Substitution

Today in grade 10 we worked on solving a system of equations by substitution. We started off with a tug of war puzzle.

4 acrobats vs 5 grandmas are a tie

1 dog vs 2 grandmas and an acrobat is a tie

we need to figure out who will win with 3 grandmas and a dog vs 4 acrobats.

Students worked out a few ways to solve the problem. I wish I had photographed them!

One method had them assign a value for the strength of each grandma and each acrobat. They decided that if a grandma had a strength of 20, then 5 grandmas have a strength of 100, and if they are tied with the 4 acrobats, each acrobat would have a strength of 25. They figured out the strength of the dog by substituting in the values into an equation: dog=2(grandma)+acrobat to calculate that a dog has the strength of 65. Then they could calculate the final situation to see that the side with the grandmas and the dog will win.

Another method was to make equations. The idea of the tie means that both sides are equal, so:

5G=4A

1D=2G+1A

when we get to the final step:

3G+1D, we can replace the dog with (2G+A)

3G+(2G+A)

5G+A

but we know before that 5G=4A so we can replace that too

4A+A

which is 5A, so now we have a final show down with the equivalent of 5A on one side, vs 4A on the other side, so we know that the 5A side (the one that originally had the dog) wins!

We then did some substitution without a context, to practice.

We made the connection to x=-1 and y=-1 being the intersection of the 2 lines using desmos.

Next we tried an interesting case:

this case led to a discussion of what happens when the final line is 2=2 and what does that mean? It means “yes” because it’s true. We are looking for the intersection of 2 lines and math says yes.

these two lines are identical and intersect at all points, so all points on one line are on the other!

We did a lot, and had a fire drill too!