This unit geometrically and numerically explores the fact that squaring and square roots are inverses. Gauss’ method of determining square roots when only squares are available is developed. Finally a powerful method of calculating square roots that quickly produces answers to any desired accuracy is shown.
This unit deals with the geometrical measuring and numerical calculation of square and cube roots, and with methods of calculating them without a scientific calculator.
Squaring a whole number gives the area of a square with that length of side. The inverse, finding the square root, gives the side length of a square with given area.
For example, 82 = 8 x 8 = 64 is the area of a square with sides of eight. √64 = 8 gives the side length of a square with area of 64 square units.
Cubing and finding the cube root are the three dimensional equivalent of this. Cubing a whole number gives the volume of a cube with that length of edge. The inverse, finding the cube root, gives the edge length of a cube with given volume.
For example, 43 = 4 x 4 x 4 = 64 is the volume of a cube with edges of four. ∛64 = 4 gives the edge length of a cube with volume of 64 cubic units.
The learning opportunities in this unit can be differentiated by providing or removing support to students, and by varying the task requirements. Ways to differentiate include:
This unit is focussed on exploring squared and cubed numbers, and their roots, and as such is not set in a real world context. You may wish to explore real world applications of squared and cubed numbers at the conclusion of the unit, for example, around dimensions of space (e.g. the area or volume of your classroom).
Te reo Māori kupu such as pūtakerua (square root), pūtaketoru (cube root), tau pūrua and (square number) could be introduced in this unit and used throughout other mathematical learning.
Session 1
Session 2
Present students with a table of squares like the one below.
Ask: How might we use the table to estimate the square root of 38?
Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Number | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
√53
√91
√41
√77
√81.4
√10.6
x | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 |
y=x2 | 0 | 0.01 | 0.04 | 0.09 | 0.16 | 0.25 | 0.36 | 0.49 | 0.64 | 0.81 | 1 |
Session 3
Extend the Gaussian algorithm to finding cube roots to a desired accuracy. Discuss how this table helps show that 4 < 3√110 < 5:
Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Number3 | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1000 |
Session 4
Dear families and whānau,
We have been exploring different method for calculating square roots. Ask your child to share their learning with you.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/square-and-cube-roots at 8:37pm on the 26th February 2024