word problems. The models use blocks but you could as well call them diagrams. The idea for the program has come from Singapore Math's usage of diagrams.
My entry was Let It Make Sense.
Friday, September 8, 2006
Living and Loving Math
X (however many) Habits of Highly Effective Math Teaching:
You are the teacher. You show the way - also with your attitudes, your way of life.
Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?
And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?
Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.
Math is not a drudgery, nor something just confined to math lessons.
Some ideas:
I've talked about all this before, so I really don't want to go into repeating myself too much. But I did want to include this principle in my "mini-series" of effective habits of math teaching.
Part 4: Living and Loving Math
You are the teacher. You show the way - also with your attitudes, your way of life.
Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?
And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?
Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.
Math is not a drudgery, nor something just confined to math lessons.
Some ideas:
- Let it make sense. This alone can usually make math quite a difference and kids will stay interested.
- Read through some fun math books, such as Theoni Pappas books, or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. And, there are even story books to teach math concepts - see a list here.
- Try including a bit about math history. This might work best in a homeschooling environment where there is no horrible rush to get through the thick book before the year is over. Julie at LivingMath.net has suggestions for math history books to buy.
- When you use math in your daily life, explain how you're doing it, and include the children if possible. Figure it out together.
I've talked about all this before, so I really don't want to go into repeating myself too much. But I did want to include this principle in my "mini-series" of effective habits of math teaching.
Living and Loving Math
X (however many) Habits of Highly Effective Math Teaching:
You are the teacher. You show the way - also with your attitudes, your way of life.
Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?
And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?
Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.
Math is not a drudgery, nor something just confined to math lessons.
Some ideas:
I've talked about all this before, so I really don't want to go into repeating myself too much. But I did want to include this principle in my "mini-series" of effective habits of math teaching.
Part 4: Living and Loving Math
You are the teacher. You show the way - also with your attitudes, your way of life.
Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?
And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?
Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.
Math is not a drudgery, nor something just confined to math lessons.
Some ideas:
- Let it make sense. This alone can usually make math quite a difference and kids will stay interested.
- Read through some fun math books, such as Theoni Pappas books, or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. And, there are even story books to teach math concepts - see a list here.
- Try including a bit about math history. This might work best in a homeschooling environment where there is no horrible rush to get through the thick book before the year is over. Julie at LivingMath.net has suggestions for math history books to buy.
- When you use math in your daily life, explain how you're doing it, and include the children if possible. Figure it out together.
I've talked about all this before, so I really don't want to go into repeating myself too much. But I did want to include this principle in my "mini-series" of effective habits of math teaching.
Thursday, September 7, 2006
Writing some proportions
Got a question in today,
This is a simple simple problem IF you know what the term "PROPORTION" means.
Proportion is simply an equation stating that one ratio is equal to another.
So to solve this, we play around and make ratios, and see if we can by happenstance come up with some equal ratios fitting the rule of using each number only once.
For example, 3:6 and 9:18 are two such ratios. So the PROPORTION then is (it needs an '=' sign)
3:6 = 9:18.
You can write the ratios using the fraction line, too.
Can you make others?
See also An idea of how to teach proportions.
How can I solve the question, "write as many true proportions as you can with the numbers 3,6,9,12 and 18. Use a number only once".
This is a simple simple problem IF you know what the term "PROPORTION" means.
Proportion is simply an equation stating that one ratio is equal to another.
So to solve this, we play around and make ratios, and see if we can by happenstance come up with some equal ratios fitting the rule of using each number only once.
For example, 3:6 and 9:18 are two such ratios. So the PROPORTION then is (it needs an '=' sign)
3:6 = 9:18.
You can write the ratios using the fraction line, too.
Can you make others?
See also An idea of how to teach proportions.
Writing some proportions
Got a question in today,
This is a simple simple problem IF you know what the term "PROPORTION" means.
Proportion is simply an equation stating that one ratio is equal to another.
So to solve this, we play around and make ratios, and see if we can by happenstance come up with some equal ratios fitting the rule of using each number only once.
For example, 3:6 and 9:18 are two such ratios. So the PROPORTION then is (it needs an '=' sign)
3:6 = 9:18.
You can write the ratios using the fraction line, too.
Can you make others?
See also An idea of how to teach proportions.
How can I solve the question, "write as many true proportions as you can with the numbers 3,6,9,12 and 18. Use a number only once".
This is a simple simple problem IF you know what the term "PROPORTION" means.
Proportion is simply an equation stating that one ratio is equal to another.
So to solve this, we play around and make ratios, and see if we can by happenstance come up with some equal ratios fitting the rule of using each number only once.
For example, 3:6 and 9:18 are two such ratios. So the PROPORTION then is (it needs an '=' sign)
3:6 = 9:18.
You can write the ratios using the fraction line, too.
Can you make others?
See also An idea of how to teach proportions.
Wednesday, September 6, 2006
Homeschooling Humor
Joyce Jackson just sent me a fun book filled with homeschooling humor, and said I can distribute it here.
The book is quite funny and inspirational. The stories, jokes, anecdotes, and other stories are by real homeschoolers that she simply compiled with background information.
You can download it here.
I liked it; I'm sure you will too. Enjoy!
The book is quite funny and inspirational. The stories, jokes, anecdotes, and other stories are by real homeschoolers that she simply compiled with background information.
You can download it here.
I liked it; I'm sure you will too. Enjoy!
Homeschooling Humor
Joyce Jackson just sent me a fun book filled with homeschooling humor, and said I can distribute it here.
The book is quite funny and inspirational. The stories, jokes, anecdotes, and other stories are by real homeschoolers that she simply compiled with background information.
You can download it here.
I liked it; I'm sure you will too. Enjoy!
The book is quite funny and inspirational. The stories, jokes, anecdotes, and other stories are by real homeschoolers that she simply compiled with background information.
You can download it here.
I liked it; I'm sure you will too. Enjoy!
Monday, September 4, 2006
Know Your Tools
Math teacher's tools are quite numerous nowadays.
First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser.
And the book you're using.
Then we also have computer software, animations and activities online, animated lessons and such.
There are workbooks, fun books, worktexts, online texts.
Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on.
And then there are games, games, games.
The choices are so numerous it's daunting. What's a teacher to do?
Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.
There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.
2) The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Keep these two things in mind:
3) Manipulatives. I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a "must".
Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to "go hog wild" over them.
Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.
Some of the most helpful manipulatives are:
Check out also some virtual manipulatives.
4) Geometry and measuring tools. These are pretty essential However, dynamic software can these days replace compass and ruler and easily be far better.
*Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.
*I would definitely use some math software if teaching graphing, algebra, or calculus. Check MathProf for example, or Math Mechanixs. I've listed a few more here.
If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing interactive math activities online (there's a menu on the right).
Let me know if I forgot something.
First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser.
And the book you're using.
Then we also have computer software, animations and activities online, animated lessons and such.
There are workbooks, fun books, worktexts, online texts.
Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on.
And then there are games, games, games.
The choices are so numerous it's daunting. What's a teacher to do?
Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.
There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.
Basic tools
1) The board and/or paper to write on. Essential. Easy to use.2) The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Keep these two things in mind:
i) Now matter what book you're using, YOU as the teacher have the control. Don't be a slave to the curriculum. You can skip pages, rearrange the order in which to teach the material, supplement it, and so on.
ii) Don't despair if the book you're using doesn't seem to be the perfect choice for your student. You can quite likely sell it on homeschool swap boards, and buy some other one.
3) Manipulatives. I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a "must".
Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to "go hog wild" over them.
Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.
Some of the most helpful manipulatives are:
A 100-bead abacus
Something to illustrate hundreds/tens/ones place value. I made my daughter ten-bags by putting marbles into little plastic bags. You can buy base ten math blocks
- Some sort of fraction manipulatives. You can just make pie models out of cardboard, even. Stores sell ready-made fraction models made of foam or plastic. The one below sells for less than $10 at LearningThings.com:
Check out also some virtual manipulatives.
4) Geometry and measuring tools. These are pretty essential However, dynamic software can these days replace compass and ruler and easily be far better.
The extras
These are, obviously, too many to even start listing.*Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.
*I would definitely use some math software if teaching graphing, algebra, or calculus. Check MathProf for example, or Math Mechanixs. I've listed a few more here.
If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing interactive math activities online (there's a menu on the right).
Let me know if I forgot something.
Know Your Tools
Math teacher's tools are quite numerous nowadays.
First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser.
And the book you're using.
Then we also have computer software, animations and activities online, animated lessons and such.
There are workbooks, fun books, worktexts, online texts.
Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on.
And then there are games, games, games.
The choices are so numerous it's daunting. What's a teacher to do?
Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.
There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.
2) The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Keep these two things in mind:
3) Manipulatives. I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a "must".
Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to "go hog wild" over them.
Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.
Some of the most helpful manipulatives are:
Check out also some virtual manipulatives.
4) Geometry and measuring tools. These are pretty essential However, dynamic software can these days replace compass and ruler and easily be far better.
*Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.
*I would definitely use some math software if teaching graphing, algebra, or calculus. Check MathProf for example, or Math Mechanixs. I've listed a few more here.
If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing interactive math activities online (there's a menu on the right).
Let me know if I forgot something.
First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser.
And the book you're using.
Then we also have computer software, animations and activities online, animated lessons and such.
There are workbooks, fun books, worktexts, online texts.
Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on.
And then there are games, games, games.
The choices are so numerous it's daunting. What's a teacher to do?
Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.
There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.
Basic tools
1) The board and/or paper to write on. Essential. Easy to use.2) The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Keep these two things in mind:
i) Now matter what book you're using, YOU as the teacher have the control. Don't be a slave to the curriculum. You can skip pages, rearrange the order in which to teach the material, supplement it, and so on.
ii) Don't despair if the book you're using doesn't seem to be the perfect choice for your student. You can quite likely sell it on homeschool swap boards, and buy some other one.
3) Manipulatives. I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a "must".
Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to "go hog wild" over them.
Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.
Some of the most helpful manipulatives are:
A 100-bead abacus- Some sort of fraction manipulatives. You can just make pie models out of cardboard, even. Stores sell ready-made fraction models made of foam or plastic. The one below sells for less than $10 at LearningThings.com:
Check out also some virtual manipulatives.
4) Geometry and measuring tools. These are pretty essential However, dynamic software can these days replace compass and ruler and easily be far better.
The extras
These are, obviously, too many to even start listing.*Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.
*I would definitely use some math software if teaching graphing, algebra, or calculus. Check MathProf for example, or Math Mechanixs. I've listed a few more here.
If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing interactive math activities online (there's a menu on the right).
Let me know if I forgot something.
Friday, September 1, 2006
Remember the Goals
Seven (?) Habits of Highly Effective Math Teaching:
Habit 2: Remember the Goals
What are the goals of your math teaching?
Are they
* to finish the book by the end of school year
* make sure the kids pass the test
or do you have goals such as
* My student can add, simplify, and multiply fractions
* My student can divide by 10, 100, and 1000.
These are all just "subgoals". But what is the ultimate goal of learning school mathematics?
Don't we want our students to be able to navigate their lives in this ever-so-complex modern world?
This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. Our youngsters need to be able to handle money wisely.
All that requires good understanding of parts, proportions, and percents.
Another very important goal of mathematics education as a whole is to enable the students to understand information aroud us. In today's world, this includes quite a bit of scientific information. Being able to read through it and make sense of it requires knowing big and small numbers, statistics, probability, percents.
And then one more. We need to prepare our students for further studies in math and science. Not everyone ultimately needs algebra, but many do, and teens don't always know what profession they might choose or end up with.
I'd like to add one more broad goal of math education: teaching deductive reasoning. Of course geometry is a good example of this, but when taught properly, other areas of school math can be as well.
The more you can keep these big real goals in mind, the better you can connect your subgoals to them. And the more you can keep the goals and the subgoals in mind, the better teacher you will be.
For example, adding, simplifying, and multiplying fractions all connects with a broader goal of understanding parts or part and whole. It will soon lead to ratios, proportions, and percent. Also, all fraction operations are a needed basis for solving rational equations and doing the operations with rational expressions (during algebra studies).
Tying in with the goals, remember that the BOOK or CURRICULUM is just a tool to achieve the goals -- not a goal in itself. Don't ever be a slave of any math book.
Habit 2: Remember the Goals
What are the goals of your math teaching?
Are they
* to finish the book by the end of school year
* make sure the kids pass the test
or do you have goals such as
* My student can add, simplify, and multiply fractions
* My student can divide by 10, 100, and 1000.
These are all just "subgoals". But what is the ultimate goal of learning school mathematics?
Don't we want our students to be able to navigate their lives in this ever-so-complex modern world?
This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. Our youngsters need to be able to handle money wisely.
All that requires good understanding of parts, proportions, and percents.
Another very important goal of mathematics education as a whole is to enable the students to understand information aroud us. In today's world, this includes quite a bit of scientific information. Being able to read through it and make sense of it requires knowing big and small numbers, statistics, probability, percents.
And then one more. We need to prepare our students for further studies in math and science. Not everyone ultimately needs algebra, but many do, and teens don't always know what profession they might choose or end up with.
I'd like to add one more broad goal of math education: teaching deductive reasoning. Of course geometry is a good example of this, but when taught properly, other areas of school math can be as well.
The more you can keep these big real goals in mind, the better you can connect your subgoals to them. And the more you can keep the goals and the subgoals in mind, the better teacher you will be.
For example, adding, simplifying, and multiplying fractions all connects with a broader goal of understanding parts or part and whole. It will soon lead to ratios, proportions, and percent. Also, all fraction operations are a needed basis for solving rational equations and doing the operations with rational expressions (during algebra studies).
Tying in with the goals, remember that the BOOK or CURRICULUM is just a tool to achieve the goals -- not a goal in itself. Don't ever be a slave of any math book.
Remember the Goals
Seven (?) Habits of Highly Effective Math Teaching:
Habit 2: Remember the Goals
What are the goals of your math teaching?
Are they
* to finish the book by the end of school year
* make sure the kids pass the test
or do you have goals such as
* My student can add, simplify, and multiply fractions
* My student can divide by 10, 100, and 1000.
These are all just "subgoals". But what is the ultimate goal of learning school mathematics?
Don't we want our students to be able to navigate their lives in this ever-so-complex modern world?
This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. Our youngsters need to be able to handle money wisely.
All that requires good understanding of parts, proportions, and percents.
Another very important goal of mathematics education as a whole is to enable the students to understand information aroud us. In today's world, this includes quite a bit of scientific information. Being able to read through it and make sense of it requires knowing big and small numbers, statistics, probability, percents.
And then one more. We need to prepare our students for further studies in math and science. Not everyone ultimately needs algebra, but many do, and teens don't always know what profession they might choose or end up with.
I'd like to add one more broad goal of math education: teaching deductive reasoning. Of course geometry is a good example of this, but when taught properly, other areas of school math can be as well.
The more you can keep these big real goals in mind, the better you can connect your subgoals to them. And the more you can keep the goals and the subgoals in mind, the better teacher you will be.
For example, adding, simplifying, and multiplying fractions all connects with a broader goal of understanding parts or part and whole. It will soon lead to ratios, proportions, and percent. Also, all fraction operations are a needed basis for solving rational equations and doing the operations with rational expressions (during algebra studies).
Tying in with the goals, remember that the BOOK or CURRICULUM is just a tool to achieve the goals -- not a goal in itself. Don't ever be a slave of any math book.
Habit 2: Remember the Goals
What are the goals of your math teaching?
Are they
* to finish the book by the end of school year
* make sure the kids pass the test
or do you have goals such as
* My student can add, simplify, and multiply fractions
* My student can divide by 10, 100, and 1000.
These are all just "subgoals". But what is the ultimate goal of learning school mathematics?
Don't we want our students to be able to navigate their lives in this ever-so-complex modern world?
This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. Our youngsters need to be able to handle money wisely.
All that requires good understanding of parts, proportions, and percents.
Another very important goal of mathematics education as a whole is to enable the students to understand information aroud us. In today's world, this includes quite a bit of scientific information. Being able to read through it and make sense of it requires knowing big and small numbers, statistics, probability, percents.
And then one more. We need to prepare our students for further studies in math and science. Not everyone ultimately needs algebra, but many do, and teens don't always know what profession they might choose or end up with.
I'd like to add one more broad goal of math education: teaching deductive reasoning. Of course geometry is a good example of this, but when taught properly, other areas of school math can be as well.
The more you can keep these big real goals in mind, the better you can connect your subgoals to them. And the more you can keep the goals and the subgoals in mind, the better teacher you will be.
For example, adding, simplifying, and multiplying fractions all connects with a broader goal of understanding parts or part and whole. It will soon lead to ratios, proportions, and percent. Also, all fraction operations are a needed basis for solving rational equations and doing the operations with rational expressions (during algebra studies).
Tying in with the goals, remember that the BOOK or CURRICULUM is just a tool to achieve the goals -- not a goal in itself. Don't ever be a slave of any math book.
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