  Teaching Strategies

### 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.

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.

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 counting by 1s

The concept of place value continues into Grade 6 with larger numbers.

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. 102 = 10 × 10 = 100 103 = 10 × 10 × 10 = 1000 104 = 10 × 10 × 10 × 10 = 10 000

By Grade 12, students are using exponents to understand geometric series and logarithms.

Mathematics 30-1

Mathematics 30-2

Demonstrate an understanding of logarithms.

Demonstrate an understanding of logarithms and the laws of logarithms.

 102 = 100 103 = 1000 104 = 10 000 105 = 100 000 log10(100) = 2 log10(1000) = 3 log10(10 000) = 4 log10(100 000) = 5 