Post on 04-Feb-2018
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Teachingtothe
BigIdeasK3
Marian Small
February 2009
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Gettingto20
You are on a number line. You can jump however youwant as long as you always take the same size jump.
How can you land on 20?
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HowWould
You
Use
Student
ResponsestoThatTask? Whats different about s way of counting than .s
way?
Do you have to start at 1 to get to 20?
Anna jumped to 2, then 4, then 6, . Ryan jumped to 4,
then 8, then 12, Who took more jumps? Why?
Lee started at 6. Could he get to 20?
Which way of counting to 20 was easiest?
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NoticeTheContrast
What I just did is in contrast to telling students how toproceed- how to count. If I do that, its much harder to
think about the fact that there are always different ways
to count.
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WhatareBigIdeas?
These are ideas that underpin a great number ofproblems, concepts, or ideas that we want students to
learn.
A big idea is NOT a topic like fractions, but might be anidea like a fraction only makes sense if you know the
whole of which it is a fraction.
Some people use language like key concepts or
enduring understandings.
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Advantages?
Simplifies your job of prioritizing and organizing
Helps you assess time and attention required by an outcome
Helps clarify what aspect of an outcome to bring into focus
Helps you look at resources critically
Helps you create appropriate assessments
Helps students build essential connections
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MakingBigIdeasExplicit
We cannot assume that students will see the big ideas ifwe do not bring them to the students attention.
Many teachers, though, do not know what the big ideas
in a lesson are, even if they know the lesson goal.
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WhatDoes
The
Teacher
Do
Differently? S/he thinks about what the big idea really is and then sets
a task and asks a question to ensure it explicitly comes
out, often toward the end of the lesson so that it stays
with the student.
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TryThis
How many are in each group? How did you know?
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BigIdea1
Numbers tell how many are in a group. You usually (butdont always) count to determine the size of a group
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BigIdea3
Choose one of these numbers: 5 or 8.
Represent it as many ways as you can.
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WhyDoesItMatter?
Why is it useful for students to be able to representnumbers in different ways? For example, what do these
four representations of 4 describe about 4?
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BigIdea4
Work in pairs.
Each of you thinks of a number between 1 and 20.
Describe two different ways to show whose number is
greater and why.
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BigIdea1
Which group is easier to count? Why?
Group 1 Group 2
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BigIdea1
1. Numbers tell how many are in a group. Countingnumbers beyond 10 or 20 is often based on creating
subgroups and counting these.
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MoreOnBI1
On some tables, put 32 counters into ten frames. On othertables, put 32 loose counters. Ask each group to tell how
many counters are there.
Then ask
Why did finish counting more quickly than ?
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YourTurn
We did not yet look at the big idea:
Students gain a sense of the size of numbers by comparing
them to meaningful benchmark numbers.
What could you have student do and what could you ask to
bring this out?
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YourTurn
One idea:
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WhatOperationWouldYouUse?
Anika had 12 cookies on a plate. If she ate them 3 at atime, how many times could she go back to get cookies?
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CreateAProblem
Create a story problem for each of these numbersentences.
Well then group them.
5 + 8 = 13
10 3 = 7
3 x 5 = 15
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DifferentMeanings
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Differentmeanings
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Differentmeanings
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DangerOfClue Words
Create a subtraction question that uses the word more.
Create an addition question that uses the word less.
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TryThisComputation
Think of 3 different ways to calculate 15 8.
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SomePossibilities
Calculate 10 8 and then add 5.
Calculate 18 8 and then subtract 3.
Calculate 16 8 and then subtract 1.
Add 2 to get to 10 and then 5 more.
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TryThisComputation
Think of 3 different ways to calculate 200 134.
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SomePossibilities
Add 6 to get to 140, then 10 more, and then 50 more.The total added is 66.
Subtract 125 and then another 9 by subtracting 10 and
adding 1.
Subtract 100, then 30, then 4.
Add 60 to get to 194 and then 6 more.
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WhatDoYouThinkOfThis?
200
- 134
134 = 100 30 4 = 66
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WhatDoYouThinkOfThis?
200 = 199 + 1
- 134 - 134
65 + 1 = 66
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WhatIsAPattern?
Write down three examples of patterns.
What makes them patterns?
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AllAbout
Patterns
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MakingNon
Patterns
Take 12 counters and start a pattern with them.
Take 12 counters and make a non-pattern.
What sorts of questions might you ask a student to focus
them on the idea that a pattern has to be aboutregularity?
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ContinuingA
Pattern
What comes next: 3, 5, 7,.?
It could be 3, 5, 7, 3, 5, 7, 3, 5, 7,
3, 5, 7, 7, 5, 3, 3, 5, 7, 7, 5, 3, ..
3, 5, 7, 9, 11,
3, 5, 7, 13, 15, 17, 23, 25, 27,
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ContinuingA
Pattern
Number of
People
Number of Eyes
1 2
2 4
3 6
4 8
What is the pattern rule?
Is there more than one
way to express it?
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ComparingPatterns
In pairs, make a pattern with your linking cubes.
Lets compare some of the patterns we made.
What do we mean by the structure of a pattern?
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ArrangingRepeating
Patterns
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ArrangingRepeating
Patterns
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FindFour
Patterns
Think about a question you
could ask to see if the
student could use the
pattern more generally.
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FindFour
Patterns
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FindFour
Patterns
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FindFour
Patterns
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FindFour
Patterns
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BuildingConjectures
Think about the pattern: 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5,
I conjecture that every 10th number is even.
Is that true? Try to explain.
I conjecture that there are more evens than odds in the
first 30 numbers.
Is that true? Try to explain.
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BuildingConjectures
Choose your own pattern.
Make two conjectures about it and then verify whether or
not they are true.
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WhatComes
Out
If
10
Goes
In?
In Out
2 8
3 11
4 14
5 17
3 x In + 2
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RepresentingRelationships
Think about 4 and 8.
You can think about 8 as two 4s (just like 10 is two fives).
You can think about 8 as four more than 4 (just like 10 is
four more than 6.)
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RepresentingRelationships
How can you represent the doubling relationship with a
picture? The picture should be useful no matter what
number you are doubling.
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RepresentingRelationships
How can you represent the +4 relationship with a picture?
The picture has to work no matter what number you are
adding 4 to.
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For example:
+4
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RepresentingEquality
8 + 4 = + 5
What goes in the blank and why?
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AndSo
So what do we need to do?
We could use words: You add a number to 5 and you
end up with the same amount you would if you added 8
and 4. What did you add? (BI 2)
We could put the answer on the left, e.g. = 3 + 4.
We could use a balance metaphor for equality, viewing
equality as a relationship.
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RepresentingEquality
2 + 1 =
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RepresentingEquality
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RepresentingEquality
2 + = 5 + 3
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RepresentingEquality
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RepresentThese
and
Solve
5 + = 8 + 3
+ = 8
Think about how youwould verbally say these
equations to your
students.
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UseThe
Balance
To
Show
That 10 is double 5.
That 8 is one less than 9.
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Modelling Subtraction
How would you use the balance to solve 8 2 = ?
How would you use the balance to solve 6 = 3?
How would you use the balance to solve 4 = 6?
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WhichDoesnt
Belong?
Which shape below doesnt belong?
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BigIdeas
in
Geometry
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Tie your yarn into a large loop. Work in groups.
Use members of the group to hold the yarn to form each
shape and be ready to tell how you arranged yourself to
make it work: a triangle, a rectangle, an isosceles
triangle, a hexagon, a circle.
Why was it more challenging to make the isosceles
triangle than the hexagon?
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BigIdeas
in
Geometry
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Draw a shape like this.
Divide the shape up into only triangles.
Can you always cut a shape into triangles? Explain.
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BigIdeas
in
Geometry
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BigIdeas
in
Geometry
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YourTurn
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We have not yet dealt with the big idea: Many of the attributes andproperties of 2-D shapes also apply to 3-D shapes.
What could you have students do and what could you ask to bring this
out?
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ForExample
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Length
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Work in pairs. One of you makes a wiggly path with
each of the two pieces of yarn you have. Dont let yourpartner see which is longer and curve the paths so its
not obvious. The partner estimates which is longer and
then tests.
How can you test to see which length is longer? Is there
more than one way to test?
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BigIdeas
in
Measurement
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BigIdeas
in
Measurement
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BigIdeas
in
Measurement
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BuildingaLine
Plot
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We are going to make a line plot to show how often
people in the room have had their hair cut in the lastyear.
What do you think it will look like?
Why is a line plot a good idea for this data?
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WhatsThe
Graph
About?
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What could thesegraphs be about?
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WatchThe
Video
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BigIdeas
in
Data
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How
Do
Teachers
Figure
Out
What
TheBigIdeasAre? Professional conversations with colleagues.
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ThenWhat?
You make sure you ask the question to bring out that big
idea.
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How Do You Figure Out What The BigIdeasAre?
Some resources help you.
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FromaResource
From
Big Ideas from
Dr. Small
F R
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FromaResource
From
Math Focus K-3
I S
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InSummary
Math really is not a set of tiny little pieces. It is a
connected whole.
It is our job to help our students seen those connections.
We have to focus in, therefore, on the big ideas, but wealso have to ask open and directed questions.
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