Friday, January 24, 2020

Multiplying Negatives

"A negative times a negative is a positive". It's a difficult 1 to explain. We all learnt it at schoolhouse in addition to practised it to the holler for of fluency, but it's non until we're asked why it plant that nosotros halt in addition to intend almost it.

Numbers lines in addition to visualisations are real helpful when didactics the improver in addition to subtraction of negative numbers. But amongst multiplication in addition to sectionalization it's non in addition to thus clear.

Let's await at a few approaches in addition to resources.

1 . Pattern Spotting
Draw a touchstone multiplication tabular array in addition to extend it backwards to include negative numbers. It's a straightforward designing that all students should hold out able to location in addition to continue. Get students to produce this using Colin Foster's activity on page v of his Negative Numbers chapter.
2. Multiplication Grids
Take 2 2-digit numbers in addition to multiply them together using grid multiplication. For simplicity, let's lead keep 12 x 11:
Here nosotros lead keep written 12 every bit 10 + 2 in addition to eleven every bit 10 + 1. But it would operate only every bit good if nosotros expressed those numbers differently. Instead, let's write 12 every bit fifteen - iii in addition to eleven every bit fifteen - 4. We should acquire the same answer:
This solely plant if -3 x -4 = 12. 

Note that this explanation requires students to commencement sympathize that positive x negative = negative. This is relatively straightforward to explicate inwards damage of repeated addition. 

3. Proof
Here's a proof that is clear in addition to accessible to us experienced mathematicians. I'm non certain how accessible it is to Year seven students, but it's worth a go.
a in addition to b are positive
a + (-a) = 0 
[a +(-a)]•b = 0•b 
a•b + (-a)•b = 0 
a•b is positive. Therefore (-a)•b is negative 

b + (-b) = 0 
(-a)•[b + (-b)] = (-a)•0
(-a)•b + (-a)•(-b) = 0
Since (-a)•b is negative, nosotros conclude that (-a)•(-b) is positive.

Perhaps start amongst a numerical instance instead of a formal proof.
3 + (-3) = 0
Multiply everything past times -4
3(-4) + (-3)(-4) = 0(-4) 
 -12 + (-3)(-4) = 0 
 (-3)(-4) must equal 12 to brand this tilt true. 

Further Reading
It's a skillful persuasion to read almost a theme earlier you lot learn it, fifty-fifty relatively unproblematic topics that you've taught many times before. Here are around helpful links:

Colin Foster suggests that you lot inquire students to brand upwards 10 multiplications in addition to 10 divisions each giving an respond of –8 (eg –2 × –2 × –2 or –1 × 8 etc).

The squaring in addition to cubing (etc) of negatives is worth discussing - students should location that an fifty-fifty ability gives a positive value (eg what is the value of (-1)100?).

It may hold out worth exploring estimator behavior likewise (ie around calculators require brackets when squaring a negative). It's of import that students know how to role their estimator properly. There's a slap-up resources from MathsPad for this - Using a Calculator: Odd One Out.

This theme is revisited inwards afterwards years when students are practising substitution. For example, if a = 3, b = -2 in addition to c = -5, abide by the values of: abc; bc2; (bc)2; a2b3 in addition to and thus on. This Substitution Puzzle from mathsteaching.wordpress.com gets quite challenging.

Do allow me know if you lot role an interesting method or resources for didactics the multiplication of negative numbers.


"Minus times minus results inwards a plus,
The argue for this, nosotros needn't discuss"
- Ogden Nash








No comments:

Post a Comment