Transcript NCTM - Marian Small Differentiation

Differentiating
Instruction in Math:
Enough Theory! How do
You Really Do It?
Marian Small
April 2008
How did She get
into DI?
What are different
developmental stages?
o Research study with many thousands of
students
o Creation of developmental maps to describe
the stages, or phases, of development for each
strand
Using Open Tasks
Start with an Answer
My answer is 15.
What’s my question?
Some Possibilities
o What is 10 + 5?
o What number comes after 14?
o What is 5 less than 20?
o What is a 2-digit factor of 30?
o What is the square root of 225?
o What number has a digital root of 6?
Number Sentences
o What number sentences does this show? Tell
as many as you can.
Some Possibilities
o 3 x 5 = 15
o 5 x 3 = 15
o 15 ÷ 3 = 5
o 5 + 5 + 5 = 15
o 10 + 5 = 15
o 3 + 3 + 3 + 3 + 3= 15
Using Percents
o 72 is ___ % of ____.
o Fill in the blanks in different ways.
Some Possibilities
o 100% of 72
o 50% of 144
o 72% of 100
o 200% of 36
o 1% of 7200
o 150% of 48
o 600% of 12
o 0.1% of 72,000
Personal estimates
o About how many meals have you eaten in your
life?
o About how many people do you see in a day?
Tell how you estimated.
The Calendar
o Pick your favorite day of the week.
o How many months can start on that day in any
one year? How do you know?
Alike and Different
o How are the numbers 10 and 45 alike?
o How are they different?
Some possibilities
o They are both 2 digits.
o They are both less than 100.
o They are both greater than 10.
o They are both multiples of 5.
o They are both amounts you can show with
nickels.
Some possibilities
o One is more than 30 and one is less.
o One is even and one is odd.
o One is in the forties and one is not.
o One has a 1 as a digit and the other does not.
o One is a multiple of 10 and the other is not.
Alike and Different
o How are the number 3/4 and 5/8 alike and how
are they different?
Some possibilities
o They are both fractions.
o They are both less than 1.
o Both have odd numerators and even
denominators.
o They can both be written as eighths.
o They can both be written as thousandths.
Some possibilities
o One is more than 7/10 and the other is less.
o One has a denominator of 8 and the other has a denominator of
4.
o One is more than 2/3 and one is less than 2/3.
o One can be written as 12ths and the other can’t. (at least not as a
simple fraction)
o The decimal for one starts with 0.6.., but the decimal for the
other starts with 0.7…
A Division
Create a division question where each of the
digits 2, 4 and 9 appears somewhere in the
question.
Some Possibilities
o 94 ÷ 2 = 47
o 249 ÷ 1 = 249
o 216 ÷ 9 = 24
o 99 ÷ 2 = 49.5
Creating a multiplication
o Create an equation involving multiplication of
fractions where the numbers 4, 9 and 10 are
used somewhere.
Some possibilities
o 4/9 x 10/12= 10/27
o 4/10 x 9/9 = 2/5
o 4/6 x 5/9 = 10/27
Build a sentence
o Use these words and numbers to make a true
sentence:
o 10, 50%, twice, difference
Some possibilities
o The difference between 10 and 50% of 80 is
twice as much as 15.
o The difference between 50% of 10 and twice as
much as 6 is that one is odd and the other is
even.
o The difference between twice as much as 50%
of 90 and 100 is 10.
Build a sentence
o Use these words and numbers to make a true
sentence:
o 4, 2, sum, almost
Some possibilities
o The sum of 4 and 2 is almost 7.
o The sum of 8 groups of 4 and eight groups of 2
is almost 50.
o The sum of the 4 numbers 2, 10, 10, and 10 is
almost 35.
A Rectangle within a Rectangle
o Draw a small rectangle.
o Create a bigger rectangle of which the small
rectangle is a part. Tell what fractional part it
is.
Possible Solutions
Questions for practice
o Replace the boxes with values from 1 to 6 five
ways to make this true.
x
Questions for practice
o Replace the boxes with values from 1 to 6
five ways to make this true.
o
234
145
262
BUT
o
x1
x3
x2
NOT
234
435
523
362
x 3
1086
Toss and Win
• Toss a beanbag 3 times.
2.8
0.9
1.5
4.2
1.0
0.8
0.7
0.6
2.1
3.1
4.5
1.6
• Add the first 2 scores.
Subtract the last one.
! Win a prize with 0.5-1.0
points!
Making Patterns
Use square tiles to create a growing pattern.
Draw it. Tell why it’s a pattern.
Some Possibilities
.
Some Possibilities
Some Possibilities
Triangle sides
o One side of a right triangle is 5 inches long.
How long could the other sides be?
Some possibilities
o 3 and 4
o 5 and 5√2
o 12 and 13
You Try!
Open these up:
o How many times as much as 16 is 48?
o How many meters long is your classroom?
o Which shape has a line of symmetry?
Parallel Tasks
Where Does It Go?
Draw a line like this:
Choose one of these sets of numbers for the ends of the line:
0 and 20
15 and 16
-8 and -10
What number describes the position of the dot? Why?
Choose your shape
Pick one of these shapes. Represent the shape
as many ways as you can. Use pictures, words,
numbers and/or objects.
cube square cylinder hexagon
Some possibilities
Cube
o Show a skeleton
o Show a net (or several nets)
o Show one face
o Show all the faces
o Draw an isometric drawing
o Hold up a cubic shaped box
Missing Digits
o Figure out where the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 go.
4x2=[]
[]6 + 49 = 8[]
[ ] x [ ] = 15
7[] + 9[] = []7[]
[] x 7 = 42
[] x 6 = 54
5 x [ ] = [][]
[] x 3 = 2[]
OR
[][] + []2 = 10[]
Missing Digits
o Figure out where the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 go.
4x2=8
36 + 49 = 85
5 x 3 = 15
76 + 94 = 170
6 x 7 = 42
9 x 6 = 54
5 x 4 = 20
7 x 3 = 21
OR
27 + 82 = 109
Make a pattern
Use pattern blocks.
o Create a shape pattern where the 10th shape is
a green triangle,
OR
o Create any shape pattern you wish.
Divide the shape
o Pick a shape. Divide it into 4 equal parts.
OR
o Divide this shape into 12 equal parts.
Draw a shape
o Draw a shape where one side is 4 cm long and
one angle is 30°.
OR
o Draw a hexagon where one side is 4 cm long,
one side is 6 cm long, one angle is 30° and
one angle is 150°.
One way to go
Probability Draws
You can put 20 square tiles in a bag- any
combo of red, blue and yellow. You will
draw one tile.
o A: What tiles would you put in so it is
possible, but not likely, to draw a red?
o B: What tiles would you put in so the
probability of red is 2/5?
Measurement Hunt
o A: Find 5 items in the room that you think have
more area than a recipe card. Check.
o B: Find 5 items in the room that you think have
an area between 20 cm2 and 30 cm2. Check.
How many groups
Choose the number of students going to the
track event and the number of buses from the
choices below. Figure out how many people
each bus needs to hold.
o 300 students in 6 buses
OR
o 435 students in 12 buses
Estimating percents
Choose a shape. Estimate the percent that is
dark.
OR
OR
Cookie prices
Find the price of the larger bag of cookies if
you know the price of the smaller one.
OR
6 for $3
15 for ?
6 for $2.50
20 for ?
In Summary
We need to change our mindset.
Celebrate Differences
They don’t all have to do the same thing at the
same time!
Download
You can download this presentation at
www.onetwoinfinity.ca (see quick links)