Transcript Why open-ended?

Creating Mathematical Conversations
using Open Questions
Marian Small
Sydney
August, 2015
#LLCAus
#LoveLearning
@LLConference
Let’s warm up
• The answer to a question is 100.
• What might the question have been?
Maybe
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What is 10 x 10?
What comes after 99?
How old are you when you are REALLY old?
What is a number in the pattern 10, 20,
30,…?
• What is a number you can represent with one
base ten block?
Agenda
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Why open-ended?
We will look at lots of examples
We will look at strategies to create them.
We will create them.
Why open-ended?
• To be accessible
• To challenge
• To evoke rich conversations
Balance
• You put yellow cubes on one side of a pan
balance.
• You put blue cubes on the other side.
• What would make the balance look like this?
We might discuss
Which might work?
• 1 and 2?
• 5 and 3?
• 9 and 1?
• What sort of relationship are we looking for?
Patterns
• The 10th shape in a pattern is
• What could the pattern be?
.
We might discuss
• Is it repeating?
• If it is, how long could the core be?
• If the core is 3, where would the triangles in
the core be?
• Could it be a growing pattern?
Representing numbers
• A two-digit number is represented with twice
as many one blocks as ten blocks.
• What number could it be?
We might discuss
• Are you allowed to have 5 ten rods and 10
ones?
• What do you notice about the possible
numbers: 12, 24, 36, 48, 60,….
Adding mentally
• Two numbers are really easy to add in your
head. What might they be?
We might discuss
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Why is it easy to add 0?
1?
10?
100?
What else is easy?
Words
• A sentence has 40 letters in it.
• What number of words do you think it has?
Why?
We might discuss
• What assumptions we make
• How valid those are
• The fact that we can’t know
Fill in the blanks
• 5 of ______ is less than 2 of ______.
• What amounts could go in the blanks?
We might discuss
• Is the first number more or less than the
second? Why?
• Is it more than half or less than half? Why?
Addition puzzle
• You add two numbers.
• The answer is twice as much as if you
subtract them.
• What might the numbers be?
We might discuss
• Why it works for so many numbers.
• What visual might show us why it is true
• What algebra might show us why it is true
Visual
10
10
10
10
Visual
5
5
5
5
Visual
8
8
8
8
Algebra
• x + y = 2y – 2x, so
• 3x = y
• x + 3x = 4x
• 3x – x = 2x
Square tiles
• Use the squares.
• Make a design that is ALMOST half red.
We might discuss
• How to start with half and half and add
either one more of another colour or subtract
one red.
Multiplication
• You multiply two numbers and the product is
ALMOST 400. What could the numbers have
been?
We might discuss
• Why we can’t be sure what almost means
• How if you start with, e.g. 2 x 200, you are
more likely to change the 200 than the 2
• How you change
• Whether you go up or down
Rectangles
• A length is four times the width of a
rectangle.
• What do you notice about the relationship
between perimeter and width?
We might discuss
• Why the perimeter is always 2 ½ of the
length.
I can see that
I can see that
I might ask….
• You add 2 fractions.
• The sum is []/15.
• What fractions might you have added?
Someone might say
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3/15 + 6/15 OR
1/3 + 2/5 OR
2/3 + 1/15 OR
3/5 + 2/15 OR
2/4 + 16/32 (if the answer is 15/15)
We would discuss
• That we usually use a common multiple as
the denominator when we add two fractions,
but not necessarily.
A problem I tried in Grade 4
We would discuss
• Some properties were easier to use than
others (e.g. a very small angle OR angle
bigger than a right angle OR symmetry) to
make the problem accessible.
• Could talk about what combos didn’t work
• Could talk about what automatically
happened when certain combos were chosen
Or
• Certain properties don’t mix (e.g. some, but
not all, equal side lengths AND four equal
side lengths).
• Some properties automatically come
together (e.g. 4 equal side lengths and some
parallel sides)
Adding fractions
• You add two fractions less than 1.
• The answer is a little less than 5/4.
• What might the fractions have been?
We could discuss
• Why 9/10 + 1/4 (or something like that
works)
• Why at least one fraction has to be more
than ½
• Why both fractions can be more than half,
but not if one is too close to 1
Consider this problem
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A sweater was on sale, 40% off.
A pair of pants was on sale, 20% off.
The sale prices were the same.
How did the original prices compare?
Could be solved numerically
• If the original sweater price was $100, the
sale price is $60.
• If $60 is 80% of the pants price, 20% is $15,
so 100% is $75.
Could be solved numerically
• If the original sweater price was $50, the sale
price is $30.
• If $30 is 80% of the pants price, 40% is $15,
so 100% is $15 + 15 + 7.50 = $37.50.
• The pants price is 3/4 of the sweater price.
Could be solved algebraically
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6/10 s = 8/10 p
6s = 8p
3s = 4p
So p/s = 3/4 , so the pants cost 3/4 of the
sweater.
Could be solved visually
We could discuss
• When you know a relationship between
percents, you can’t know the absolute values,
only relative values.
Right triangles
• One side of a right triangle is 10 cm long.
What might be the lengths of the other two
sides?
We might discuss
• Whether or not the triangle could be
isosceles
• The idea that the 10 could be a leg or a
hypotenuse
Multiplying integers
• You multiply two integers. The result is about
50 less than one of them.
• What might they be?
We could discuss
• Why it is likely one is positive and one is
negative, but
• Why it could be 0 and 50 or 0 and 49, etc.
• Why there can’t be a really big positive
integer
Algebra
• Create a story that you would represent with
the equation 3x + 5y = 60.
Maybe
• I build triangles and pentagons with toothpicks.
I used 60 toothpicks. How many of each?
• Some kids were in groups of 3 and some in
groups of 5. There were 60 kids. How many of
each size group?
• I bought a bunch of notebooks that each cost $3
and a bunch of packs of paper that each cost $5.
I spent $60. How many notebooks and how
many packs of paper?
Maybe
• I bought 3 identical pairs of shorts and 5
identical t-shirts and spent $60. How much
did each cost?
• There were 3 identical groups of girls and 5
identical groups of boys. There were 60 kids.
How many girls were in a group? How many
boys in a group?
We could discuss
• that 3x can either mean a lot of 3s (x of
them) or 3 of the same thing.
Volume
• A cone and cylinder have the same volume.
• The cylinder is taller.
• Create possible dimensions.
We could discuss
• The relationship forced by the formulas
between the radii if the heights are the same
• The relationship forced by the formulas
between the heights if the radii are the same
Powers
• Write 88 as the sum of powers in different
ways.
We could discuss
• How using the exponent 1 can be very helpful
Linear Relations
• A system of equations is graphed. The
solution is a point in Quadrant II.
• What might the equations be?
We might discuss
• Why if you know the solution is, e.g. (–3,1), it
is easy to come up with equations
numerically.
• e.g. since –3 = 3 x (–1), one equation is x = 3y.
• Since 2 x (–3) + 4 x (1) = –2, another equation
is 2x + 4y = –2, etc.
Mathletics tasks
Task
Video and interactive
• Let’s check them out
Opening up Questions
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Start with the answer. Ask for the question.
The answer is “a square”.
The answer is “2/3”.
The answer is √10.
The answer is 4x – 2.
Opening up Questions
Similarities and differences
• How are 7 and 10 alike and different?
• 350 and 550?
• 3x and 2x
• Adding and subtracting
Opening up Questions
Choose your own values for the blanks.
• Add 5[] + []9
• [][] is a little less than 4[]. What can go in the
blanks?
• [] is about 25 less than []. What values can go
in the blanks?
Opening up Questions
Use “soft” words.
• The square root of a number is ABOUT 30.
What could it be?
• The quotient of two numbers is a LITTLE LESS
than 20. What could they be?
• A line is VERY steep. It goes through (4,2).
What could the equation be?
Your turn
• Choose three curriculum topics.
• Create different sorts of rich open questions
you could use.
• We will share!
Download
• You can download this at
www.onetwoinfinity.ca
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