Some Essential Ideas in
Number and Algebra
in Grades 7 – 10
Marian Small
October 2016
Does this picture show addition or
subtraction or both?
What are some CRITICAL big ideas you
need to bring out in operations in 7 – 10?
u
Addition and subtraction are always related.
u
Both involve parts and a whole; all that is
different is which information you have.
I might use the same diagram
part
Whole
part ….. part
What might the story be?
5/6
????
3/8
What might the story be?
????
5/6
4/5
A lesson
u
With the learning goal that if subtraction is
happening, so is addition (or vice versa), I might
create an action task:
u
Make up a subtraction problem (or draw a
subtraction picture) that includes the numbers
2/3 and 8/5. Tell what additions are also
involved.
Consolidation
u
What did you subtract? Why?
u
What did you add? Why?
u
Could a subtraction story involving 8/5 and 2/3
have involved a different addition?
1 2/3 – 3/4
u
Create three different problems that are solved
by subtracting 3/4 from 1 2/3.
u
Try to make them sound REALLY DIFFERENT.
Why are the meanings equivalent
u
Why is how much is left when you take 2/3 from
9/10 the same as how much more 9/10 is than
2/3?
What are some CRITICAL big ideas you
need to bring out in operations in 7 – 10?
u
In particular, subtraction can be used in either
take away, missing addend or comparison
situations.
This is equally true with polynomials
u
I can show (4x + x2) – (2x2 + 5) as take-away
This is equally true with polynomials
u
I can show (4x + x2) – (2x2 + 5x) as take-away
This is equally true with polynomials
u
I can show (4x + x2) – (2x2 + 5x) as take-away
This is equally true with polynomials
u
I can show (4x + x2) – (2x2 + 5x) as missing addend
This is equally true with polynomials
u
I can show (4x + x2) – (2x2 + 5x) as comparison
A problem
u
Jane said that 3/5 + 2/3 = 5/8. (or that 3x + 2x2 =
5x2)
u
Do you agree or disagree? Why?
What are some CRITICAL big ideas you
need to bring out in operations in 7 – 10?
u
When you add things or subtract things, you need
to think about whether they are the same kind of
things to see if your answer makes sense.
What are some CRITICAL big ideas you
need to bring out in operations in 7 – 10?
u
Addition and subtraction involving fractions or
integers or polynomials completely parallel in
usage and in strategies addition and subtraction
with whole numbers.
Look at this equation
u
WITHOUT SOLVING IT, tell how you know the
answer has to be negative:
u
38x + 29 = 17x + 8
What are some CRITICAL big ideas you
need to bring out in algebra in 7 – 10?
u
Equality is an expression of balance.
So…
u
My learning goal focuses on the kinds of numbers
(e.g. negative, large, fractional, etc. that would
make a balance)
u
E.g. If 2/3 x = 6/13, is x positive? Negative? A
fraction or not?
u
What if 2/3x = 6/3? Or 2/3 x + 8 = –6/3?
So to create a lesson
u
I ask students to interpret equations and not just
solve them.
u
For example: –3x + 5 = 7 + x MEANS that I took a
number. First I added to 7. But then I subtracted 3
of them from 5 AND I GOT THE SAME THING.
So here’s a problem
u
I solved a problem and wrote: 5x + 2y = 100.
u
What problem could I have been solving to lead
me to write that equation?
u
Is there more than one possibility?
What are some CRITICAL big ideas you
need to bring out in algebra in 7 – 10?
u
Any equation can represent many situations.
So…
u
I build a lesson around creating a variety of
scenarios that might describe various situations,
e.g. 5.20x + 4.50 = 20.1
u
OR 50 – 3.90x = 22.7
u
OR 223.1÷ x = 9.7
Try this!
u
Choose a number.
u
Write the number that is 1 more and 1 less.
u
Calculate the product of the 1 more and 1 less
number.
u
Calculate the square of the original number.
u
What do you notice?
Try this!
u
Now show the product of those two numbers in an
array.
u
For example, if I chose 4, I would show 3 rows of
5.
Rearrange the counters
Now
u
How does that relate to what you did numerically?
How can you write that algebraically?
u
Maybe (n + 1)(n – 1) = n2 – 1
What are some CRITICAL big ideas you
need to bring out in algebra in 7 – 10?
u
The use of algebraic notation often simplifies the
description of a numerical relationship.
So
u
I build lessons around how algebra describes
number properties, e.g.
u
4x + 3x = 7x tells me…
u
x ÷ 5 = 2x ÷ 10 tells me
u
2x+ 3 = 151 asks me….
A problem
u
We explore lines of the form y = mx + b in Grade
9.
u
What happens when you change the m (e.g. from
1 to 20)?
A problem
u
What happens when you change the b (e.g. from –
8 to 20)?
A problem
u
What happens when m = b?
What are some CRITICAL big ideas you
need to bring out in algebra in 7 – 10?
u
The parameters in an algebraic situation always
have a geometric interpretation.
The point is…
u
When creating lesson goals, activities to bring
them out and consolidation, it is important to
focus more on essential understandings than
simply on just topics or skills mentioned in
expectations.
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