Big Ideas & Better Questions, Part II Marian Small May, 2009 1©Marian Small, 2009

Big Ideas & Better Questions, Part II

Marian SmallMay, 2009

1©Marian Small, 2009

Agenda

• What you tried• Integers• Powers and roots• Measures of central tendency• Linear and quadratic relations• Solving equations


You were going to…

• Try some of the questions from last time or create a question for a big idea you are teaching.

• Discuss what happened at your table.

©Marian Small, 2009 3

Working with Integers


What is most important for students to understand about integers?


Maybe• The negative integers are the “opposites”

of the positive integers on the number line.

• Integers are more like whole numbers than they are like fractions or decimals.


Maybe• The zero property plays a critical role in

many integer operations.• The meanings for the operations that are

used with whole numbers don’t change for integers.


Questions you might use

• I am thinking of an integer less than -2? What might it be? What else do you know about it?

• An integer is more than 10 units away from 0 on a number line. What might it be?


Questions you might use

• How many integers are between 3 and 10? Between -3 and -10?

• How many fractions are between 1/3 and 1/10? How do you know?

• Why might you think that integers are more like whole numbers than fractions?


Or..

• Two integers are 8 apart. Are their opposites 8 apart? How do you know?

• Can two opposite integers be 15 apart? How do you know?


Integer operations

• You add -3 to another integer. Then you add -4 to that same other integer. Which answer is greater? How do you know?

• You subtract -3 from another integer. Then you subtract -4 from that same other integer. Which answer is greater? How do you know?


Integer operations

• The sum of two integers is less than their difference. What might they be? What can’t they be?

• Why do you need to know that -1 + 1 = 0 to add -4 and +2? Why don’t you need to know that to add -4 and -2?


Integer operations

• Write down three questions where you need to know the zero principle to solve them and three where you don’t. Explain which is which and why.

• Why is it easier to model 2 x (-4) than to model (-2) x (-4)?

• Why is it easier to model (-4) ÷ (-2) or (-4) ÷ 2 than 4 ÷ (-2)?


Integer operations

• You are working with two negative integers. Which is usually the greatest: the sum, product, quotient, or difference? Why? Which is usually the least? Why?

• Are there more integers with a quotient of -20 or a product of -20? Explain.


Working with powers


What do you think matters most?• What would you list as the big ideas about

powers (and/or roots)?• Work with people around you to come up

with no more than 3 ideas.


Just one opinion

• There is often, but not always, an alternate way to represent a number as a single whole number power, but always many ways to represent it as the sum of powers.

• Representing a power a different way can sometimes simplify calculations with it.


Just one opinion

• Powers sometimes have geometric, as well as numeric, meanings.

• Taking powers and taking roots are opposite operations.


Questions to ask

• Represent 81 in each of these ways:- As a power- As the product of powers- As the sum of powers• Which was easiest for you?• Would you say the same thing if you were

representing 82?


Questions to ask

• Why can you describe 83 x 162 as a single power? Why can you not describe 533 x 352 as a single power?

• How could you use mental math to calculate 54 x 26?


Questions to ask

• What picture could you draw to show why √17 is about 4.1?

• How could you use a similar picture to estimate √90?

• How is a model for 53 different from a model for 52? How is it similar?


Questions to ask

• Can the square root of a number be greater than the number? When?

• Why don’t we take square roots of negative numbers?

• Is it always easier to calculate a square than a square root?


Measures of central tendency


Big ideas?

• What might you suggest for the big ideas related to measures of central tendency?


One opinion

• Sometimes a whole set of data can be “summarized” using a single statistic, such as a measure of central tendency.

• You can predict how measures of central tendency are related when you know how data sets are related.


Questions to ask

• A team scores these numbers of goals in 10 games: 3, 2, 3, 1, 0, 4, 1, 2, 1, 2. If you had to report their performance using a single number, what number would you choose? Why?

• How do you know there are as many sets of data with a mean of 10 as with a mean of 3?


Questions to ask

• To calculate the mean of 43, 52, 47, 55, and 40, Jane calculated the mean of -7, 2, -3, 5 and -10 instead. Why did she do that? How will it help her?


Questions to ask

• Ian said that to calculate a mean you can just increase some numbers and reduce others by the same amount. For example, for 16, 17, 29, 35, he would change them to 16, 18, 30, 33, then 20, 20, 28, 29, then 23, 23, 25, 26 and then 24, 24, 25, 24. Is he right? How does it help?


Questions to ask

• Can you always create a set of 4 numbers with the same mean as a given set of 5 numbers? Explain.


Your turn

• Work together to create a couple of questions to focus on big ideas related to measures of central tendency.


Linear and quadratic relations


What are the big ideas?

• Linear and quadratic functions differ in how the y-values change for given x-values.

• Linear and quadratic functions are both “predictable”. If you know the x-value, you can predict the y-value.



• A linear function is fully determined by two pieces of information.

• A quadratic function is fully determined by three pieces of information.

• Alternate representations of each function reveal different information about it.


Linear relations

• A table of values begins as below:

• Could it represent a linear relation? How do you know? A quadratic relation? How do you know?


Linear relations

• Describe the graph of the relation.• Does it make sense that (10,20) is a point

on the graph? Why?• Pauline said she graphed the relation and

the graph was not very steep. Is that possible?


Linear relations

• Could a different linear relation pass through the ordered pair in the first column? The ordered pairs in both of the first two columns?

• Brad says that if you write the relation in the form Ax + By + C = 0, you can tell what the slope and intercepts are right away. Do you agree? Explain.


Linear relations

• A different linear relation goes through the points (2,5), (8,k) and (-k, -19). What is the value of k?


Quadratic relations

• How does writing a quadratic relation in factored form make it easier to graph it?

• Is the graph of y = 3x2 + 4x – 5 more like the graph of y = 3x2 -5 or more like the graph of y = 4x – 5? Explain.

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©Marian Small, 2009

Quadratic relations

• You graph y = 6x2 + 5x + 1. Does the graph change more if you increase the 6 to 7, the 5 to 6, or the 1 to 2?

• Jane said that you need to know 3 points on a parabola to graph it; Aaron said that you need to know 2 points and Amanda said that you need to know 4 points. Who is right? How do you know?

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Quadratic relations

• Describe a real-life situation that could be described by a quadratic relation. How did you know that a quadratic would make sense?

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Your turn

• Create two of your own interesting questions on this topic that lead to a focus on the big ideas.


Solving linear and quadratic equations



• Solving an equation means determining values that can be substituted to balance the left and right sides.

• There are many methods to determine the solution to an equation.

• The number of solutions of a linear or quadratic equation can vary.


Linear equations

• How do you know that there cannot be any solutions to x + 4 = x + 5?

• Write a linear equation to which there is more than one solution.

• Write a linear equation where the solution is 5.


Linear equations

• Describe two different ways to solve the equation 2x – 4 = 18 + x.


Guess and test

• 2x – 3 = 18 + x• Try 0 and get -3 and 18. No• Try 10 and get 17 and 28. No, but closer.• Try 20 and get 37 and 38. Close!• Try 21 and get 39 and 39. Done.


Visual model

• 2x – 3 = 18 + x

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18

3


Opposite operations

• 2x – 3 = 18 + x• X – 3 = 18• X = 21


Questions to ask

• Why is it useful to isolate the term involving the variable to solve an equation? Why does it not change the solution to the equation?

• If you have an equation like 3x – 2 = 4x + 12, how do you know that the answer is probably going to be negative?

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Questions to ask

• How do you know that there are a lot of equations with a solution of x = -3?

• How is solving -3x – 2 = 4x + 12 like solving 3x – 2 = 4x? How is it different?

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Quadratic equations

• How is solving 3x2 = 18 like solving 3x2 – 2x + 2 = 0. How is it different?

• Could a quadratic equation have solutions that are exactly 3 apart? What could the equation be?

• How do you know that the solutions for x2 + 5x + 6 = 0 have to be negative before solving the equation?


Quadratic equations

• Can a linear equation and a quadratic equation have exactly the same solutions?

• Can a quadratic equation have an infinite number of solutions? When?


Your turn

• Create two of your own questions on this topic that focus on the big ideas that you associate with the topic.


Planning a lesson

• In planning a lesson, you should be saying to yourself:

• What specific idea am I trying to bring out?

• What big idea does this link to and how am I bringing it out?

• Did I plan questions that focus on these important ideas?

©Marian Small, 2009 54

In summary

• Focusing on what’s really important will help you ensure that your students learn those ideas.

• This was a first venture. It gets easier and easier.


Download

• You can download this presentation for about a week or so at:

• www.onetwoinfinity.ca • Quick Links/Renfrew 2


http://www.onetwoinfinity.ca/

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Big Ideas & Better Questions, Part II Marian Small May, 2009 1©Marian Small, 2009