UNDERSTANDING RATIONAL NUMBERS
LEARNING PROGRESSIONS BY CHALLENGE LEVEL
The overarching idea for this learning progression is the "need for quantification to organize the world.”
(De Lange, n.d.) This includes understanding relative size, recognition of numerical patterns, and the ability to use numbers to
represent quantifiable attributes of real world objects. Students will need to process and understand numbers that are represented in
various ways with connections to quantitative reasoning. This learning progression connects to GLE stems relating to number
sense, relative magnitude, mental arithmetic (fluency), and representations of numbers.
Advanced
Challenge
SubLevel
Level
Tower Stem→
A2
A1
I5
Types of Rational Numbers
Using models, explanations, or other representations student demonstrates conceptual understanding of…
real numbers and common irrational
and
numbers
1] solves problems involving comparing and ordering real numbers and
common irrational numbers using number lines, and equality and
inequality symbols.
scientific notation
and
1] compares and orders numbers represented in scientific notation using
number lines, and equality and inequality symbols.
exponents, square roots and cube roots and
1] compares and orders exponents, square roots and cube roots using
number lines, and equality and inequality symbols.
integers and absolute values
I4
Intermediate
Comparisons and Relationships of Rational numbers
percents
and
1] compares and orders integers and absolute values using number lines,
and equality and inequality symbols.
1] as a way of describing change (percent increase, percent decrease)
I3
ratios and percents
1] as a means of comparing parts when the wholes vary in magnitude; &
2] as a way of expressing multiples of a number.
A] percents relating to corresponding
positive fractional numbers, and
decimals,
B] ratios (comparison of two whole
numbers by division a/b, a:b , a÷b,
where b≠0) ,and
rates (e.g., a out of b, 25%)
and
A1] compares and orders positive fractional numbers , decimals, and
percents (across number formats) using number lines and equality and
inequality symbols;
B1] uses ratio and rates to describe relationships; &
B2] identifies equivalent ratios
I2
I1
Coding Alignment
M:N&O:10:1
M:N&O:10:2 (B)
1 SU
M:N&O:8:2 (B)
M:N&O:10:2 (A)
1 SU
M:N&O:6:2 ('C)
M:N&O:7:1 ('C)
M:N&O:8:1 (B)
M:N&O:8:2 (A)
1 SU
M:N&O:7:2 (B)
1 SU
M:N&O:7:1 (B)
M:N&O:8:1 (A)
1 SU
M:N&O:6:1 (B)
M:N&O:7:1 (A)
M:N&O:7:2 (A)
2 SU's: 1, 2
M:N&O:6:1 (A)
M:N&O:6:2 (B)
3 SU's: A1, B1, B2
Challenge
SubLevel
Level
Tower Stem→
T7
T6
Transitional
T5
T4
T3
T2
T1
Types of Rational Numbers
Comparisons and Relationships of Rational Numbers
Using models, explanations, or other representations student demonstrates conceptual understanding of…
percents relating to corresponding
and
positive fractional numbers, and
1] compares and orders positive fractional numbers, decimals, and
decimals
percents (across number formats) using area and set models
A] benchmark percents (10%, 25%,
and
50%, 75%, and 100%) relating to
A1] compares and orders benchmark percents as a part to whole
corresponding positive fractional
relationship using area and set models; &
numbers, and
B1] compares and orders decimals using area, set or linear models.
B] decimals
and
A] whole numbers
A1] compares and orders whole numbers, & A2] applies concepts of
B] positive fractional numbers (a/ 2,
equivalency in composing and decomposing numbers; &
a/3, a/4, a/5, a/6, a/8, a/12),
B1] compares and identifies equivalent positive fractional numbers using
C] whole number powers of ten,
area and set and linear models,
D] decimals as thousandths
C1] compares and orders whole number powers of ten, &
D1] compares and orders decimals using area, set or linear models.
A] whole numbers 0 - 999,999 using
and
place value,
A1] compares and orders whole numbers; A2] applies concepts of
B] positive fractional numbers (a/5,
equivalency in composing and decomposing numbers; &
a/10 where a is a whole number
B1] compares and identifies equivalent positive fractional numbers using
greater than zero and less than or
area and set and linear models; &
equal to the denominator),
C1] compares and orders decimals using area, set or linear models.
C] decimals as tenths and hundredths
A] whole numbers 0 - 999,999 using
and
place value,
A1] compares and orders whole numbers & A2] applies concepts of
B] positive fractional numbers (a/5,
equivalency in composing and decomposing numbers;
a/10 where a is a whole number
B1] compares and identifies equivalent positive fractional numbers using
greater than zero and less than or
area and set and linear models, &
equal to the denominator,
C1] compares and orders decimals using area, set or linear models.
C] decimals as hundredths
A] whole numbers 0 - 999 using place
and
value, and
A1] compares and orders whole numbers; A2] applies concepts of
B] positive fractional numbers (a/6, a/8
equivalency in composing and decomposing numbers; &
where a is a whole number greater than
B1] compares and identifies equivalent positive fractional numbers using area
zero and less than or equal to the
and set and linear models; &
denominator) ,
C1] links collection of coins with their value using monetary notation.
C] decimals within a context of money
A] whole numbers 0 - 999 using place
and A1] compares and orders whole numbers, A2] applies concepts of
value, and B] positive fractional
equivalency in composing and decomposing numbers, &B1] compares and
numbers (a/6, a/8 where a is a whole
identifies equivalent positive fractional numbers using area and set models.
number greater than zero and less than
or equal to the denominator)
Coding Alignment
M:N&O:5:1 ('C)
M:N&O:6:2 (A)
1 SU
M:N&O:5:1 (B)
M:N&O:5:2 (B)
2 SU’s: A1, B1
M:N&O:5:1 (A)
M:N&O:5:2 (A)
5 SU's: A1, A2, B1, C1, D1
M:N&O:4:1 (C)
M:N&O:4:2 (B)
4 SU's: A1, A2, B1, C1
M:N&O:4:1 (B)
M:N&O:4:2 (A)
4 SU's: A1, A2, B1, C1
M:N&O:3:2 (B)
M:N&O:4:1 (A)
4 SU's: A1, A2, B1, C1
M:N&O:3:1 M:N&O:3:2 (A)
3 SU's: A1, A2, B1
Challenge
SubLevel
Level
Tower Stem→
B5
Beginning
B4
B3
B2
B1
Types of Rational Numbers
Comparisons and Relationships of Rational Numbers
Using models, explanations, or other representations student demonstrates conceptual understanding of…
A] whole numbers 0 - 199 using place and
value,
A1] recalls the sequence of numbers; A2] compares and orders whole
B] positive fractional numbers (a/2,
numbers using previously learned strategies including skip counting; &
a/3, a/4 where a is a whole number
A3] applies concepts of equivalency in composing and decomposing
greater than zero and less than or
numbers; & B1] identifies, compares and orders the fractional numbers
equal to the denominator)
using area and set models
A] whole numbers 0 -199 using place
and
value,
A1] recalls the sequence of numbers; A2] compares and orders whole
B] positive fractional numbers (a/2,
numbers using previously learned strategies including skip counting; &
a/3, a/4 where a is a whole number
A3] applies concepts of equivalency in composing and decomposing
greater than zero and less than or
numbers; & B1] identifies, compares and orders the fractional numbers
equal to the denominator)
using set models
A] whole numbers 0-100 using place
and
value,
A1] skip counts by a variety of numbers forwards and backwards using
B] positive fractional numbers (a/2,
benchmark and non-benchmark starting numbers (e.g. skip count by 10
a/3, a/4 where a is a whole number
staring at 17); A2] applies concepts of equivalency in composing and
greater than zero and less than or
decomposing numbers; & A3] connects number words and numerals to
equal to the denominator)
the positions they represent on number lines; &B1] identifies and orders
the fractional numbers using area models
whole numbers 0-100 using place
and
value
1] compares and orders whole numbers using "5 more", "5 less", "10
more", and "10 less"; 2] demonstrates that to count a collection the
objects can be touched in any order; & 3] that neither the starting point nor
the arrangement of the objects affect how many there are (counting
principles).
whole numbers 0-100 using place
and
value
1] recalls the sequence of number names; 2] compares and orders whole
numbers using "1 more" or "1 less"; & 3] connects numbers orally and
written as numerals to the quantities they represent.
Coding Alignment
M:N&O:2:1 (B)
M:N&O:2:2 (B)
4 SU's: A1, A2, A3, B1
M:N&O:2:1 (A)
M:N&O:2:2 (A)
4 SU's: A1, A2, A3, B1
M:N&O:1:1 (C)
M:N&O:1:2 (C)
M:N&O:1:6
4 SU's: A1, A2, A3, B1
M:N&O:1:1 (B)
M:N&O:1:2 (B)
3 SU's: 1, 2, 3
M:N&O:K:2 (B)
M:N&O:1:1 (A)
M:N&O:1:2 (A)
3 SU's: 1, 2, 3
Compare and Order: These are very solid examples.
 Teacher stimulus presents an unordered group of numbers; credit given if the student puts them in order and describes why “a” comes before “b”.
 Teacher stimulus presents an unordered group of numbers; credit given if the student puts them in order smallest to largest or largest to smallest.
 Teacher stimulus presents an unordered set of numbers; credit given if the student takes just two numbers and finds bigger/smaller (compares), and then
puts them in order (sequence).
 Teacher stimulus presents an unordered group of numbers; credit given if the student puts them in order and demonstrates understanding of one greater than
the stimulus sample.
Challenge
SubLevel
Level
Tower Stem→
Types of Rational Numbers
Emergent
Coding Alignment
Using models, explanations, or other representations student demonstrates conceptual understanding of…
A] whole numbers 0-20,
B] 1/2 as fair share
and
A1] compares two collections deciding how much bigger/smaller;
A2] demonstrates that to count a collection each object must be touched or
'included' exactly once as the numbers are said;
A3] the numbers must be said in the conventional order (no skips/no duplicates);
A4] the last number said tells 'how many' in the whole collection, it does not
describe the last object touched. (counting principles); &
B1] shares a quantity (1-20) of objects evenly.
M:N&O:K:2 (A)
5 SU's: A1, A2, A3, A4, B1
A] whole numbers 0-12,
B] 1/2 as fair share
and
A1] compares different collections based on magnitude (bigger, smaller,
same); A2] indicates if a change to a quantity will make it
bigger/smaller/stay the same (a focus on quantity versus size), &
B1] shares a quantity (1-12) of objects evenly.
and
1] distinguishes numerals from other written symbols (e.g. ordinal
numbers); 2] uses number names to count; & 3] recalls the sequence of
number names
NCTM Focal Points K
M:N&O:K:1
3 SU's: A1, A2, B1
E4
E3
whole numbers 1-10
E2
whole numbers 1-6
E1
Comparisons and Relationships of Rational Numbers
and
1] know at a glance [or at a touch] how many are in small collections
(subitize); & 2] attach correct number names to such collections.
When number parameters are given, students must demonstrate
numbers beyond the previous level in order to receive credit for
scorable units.
NCTM Focal Points:
K Numbers & Operations
(A)
NCTM Research * (B)
3 SU's: 1, 2, 3
NCTM Research * (A)
2 SU's: 1, 2