Double and halve
This is a level 3 number activity from the Figure It Out series. It is focused on using doubling and halving strategies. A PDF of the student activity is included.
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:
- Use doubling and halving strategies.
Double and halve
Achievement objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
Required materials
- Figure It Out, Link, Number Sense, Book One, "Double and halve", page 11
See Materials that come with this resource to download:
- Double and halve activity (.pdf)
Activity
The strategy of doubling and halving can be used to show the connection between division and multiplication by a basic fraction, such as a half or a quarter.
In this activity, the strategy of halving also shows the connection between the common fraction of one-half and its decimal equivalent of 0.5 because this will be the result of halving any whole number until a number with a decimal point is reached.
The interest in this activity is in the number of steps it takes to get to the number with a decimal point. Some students may assume that larger numbers will require more steps, so be sure to compare the number of steps it takes when we start from 2 000 (five steps) with the number of steps it takes from 2 500 (three steps).
Encourage the students to explore the number of steps it takes to reach a number with a decimal point for numbers that end in the same quantity, for example, 7 500, 6 500, 3 500, and 500. This way, they will see that the number of steps depends on the first significant figure (that is, the first non-zero figure, in this case, the 5). All the other significant figures have no influence.
The students could then investigate putting any odd number in the ones place, then in the tens place, and then in the hundreds place. They will find that it is the place the odd number is in that decides how many steps it takes to reach a decimal fraction.
An investigation into even numbers is quite a different matter because for any even number, you first have to halve the number until you produce a number where the first significant figure is odd and then you proceed with the steps to a decimal fraction.
By comparing the number of steps for 1 600 and 50, we can see the effect:
Steps for 1 600 |
Steps for 50 |
|---|---|
800 |
25 |
400 |
12.5 two steps |
200 |
|
100 |
|
50 |
|
25 |
|
12.5 steven steps |
|
Here the process is reversed so that the students can see the effect of doubling. Note that, in the first question, Jarod will not reach exactly 1 000 000 by doubling from 1 000, but it will take him 10 steps to go past that number.
A good context that many students will relate to is the way the size of the memory on computers usually grows in doubles. A chart such as the one below may be helpful. As an extension, the megabyte increases could be shown as powers of 2.
1 megabyte = 21
2 megabytes = 22
4 megabytes = 23
8 megabytes = 24
16 megabytes = 25
32 megabytes = 26
64 megabytes = 27
128 megabytes = 28
256 megabytes = 29
512 megabytes = 210
Activity 1
1.
a.
2 000
1 000
500
250
125
62.5
b.
2 500
1 250
625
312.5
c.
32
16
8
4
2
1
0.5
2.
Numbers will vary. They are all odd numbers.
3.
Numbers will vary. They all end up as an odd number after the second halving.
Activity 2
1.
10 times. (On the tenth doubling, he got 1 024 000.)
2.
9 days (counting the $40 as 1 day.)
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