October 22, 2018

Archives for July 2010

Geometry: Translations, Reflections, Rotations, and Dilations

One unit covered in Geometry deals with the concept of translations, reflections, rotations, and dilations.  This article will serve to summarize some the major points for students studying these topics.

Prerequisite:  The student needs to come into the lesson with some basic understanding of matrices.  Given a shape with points on a coordinate plane they need to be able to write those in matrix form.  It is very simple actually, you take the x-coordinates and make them the first row of your matrix and take the y-coordinates and make them the second row of your matrix.

Example: A quadrilateral with points (-4,-3), (-1,0), (1,-3), and (-3, -5) would be written as the following matrix:

$$ \left[
\begin{array}{ c c c c }
-4 & -1 & 1 & -3\\
-3 & 0 & -3 & -5
\end{array} \right]
$$

Students also need to know the identity matrix when multiplied by a matrix gives back the original matrix.  It is like multiplying a number times 1.  The identity matrix is:

$$ \left[
\begin{array}{ c c }
1 & 0 \\
0 & 1
\end{array} \right]
$$

Translations: Translations simply slide your figure around.  It is the easiest to work with since it just involves adding a value to the x-coordinates and a value to the y-coordinates.

Example: Translate the example matrix above by moving it to the RIGHT four and DOWN 1.  This would mean we just add 4 to the top numbers and subtraction 1 from the bottom numbers:

$$ \left[
\begin{array}{ c c c c }
-4 & -1 & 1 & -3\\
-3 & 0 & -3 & -5
\end{array} \right] +
\left[ \begin{array}{ c c c c }
4 & 4 & 4 & 4\\
-1 & -1 & -1 & -1
\end{array} \right]=
\left[
\begin{array}{ c c c c }
0 & 2 & 5 & 1\\
-4 & -1 & -4 & -6
\end{array} \right]
$$

Dilations: Dilations make an object bigger or smaller.  If the dilation is a number bigger than 1, the object will increase in size; if it is less than 1, it will get smaller.  Dilations require you multiple each number in the given matrix by the dilation value.

Example: Dilate the given quadrilateral by 3.

3 * $$ \left[
\begin{array}{ c c c c }
-4 & -1 & 1 & -3\\
-3 & 0 & -3 & -5
\end{array} \right] =
\left[
\begin{array}{ c c c c }
-12 & -3 & 3 & -9\\
-9 & 0 & -9 & -15
\end{array} \right]
$$

The “slightly” harder problems involve ROTATION and REFLECTION.

These simple “adjust” the coordinates according to a specific matrix.  Let’s look at some different matrices and see what they do:

Identity: Doesn’t change the value of the matrix.

$$ \left[
\begin{array}{ c c }
1 & 0 \\
0 & 1
\end{array} \right]
$$

Matrix 1: Notice this looks just like the identity matrix except it has negative 1’s rather than positive ones.  This means that it will change each sign to its opposite in the matrix.

$$ \left[
\begin{array}{ c c }
-1 & 0 \\
0 & -1
\end{array} \right]
$$

Matrix 2: This matrix looks like the identity but has a negative only in the top 1.  This means only the top row will change to their opposite signs but the bottom row will stay the same.

$$ \left[
\begin{array}{ c c }
-1 & 0 \\
0 & 1
\end{array} \right]
$$

Matrix 3: Matrix 3 is similar to Matrix 2 but the negative is on the bottom instead of the top.  This means the bottom row will change to its opposite sign and the top row stays the same.

$$ \left[
\begin{array}{ c c }
1 & 0 \\
0 & -1
\end{array} \right]
$$

Matrix 4: This matrix is a little different from the identity.  The 1’s and the 0’s have changed places.  When this happens, the whole row changes places.  Since both are positive, the numbers keep their original signs.

$$ \left[
\begin{array}{ c c }
0 & 1 \\
1 & 0
\end{array} \right]
$$

Matrix 5: Can you guess what happens in this matrix?

$$ \left[
\begin{array}{ c c }
0 & -1 \\
1 & 0
\end{array} \right]
$$

Matrix 6: How about this one?

$$ \left[
\begin{array}{ c c }
0 & 1 \\
-1 & 0
\end{array} \right]
$$

Matrix 7: And this one?

$$ \left[
\begin{array}{ c c }
0 & -1 \\
-1 & 0
\end{array} \right]
$$

Matrix 5 – the rows switch places and the top row has opposite signs.

Matrix 6 – the rows switch place and the bottom row has opposite signs.

Matrix 7 – the rows switch places and both rows also change signs.

Each of these matrices are multiplied times the matrix defined by the shape in the problem  Note that the “identity” type matrix always comes first, then the other matrix so that the dimensions match for multiplying.

Here is a summary of when to use each matrix:

Reflection over y = x: use matrix 

$$ \left[
\begin{array}{ c c }
0 & 1 \\
1 & 0
\end{array} \right]
$$

Reflection over x-axis: use matrix   

$$ \left[
\begin{array}{ c c }
1 & 0 \\
0 & -1
\end{array} \right]
$$

Reflection over y = -x: use matrix 

$$ \left[
\begin{array}{ c c }
0 & -1 \\
-1 & 0
\end{array} \right]
$$

Reflection over y-axis: use matrix 

$$ \left[
\begin{array}{ c c }
-1 & 0 \\
0 & 1
\end{array} \right]
$$

Rotation of 90 degrees: use matrix   

$$ \left[
\begin{array}{ c c }
0 & -1 \\
1 & 0
\end{array} \right]
$$

Rotation of 180 degrees (same as rotation over Ho): use matrix

$$ \left[
\begin{array}{ c c }
-1 & 0 \\
0 & -1
\end{array} \right]
$$

Rotation of 270 degrees: use matrix

$$ \left[
\begin{array}{ c c }
0 & 1 \\
-1 & 0
\end{array} \right]
$$

Example: Given A(2,5) and B(1, -2) and C(-2,3).

Find a rotation of 270 degrees:

$$ \left[
\begin{array}{ c c }
0 & 1 \\
-1 & 0
\end{array} \right] *
\left[
\begin{array}{ c c c }
-2 & 1 & -2 \\
5 & -2 & 3
\end{array} \right] =
\left[
\begin{array}{ c c c }
5 & -2 & 3\\
-2 & -1 & 2
\end{array} \right]
$$

Note:  The two rows switched, then the bottom row switches signs.

Find a reflection over the y-axis:

$$ \left[
\begin{array}{ c c }
-1 & 0 \\
0 & 1
\end{array} \right] *
\left[
\begin{array}{ c c c }
2 & 1 & -2 \\
5 & -2 & 3
\end{array} \right] =
\left[
\begin{array}{ c c c }
-2 & -1 & 2\\
5 & -2 & 3
\end{array} \right]
$$

Note: The rows did not switch, but the signs on the top row changed to their opposites.

To summarize:

Dilations: Multiply matrix through by the amount of the dilation.

Translation:  Adjust each x by the change in the x-axis of the translation and adjust each y by the change in the y-axis  of the translation.

Reflections and Rotations:  Find the corresponding matrix for each reflection or rotation and multiply the matrix by the correct “identity-type” matrix listed above.

Happy Transformations!

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Teaching Multiplication Facts

Teaching Multiplication Facts

Multiplication facts are part of the North Carolina Standard Course of Study curriculum for the third grade.   Eventually we hope that students will just “know” their facts – in other words, they are memorized but when first learning facts, it is best to teach students strategies for finding facts.  This allows students to always have a fall back plan in case they “forget” the fact and it makes them quicker to learn. The order in which facts should be taught is given below.  The reason for this is that the easier ones get learned first and then they can rely on their “partner fact” (3X4=4X3) for some of the harder facts.

Here are some methods for each fact:


X0 Fact: Anything X0 is 0. This is a very easy fact since students just need to learn that anything times 0 is 0.  Remind them that 0 sets of something is 0.

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X1 Fact: Anything X1 is itself. Again, another very easy fact.  For example,  1 X 7, this means 1 set of 7, which is just 7 items.

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X2 Fact: Circle the number that is not 2, double that number. Since the student is very good at doubling, this is an easy fact.

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X4 Fact: Circle the number that is not 4, double the number and double again.   The student should have doubling down well before starting, so X4 facts will come very easy to them.  The only one that might be difficult is X9 – tell them that they can wait and use the “9’s trick” on that one instead of the 4’s trick if they don’t know 18 + 18 since we didn’t really drill that double.

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X10 Fact: Add 0 to the number that is multiplied by 10.

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X5 Fact: There are two strategies that work here.  Lots of kids like to just count by 5’s because they are good at it.  That takes longer when they are counting by 5 eight times for numbers like 5 X 8.  For these bigger numbers (especially the even ones) they can try this other strategy:   multiply the number times 10 (see 10 fact) and then take half of that number:  so for 5 X 8 we would do 10 X 8 = 80 then take half = 40.

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X11 Fact: Circle the number that is not 11, write the number twice – 11 X 3 = 33.

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X9 Fact: Use the finger trick – hold up all ten fingers, count from the left and bend down the finger of the number you are multiplying by 9. For example, for 9X3, bend down your third finger. Now count the number of fingers on the left side of the bent finger, write that number down – then count the number of fingers on the right side of the bent finger, that is your second digit. With the 9X3 example, you have 2 fingers to the left of the bent finger and 7 fingers on the right side of the bent finger, the answer is therefore 27.  Also show your child all the different patterns in the 9’s.  The sum of the digits of all the 9 facts add to be 9.  The tens place of the 9’s facts is always 1 less than the number you are multiplying by:  if you multiply 9 X 6 then you know that your answer is going to be Fifty – something since 5 is one less than 6.

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X3 Fact: This is where we get the harder ones – students should know the commutative property and therefore only need to learn: 3X3, 3X6, 3X7 and 3X8. There is no good trick for these, tell the child to double the number and add one more. 3X3 = (double 3) + 3 = 6 + 3; 3X7 = (double 7) + 7 = 14 + 7 = 21.  They can also learn to count by 3′s.

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X6 Fact: With the commutative property, you will only need to learn 6X6, 6X7, and 6X8. 6X6=36 and 6X8 can be taught using rhythm (tap them out on the table as you say them).

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X7 Fact: For this group, you will need to learn 7X7 and 7X8. To memorize 7 X 8, I use the visual:  5 6 7 8 these numbers go in order of counting and you can just put = and x in the middle and get your fact 56 = 7 x 8. Help your child see this visual and use it as a cue to help them to remember the fact.  7 X 7 = 49 – sorry, just got to memorize that one.

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X8 Fact: You only need to learn 8X8=64. You can use “bend down touch the floor, eight times eight is 64” or “Skate X Skate = Sticky Floor.”  You can also do “double, double, double if the student can double 3X.  This triple double works really well for 8 X 3 as it is a harder 8 fact but is easy to double 3X.  Double 3 gets to 6, double 6 gets to 12, double 12 gets to 24.

You will want to work on 1 fact at a time, do lots of practice with that fact before moving to the next one. Once you move to the second fact, you should work on that alone and then provide a mix of that with all previously learned facts, and so on.

TABLE of FACTS – shows strategies for each of the different facts:

1 2 3 4 5 6 7 8 9 10
2 Double Double Double Double Double Double Double Double Add Zero
3 Double Count by 3’s Double Double Again Count by 5’s Double 6 + 6 Double 7 + 7 Double 3 times Nines Trick Add Zero
4 Double Double Double Again Double Double Again Double Double Again Double Double Again Double Double Again Double Double Again Double Double Again Add Zero
5 Double Count by 5’s Double Double Again Count by 5’s Count by 5’s Count by 5’s Count by 5’s Nines Trick Add Zero
6 Double Double 6 + 6 Double Double Again Count by 5’s Rhythm Memorize Rhythm Nines Trick Add Zero
7 Double Double 7 + 7 Double Double Again Count by 5’s Repeat Memorize 5678 Nines Trick Add Zero
8 Double Double 3 times Double Double Again Count by 5’s Rhythm 5678 Rhyme Nines Trick Add Zero
9 Double Nines Trick Double Double Again Nines Trick Nines Trick Nines Trick Nines Trick Nines Trick Add Zero
10 Double Add Zero Add Zero Add Zero Add Zero Add Zero Add Zero Add Zero Add Zero
    ANY CHARACTER HERE

You will want to work on 1 fact at a time, do lots of practice with that fact before moving to the next one. Once you move to the second fact, you should work on that alone and then provide a mix of that with all previously learned facts, and so on.  Note that the facts above are written in the order in which they should be taught.  Don’t teach them in numerical order but instead from easiest to hardest so that students can use the commutative property to their advantage and have a feeling of accomplishment.

We are getting ready to launch our Online Multiplication Curriculum. This online program will be interactive and affordable. It will teach your child each strategy mentioned and give them practice with instant feedback using the strategies. We hope to post it on our site soon but for those who can’t wait, feel free to contact us about our pre-release version.

How to Teach Division to Children

Teaching Division

In order to learn division, the student must first have a good understanding of multiplication. The child doesn’t need to be perfect but should know the majority of the facts or have a reasonably quick strategy to figure out the answer.  

When teaching division, we will be going in steps.

Step 1: Have the student understand the concept of division and be able to solve division problems with manipulatives.

For example: Given 12 pennies. Have the student evenly divide the 12 pennies among 2 people. From that practice writing the division statement 12 / 2 = 6 and have student explain the meaning: “12 pennies divided into 2 groups gives 6 in each group.” Then have the student divide the 12 into 3 groups and repeat the math sentence and the explanation. Next, divide 12 into 4 groups, then 6 groups. Repeat with other numbers such as 16 (divide into 2, 4, and 8 groups) until the student shows mastery of the concept and writing the corresponding math sentence.

Step 2: Have the student understand the concept of a remainder. You will continue with manipulatives in this exercise. Give the child 5 pennies. They have to share the pennies among 2 people. Let them try, if they split the pennies into 2 and 3 then discuss how that isn’t fair. If they divide it 2 and 2 with 1 left over, explain that happens sometimes with division: we don’t have an even amount to divide and therefore get a remainder. Have the student write the problem as: 5 / 2 = 2 with Remainder 1. Continue this process with other numbers: 9 divided by 2. 11 divided by 3, etc.

Step 3: Have the student understand the link between multiplication and division. Go back to your manipulatives and have them show you 3 x 4 = 12 with manipulatives. Remember that multiplication should have been taught as the x means “groups of” so 3 x 4 means 3 groups of 4. Put your 3 groups of 4 equals 12 to the side. Now have them take 12 and divide it 3 groups as you did in step 1. 12 / 3 = 4. Show them that their result is the same as their multiplication piles they made. In the end they have 3 groups of 4 items. Have the student write the fact family: 12 / 3 = 4; 12 / 4 = 3; 3 x 4 = 12; 4 x 3 = 12. Show them how if you read a division problem “backwards” you have a multiplication problem. Also show them that they can think of 12 / 3 as “what times 3 equals 12?” Continue with practice – 1) writing fact families and 2) finding missing factors: 4 X ___ = 16 (After they get the answer, convert to a division problem – 16 / 4 = 4).

Step 4: The next step is to teach the long division process. The key here is that we are focusing on the process, not on learning division – although doing the practice will help reinforce the concepts of division. Print out and cut the division cards, you will need to use these are you teach the process.

Example: 215 / 5.

We actually start with a 3 digit number because it shows the repetitive process. Get out the X5 division card. On a white board, write the problem. The steps to the division process are:

1. Look at the first number, 2, does 5 go into 2? No, 5 is too big.

2. Look at the first 2 numbers together: 21. Looking at the division card, find the number closest to 21 without going over. You see that the number is 20.

3. Write the number to the left (blue number) above the 1 on top. Write the number to the right (red number) below the 21. So, a 4 goes on top and the 20 goes below.

4. Now, subtract 21-20. You get 1. Using an arrow (make sure they use the arrow) bring down the 5 so it is next to the 1, making 15.

5. Repeat process: find a number as close to 15 without going over. We find 15 in the table. The red number (3) goes on top above the 5. The blue number (15) goes below the 15, now subtract. We get 0. Note, there are no more numbers to bring down and since we ended with 0, we have no remainder.

Example: 3426 / 5

1. 5 doesn’t go into 3.

2. Find closest to 34 on chart without going over: 30 (5X6=30).

3. Put 6 on top and 30 under the 34.

4. Subtract, get 4.

5. With arrow, bring down the 2, to get 42.

6. Find the closest to 42 without going over: 40 (5X8=40)

7. Put the 8 on top and the 40 under the 42.

8. Subtract, get 2.

9. With arrow, bring down the 6, to get 26.

10. Find the closest to 26 without going over: 25 (5X5 =25)

11. Put the 5 on top and the 25 under the 26.

12. Subtract, get 1.

13. Since that was our last number, 1 is the remainder.

14. The answer is 685 Remainder 1.

Keep practicing with different divisors until the student can do it independently with the division cards.

Step 5: The student now knows the concept of division and the process of division. Now they need to practice division without the help of the division strips. First give them problems with one of the easy divisors such as 2 or 5 and a 2 digit dividend. Even if it goes in evenly such as 5 into 25, make sure they write out the 25 underneath and show the remainder goes to 0. Give them remainders that go to zeros and ones that don’t so they get practice taking the problem to the end.

Step 6: Once the student is successful with 2 digit dividends with divisors of 2 and 5, have them work with 3 and 4 digit dividends but still use the divisors of 2 and 5.

Step 7: Expand with new divisors one at a time. Do 2 digit dividends first and then expand to 2or 4 digit dividends before moving onto a new divisor.

To view division strips visit:  http://www.apex-math.com/teaching-division-to-children