What Does it Mean to Do Math?

Gurps1.jpgThe GURPS Basic Set which contains rules for the paper and pencil roleplaying game designed by Steve Jackson describes an ability that you can bestow on a character you are creating.

Lightning Calculator – You have the ability to do math in your head, instantly. If you have this talent, then you (the player) may use a calculator at any time, to figure anything you want – even if your character is fleeing for his life at the time!

This description seems to imply that doing math and calculating are the same thing. Do you agree or disagree? Make your case.

How Tall is a Giant?

imgresINTRODUCTION TO SCALING – Prerequisite for 5.NF.B.5a/b – This lesson addresses only one dimension when considering scale thus it may be used as a precursor to thinking about multiplication as scaling using a pair of factors.

CONTEXT – The height of a giant can vary greatly. The evidence of this comes from Giants, a tabletop RPG supplement written by Bruce Humphrey and published in 1987 by Mayfair Games Inc. According to this “reference for the society of giants,” there are many races of giants and though scholars have scoffed at their physical impossibility, their existence cannot be denied. Instead of delving into the biology of giants (see Giants pages 5 and 6 if you must), our focus today is on their height.

Fire Giants, 12 feet tall, impetuous and powerful warriors

Titans, 24 feet tall, first giants and progenitors of the other giants

Sea Giants, 18 feet tall, noble and personable giants from the depths

Frost Giants, 15 feet tall, resilient hermits from the icy realms

Hill Giants, 10 feet tall, aggressive and stubborn hunters

Investigate

Q1:  How many feet taller is each race of giants when compared to a 6 foot tall human?

Q2:  How many times taller is each race of giants when compared to a 6 foot tall human?

Q3:  How many times shorter is each race of giants when compared to a 60 foot tall sauroposeidon?

Q4:  How many times taller is each race of giants when compared to a 6 inch tall tufted titmouse?

Activity:  Draw a picture of yourself and a representative of each giant race, all to scale in regards to height.

Create

Activity:  Make a new race of giants. What will they be called? What characteristics will they have? How tall will they be? How will the height of the new race compare to the heights of other giants and animals?

I Can’t Carry Anymore!

UnknownEncumbrance, in role playing games, means how much you can carry. This is usually determined by the total weight of the items you are carrying. A common definition of encumbrance is “a burden” and for most tabletop role playing game enthusiasts, it’s an apt definition in more ways than one. So, it’s often ignored. I mean, who wants to limit how much of the dragon’s treasure you will bring back to your own den? In role playing videogames, it often can’t be ignored because it’s an integral part of the game. Champions of Norrath (2004) attempts to give players the best of both worlds, a limiting factor for realism and an unrealistic way to horde your winnings.

Champions of Norrath is an action oriented role playing videogame. The action comes from hitting and shooting things while the role playing comes, in part, from collecting gear to improve your character’s performance. Often when I compel my warrior, Morg, to pick up something, he will bleat, “I can’t carry anymore!” In which case, I transport Morg to the store and lighten his load in exchange for gold coins. In this game although weapons, armor and other equipment have weight, gold coins do not. This is a treat since Morg is currently holding 1,114,064 gold coins. But it’s also a bit silly.

Grade 6 Content (6.NS.B.2)

Investigate

Q1:  If 1 gold coin weighs 1 ounce, what is the weight of the gold Morg is carrying in pounds?

Q2:  If a car weighs 3,000 pounds, about how many cars would it take to equal the weight of the gold Morg is carrying?

Key

If 1 gold coin weighs 1 ounce, then the weight of 1,114,064 gold coins is 1,114,064 ounces.  There are 16 ounces in a pound so,

1,114,064 ounces / 16 ounces per pound = 69,629 pounds, the number of pounds in gold coins that Morg is carrying.

If a car weighs 3,000 pounds then Morg is carrying,

69,629 pounds / 3,000 pounds per car = the weight of more than 23 cars in gold coins.

That’s one way to bury the local cutpurse.

No Squares Here

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Recognize and Categorize Shapes – Grade 4

Everyone wants to count triangles in this figure but what about rhombi, parallelograms and trapezoids? Hexagons and pentagons? Dodecagons anyone? The abundance of parallel lines make this a good figure for 4th graders to study, both as a review of some of the shapes addressed in previous grades and as an opportunity to classify quadrilaterals based on the presence of one or two sets of parallel lines.

2.G.A.1 – recognize triangles, quadrilaterals, pentagons and hexagons

3.G.A.1 – recognize rhombi as examples of quadrilaterals

4.G.A.2 – recognize parallelograms and trapezoids* by their attributes

Investigate

Q1:  What shapes do you see?

Q2:  How many of each shape are there? Count all same sized shapes as one shape. [For example, there are 4 different sized triangles, all equilateral.]

Q3:  How would you organize each of these shapes into categories?

Q4:  How many categories do the rhombi belong to? Make your case.

*Although students are composing trapezoids as early as grade 1, it’s not until grade 4 that students are expected to consider its defining attributes. Use the inclusive definition of the trapezoid, a quadrilateral with at least one set of parallel sides.

Faster is Not Smarter

LaurentSchwartz.jpghttps://en.wikipedia.org/wiki/Laurent_Schwartz

Mathematician Laurent Schwartz won the Fields medal in 1950. The following is an excerpt from his autobiography, A Mathematician Grappling With His Century (2001).

I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent. And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Even when I was the first to answer the teacher’s questions, I knew it was because they happened to be questions to which I already knew the answer. But if a new question arose, usually students who weren’t as good as I was answered before me. Towards the end of the eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time. Not only did I believe I was stupid, but I couldn’t understand the contradiction between this stupidity and my good grades. I never talked about this to anyone, but I always felt convinced that my imposture would someday be revealed: the whole world and myself would finally see that what looked like intelligence was really just an illusion. If this ever happened, apparently no one noticed it, and I’m still just as slow. (…) At the end of the eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn’t have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn’t really relevant. Naturally, it’s helpful to be quick, like it is to have a good memory. But it’s neither necessary nor sufficient for intellectual success.

Understanding the Equal Sign

Challenge

Some children misperceive the equal sign as an instruction to compute rather than understanding the equal sign as a symbol that shows two expressions have the same value. These children will often place a 5 in the blank when given 2 + 3 = _ + 4. Practice with addition facts may actually strengthen this misconception if children are repeatedly given left to right “4 plus 2 makes 6” formatted facts. This can become a significant stumbling block in a child’s development of algebraic reasoning.

Teachers

  • Explicitly teach the equal sign means “the same value as.” In lower grades you may say, “6 marbles is the same as 4 marbles and 2 marbles.”
  • Avoid the input-output model to describe equations. Numbers are not placed into a machine with the answer spit out. The equal sign is not an operation that “makes” a number.
  • Use a variety equation formats. For example: 3 = 5 – 2 or 9 = 9 or 2 + 6 = 10 – 2

Students

  • Represent a variety of equation formats with concrete objects and pictorials.
  • Describe their own representations of equations using “same value as” to express equality.
  • Solve for unknowns in equations such as 8 + _ = 3 + 9.

Source: Teaching the Meaning of the Equal Sign to Children with Learning Disabilities: Moving from Concrete to Abstractions by Ruth Beatty and Joan Moss at the University of Toronto. Published in The Learning of Mathematics, NCTM’s 69th Yearbook, 2007.

 

Variations on the Hundreds Chart

Splat Square 1 – 100 and Splat Square 0 – 99 – Use 7 different colors in these two versions of a hundreds chart. Only 1 color may be used per number. Also, there’s Splat Square Reveal 1 – 100 and Splat Square Reveal 0 – 99.

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Hundreds Chart – Print a hundreds chart from this pdf file.

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Blank Hundreds Chart – Print a blank hundreds chart from this pdf file.

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Blank 1 to 120 Chart – Print a blank 1 – 120 chart from this pdf file.

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