This is just a reminder for myself from the tutoring I did tonight.
When you solve an absolute value equation or inequality you usually eliminate the absolute value sign by creating 2 new equations, one that mirrors the original and one that is the inverse of the original. The inverse equation is usually made by making the "answer" negative, but you could also do it by making everything inside the absolute value symbols negative leaving the answer as it is.
If you teach it this way to begin with then you are learning one process that you use again and again in absolute value equations and inequalities AND functions. Doing it this way helps you know how to find the x-intercepts and y-intercepts of an absolute value function. It helps in graphing an absolute value function. etc.
Wednesday, March 2, 2011
Tuesday, February 22, 2011
Kinesthetically graphing a line on a grid
This activity only works if you have a rather large group of people. Draw a coordinate system on the ground (with chalk outside or tape inside). Ask each person to sit at a point on the grid and identify the coordinates of their point.
Then hold up a simple linear equation. Ask each person whose coordinates make that equation true to stand up. You now have a living line.
You can do with with an equation depending on the ability level of the students.
Then hold up a simple linear equation. Ask each person whose coordinates make that equation true to stand up. You now have a living line.
You can do with with an equation depending on the ability level of the students.
Fact Practice to Introduce Graphing Linear Equations
Create a "fact practice" game that is self-correcting - they know they got the right answers if the answers all line up. And they learn graphing linear equations on the sly.
Make a Cartesian Coordinate System, just the first quadrant (positive/positive) if you are working with only positive numbers, all 4 quadrants if you include negative numbers. Make it nice and big, especially for the younger kids.
Make a stack of number cards so that two of the numbers would fit nicely at each point on the coordinate system. Make 6 or more of each number.
If the child is unfamiliar with finding points on a grid explain how it works. You can use games like battleship and cross word puzzles to teach them the basics. Explain for this grid that they will have 2 numbers and to find a point on the grid, go to the first number on the bottom and the second number on the side and find there they meet. Remind them that the first number is the "across" number and the second number is the "up/down" number. You may even want to write that on the grid somewhere.
Stack half of the number cards on the x-axis or bottom grid line. Put the ones cards at the first mark, the 2's at the second mark, the 3's at the third mark, etc. Do the same with the y-axis or vertical grid line.
Give your student one of the "problems". Then ask them to take any number from the bottom grid and put it into the first blank in the problem. Once they've solved the problem they will take a number from the vertical grid line and put it into the answer slot. Then take the two numbers (remembering which is first and which is second) and put them at their "point" on the coordinate system.
Repeat this process 3 or 4 times and they should see a "line" of solution points start to appear. If they get an answer that does not line up with the others, they know they've done it wrong.
Here is an example.
Problem:
The student chooses a 2 from the x-axis and puts it into the first box.
Then they read the equation 8 + 2 = and solve the equation, taking a 10 from the y-axis and putting it into the second box.
They then find the point on the grid that is (2,10) and put their number cards 2 and 10 at that point.
They repeat this process with the same equation using different numbers from the x-axis and different answers from the y-axis. Each time they find a "solution point" on the coordinate system it should line up with the other points.
While it may sound like a lot of prep work, it's worth the underlying principles they learn while playing the game.
Make a Cartesian Coordinate System, just the first quadrant (positive/positive) if you are working with only positive numbers, all 4 quadrants if you include negative numbers. Make it nice and big, especially for the younger kids.
Make a stack of number cards so that two of the numbers would fit nicely at each point on the coordinate system. Make 6 or more of each number.
If the child is unfamiliar with finding points on a grid explain how it works. You can use games like battleship and cross word puzzles to teach them the basics. Explain for this grid that they will have 2 numbers and to find a point on the grid, go to the first number on the bottom and the second number on the side and find there they meet. Remind them that the first number is the "across" number and the second number is the "up/down" number. You may even want to write that on the grid somewhere.
Stack half of the number cards on the x-axis or bottom grid line. Put the ones cards at the first mark, the 2's at the second mark, the 3's at the third mark, etc. Do the same with the y-axis or vertical grid line.
Give your student one of the "problems". Then ask them to take any number from the bottom grid and put it into the first blank in the problem. Once they've solved the problem they will take a number from the vertical grid line and put it into the answer slot. Then take the two numbers (remembering which is first and which is second) and put them at their "point" on the coordinate system.
Repeat this process 3 or 4 times and they should see a "line" of solution points start to appear. If they get an answer that does not line up with the others, they know they've done it wrong.
Here is an example.
Problem:
The student chooses a 2 from the x-axis and puts it into the first box.
Then they read the equation 8 + 2 = and solve the equation, taking a 10 from the y-axis and putting it into the second box.
They then find the point on the grid that is (2,10) and put their number cards 2 and 10 at that point.
They repeat this process with the same equation using different numbers from the x-axis and different answers from the y-axis. Each time they find a "solution point" on the coordinate system it should line up with the other points.
While it may sound like a lot of prep work, it's worth the underlying principles they learn while playing the game.
Introduction to Integers
I LOVE coming at math backwards - doing an activity that at first seems unrelated to math but by the end the learner has "figured out" an easier way or a short cut - which is the math I wanted them to learn in the first place. I did this activity with my 5 year old and my 8 year old together. They both enjoyed it and had NO idea they were learning "Negative Numbers."
I drew a picture of the ground with a ladder going from below the ground to above the ground. The ladder had 5 rungs below and these were blue, and 5 rungs above and these were red. See the image below. Then I put a lego star wars storm trooper "ground level" or 0.
Then I told the boys that a creature was chasing the storm trooper, one was coming down from the top and one coming up from the bottom.
me: "If the storm trooper moved up 2 and then down 4, which rung would he be at?"
boys: "At 2!"
me: "Which 2?"
boys: "The 2 that is below. The "below 2"!"
me: GOOD!
I continued to ask them these kinds of questions and the storm trooper moved up and down that ladder until my 5 year old was done and left the game. Then I told my 8 year old a little secret. I told him that mathematicians use "below numbers" all the time, but they call them negative numbers. And they use "above numbers" but call them positive numbers. Then I did a few more problems with him using the new terms.
me: "If the storm trooper moves positive 2 and then negative 5, where will he be?"
boys: "below 3. Oh! I mean negative 3."
SO FUN to see him adding integers and not even know it!
The next step will be to write down a few "subtraction" or integer problems and see if he can move the storm trooper to the right answer just using the numbers and not the words. (That's tomorrow's game:)
I'll start with simple subtraction problems, and then introduce the subtraction problems he thinks he "can't" do - the ones that are really integer problems. As below.
8-3
2-1
4-6
2-5
We'll see what happens! Hopefully this won't backfire with him correcting his 2nd grade teacher telling her "yes you can subtract 5 from 3! It's negative 2!" (Of course, I couldn't be more proud if he did ;)
I drew a picture of the ground with a ladder going from below the ground to above the ground. The ladder had 5 rungs below and these were blue, and 5 rungs above and these were red. See the image below. Then I put a lego star wars storm trooper "ground level" or 0.
Then I told the boys that a creature was chasing the storm trooper, one was coming down from the top and one coming up from the bottom.
me: "If the storm trooper moved up 2 and then down 4, which rung would he be at?"
boys: "At 2!"
me: "Which 2?"
boys: "The 2 that is below. The "below 2"!"
me: GOOD!
I continued to ask them these kinds of questions and the storm trooper moved up and down that ladder until my 5 year old was done and left the game. Then I told my 8 year old a little secret. I told him that mathematicians use "below numbers" all the time, but they call them negative numbers. And they use "above numbers" but call them positive numbers. Then I did a few more problems with him using the new terms.
me: "If the storm trooper moves positive 2 and then negative 5, where will he be?"
boys: "below 3. Oh! I mean negative 3."
SO FUN to see him adding integers and not even know it!
The next step will be to write down a few "subtraction" or integer problems and see if he can move the storm trooper to the right answer just using the numbers and not the words. (That's tomorrow's game:)
I'll start with simple subtraction problems, and then introduce the subtraction problems he thinks he "can't" do - the ones that are really integer problems. As below.
8-3
2-1
4-6
2-5
We'll see what happens! Hopefully this won't backfire with him correcting his 2nd grade teacher telling her "yes you can subtract 5 from 3! It's negative 2!" (Of course, I couldn't be more proud if he did ;)
Number Comprehension
I can't take credit for this activity - I got it from some home school curriculum my friends uses, but I LOVE it.
Once a child understands how to count and can combine simple numbers (1 + 1) start teaching them the meaning of numbers in terms of 5.
You can start out with a little saying and finger play:
6 is 5 and 1. See the 1 run.
7 is 5 and 2. See them tie the shoe.
8 is 5 and 3. See them climb a tree.
9 is 5 and 4. See them shut the door.
10 is 5 and 5. See them come alive.
(or whatever rhyme you like)
Then build on this by showing them 8 fingers and asking how many there are but DON'T let them count the 5. Point out that they know that one hand is 5. They can count on from there. OR they can remember their rhyme and since you are holding up 5 and 3, then it is 8.
You can model these numbers with base 10 blocks, or m&m's, or beans, or bites they have to eat at dinner - show them 7 in terms of 5 and then 2. They can count out money - 8 pennies is 5 pennies and 3 pennies. Build on this knowledge with lots of different number games. Use dice, playing cards, etc.
When they are old enough and mentally ready to move on, higher addition becomes MUCH easier. When adding 8 and 9 they have a mental "visual" of 8 as 5 and 3 and of 9 as 5 and 4. So adding 8 and 9 easily becomes adding 5 & 5 & 3 & 4 which is MUCH easier then counting up 9 more than 8.
Once a child understands how to count and can combine simple numbers (1 + 1) start teaching them the meaning of numbers in terms of 5.
You can start out with a little saying and finger play:
6 is 5 and 1. See the 1 run.
7 is 5 and 2. See them tie the shoe.
8 is 5 and 3. See them climb a tree.
9 is 5 and 4. See them shut the door.
10 is 5 and 5. See them come alive.
(or whatever rhyme you like)
Then build on this by showing them 8 fingers and asking how many there are but DON'T let them count the 5. Point out that they know that one hand is 5. They can count on from there. OR they can remember their rhyme and since you are holding up 5 and 3, then it is 8.
You can model these numbers with base 10 blocks, or m&m's, or beans, or bites they have to eat at dinner - show them 7 in terms of 5 and then 2. They can count out money - 8 pennies is 5 pennies and 3 pennies. Build on this knowledge with lots of different number games. Use dice, playing cards, etc.
When they are old enough and mentally ready to move on, higher addition becomes MUCH easier. When adding 8 and 9 they have a mental "visual" of 8 as 5 and 3 and of 9 as 5 and 4. So adding 8 and 9 easily becomes adding 5 & 5 & 3 & 4 which is MUCH easier then counting up 9 more than 8.
Monday, February 21, 2011
Algebra - Modeling Polynomials using Algeblocks
I like to introduce algebra using Algeblocks. They help build a visual and physical understanding on which the math easily builds. See the previous post on what Algeblocks are if needed.
1. Have the students measure the small square. Since it is 1 by 1, the area is 1 cm squared. I just name it a "1". Write 1 on one of the small squares so they don't forget it's name.
2. Have the students measure the rectangle. They can see that it's width is 1 cm but can't measure it's exact length. So the length is x since it is "unknown". The area then is 1 by x which is x cm squared. I just name this shape x. Write x on one of the small squares so they don't forget it's name.
3. Have the students measure the large square. They can't get an exact measurement for the length or the width, so it is x by x or x2. Write x2 on a large square so they don't forget it's name.
Once you have all 3 shapes "named" you can then model a polynomial.
4 large squares = 4x2
2 rectangles = 2x
3 small squares = 3
To "add" these shapes up you simply connect them with and addition symbol.
4x2 + 2x + 3
Model several different polynomials. Have the student make the model, and name the polynomial.
1. Have the students measure the small square. Since it is 1 by 1, the area is 1 cm squared. I just name it a "1". Write 1 on one of the small squares so they don't forget it's name.
2. Have the students measure the rectangle. They can see that it's width is 1 cm but can't measure it's exact length. So the length is x since it is "unknown". The area then is 1 by x which is x cm squared. I just name this shape x. Write x on one of the small squares so they don't forget it's name.
3. Have the students measure the large square. They can't get an exact measurement for the length or the width, so it is x by x or x2. Write x2 on a large square so they don't forget it's name.
Once you have all 3 shapes "named" you can then model a polynomial.
4 large squares = 4x2
2 rectangles = 2x
3 small squares = 3
To "add" these shapes up you simply connect them with and addition symbol.
4x2 + 2x + 3
Model several different polynomials. Have the student make the model, and name the polynomial.
Algeblocks Toys (Manipulatives) for Algebra
An AWESOME manipulative for any algebra concept is ALGEBLOCKS. They are like base 10 blocks but without the lines. You can use them for everything from solving simple equations to completing the square and the quadratic equation. You can either purchase them or make them. I have several sets I've made out of paper that I give to my students and they work great.
What are Algeblocks? Here is the website if you want more info.
How do you make a paper set of algeblocks?
1. Cut 1 cm x 1 cm squares of paper (I don't actually measure, just eyeball it).
2. Cut 1 cm x 3.5 ish cm rectangles of paper. It's best if the rectangles are the same width as the small squares, and are NOT a whole number length. Again, I just eyeball it. They can really be as long as you want. It helps to have these be a different color than the small squares.
3. Cut large squares the same length and width as the length of the rectangles. So if you are following the measurements I give these squares would be 3.5ish cm x 3.5ish cm. Make them the same color as the rectangles.
I usually stop here. These are enough shapes to solve all one-variable equations and model one-variable polynomials. I call them the X blocks. You can go on to make manipulatives for 2 variables as well. To do this...
4. Cut rectangles 1 cm x 8.2 cm (or whatever length as long as it's longer than the first rectangles you made).
5. Cut squares 8.2 cm x 8.2 cm (or whatever length you made the rectangles.)
These would be the Y blocks.
What do you do with them? See the algebra posts to find out!
What are Algeblocks? Here is the website if you want more info.
How do you make a paper set of algeblocks?
1. Cut 1 cm x 1 cm squares of paper (I don't actually measure, just eyeball it).
2. Cut 1 cm x 3.5 ish cm rectangles of paper. It's best if the rectangles are the same width as the small squares, and are NOT a whole number length. Again, I just eyeball it. They can really be as long as you want. It helps to have these be a different color than the small squares.
3. Cut large squares the same length and width as the length of the rectangles. So if you are following the measurements I give these squares would be 3.5ish cm x 3.5ish cm. Make them the same color as the rectangles.
I usually stop here. These are enough shapes to solve all one-variable equations and model one-variable polynomials. I call them the X blocks. You can go on to make manipulatives for 2 variables as well. To do this...
4. Cut rectangles 1 cm x 8.2 cm (or whatever length as long as it's longer than the first rectangles you made).
5. Cut squares 8.2 cm x 8.2 cm (or whatever length you made the rectangles.)
These would be the Y blocks.
What do you do with them? See the algebra posts to find out!
Sunday, February 20, 2011
Fractions
I love the pattern blocks for an introduction to fractions. You can buy them at any learning store in the math section, you can download a pattern and make paper ones for free (http://www.teachervision.fen.com/geometry/printable/6175.html). Do a quick search for pattern blocks activities fractions and you'll find a bazillion different activities to do with your kids that relate pattern blocks to fractions.
Ideas with pattern blocks
1. Modeling Fractions:
The hexagon is a whole. Ask your child to tell you how many trapezoids (red pieces) it takes to cover the hexagon (answer: 2). Then name the red piece 1/2. Repeat this process for the parallelogram (blue piece = 1/3) and triangle (green piece = 1/6). Explain that the top number tells you how many pieces you have and the bottom number tells you how many pieces it takes to make a whole. Also tell them that the bottom number could also tell you what shape or size the piece is. Model different fractions for them and ask them to name the fraction. Hold up 3 green pieces - and tell them it is 3/6. Hold up 2 blue pieces and ask them to name the fraction (2/3). Hold up 3 red pieces and have them name the fraction (3/2).
2. Adding like fractions:
Once the kids know the names of the shapes it's easy to model adding like fractions. Show them 2 green triangles and 3 green triangles and ask how many there are. Answer: 5 sixths or 5/6. Then show them that when adding fractions you add the top number (numerator) but not the bottom number (denominator). The number of pieces that fit in a whole does not change, only how many of them you have changes.
4. Equivalent fractions
Show your child 1/3 and ask them how many triangles (1/6) it would take to cover it. (answer 2). Then 1/3 - 2/6. Repeat the process: How many greens (1/6) would it take to cover a red (1/2)? (answer 3) Then 1/2 = 3/6. Model it several times before you decide to move into doing just with numbers.
3. Equivalent fractions - mixed numbers:
Model converting mixed numbers into fractions. Show 2 hexagons and 1 blue parallelogram (1/3). Ask them to write the number (2 and 1/3). Then ask them how many blues (1/3) it would take to cover all of these pieces. Since it would take 7 blues, then 2 and 1/3 equals 7/3. Do several of these with the shapes to show them the concept. If you want to move into the mathematical procedure you can later. Let them SEE it first.
Ideas with pattern blocks
1. Modeling Fractions:
The hexagon is a whole. Ask your child to tell you how many trapezoids (red pieces) it takes to cover the hexagon (answer: 2). Then name the red piece 1/2. Repeat this process for the parallelogram (blue piece = 1/3) and triangle (green piece = 1/6). Explain that the top number tells you how many pieces you have and the bottom number tells you how many pieces it takes to make a whole. Also tell them that the bottom number could also tell you what shape or size the piece is. Model different fractions for them and ask them to name the fraction. Hold up 3 green pieces - and tell them it is 3/6. Hold up 2 blue pieces and ask them to name the fraction (2/3). Hold up 3 red pieces and have them name the fraction (3/2).
2. Adding like fractions:
Once the kids know the names of the shapes it's easy to model adding like fractions. Show them 2 green triangles and 3 green triangles and ask how many there are. Answer: 5 sixths or 5/6. Then show them that when adding fractions you add the top number (numerator) but not the bottom number (denominator). The number of pieces that fit in a whole does not change, only how many of them you have changes.
4. Equivalent fractions
Show your child 1/3 and ask them how many triangles (1/6) it would take to cover it. (answer 2). Then 1/3 - 2/6. Repeat the process: How many greens (1/6) would it take to cover a red (1/2)? (answer 3) Then 1/2 = 3/6. Model it several times before you decide to move into doing just with numbers.
3. Equivalent fractions - mixed numbers:
Model converting mixed numbers into fractions. Show 2 hexagons and 1 blue parallelogram (1/3). Ask them to write the number (2 and 1/3). Then ask them how many blues (1/3) it would take to cover all of these pieces. Since it would take 7 blues, then 2 and 1/3 equals 7/3. Do several of these with the shapes to show them the concept. If you want to move into the mathematical procedure you can later. Let them SEE it first.
Friday, February 11, 2011
Long Division
QUICK: For the problem 4 into 432 have your child first write the multiples of 4 on the side of the page. Then when they are "dividing into" they have a list to choose from, and when they are "multiplying" the multiplication is already done.
This same process can be used for double digit division. It will take a little longer to do the problem but there is a much higher chance of getting it right. If dividing 43 into 86543 first write the multiples of 43 from 1-9 on the side of the page. Then use these multiples to do the problem.
LONG EXPLANATION: Long division can be confusing and difficult. Here are a few tips to help your child master long division.
We'll use the following problem in the examples below.
43 INTO 86543
There are several steps to remember when doing long division.
The main 4 steps are:
1 - divide
2 - multiply
3 - subtract
4 - bring down
You can use lots of different sayings to help your child remember these steps (dad, mom, sister, brother OR does my sister bite OR make up one of your own). But if they don't understand how to do these steps, remembering the order won't help much.
Have them do a problem and watch each step closely to discover which one they struggle with the most.
If they struggle with the first step - the division step - you can...
1. Change it into a "multiplication question" step.
Rather than
First - Have them create a multiplication table on the side before they begin the problem. They can then refer to this table for both the division step and the multiplication step.
This same process can be used for double digit division. It will take a little longer to do the problem but there is a much higher chance of getting it right. If dividing 43 into 86543 first write the multiples of 43 from 1-9 on the side of the page. Then use these multiples to do the problem.
LONG EXPLANATION: Long division can be confusing and difficult. Here are a few tips to help your child master long division.
We'll use the following problem in the examples below.
43 INTO 86543
There are several steps to remember when doing long division.
The main 4 steps are:
1 - divide
2 - multiply
3 - subtract
4 - bring down
You can use lots of different sayings to help your child remember these steps (dad, mom, sister, brother OR does my sister bite OR make up one of your own). But if they don't understand how to do these steps, remembering the order won't help much.
Have them do a problem and watch each step closely to discover which one they struggle with the most.
If they struggle with the first step - the division step - you can...
1. Change it into a "multiplication question" step.
Rather than
First - Have them create a multiplication table on the side before they begin the problem. They can then refer to this table for both the division step and the multiplication step.
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