Quacky questions

These are level 2 number, algebra, and geometry problems from the Figure It Out series. A PDF of the student activity is included.

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About this resource

Figure It Out is a series of 80 books published between 1999 and 2009 to support teaching and learning in New Zealand classrooms.

This resource provides the teachers’ notes and answers for one activity from the Figure It Out series. A printable PDF of the student activity can be downloaded from the materials that come with this resource.

Specific learning outcomes:

Find rectangles within a shape (Problem 1).
Solve problems using additive strategies (Problem 2).
Continue a spatial pattern (Problem 3).

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Quacky questions

Achievement objectives

GM2-3: Sort objects by their spatial features, with justification.

NA2-1: Use simple additive strategies with whole numbers and fractions.

NA2-8: Find rules for the next member in a sequential pattern.

Required materials

Figure It Out, Levels 2–3, Problem Solving, "Quacky questions", page 14

See Materials that come with this resource to download:

Quacky questions activity (.pdf)

Activity

Problem 1

This is a similar problem to Problem 1 on page 5. It can be solved using the same strategies.

A square with with two lines intersecting it horizontally and vertically. The intersection are labelled from left to right, top to bottom: A, B, C, D, E, F, G, H, I.

Students can label the corners and make an organised list.

ABED
ABHG
ACFD
ACIG
BCFE
BCIH
DEHG
DFIG
EFIG

Similarly, they might classify the rectangles as made up of one, two, or four smaller rectangles:

One-unit rectangles	Two-unit rectangles	Four-unit rectangles	Total
4	4	1	9

Extend the problem by adding another column or row to the figure.

Problem 2

Knowing that, in general, ducks have two legs and sheep have four legs is important. Students may use a variety of strategies.

Draw a table:

Ducks	Sheep	Legs
8	0	16
7	1	18
6	2	20
5	3	22
4	4	24

A diagram of 8 ducks and another diagram below of 4 sheep and 4 ducks. The text instructs to make every animal a duck, then change them to a sheep one by one until there are 24 legs altogether.

Change the conditions of the problem to see whether students have generalised the process. For example:

There are nine animals and 30 legs. How many ducks are there?

Problem 3

Making a table or writing an equation are useful strategies.

Letter	1st	2nd	3rd	4th	5th
Number of tiles	7	10	13	16	19

7 + 3 + 3 + 3 + 3 = 19

Students should recognise that three tiles are added each time to increase the letter size. This can be highlighted by building the letters on an overhead projector.

Students might solve other letter-building sequence problems, such as:

Three versions of the letter H are made of a series of squares. They increase in size by adding more squares to the arrangement.

Problem 4

Students may solve the first two cutting problems randomly, but they will need to be systematic with the pizzas containing six and seven olives:

Three pizzas with lines on them indicating the cuts that are made in order to evenly distribute the olives between the slices.

For example, with the seven-olive pizza:

Make the first cut so that four olives are on one side and three olives on the other.

A pizza with a line on it indicating a slice that separates three olives from the other four.

Make the second cut so that the three olives are divided into one and two and the four olives are divided into two and two.

A pizza with two lines through it indicating cuts, which separate 1 olive from the rest, and three slices each with 2 olives.

This leaves three sections. The three olives are divided with two olives in each, which need to be divided by the third cut:

A pizza with three lines on it indicating slices, with a single olive on each slice.

As an extension, ask the students to draw a pizza with eight olives and to make three cuts that leave each olive in a piece by itself. This is impossible and should lead students to realise that seven olives is the maximum number possible with three cuts.

Answers

9 rectangles

4 ducks

Pizzas a and b: various answers, for example:

2 pizzas with slices separating the olives, so there is only 1 per slice.

Pizza c:

A pizza with three lines separating the olives so that there is a single olive per slice.

Pizza d: 3 steps:

Three pizzas with an increasing number of lines on them, which eventually separate the olives into an olive per slice.

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