Cycle 2 -Math -Competency 2 – To Reason using Mathematical Concepts and Processes
End of Cycle: The student continues developing and applying his/her own computational processes, but uses the four operations. They become familiar with conventional processes with written computations that involve adding and subtracting natural numbers and decimals. They can describe
plane figures and solids. They begin to estimate, measure or calculate lengths, surface areas, and time. They can produce frieze patterns and tessellations by means of reflections. They can do simulations related to activities involving chance and interpret and draw broken-line graphs. Without being
able to explain why, they can recognize situations in which it is appropriate to use technology. In diverse situations, students express their reasoning either orally or in writing. By questioning: “Why did you write that?” or “Why did you perform that step?” and having students justify their answers,
students are able to explain their line of reasoning. In this way, we are able to understand the mathematical concepts, processes, and connections the student has made. This information allows us to understand the child’s development in the competency of mathematical reasoning.
ARITHMETIC
BG
DV
AP
In problem solving situations, the student works with numbers ≤ 1000.
Using problem solving strategies such as using manipulatives, representations,
modeling, organized lists, or patterns, the student can demonstrate, estimate, and
explain counting strategies through:

Varying representations of numbers: in words, by digits, expanded form (e.g. 516
= 5 hundreds 16 ones, 51 tens 6 ones, 516 ones, 50 tens 16 ones etc., 4 hundred 11
tens 6 ones…, 500 + 10 + 6)

Counting by 1s, 10s, 100s, 2s, 3s, 4s, and 5s forwards and backwards

Comparison of numbers (2-5) in increasing or decreasing order or using <, >, or =.

Identifying numbers that are divisible by 2s, 5s, and 10s by following a pattern.

Following a pattern (e.g. 1, 2, 4, 7, 11. or 0, 2, 5, 7, 10…)

Counting on or off from the largest number

Counting more than two numbers and being able to combine them in different
ways in order to facilitate computation e.g. (a + b + c = a + c + b)

Subtracting using own processes from a number that has either 1 (450 or 706) or 2
(100, 200) zeroes in it. Once this concept is mastered, standard algorithms
should start to be introduced.

Combining two numbers and then subtracting them from a third

Using hundreds grids

Using mental processes
By using the problem solving strategies such as guessing and testing, representations,
patterns, organized lists or modeling, the student can demonstrate, estimate and explain
counting strategies through:

Relating words and expressions to the use of operations.

Demonstrating that multiplication is repeated addition through patterns or
Cartesian graphs

Solving: Helen has 5 bean plants. There are 4 beans on each plant. How many
beans do Helen’s plants have?

Solving: Helen has some beans. There were 6 beans on each of her plants. She has
24 beans altogether. How many plants did Helen have?

Solving: Kyle has 35 tomatoes on some tomato plants. He has the same amount of
tomatoes on each plant. He has 5 plants. How many tomatoes were on each plant?
The student knows due to extensive practice the basic facts of the 1, 10, 5, and 2 times
tables and retains knowledge of basic addition and subtraction facts.
The student is able to write own problems.
In problem solving situations, the student works with numbers ≤
10 000.
Using problem solving strategies such as using manipulatives, patterns,
guessing and testing, representations, and/or drawing, the student can
estimate, demonstrate and explain counting strategies through:

Understanding the concept of place value, meaning of a digit, and
the patterns of the base 10 number system

Different representations of numbers, (9034 = 9 thousands 3 tens 4
ones = 8 thousand 10 hundreds 3 tens 4 ones… = 90 hundreds 34
ones = 903 tens 4 ones…= 9000 + 30 + 4) in words and symbols

Identifying prime numbers under 20

Identifying compound numbers, factors and prime factors of
numbers when using arrays.

That 356 + 117 + 247 = 117 + 356 + 247

3 x 4 = 4 x 3; 20 x 7 = 7 x 20 = 7 x 10 + 7 x 10

Extending the multiplication strategies and processes developed in
“BG” to include two digits x one digit numbers and two digit
numbers divided by one digit numbers.
(24 x 6 = 120 + 24 OR 20 x 5 + 20 x 1 + 6 x 4 = 100 + 20 + 24).
Or 48 ÷ 9 means 9, 18, 27, 36, 45.1, 2, 3. (5 groups)

Working with uneven numbers in multiplication and division (i.e.
James had three boxes of twelve crayons and 7 more. How many
crayons did he have? Julio was giving out books for his teacher.
He had 74 books and had to give them out in groups of 8. How
many groups did he give out?
Leona asked her father to buy some cupcakes for her class of 26
students. If cupcakes come in packages of 8, how many packages does
her father need to buy?

Learning standard algorithms for addition and subtraction ≤ 9999.

Using money – dimes, nickels, pennies, quarters, loonies,
“twonies”, $5, $10, $20, $50, $100, and $1000 – to compute with.
In problem solving situations, the student works with numbers
≤ 100 000.
Continuing the problem solving strategies already developed in BG and
DV, the student can estimate, demonstrate and explain either verbally or
in writing, counting strategies through:

Understanding the concept of hundreds thousands, ten
thousands, thousands, hundreds, tens, ones, place value,
meaning of the digits, and the patterns of the base 10 number
system

Different representations of numbers in expanded form, words
and symbols

Expanding the standard algorithm to include computing with 4
digit whole numbers with 2 decimal places.

Using own processes to multiply a three-digit number by a 1digit number and dividing a 3-digit number by a 1 digit one
with and without “remainders”.

Remainders are demonstrated as parts of a fraction (i.e. 28 ÷ 5 =
5 and 53 left over.
The student can recall number facts, due to repeated use, of 1, 2, 3, 4, 5,
and 9, 10 times tables.
The student is able to write own problems.

Comparing and ordering decimal and whole numbers and using
the symbols <, ≤, ≥, >, ≠, and =.

Recognizing that subtraction is the inverse of addition and
multiplication is the inverse of division.

Recognizing the different ways of expressing the 4 operations
(e.g. increase, remove, sharing, rectangular array, etc.)

Calculate prices using money

Using mental and written numbers to add, subtract, multiply, or
divide with using their own mental processes

Relating words and expressions to the use of different
operations
The student can recall number facts up to and including 10 x 10 to 100 ÷
10.
The student is able to write own problems.
TR
In problem solving situations:

Students are able to demonstrate and work with
numbers larger than 100 000.

Students are able to estimate, explain, and perform
operations (addition and subtraction) using
conventional processes on numbers larger than
9999.

Students are able to estimate, explain, and perform
operations (multiplication and division) such as 24
x 12 and 1536 ÷ 32 using their own processes or
conventional methods.

Students use the distributive method to help
perform operations – (24 x 12 = 20 x 12 + 4 x 12)
or
1536 ÷ 32 could be:
32 x 30 = 960
30
+ 32 x 10 = 320
10
1280 with 256 missing
(3 groups 0f 32) 96 + 96 (192) (3 + 3) = 6
64 (1 + 1) = 2
Grand total: 30 + 10 + 3 + 3 + 2 =
48
FRACTIONS AND DECIMALS
BG
DV
In problem solving situations, students can identify and
demonstrate use of the denominator and numerator of a fraction in
parts of an object (pizza) or parts of a set (12/25 of the class are
girls.) They can represent either as a fraction in words or in
numbers.
In problem solving situations, students can demonstrate
knowledge of different fraction names, different ways of
representing them, and can show equivalent parts. (e.g. Michael
bought a pizza that was cut into 8 slices. He ate ¾ of the pizza.
How many pieces did he eat?)
Through the use of manipulatives, drawings, representations, and
1
dimes, students make connections between the fraction 10
, 0.1,
Through drawings, hundredths grids, pennies, and dimes, students
show development of the concept of the hundredths place. They
can represent it by words, equivalent expressions, expanded form,
fractions, and decimals.
They can order decimals, fractions, and words in tenths and
hundredths ≤ 2.
They can mentally perform addition and subtraction operations
with decimal numbers (hundredths place) ≤ 1.
and the word tenths. They are able to order them, represent them,
and write or draw equivalent expressions (0.7 = 0.5 + 0.2) or 0.7
7
= 10
= 7 tenths.
They are able to add or subtract using own processes on decimal
numbers (tenths place) under or equal to 2 by first estimating the
result of an operation and then performing the operation.
They can mentally perform addition and subtraction operations on
decimal numbers (tenths) ≤ 2.
AP
TR
In problem solving situations, students continue to demonstrate different
ways of expressing fractions. (Mary ordered a cake that was cut into 12
pieces, Fernando ordered a cake that was cut into 16 pieces, and Georgia
ordered a cake that was cut into 8 pieces. Exactly half of each cake was
eaten. A) Which cake had the largest pieces and B) how many pieces of each
cake were eaten?
In problem solving situations:

Students are able to estimate, explain, and perform
operations on decimal numbers (e.g. 315
18
hundredths + 6 tenths + 1 ½, - 100
)
They are able to order tenths and hundredths using whole numbers (10.7,
10.17, 10.03, 10.14).
3
34
4
 100
They are able to represent 100
= 0.34 = 10
= 0.04 + 0.3 = thirty four


hundredths, = 3 tenths 4 hundredths
The students mentally solve addition and subtraction problems including
decimals (hundredths place).
The students are able to estimate and perform addition and subtraction
operations on decimal numbers (hundredths place) using standard
algorithms.
Working Document
Becky Thompson, Educational Consultant, Sir Wilfrid Laurier School Board, 2002
Students demonstrate through estimation,
explanation, and performance the ordering of
fractions with differing denominators
Students can estimate, explain and perform
operations on fractions using pictures and
representations.
Cycle 2 – Math – Competency 2 Cont’d)
BG
In problem solving situations, students describe pyramids and
prisms (including pentagon, hexagon, and octagon) according to
their vertices, faces, and edges. They continue to classify them
according to their attributes and make connections between the
similar patterns of their composition. (Pyramids have some
triangular faces, prisms, except cubes, have some rectangular
faces, or Prisms have 3x the amount of edges than that of the its
identifying plane figure i.e. a square has 4 sides, a cube or
rectangular prism has 12 edges (4 x 3)).
Students can construct and/or identify prisms and pyramids using
nets. They can differentiate between similar and congruent
shapes and figures.
GEOMETRY
DV
AP
In problem solving situations, students can place or locate a
point or a figure on the positive quadrant of a Cartesian graph or
a plane.
In problem solving situations and through the use of drawings
and words, students can describe convex and non-convex (
)
polygons. They can produce more advanced frieze and
tessellation patterns and describe or use a reflection line. They
can demonstrate knowledge that (e.g.) a hexagon is made up of 6
triangles, or two triangles can make a rectangle.)
Through the use of parallel, perpendicular, congruent, and/or
non-congruent lines, and right, acute, and/or obtuse angles
students construct, observe, describe, compare and classify
quadrilaterals which include trapezoids, parallelograms, squares,
rectangles, and rhombi.
Using reflections a student can produce and/or describe frieze
patterns. ▪►▪▪◄▪▪►▪▪◄ ▪
Using reflections, students can produce
and/or describe tessellations
Using geometric solids and polygons, students can produce and
observe patterns. (e.g. ▲◊∆◊▪◊▫◊▲◊…)
TR
In problem solving situations, students can produce
tessellations through the use of translations (slides).
They examine, identify, describe, classify, and construct
triangles using the words congruent, acute, and obtuse. They
can’t name, but notice and are able to classify, triangles that are
equilateral, isosceles, and scalene.
Within different polygons, they identify different shapes that
could be used to construct them and the fractional parts they
occupy.
MEASUREMENT
BG
In problem solving situations, students begin estimating and
then measuring with conventional units of measurement (cm, m)
and time (h, min).
They expand their understanding of daily, weekly, monthly, and
yearly cycles through the use of bar graphs, calendars,
rudimentary schedules and time lines.
They solve problems such as:
 Rejean took a walk around the block. He found out that he
could walk 500 metres in 10 minutes. If he walked at the
same speed for 30 minutes, how far would he go?
 Laura had to cut a ribbon into 3 parts. If each part was 20
cm long, how long was the ribbon to begin with?
 Arianna walked 80 m in 10 minutes. How far could she walk
in 5 minutes at the same speed?
 Ted had to divide his paper into four equal parts. If the paper
was 32 cm long, how big was each part?
DV
AP
In problem solving situations and building on the strategies
developed in “BG”, students expand their understanding of the
metric system by measuring in mm, cm, dm, and m. Through
regular exposure and practice, they begin to be able to explain
appropriate situations when each of these units should be used.
TR
In problem solving situations, students estimate, measure, or calculate lengths (including
perimeter) using conventional units. They use the most appropriate unit of measurement
when measuring a length. They are able to demonstrate the relationship between mm, cm,
dm, and m.
In conjunction with developing multiplication strategies, students
begin measuring surface areas through the use of arrays and
unconventional units. They experiment with different
representations of the same “area”. (E.g.. 12 can be 1 x 12, 2 x 6, or
3 x 4). They demonstrate an understanding of perimeter using
unconventional units.
Students estimate, measure, or calculate areas and volumes using unconventional units of
measurement.
As a natural extension of comparing quadrilaterals, the student demonstrates, compares and
explains the differences between right, acute, and obtuse angles.
Students increase their understanding of time to include seconds.
PROBABILITY AND STASTITICS
BG
DV
The students formulate survey questions that involving a yes, no,
or maybe answer. They collect, display, and organize their data
and, based on their findings, come to simple conclusions.
The students simulate simple probability experiments. They
gather data and make observations (e.g. in tossing a die 20 times,
the number 6 came up 3 times). They display this data using a
pictograph, bar graph, or table.
The students interpret the results obtained from a study using a
broken-line graph.
Students expand survey questions to include a wider choice of
responses (What is your favourite breakfast food?) They collect,
organize, and display their data and make simple conclusions that
use the words more, less, or just as likely.
Students demonstrate an understanding of chance by using such
terms as equally probable and less probable.
They represent the data for a study using a broken-line graph.
CULTURAL REFERENCES *One of
AP
TR
Students enumerate the possible results of a probability experiment using
simulations (e.g. tossing two coins: HH, HT, TH, TT).
They gather data (questionnaire, measuring instrument, documentation) and
organize them into a table.
They represent the data with the help of diagrams and interpret the results.
these topics must be covered per cycle.
* Calculator keys: 0 to 9, +, -, x, ÷, ON, OFF, C, AC, CE

Origin and creation of number systems (e.g. Arabic, Babylonian,
Roman, Mayan) characteristics, advantages, disadvantages

Social context (date, price, telephone, address, age, quantity, mass, size)

Origin of the symbols, their need, development, and mathematicians involved
of +, -, >, <, =, x, ÷, ∞ (infinity), ,  , //, 

Origin, development of technology (sticks, strokes, abacus, calculator,
software) limitations, advantages, and disadvantages

Social context of maps

Historical aspects of systems of measurement

Origin of units of measurements, development according to society’s
need of instruments (hourglass, clock), m, dm, cm, mm

Origin, development, need of h, min, s (Writing time 2:10, 2 h 10 min)
Becky Thompson, Educational Consultant
Sir Wilfrid Laurier School Board, 2002
Working Copy

Numbers written using digits

Writing fractions

Writing decimals using a period as a decimal
marker)

$, ¢