Understanding Progression of Mathematical Understanding
Knowledge of learning objectives from previous and following grades is necessary for teachers to develop trajectories of learning that lead to conceptual growth and a deepening of understanding. In the Alberta Mathematics K–9 Program of Studies, grades share the same general learner outcomes and mathematics ideas are connected through the grades. For example, when students are learning about counting by 10s in the early grades, they are beginning to connect to concepts like addition, multiplication, arithmetic sequences, exponentials and logarithms. These new understandings form a foundation for complex mathematical concepts.
The following chart summarizes grade specific outcomes for base ten numbers to illustrate the vertical connection of number sense to exponentials using the number 10.
Grade 3: 
Say the number sequence 0 to 1000 by 10s.
10 20 30 40 50 60 70 80 90 100 … 1000
10 + 10 = 20
20 + 10 = 30
30 + 10 = 40
...
990 + 10 = 1000 
The number sequence can be generated by repeated additions. 
Relate multiplication to repeated addition.
10 + 10 + 10 + 10 + … + 10 = 10 × 10 = 1000
100 + 100 + … + 100 = 100 × 10 = 1000
Illustrate, concretely and pictorially, the meaning of place value for numerals to 1000.
Place Value Position


1000 
100 
10 
1 

1 
0 
0 
0 
 1000 in terms of place value

In the examples above, the number 1000 has been constructed in several different representations: a sequence, repeated addition and multiplication. Each of these constructions can be represented by different positions in the place value category.
Place Value Position 

1000 
100 
10 
1 


10 


 100 + 100 + … + 100 = 100 × 10
 repeated addition and multiplication
 counting by 100s



100 

 10 + 10 + 10 + … + 10 = 10 × 10
 sequences, repeated addition and multiplication
 counting by 10s




1000 


The concept of place value continues into Grade 6 with larger numbers.
Grade 6: 
Demonstrate an understanding of place value, including numbers greater than one million.
1000,000 
100 000 
10 000 
1000 
100 
10 
1 
1 
0 
0 
0 
0 
0 
0 

In Grade 9, the concept of exponents is introduced. Exponents build on an understanding of multiplication.
Grade 9: 
Demonstrate an understanding of operations on powers with integral bases and whole number exponents.
10^{2} = 10 × 10 = 100
10^{3} = 10 × 10 × 10 = 1000
10^{4} = 10 × 10 × 10 × 10 = 10 000

By Grade 12, students are using exponents to understand geometric series and logarithms.
Grade 12:
Mathematics 301
Mathematics 302 
Demonstrate an understanding of logarithms.
Demonstrate an understanding of logarithms and the laws of logarithms.
10^{2} = 100
10^{3} = 1000
10^{4} = 10 000
10^{5} = 100 000 
log_{10}(100) = 2
log_{10}(1000) = 3
log_{10}(10 000) = 4
log_{10}(100 000) = 5 

