# Posing Questions and Making Conjectures - Lower Secondary

Posing Questions and Making Conjectures is part of our Thinking Mathematically collection.

### Summing Consecutive Numbers

##### Stage: 3 Challenge Level:

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### Pick's Theorem

##### Stage: 3 Challenge Level:

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

### Producing an Integer

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 46 - 2011
Multiply a sequence of n terms together. Can you work out when this product is equal to an integer?

### Pair Products

##### Stage: 4 Challenge Level:

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

### Tilted Squares

##### Stage: 3 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

### Squarely in the Middle

##### Stage: 3 Short Challenge Level:

Weekly Problem 20 - 2013
Can you calculate the answer to a large sum?

### Odds and Evens

##### Stage: 3 and 4 Challenge Level:

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

### Interactive Spinners

##### Stage: 3 and 4 Challenge Level:

This interactivity invites you to make conjectures and explore probabilities of outcomes related to two independent events.

### White Box

##### Stage: 2, 3, 4 and 5 Challenge Level:

This game challenges you to locate hidden triangles in The White Box by firing rays and observing where the rays exit the Box.

### Difference Dynamics

##### Stage: 4 and 5 Challenge Level:

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

### What Numbers Can We Make?

##### Stage: 3 Challenge Level:

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

### Magic Letters

##### Stage: 3 Challenge Level:

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

### Supercomputer

##### Stage: 4 Short Challenge Level:

Weekly Problem 28 - 2014
What is the units digit of the given expression?

### Little Difference

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 30 - 2014
What is the value of $2006 \times 2008 - 2007 \times 2007$?

### Eulerian

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 37 - 2014
Which of the five diagrams below could be drawn without taking the pen off the page and without drawing along a line already drawn?

### Last-but-one

##### Stage: 3 Short Challenge Level:

Weekly Problem 1 - 2016
What is the last-but-one digit of 99! ?

### Ones, Twos and Threes

##### Stage: 3 Short Challenge Level:

Weekly Problem 5 - 2017
Each digit of a positive integer is 1, 2 or 3, and each of these occurs at least twice. What is the smallest such integer that is not divisible by 2 or 3?

### Leaning Over

##### Stage: 3 and 4 Short Challenge Level:

Weekly Problem 31 - 2017
The triangle HIJ has the same area as the square FGHI. What is the distance from J to the line extended through F and G?