Reasoning Algebraically

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Thinking Skills
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Children should be led to make their own
investigations, and to draw upon their own
inferences. They should be told as little as
possible which can produce unlimited learning
potential. Herbert Spencer Intellectual Moral
and Physical 1864
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Research Based Curriculum

• Mathematics is more meaningful when it is rooted
  in real life contexts and situations, and when
  children are given the opportunity to become
  actively involved in learning.
• Children begin school with more mathematical
  knowledge and intuition than previously believed.
  
• Teachers, and their ability to provide excellent
  instruction, are the key factors in the success
  of any program.

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Think Algebraically
Math could be spark curiosity!
Is there anything interesting about addition and
subtraction sentences?
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Write two number sentences

• To 2nd graders see if you can find some that
  dont work!

4 2 6
3 1 4


10
3
7
How does this work?
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Math could be fascinating!
Is there anything more exciting than memorizing
multiplication facts? What helps people
memorize? Something memorable!
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Lets multiply 53 x 47
about 50
2500

• OK, 53 is near 50
• OK, 47 is also near 50
• Actually, they are both 3 units away!
• To do
• 53? 47

I think 50 ? 50 (well, 5 ? 5 and ) 2500 Minus
3 ? 3 9 2491
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But nobody cares if kids can multiply 47 ? 53
mentally!
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What do we care about, then?

• 50 ? 50 (well, 5 ? 5 and place value)
• Keeping 2500 in mind while thinking 3 ? 3
• Subtracting 2500 9
• Finding the pattern
• Describing the pattern

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5 ? 9 (7 2) ? (7 2) 7 ? 7 2 ? 2
49 4 45
(n d) ? (n d) n ? n
(n d) ? (n d) n ? n d ? d
(n d) ? (n d)
(n d)
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We also care about thinking!

• Kids feel smart!
• Teachers feel smart!
• Practice.Gives practice. Helps me memorize,
  because its memorable!
• Something new. Foreshadows algebra. In fact,
  kids record it with algebraic language!
• And something to wonder about How
  does it work?

It matters!
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What could mathematics be like?
It could be surprising!
Surprise! Youre good at algebra!
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A number trick

• Think of a number.
• Add 3.
• Double the result.
• Subtract 4.
• Divide the result by 2.
• Subtract the number you first thought of.
• Your answer is 1!

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How did it work?

• Think of a number.
• Add 3.
• Double the result.
• Subtract 4.
• Divide the result by 2.
• Subtract the number you first thought of.
• Your answer is 1!

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How did it work?

• Think of a number.
• Add 3.
• Double the result.
• Subtract 4.
• Divide the result by 2.
• Subtract the number you first thought of.
• Your answer is 1!

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Why a number trick? Why bags?

• Computational practice, but much more
• Notation helps them understand the trick.
• invent new
  tricks.
• undo the trick.
• But most important, the idea that
• notation/representation is powerful!

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3rd grade detectives!
I. I am even.
II. All of my digits lt 5
h
t
u
III. h t u 9
1 4 4
432 342 234 324 144 414
IV. I am less than 400.

• 0 0
• 1 1
• 2 2
• 3 3
• 4 4
• 5 5
• 6 6
• 7 7
• 8 8
• 9 9

V. Exactly two of my digits are the
same.
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Representing ideas and processes

• Bags and letters can represent numbers.
• We need also to represent
• ideas multiplication
• processes the multiplication algorithm

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Combinations

• Four skirts and three shirts how many outfits?
• Five flavors of ice cream and four toppings how
  many sundaes? (one scoop, one topping)
• How many 2-block towers can you make from four
  differently-colored Lego blocks?

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Lesson Components

• Math Messages
• Alternative Algorithms
• Mental Math and Reflection
• Explorations
• Games

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Arithmetic Tricks

• Multiply by 11
• Take the original number and imagine a space
  between the two digits
• 52 x 11
• 5 _ 2
• Now add the two numbers together and put them in
  the middle
• 5_(52)_2
• That is it you have the answer 572.

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Arithmetic Tricks

• If you need to square a 2 digit number ending in
  5, you can do so very easily with this trick.
• Multiply the first digit by itself 1, and put
  25 on the end. That is all!
• 25 x 25 (2 x (21)) 25
• 2 x 3 25
• 6 25
• 625

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Arithmetic Tricks

• Multiply by 5
• Take any number, then divide it by 2 (in other
  words, halve the number). If the result is whole,
  add a 0 at the end. If it is not, ignore the
  remainder and add a 5 at the end. It works
  everytime

• 5887 x 5
• 2943.5 (fractional number (ignore remainder, add
  5)
• 29435

2682 x 5 (2682 / 2) 5 or 0 2682 / 2 1341
(whole number so add 0) 13410
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Arithmetic Tricks

• Divide by 5
• Dividing a large number by five is actually very
  simple. All you do is multiply by 2 and move the
  decimal point
• 195 / 5 Step1 195 2 390
• Step2 Move the decimal 39.0 or just
  39
• 39
• 2978 / 5 Step 1 2978 2 5956
• Step2 595.6
• 595.6

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Puzzle

• Suppose you have a list of numbers from zero to
  one hundred. How quickly can you add them all up
  without using a calculator?
• HINT There is a swift way to add these numbers.
  Think about how the numbers at the opposite ends
  of the list relate to each other.

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Putting It Together Solution

• The list contains fifty pairs of numbers that add
  to 100
• (1000, 991, 982, 973, etc.)
• with the number 50 as an unpaired leftover
• 50 X 100 50 5,050

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FOUR 4's Puzzle

• Challenge
• Using four 4's and any operations, try to
  write equations to produce the values 0 to
  10.
• Example 0 44 44
• 1 ?
• 10 ?

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FOUR 4s Puzzle Solution
0 (44) (44) 1 (444)/4 44/44 2
(44)/(44) 3 (444)/4 (444)/4 4
(44)44 5 (444)/4
6 ((44)/4)4 7 (44) (4/4) 44/44 8
(44) (44) 4444 9 (4/4)44 10 (444)/
4
try to write equations to produce the values 0 to
100.