## Solving tricky triangle problems

Grade 9s are working on solving some challenging problems.

We are asked to prove whether triangle PQR is a right triangle or not.

We looked at calculating missing sides with the pythagorean theorem.

Some of us were thrown off by the potential hypotenuse being the base of the triangle, or by the fact that we were having to prove something instead of solving something.

We needed to use the pythagorean theorem again to show that 10^2+8.2^2=11^2 was not true, so it is not a right angle triangle.

The pythagorean theorem is only true for right triangles, so if the pythagorean theorem doesn’t work, we can assume the triangle is not a right triangle.

## So many strategies!

We are working on multiplying, and using many strategies to solve problems.

We looked at 6×8 today, which we can break into 5×8 and add that to 1×8, or also by using subtraction find 8×8 and subtract 2×8.

We can also double the 8 and halve the 6 and multiply 3×16 to get 48.

Another way is to halve both the 6 and the 8 and then multiply our resultant by 4 (by 2 twice).

We can decompose the numbers into factors and multiply them in any order.

We can write down 1+1 so many times! Actually, we made it into an array so we could keep track. The array was 6 by 8. We can divide this array up into rectangles with friendly numbers and then add up the rectangles. Here it would be 6×4 and another 6×4 each one is 24 and 24+24 is 48.

Another really neat observation was that when the numbers hat we multiply are different by 2, we can say that the answer will be one less than the middle number’s square.

E.g. since 8-6=2 we know the answer to 6×8 is 7×7-1. We tried it with smaller models, and showed how it works with a visual model.

## Area and perimeter

Today grade 9s got their formula sheet which can be used on all tests, quizzes and the eqao test. We practiced how to calculate area and perimeter of composite shapes.

we looked at the different ways to break up a composite shape to calculate the area. We also looked at how to calculate the length of missing sides. Sometime we’ll need to use the pythagorean theorem to help us.we learned the area of a trapezoid song, and sang it in a round!we looked at circles, and circumference and area calculations as well.and finally we looked at how to determine the perimeter of a shape when we know the area. In this case we needed to divide the area by a side length to find the missing side length. We worked with fractions and saw how to divide, multiply, and add fractions to solve.

Here’s the trick for dividing fractions!

we know that the second fraction in the division question is named the “divisor”, so we can flip the divisor, or flip da-visor…and then we multiply.

## Systems of equations

Grade 10s are working on solving systems of equations. We’ve looked at using substitution and elimination to solve problems within a context. Today we looked at equations out of context, and we made the connection that the solution that is the end result of elimination or substitution represents the intersection of the two linear equations.

Most linear equations will intersect once, and that point can be found by substitution, elimination or graphing (desmos makes it really quick to find).

Some systems will not be easy to solve. Some will end up with an equation like 0=0 at the end. Others will end up with something like 0=5.

In the case of the first, 0=0, we know that is true, so it’s like the math says…yes. As in every point is a solution. When two lines are identical we have that case.

When we see something like 0=5 that is not true, so it’s like the math says…no. As in there is no solution. This means the lines are parallel and do not intersect.

## Pythagorean theorem in grade 9

We looked today at a quilt, and made a list of math questions we could solve about the quilt. We calculated area, volume, and perimeter, looked at the number of squares and number of triangles, we talked about types of triangles, and angles. We converted measurements from imperial to metric, and then worked together to figure out how long the diagonal would be.

We looked at pythagorean theorem, and watched the video that shows the area method of solving.

We can use this method to help us solve for any missing side, the hypotenuse or the legs (cathètes).

We check that our answer makes sense…the hypotenuse should always be the longest side.

## Planting beans in grade 9

We are starting an experiment in grade 9. We are germinating beans and will be planting them after they sprout, and measuring their growth over a few weeks.we wrapped the beans in paper towel and dunked the packet in water, and put it in baggies to germinate.we put our names on the baggies and left them on the windowsill.

## Friday in grade 10

We explored the 3act task to find out which rectangle had the biggest area. Many of us thought that the biggest area would be a square. We didn’t notice that the rectangle didn’t always have the same perimeter, so this case was very different to our experience from grade 9.

We made tables and looked for patterns. We know how to calculate area (length times width), and we noticed how the length and width were changing.Some groups noticed that the area has a constant 2nd difference, so we know it’s not linear.some of our groups started to use variables, and develop expressions.

We will come back to these explorations in the future. Good work this week grade 10s. There will be a practice sheet posted to the google classroom, along with the quiz solutions.