Building Powerful Numeracy
from Whole Numbers to
Ratios, 5-7
Pamela Weber Harris
@pwharris
pamelawharris.com
pharris@byu.net
facebook.com/numeracy
Math is
figure-outable!
pharris@byu.net
0.26 x 24
pharris@byu.net
Counting Strategies
Additive Thinking
Multiplicative
Thinking
Proportional
Reasoning
Algebraic
Reasoning
pharris@byu.net
15 - 7
pharris@byu.net
Counting Rack
Rekenrek
pamelawharris.com
Counting Rack
Rekenrek
Problem Strings
As Close As It Gets
409 + 1.19
a) 400
b) 410
c) 419
d) 500
As Close As It Gets
13.2 + 1,989
a) 14
b) 1,900
c) 2,000
d) 2,100
As Close As It Gets
a) 0
b) 0.5
c) 0.75
d) 1
Relational Thinking
36 + 97 = ______ + 100
3,554+2,849 = 3,403+ ____
True or False?
19.8 + 2.03 = 20 + 0.03
As Close As It Gets
14,009 - 9.19
a) 14,000
b) 5,000
c) 13,000
d) 4,000
As Close As It Gets
1,002 - 899
a) 3
b) 100
c) 200
d) 300
As Close As It Gets
a) 5
b) 6
c) 4
d) 3
Relational Thinking
36 - 19 = ______ - 20
3,554 - 2,849 = ______ - 3,000
True or False?
6.2 - 3.8 = 6 - 4
19.8 - 1.98 = 20 - 2
Counting Strategies
Additive Thinking
Multiplicative
Thinking
Proportional
Reasoning
Algebraic
Reasoning
pharris@byu.net
11 x 12
pharris@byu.net
132 ... Um, I knew 10 x 12 = 120, and um, then I just
add 12 because 1 x 12 = 12 and um I just add 12 to
120 and I got 132.
I didn’t ask you if you know
it, I asked you what it is.
What do you know...
and figure it out from there!
pharris@byu.net
ZERO FACTS
ONE
FACTS
ZERO FACTS
ONE FACTS
DOUBLE
ZERO FACTS
ONE FACTS
DOUBLE
DOUBLE + ONE
ZERO FACTS
ONE FACTS
DOUBLE
DOUBLE + ONE
DOUBLE, DOUBLE
ZERO FACTS
ONE FACTS
DOUBLE
DOUBLE + ONE
DOUBLE, DOUBLE
DOUBLE, DOUBLE
DOUBLE
ZERO FACTS
ONE FACTS
DOUBLE
DOUBLE + ONE
DOUBLE, DOUBLE
DOUBLE, DOUBLE
DOUBLE
TENS
ZERO FACTS
ONE FACTS
DOUBLE
DOUBLE + ONE
DOUBLE, DOUBLE
DOUBLE, DOUBLE
DOUBLE
NINES
TENS
ZERO FACTS
ONE FACTS
DOUBLE
DOUBLE + ONE
DOUBLE, DOUBLE
5 IS HALF OF 10
DOUBLE, DOUBLE
DOUBLE
NINES
TENS
ZERO FACTS
ONE FACTS
DOUBLE
DOUBLE + ONE
DOUBLE, DOUBLE
5 IS HALF OF 10
5 AND ONE MORE
DOUBLE, DOUBLE
DOUBLE
NINES
TENS
ZERO FACTS
ONE FACTS
DOUBLE
DOUBLE + ONE
DOUBLE, DOUBLE
5 IS HALF OF 10
5 AND ONE MORE
DOUBLE, DOUBLE
DOUBLE
NINES
TENS
Practice with a Game
pharris@byu.net
TI-Nspire, 84, 83 and
APPS
pharris@byu.net
How to Build Multiplicative &
Proportional Reasoning at the Same
Time?
• Good, rich contexts - See Cathy Fosnot
• Mini-lesson: Problem Strings
pharris@byu.net
Proble
m
Strings
pharris@byu.net
2 x 27
4 x 27
8 x 27
10 x 27
9 x 27
5 x 27
100 x 27
99 x 27
324/27
405/27
Consider the
Multiplication and
Division Algorithms
pharris@byu.net
Graph it
pharris@byu.net
What kind of relation?
•Open:
•packs sticks.tns
•Enter the values in lists
•Enter the equation in
f (x)1 
•What questions could
•you ask?
pharris@byu.net
2 x 3.24 x 3.28 x 3.210 x 3.29 x 3
x 3.2
15 x 3.2100 x 3.299 x 3.2
44.8/3.2
323.2/3.2
313.6/3.2
480/3.2
Proble
m
Strings
pharris@byu.net
PB
CC
2
4
Proble
m
Strings
8
10
9
6
5
3
400
450
456
18
25
pharris@byu.net
270 18
pharris@byu.net
Proble
m
Strings
pharris@byu.net
25 x 36
1/4 x 36
0.25 x 36
0.26 x 36
0.24 x 36
0.25 x 24
0.75 x 24
0.76 x 24
pharris@byu.net
Math is Figure-outable
• Look to most efficient strategies and
build proportional reasoning at the same
time
• Teachers build own numeracy
• It’s about relationships:
-
Among numbers to solve problems
Between teachers and students to
build young mathematicians
pharris@byu.net
Building Powerful Numeracy
from Whole Numbers to
Ratios, 5-7
Pamela Weber Harris
@pwharris
pamelawharris.com
pharris@byu.net
facebook.com/numeracy