SHS2 Mathematics · Semester 1, Week 6
Real Number and Numeration System
Lesson notes
Learning Objectives
Indicator: 2.1.1.LI.1 - Investigate the concept and existence of modulo arithmetic in learners’ environment and introduce it as the arithmetic of remainders.
By the end of the lesson, learners can:
- Identify and describe at least three real-life situations in their environment where remainders are used to sort or classify numbers (such as days of the week, clock faces, or market groupings).
- Sort a given set of integers into distinct subsets according to modulo 2 and modulo 3, using number arrays and the divisibility rules already known from previous lessons.
- Compute the remainder when a positive integer is divided by a given modulus, and express the result in the form a (mod m) = r.
- Determine the modulo of negative numbers using repeated addition or subtraction to reach a non-negative remainder.
- Explain why modular arithmetic is “the arithmetic of remainders” and connect it to the idea of place value and grouping that was studied earlier in the term.
Sign in with your phone number to read the full note and download the GES plan - free.
Sign in with phone numberCurriculum details
- Strand
- Numbers for Everyday Life (Strand 1)
- Sub-strand
- Real Number and Numeration System (1.1)
- Content standard
- 2.1.1.CS.3 - Demonstrate understanding of the concepts of modulo arithmetic and solve real life problems on them. 2.1.1.LO.3 Establish the connections between number bases and modular arithmetic and apply these relationships to the concept of place value.
- Indicator
- 2.1.1.LI.1 - Investigate the concept and existence of modulo arithmetic in learners' environment and introduce it as the arithmetic of remainders.
- Suggested placement
-
Semester 1, Week 6
(Week 6 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
- Curriculum reference
- NaCCA curriculum document, p. 171
Exemplars (from the NaCCA curriculum)
In an interactive and differentiated learning environment, undertake a brief review of sorting and divisibility rules of integers and determine their connections to modular arithmetic using models such as number arrays.
Examples: i. Sort integers into odd and even blocks (mod2 and mod3): i.e., For modulo 2, the integers are sorted into 2 distinct subsets as,
- {. . . , −4, −2, 0, 2, 4, 6, 8, . . .} and
- {. . . , −3, −1, 1, 3, 5, 7, . .}. (Remember that the set of integers is used in its entirety). Similarly, in modulo 3, the set of integers is sorted into 3 distinct subsets, such as
- {. . . , −6, −3, 0, 3, 6, 9, 12, . . .},
- {. . . , −5, −2, 1, 4, 7, 10, 13, . . .}, and
- {. . . , −4, −1, 2, 5, 8, 11, 14, . . .}.
ii. Use repeated subtraction/addition to determine the modulo of negative numbers. I.e., −8 (𝑚𝑜𝑑 3) −8 (𝑚𝑜𝑑 3) = −8 + 3 = −5 = −5 + 3 = −2 = −2 + 3 = 1
Teaching and Learning Resources:
- Fractional boards
- Square or grid paper
- Multi-base blocks
- Algebraic tiles
- Video clips
- Cardboards
- Models
- SHS Mathematics curriculum, etc.
- Technology tools
- computer
- mobile phone
- calculator
- Youtube videos, etc.
- Number line
- Number track
- Wheel of Theodorus
- twine, cylinders
- Tape measure or rule
- GeoGebra (free software)
- video clips cardboards
- models
- SHS Mathematics curriculum, etc.
Assessment (2.1.1.AS.1). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning.