Level 3 Teacher Notes and Solutions
Problem Title
1 Basketball
Teams
Concept
Factors &
Multiples
Teacher Notes
Students need to partition into the required multiples and recall the
factors to answer the problem.
Baskets of
Apples
Proportional
Reasoning
Buses
Multiplicative
Thinking
Addition &
Difference
Students need to select a reference to which all the others can be
proportional compared. The most obvious reference is to use the
number 1 (1kg)
The context of the problem is as important as the numbers will
create halves and having half people and half buses is not real!
Drawing a diagram of the problem assists students to comprehend
the problem and realise what needs to be done with the numbers in
the problems.
This is a two-step problem so again problem comprehension is
essential. Reliance on keyword approach to problem comprehension
is insufficient.
(For further information on mathematical literacy see the chapter
on Developing Mathematical Literacy in the Teacher Handbook
Fractions Decimals & Percentages available from the online store
www.thewilkieway.co.nz )
Paying attention to the unit part of each number is essential.
Students need to realise the words are as important as the number.
In the context of this problem a week is 5 days not 7 days.
Students should be using recall of multiplication facts once they
have worked out what factors they need to use.
In understanding the size of the fraction depends on the size of the
whole, the reverse is if the whole changes and the result of the
fraction remains the same then the fraction must have changed.
This is called re-unitising. The problem also calls for use of
equivalent fractions in simplifying fractions.
Car breakdown
Jason & Tom’s
Earnings
Addition &
Subtraction
Packets of
Chippies
Sam and Ken
Cycling
Four
Operations
Multiplication
Steeds Family
Fractions
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Solution
2 netball teams and 12 basketball
teams
8 netball teams and 5 basketball
teams
1kg for $1.50
48 people
15 buses
432km
Jason has $299
Tom has $267
68 packets left
Ken cycles 70km per week
Sam cycles 50km per week
9 people live in Springfield Road
18 people live in Herbert Street
8 people live in Sala Street
1 person lives in Lynford Avenue
9/40 of the family live in Springfield
9/20 of the family live in Herbert
1/5 of the family live in Sala
1/40 of the family live in Lynford
1/10 of the family live in Anne
Street
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Three
daughters
Creating
linear
equations
Three Sons
Creating
linear
equations
Bike Ride 2
Proportional
Thinking
Proportional
Thinking
Making Cookies
While the numbers in the problem are small enough for students to
answer the problem through trial and error this is an algebra
problem and students should be taught how to use the number
relationships to form a simple equation.
The common element in this problem is Emily (E)
Jane = E + 6 Kim = E + 3
E + E + 6 + E + 3 = 30 (The total of their ages)
3E + 9 = 30
An understanding of = as a symbol of equality and using the
inverse properties of addition and subtraction brings us to: 3E = 21
Using the inverse properties of multiplication and division
E=7
As for the previous problem, the common element is Pete so the
ages of Sam and Jack can be described in terms of Pete.
P + P + 4 + P + 2 = 24
3P + 6 = 24
3P = 18
P=6
Finding common factors and looking for a relationship between
factors of the same unit is essential to thinking proportionally. A
guide to solving proportional problems is available under Wilkie
Way Guides on the Professional Learning page of the subscription
area of the website. www.thewilkieway.co.nz
Understanding the relationship between multiplication, division and
fractions is essential to building a deeper conceptual understanding
of fractions. This problem is an opportunities to explore and explain
the relationships.
Emily is 7 years old
Jane is 13 years old
Kim is 10 years old
3 4 11
3 5 10
369
378
$4.50
Ratu’s Share
Fractions &
Proportions
Jack’s
Pumpkins
Working
systematically
The constraints of the problem mean there are is limited number of
possible combinations. Being systematic enables students to ensure
they have all possible combinations
Buying
Pumpkins
Proportional
Thinking
Students can either find the cost of a whole pumpkin and then
divide by four or divide the weight of the pumpkin by four and
multiply by the cost per kg. Again problem comprehension is
essential.
©2017 NCWilkinsons All rights reserved
Pete is 6 years old
Sam is 10 years old
Jack is 8 years old
54km
750g
Nephews received $95 each
Nieces received $95
2/3 = 4/6 and 1/3 = 2/6 so they
each received 1/6 of the total
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459
468
569
Kristina’s
Lollies
Multiplicative
Thinking
©2017 NCWilkinsons All rights reserved
Reading and understanding the problem is required to find the
numbers to calculate with. You cannot solve the problem direct
from the numbers in the problem.
$17.50
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