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Counting Students
Description
Written (or spoken)
explanations
Counting students are preoccupied with the number one.
Students can explain their thinking orally or can record their thinking using written
explanations.
e.g. I counted all the objects - 1, 2, 3, 4, 5, 6, 7, 8, 9…
Equations
They can use simple written equations to record their solutions.
8 + 6 =14
14 - 6 = 8
Calculators
Students can use a calculator to explore patterns in counting and problems where
the numbers are too difficult for them to solve mentally.
Diagrams (formal)
Students have two main strategies for solving problems.
i.
number strip
1
ii.
2
3
4
5
6
number line
8
13
Diagrams (informal) Students can use marks or drawings to represent the items being counted
e.g. ||||| ||||
or ℡℡℡℡℡ ℡℡℡℡
1
Additive Students
Description Additive students use primarily adding strategies, though they combine these with
counting and simple multiplication on occasions.
Written (or
Students can explain their thinking using words such as “sum” and “difference” and
spoken)
strategy descriptors such as “place value”, “tidy numbers” and “adding on (reversing)”
explanations
e.g. Frank has $63 and Mary has $49. How much more money does Frank have than
Mary?
“I found the difference between 49 and 63. Forty-nine and 11 is 60 and three more is
sixty-three. That is 14 altogether.”
Equations
Students can record their solutions using a range of equation types such as those below:
24 + 8 = 32
11
31
256
+567
823
63 – 29 = 63 – 30 + 1
= 34
427
- 285
142
4 x 8 → 8 + 8 = 16
16 + 16 = 32
Calculators
Diagrams
(formal)
1
1
4 of 28 → 2 of 28 = 14
1
→ 4 of 28 = 7
Students can use a calculator to solve addition, subtraction, multiplication and division
problems when the numbers are too difficult to solve mentally. They can use their
knowledge of basic facts and place value to estimate answers to check for reasonableness,
e.g. 1203 – 798 = is about 1200 – 800 = 400.
Students use three main types of diagram:
i.
iii.
Dotty arrays
6x5
6x3
+1
33 34
69
Ratio tables
Triangles
Sticks
1
3
2
6
3
9
Students can use diagrams to represent problems so they know what calculations to
perform.
ii.
Diagrams
(informal)
Number Lines
-30
ten
ten
ten
ten
ten
For 25 + 38 = 63
2
Multiplicative Students
Description Multiplicative students choose appropriately between multiplication and addition
strategies.
Written (or
Students can explain their thinking using words such as “factor”, “multiple”, “divisible”
spoken)
and strategy descriptors such as “place value”, “reversing”, and “doubling and halving”.
explanations
e.g. Each container has 24 beans. How many beans are there in 6 containers?
“I changed 6 x 24 to 12 x 12 by doubling and halving. 12 x 12 is 144.”
Equations
Students can record their solutions using a range of equation types such as those below:
20 x 6 = 120 or
4 x 6 = 24
144
24
x6
144
5 6
5 11
3
4 + 8 = 8 + 8 = 8
2
3 x 36 = 24
3.2 – 1.95 = 3.2 – 2.0 + 0.05
= 1.2 + 0.05
= 1.25
24
144
6
or 144 ÷ 6 = 24
Calculators
Students can use a calculator to explore patterns and solve addition, subtraction,
multiplication and division problems where the numbers are too difficult for their mental
strategies. They are able to check the validity of the calculator answer using rounding and
place value understanding of whole numbers and decimals, e.g. 0.8 x 3507 must be
between 1750 (half of 3507) and 3507 (1 x 3507).
Diagrams
(formal)
Students use three main types of diagram:
iii.
Arrays:
20
i.
4
3
6 6 x 20 = 120 6 x 4
= 24
8
x5
x5
15 40
Double number lines
ii.
Ratio tables
Peanuts Total nuts
x5
0
3
15
0
8
40
x5
Diagrams
(informal)
Students can use diagrams to represent problems so they know what calculations to
perform.
T
T
T
T
T
T
T
T
T
T
T
T
3
Proportional Students
Description
Proportional students can use fractions as numbers and operators appropriately.
Written (or
Students can explain their thinking using words such as “common factor” and “common
spoken)
multiple”.
explanations e.g. Joel got 12 out of 19 shots in.
2
12
If Joel had shot 18 then that would be 66.6% because 18 = 3 (12 and 18 have a common
factor of six). If Joel had shot 20 then that would be 60% because12 out of 20 is 60%. So
he shot between 60% and 66.6%.
Equations
Students can record their solutions using a range of equation types such as those below:
Calculators
Diagrams
(formal)
6 3
4 10
2:3 = 40:60 (ratios)
6 = 15 (fractions)
8 = 4 = 0.75 = 75% (converting)
Students can use a calculator to find answers to a variety of problems involving decimals,
fractions and percentages. They calculate with a sense of the possible size of the answer
and express answers in ways that are appropriate to the problem. This includes division
with remainders, e.g. Nathan cleaned 8 cars in 59 minutes. How long did he take per car?
3
59 ÷ 8 = 7.375 = 7 8 ≈ 7 minutes and 23 seconds
Students should also use units of measure correctly in answers to problems, e.g. 3.75 litres
or 24.3 kilograms.
Students use two main types of diagram:
i.
ii.
Diagrams
(informal)
Ratio tables
Dollars Oranges
6
14
?
35
Double number lines
0
6
?
Dollars
0
35
Oranges
14
Students can use diagrams to represent problems so they know what calculations to
perform.
9
3 1
e.g. 4 ÷ 3 = 4 (how many one thirds go into three quarters)
1
1
1
1
1
1
1
1
1
12
12
12
12
12
12
12
12
12
1
1
1
4
4
4
1
1
3
3
4
1
3