Real Number and Numeration… · SHS2 Mathematics

Lesson notes

Learning Objectives

Indicator: 2.1.1.LI.2 - Model and solve real-life problems involving modular arithmetic

By the end of the lesson, learners can:

Identify real-life situations (such as market days, clock time, and vehicle scheduling) that can be modelled using modular arithmetic.
Set up a modular arithmetic equation for a given real-life problem, correctly identifying the modulus and the values involved.
Solve problems involving modular arithmetic in contexts such as time, market cycles, and capacity planning, using number lines or tables where helpful.
Verify solutions to modular arithmetic problems by checking remainders and justifying their answers in words.
Work collaboratively in mixed groups to present and explain a modelled real-life problem to the class using appropriate mathematical vocabulary.

Sign in with your phone number to read the full note and download the GES plan - free.

Sign in with phone number

Curriculum 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.2 - Model and solve real-life problems involving modular arithmetic
Suggested placement: Semester 1, Week 7 (Week 7 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.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

exemplars - p.171: the equation text on this page is set in a CambriaMath subset whose /ToUnicode map doubles every letter and gets many of them wrong, and the glyph ids could not be lined up with the extraction to repair it, so any italic variable here may be doubled or be the wrong letter; read the page
exemplars - p.172: this exemplar sets a two-dimensional construct - a stacked fraction, an index or a column vector - which the text layer flattens into separate lines, so the parts are present but their vertical arrangement is lost; read the page
Curriculum reference: NaCCA curriculum document, p. 171

Exemplars (from the NaCCA curriculum)

Experiential Learning, Problem-based Learning, Group Work/Collaborative Learning.
In an interactive and differentiated learning environment, model simple situations involving modular arithmetic concepts, connect the ideas to real world problems and solve them using
appropriate models and technology. Employ differentiated assessment and ensure values such as tolerance, truth, honesty, respect for others' views, etc., among learners.
Example 1: Determine the appropriate connections between number bases and modular arithmetic.
In GESI-responsive group activities, initiate reflective thinking (or discourse) on the application of modulo-arithmetic concepts in everyday business:
Modulo in Everyday Activities

List of everyday uses of modulo, a five-column array from minus 9 to 20, and the working 9 + 7 congruent to 1 mod 5. — A list headed "Modulo in Everyday Activities" naming travelling and vehicle capacity, modelling time on a 12-hour clock, market day arithmetic, restaurants and feeding capacity, and modulo arithmetic in music. Below it a five-column number array runs from minus 9 to 20, and beneath that the working "Knowing that 9 is congruent to 4 (mod 5), and 7 is congruent to 2 (mod 5). Then, 9 + 7 is congruent to 4 + 2 is congruent to 1 (mod 5)".

 1. Travelling and Vehicle capacity, 2. Modelling time (using a 12-hour clock), 3. The market day arithmetic, 4. Restaurants and feeding capacity. 5. Modulo arithmetic in music (i.e., base 8 for octave).
The idea is that 𝑎 and 𝑏 are "equivalent" when they leave the same remainder upon division by mod number. Create number arrays using spreadsheets.
-9 -8 -7 -6 -
-4 -3 -2 -1 0
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
Knowing that 9 ≡ 4(𝑚𝑜𝑑5), and 7 ≡ 2(𝑚𝑜𝑑5) Then, 9 + 7 ≡ 4 + 2 ≡ 1 (𝑚𝑜𝑑5)
Example 2: Explore the relationships that will lead to the following generalisation. 𝑎𝑎 ≡ 𝑏𝑏 (𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚), 𝑐𝑐 ≡ 𝑑𝑑 (𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚) 𝑎𝑎 + 𝑐𝑐 ≡ 𝑏𝑏 + 𝑑𝑑 (𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚) 𝑎𝑎 − 𝑐𝑐 ≡ 𝑏𝑏 − 𝑑𝑑 (𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚) 𝑎𝑎 × 𝑐𝑐 ≡ 𝑏𝑏 × 𝑑𝑑 (𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚)
Example 3: Investigate the following properties of operation involving modulo arithmetic. Commutativity: Associativity: Distributive Existence of Identity Elements Existence of Additive Inverses: E.g. 02 = 0 ≡ 0 (mod 4) 12 = 1 ≡ 1 (mod 4) 22 = 4 ≡ 0 (mod 4) 32 = 9 ≡ 1 (mod 4) 2 42 = 16 ≡ 0 (mod 4) 𝑛𝑛 2 = 0(𝑚𝑚𝑚𝑚𝑚𝑚4) ≡ 𝑜𝑜𝑜𝑜 𝑛𝑛 2 = 1(𝑚𝑚𝑚𝑚𝑚𝑚4)
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.2). The document marks no depth-of-knowledge level for this indicator.
rational and irrational numbers