Lesson 126 of Doodles Do Algebra

Today we teach your child about the concept of a negative solution to a problem. What does it mean for a number to be negative in real-life situations?
As DoodleCat explains on the worksheet, if you find the solution to an unknown is a negative number, say -5, that means you have a deficit of 5 or you owe 5, or you need 5 of something to get back to the start.
If your child has a hard time internalizing the concept of negative, you can pull out an ice cube tray. This works best if there are ice cubes in the tray to start with. Point out to her that there are 12 (that is the size of our ice cube trays) cubes of ice in a full tray. So one tray equals “plus, or positive, twelve” ice cubes. Now dump the tray out onto the kitchen table. Depending on her age, let her play with the ice a bit (my daughter always wants to eat the ice…but that is a totally different issue). After a bit, ask her how many ice cubes are in the tray. She will probably answer, “none.” Now point out that actually there are spots for 12 ice cubes which are empty in the tray so you could say that there are negative twelve ice cubes in the tray. This is especially true since you just took the ice cubes out of the tray while you were sitting there. You can explain to her that if you were both aliens from another planet who had never seen an ice cube tray before and if you found the tray empty, you would have no context for understanding that there were negative 12 ice cubes in the tray. In that way, many word problems and much of applied math and physics and chemistry is actually dependent on a shared culture and language. It is one of those questions that leads to much discussion if you have teenagers about the invalidity of relativistic ideals and morals, but if you have younger children, it is probably best to just stick with the ice cubes.
Once your child gets the idea, you can move her through the problems on the worksheet, or let her do them by herself.

1. The equation that comes out of the statement of the problem is $x+8=5$ and the answer is $x=-3$
the first equation is $45-x=15$ so dad was 20 when son was born, or you can rewrite it as $s+20=d$
the second equation is that at some point in time, the son will be 1/4 of dad’s age, or $s=d*1/4$
$d*1/4+20=d$ or $d*3/4=20$ or $d=80/3=26 2/3$ So if Dad was 26 and 2/3 years old when his son was 1/4 as old as he was, then that occurred 6 and 2/3 years after his son was born, since dad was 20 when son was born. That means that it happened 15-6 2/3, or 9 1/3 years ago since son is currently 15, or the answer is that son will be 1/4 as old as dad in -9 1/3 years (recall the empty ice tray).