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.
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.
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:
- The FAQ pages on Math Forum ever render interesting answers: Why is a negative times a negative a positive?
- I taste James Tanton's curriculum essays: Directed Number Gaps' activity - it plant good amongst whatsoever twelvemonth group
- MathsPad has a slap-up hit of negative release puzzles including arithmagons (some of these resources are solely available to subscribers)
- CIMT's negative numbers chapter has activities for practising multiplying negatives.
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
