Visual Lessons

Sum & difference lab

Find a pair with sum 25 and first − second = 11. Change one value, then check both clues.

$10+15=25;10-15=-5$

Compare each result with its clue.

Carrom number line

Begin at zero. The second movement starts where the first ends. The first arrow is solid; the second is dashed.

$5+\left(-7\right)=-2$

Endpoint: -2. Distance from zero: 2 units. A zero movement has no arrow.

Zero-pair subtraction

Start with the given positive tokens.

$\mathrm{Value}\mathrm{before}\mathrm{removal}=7$

Integer token bag

Start with an empty bag. Predict which tokens will remain.

Continue the product pattern

Move the first factor down one step. Predict the next product before moving again.

1. $5\times \left(-3\right)=-15$
2. $4\times \left(-3\right)=-12$
3. $3\times \left(-3\right)=-9$

For each decrease of 1 in the first factor, the product changes by 3.

The −1 mirror

$1\times \left(-5\right)=-5;\left(-1\right)\times \left(-5\right)=5$

Multiplying by −1 gives the additive inverse. Zero stays zero.

Division: find the missing factor

$\left(-4\right)\times ?=\left(-12\right)$

A signed distributive array

$4\times (2+\left(-3\right))=8+\left(-12\right)=-4$

This array models a positive number of groups. For a negative first factor, add the inverse of each group that many times.

Discover the machine

Source Machine 1: (5, 8, 3) → 10; (−4, −1, −6) → 1; (−3, 10, 2) → 5. Find a rule that fits.

Output: -13