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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 Is there anything interesting about addition and subtraction sentences? Math could be spark curiosity!

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Write two number sentences… To 2nd graders: see if you can find some that don’t work! = = = 7 3 How does this work?

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Is there anything more exciting than memorizing multiplication facts? What helps people memorize? Something memorable! Math could be fascinating!

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“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 – Let’s multiply 53 x about 50

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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 Algebraic thinking Algebraic language Science

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(n – d) (n + d) = n n – (n – d) (n + d) = n n – d d 5 9 = (7 – 2 ) (7 + 2 ) = 7 7 – 2 2 n – d n + d n (n – d) (n + d) (n – d) = 49 – 4 = 45

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We also care about thinking! Kids feel smart! Teachers feel smart! Practice. Gives practice. Helps me memorize, because it’s 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? Surprise! You’re good at algebra! It could be surprising!

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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. htu II. All of my digits < 5 III. h + t + u = 9 IV. I am less than 400. 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_(5+2)_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 (2+1)) & 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: 2682 x 5 = (2682 / 2) & 5 or / 2 = 1341 (whole number so add 0) x (fractional number (ignore remainder, add 5) 29435

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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 / 5Step1: 195 * 2 = 390 Step2: Move the decimal: 39.0 or just 39 = / 5Step 1: 2978 * 2 = 5956 Step2: = 595.6

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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. Puzzle

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Putting It Together Solution The list contains fifty pairs of numbers that add to 100 (100+0, 99+1, 98+2, 97+3, etc.) with the number 50 as an unpaired leftover: 50 X = 5,050

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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 4’s Puzzle Solution 0 (4+4) – (4+4) 1 (4+4–4)/4 44/44 2 (4*4)/(4+4) 3 (4+4+4)/4 (4*4–4)/4 4 (4–4)*4+4 5 (4*4+4)/4 6 ((4+4)/4)+4 7 (4+4) – (4/4) 44/4–4 8 (4*4) – (4+4) 4+4+4–4 9 (4/4) (44–4)/4 try to write equations to produce the values 0 to 100.

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