Introducing the Math Icons 4th -8th Copyright © 2006 Melanie Montgomery

Applications Copyright © 2006 Melanie Montgomery

Balance Copyright © 2006 Melanie Montgomery

Conversion Copyright © 2006 Melanie Montgomery

Expressions Copyright © 2006 Melanie Montgomery

Extensions Copyright © 2006 Melanie Montgomery

Imbalance Copyright © 2006 Melanie Montgomery

Inquiry Copyright © 2006 Melanie Montgomery

Proofs Copyright © 2006 Melanie Montgomery

Strategies Copyright © 2006 Melanie Montgomery

What are Proofs? Oftentimes in math we are asked to provide an answer to a question or questions. Inverse operations can validate our assertions with respect to calculations that are right on target. When we provide evidence to substantiate a claim, that’s proof. Our solutions and our checks are our proofs. Copyright © 2006 Melanie Montgomery

What are Strategies? Every problem necessitates a strategy. How are we going to solve this? What steps do we need to take if we want to arrive at a correct or reasonable conclusion? As we are deciding how to attack a problem, we are strategizing. We’re getting our troops in order. We’re recalling other battles, remembering what worked and what didn’t, and thinking about which plans were most efficient and which plans have landed us on long roads to nowhere. A good strategy gets us from A to B (without having to visit C or D). Copyright © 2006 Melanie Montgomery

Expressions Related Terms constant define label representation symbol value variable Copyright © 2006 Melanie Montgomery

Inquiry Related Terms challenge debate examine exploration hypothesize postulate research theorize Copyright © 2006 Melanie Montgomery

Applications Key Questions: How do _____ applications relate to the real world? Which real life activities rely on ________? Why might it be important/beneficial to understand ____? Which professions require a working knowledge of _____? When might I use _____ to _____? Copyright © 2006 Melanie Montgomery

Balance Key Questions: What are the equal values? Are ____ and ____ worth the same amount? How does symmetry manifest itself in ____? Can we make (2? 3? 4?) groups each with the same value? How many of each ____ would produce balance? What makes _____ symmetrical? Copyright © 2006 Melanie Montgomery

Conversion Key Questions: Why might we express ____ as ____? How else can we express ____? When might ____ be most useful? How are ____ and ____ related? Copyright © 2006 Melanie Montgomery

Expressions Key Questions: Can you give ____ a nickname? Why might a person use ____ to represent ____? How else might you represent this value? How are ____ and ____ related? What does the term ____ assume? How does ____ help us to ____? Copyright © 2006 Melanie Montgomery

Extensions Key Questions: Where did ____ originate? How much further can we take ____? What comes before ____? After ____? What is the purpose behind ____? How does additional data influence? What conclusions can be drawn from ____? Copyright © 2006 Melanie Montgomery

Imbalance Key Questions: Which amount is greatest? Which ____ has the least value? Is there a remainder? What does a remainder imply? Why are there (2? 3? 4?) left over? How many ____ make one ____? How are ____ and ____ related? Copyright © 2006 Melanie Montgomery

Inquiry Key Questions: What does ____ assume? Why does ____ work this way? Will ____ work in all situations? What other ways are there to arrive at the same answer? What conclusions can be drawn from ____? How are ____ and ____ related? I wonder if .... Copyright © 2006 Melanie Montgomery

Proofs Key Questions: How do you know ____? Does ____ make sense? What evidence is there to support ____? How does ____ compare with ____? How are the question and the answer related? Why don’t we arrive at a correct answer when we ____? How can we check for ____? Copyright © 2006 Melanie Montgomery

Strategies Key Questions: Which information is relevant? Irrelevant? Is there something missing? Something we still need to find out? Have we solved other problems like this one? How did we approach those problems? Which cue words help us to know how to solve ____? Do you see a pattern? Explain. What steps do we need to take to solve ____? Copyright © 2006 Melanie Montgomery

Balance Related Thinking Skills classify describe generalize group make analogies observe relate Copyright © 2006 Melanie Montgomery

Balance Related Thinking Skills classify describe generalize group make analogies observe relate Copyright © 2006 Melanie Montgomery

Imbalance Related Symbols < > + - x ÷ ≠ ~ ~ Copyright © 2006 Melanie Montgomery

Inquiry Related Resources dictionary encyclopedia graph Internet periodical map thesaurus Copyright © 2006 Melanie Montgomery

4+2 Copyright © 2006 Melanie Montgomery

4+2 + = 4 + 2 = 6 + = Copyright © 2006 Melanie Montgomery

4+2 4 + 2 5 4 + 2 > 5 = 4 + 2 = 7 4 + 2 < 7 + = Copyright © 2006 Melanie Montgomery

4+2 I can use my fingers I can use a number line I can use counting bears I can use tally marks Copyright © 2006 Melanie Montgomery

4+2 I will set the table for my family of four plus two guests. I need six plates, six forks, and six cups. My mom says that I can invite four friends from school and two friends from soccer to my birthday party. I need to make six invitations. Copyright © 2006 Melanie Montgomery

4+2 6 cm. 2 6 cm. 2 Copyright © 2006 Melanie Montgomery

Jack is breeding fruit flies Jack is breeding fruit flies. At the end of the second day, double the first day’s population has been added to the total number of flies. On every subsequent day, twice the previous day’s increase is added to the total population. At the end of the sixth day Jack has 189 fruit flies. How many fruit flies did Jack start with? Copyright © 2006 Melanie Montgomery

x= fruit flies Jack started with x+2x+4x+8x+16x+32x=189 63x=189 x=189 Copyright © 2006 Melanie Montgomery

3 6 12 24 48 +96 189 Copyright © 2006 Melanie Montgomery

189 - 96 93 - 48 3 45 6 - 24 12 21 24 - 12 48 9 +96 - 6 189 3 Copyright © 2006 Melanie Montgomery

63x 189 - 32x - 96 31x 93 - 16x - 48 15x 45 x 3 - 8x - 24 63 2x 6 3 189 4x 7x 12 21 8x - 4x 24 - 12 18 3 16x 09 63 189 3x 48 9 + 32x - 2x +96 - 6 9 189 63x x 189 3 Copyright © 2006 Melanie Montgomery

Given that Jack started with three fruit flies, how many days would it be before his population numbered over 1,000,000? How many fruit flies would Jack have had on the sixth day if he’d started with only two fruit flies? What if he’d started with ten fruit flies? Copyright © 2006 Melanie Montgomery

How long does a fruit fly live. Do fruit flies lay eggs How long does a fruit fly live? Do fruit flies lay eggs? Do they birth live young? Why might a Biologist breed fruit flies? Are there populations that double on a daily basis? Copyright © 2006 Melanie Montgomery

Copyright © 2006 Melanie Montgomery

A circle has a 6 cm. diameter. Find the area of the circle. Copyright © 2006 Melanie Montgomery

diameter d area A area of a circle r 2 pi (3.14, 22/7) radius r Copyright © 2006 Melanie Montgomery

d 2 r = d = 2r 6 cm. 2 r = r = 3 cm. Copyright © 2006 Melanie Montgomery

A = r 2 A ~ 3.14 (3 cm.) 2 . A ~ 3.14 9 cm. 2 A ~ 28.26 cm. 2 Copyright © 2006 Melanie Montgomery

= the relationship between the. diameter of a circle and its = the relationship between the diameter of a circle and its circumference ~ 3.14 or 22/7 Copyright © 2006 Melanie Montgomery

- buying or cutting a circular pool cover - sewing a circular tablecloth Copyright © 2006 Melanie Montgomery

A circular pool has a diameter = 20 m A circular pool has a diameter = 20 m. It is surrounded by a rectangular pool deck measuring 40m. x 30m. What is the area of the deck? Given what we know about pi and calculating the area of a circle, how might we go about calculating the volume of a sphere? Copyright © 2006 Melanie Montgomery

diameter d area A area of a circle r 2 pi (3.14, 22/7) radius r - buying or cutting a A circular pool has a diameter = 20 m. It is surrounded by a rectangular pool deck measuring 40m. x 30m. What is the area of the deck? Given what we know about pi and calculating the area of a circle, how might we go about calculating the volume of a sphere? circular pool cover - sewing a circular A circle has a 6 cm. diameter. Find the area of the circle. tablecloth d r = A = r 2 2 A ~ 3.14 (3 cm.) 2 d = 2r A ~ 3.14 9 cm. 2 r = 6 cm. 2 A ~ 28.26 cm. 2 = the relationship between the diameter of a circle and its circumference r = 3 cm. ~ 3.14 or 22/7 3.14 or 22/7 Copyright © 2006 Melanie Montgomery

Melanie Montgomery melanie.n.montgomery@gmail.com Copyright © 2006 Melanie Montgomery