Ms. Bearse's Site

Grade 10s experienced some productive struggle today as we attempted to determine the shortest distance between a point (4,5) and a line (y=2/5x-6).  Many could guess approximately where the shortest distance would be…. and a few lightbulb moments happened when we realized it would be perpendicular to the line….and we know perpendicular lines have slopes that are negative reciprocals (inverse négative)…so we made an equation for the perpendicular line….and then we needed to solve for the intersection of the original line, and the new line.  We substituted, and solved….and then….and only then….could we use the distance formula.

It was not easy to wrestle with this at first.  Most groups got half way there, and then we looked at the process as a class to be sure we are all on the same page.  

Today we made some visual patterns using our square tiles.

Here’s an example of 2n+1.  We see the 1 is a constant term, always there (the yellow square).  We are looking for groups of n when we look at the blue.  See them? Horizontally we have 2 groups of n (when n is 1 we have 2 rows of 1, when n is 3 we have 2 rows of 3).  We are using n to represent the figure number.

We experimented with what to do if the constant term is negative….how we can show that.

Our expression is 6-2n.  We know that the constant is the number that isn’t beside the n.  It is not dependent on n.  It does not change…it is CONSTANT.  In this image blue is representing positive, red represents negative and a sandwich of red on top of blue is equivalent to zero.  We start with a constant of 6 blue tiles.  And each figure number will have 2 groups of n that are negative (red) placed on the existing blue tiles.  Figure 1 then is equivalent to 4, figure 2 is equivalent to 2 and figure 3 is 0 and figure 4 is negative 2.

This is not the only way to see this…here’s another way to show negatives.

In this example the constant is -1 which can be interpreted as taking 1 away each time.  This group shows 2 groups of n with one removed (dotted lines)

Each group was given 2 different expressions to build on their graph.  We used the big chart paper because our tiles are the same as the squares.  We lined up all the tiles from each figure vertically, and put a dot with marker on the top left corner of each column.

These dots (when the squares are removed) show the linear trend described in the expression.  In this case it is 3n+4.  The colours are being used to show the constant (yellow) and the growth.  We are seeing 3 groups of n each time.  We also noticed that the pattern looks like stairs, going up by 3 each time.

We posted all of our work up to admire and examine.  We are really good at naming the trends we see in the graphs.  We continued to use our new word “ordonnée à l’origine” (y intercept) and we connected that to the constant in the expression and the pattern.  That is the same as figure zero.  

We looked at positive and negative trends, we looked at parallel lines…those have the same coefficient in the expression….the word we use for that is “taux” (rate)

We also noticed that the bigger the rate, the steeper the line, because the stairs are each higher.

We are now pretty comfortable working with visual patterns and extending our patterning to a graph.  We’ll get to show what we know tomorrow on a quiz.

If you missed class today please give this video 11 minutes of your time.  The message is worth hearing, maybe a few times!!

We are working on developing a growth mindset.

Today we got our first summative back in grade 9.  We are working on error analysis sheets now to dive into our errors, and get to the bottom of things.  We will classify the errors that we make as “inattention”–the “oops” kind of errors where you just don’t think.  “Calculation” where you went wrong with integers, order of operations, or exponents etc. “Precision” where you don’t communicate things clearly, miss phrases/headings/units, or things are a general mess, and “Problem Solving” which would be errors like not having a plan, not knowing where to start, getting stuck and not having a strategy.

We also noticed that some of us had trouble with other test taking skills like time management, or knowing that there was a backside to the page!  We have called these errors “strategy errors” and we’re making note of things like–“I didn’t have a plan and got stuck on a question and couldn’t move on…” or “I wrote WAAAAAY too much for one question and gave maybe more than was needed and ran out of time”… or…. “I didn’t read the question and I calculated area instead of volume, and then I realized and I had to do it all over again”…. or “I used trial and error and it took me forever to do one question…so I ran out of time”

There’s also a column to write the correct solution to the problem.  It’s important to face our errors, and learn from them.  They help us see what things we overlook, or what things we don’t know well enough….YET.

We have been making visual patterns today.  We produced figure 1, figure 2 and figure 3 and are working on identifying expressions to explain the pattern.

We are identifying constants (here in red) that never change.  We are also looking for groups of “n” (eg in figure 3 we are looking for groups of 3).  Sometimes we extended the pattern to show figure 0 and figure 4

Patterns can grow in 2 directions at once

Or sometimes they grow on an angle.

These patterns were all built from the same question (figure 2 was shown and we creatively made patterns that could work).

We used some patterns from visualpatterns.org and analyzed them to find the expression.

What is neat about these patterns is that there are sometimes different ways to see them.

We built the patterns

We looked at how many pieces there’d be in each term…and made some difference tables.

And then we colour coded some areas to correspond to some algebra.

We can see squares and rectangles and a constant of 3.

What we will see later is that the algebra works out to the same thing!

We learned that the balance point, or center of gravity of a triangle is the exact point where the medians intersect.  We are good at finding intersection points, and we are still working on medians, so today we gave it all a try.

For some groups we noticed that there certainly were a lot of steps, and without care, it is easy to get lost!  It’s sometimes hard to find errors at the end of a problem.  We worked through the solution checking each step to see if what we calculated was reasonable.  Our method was a graphical one.

At the end of this exercise we learned that there’s a bit of a shortcut to finding the center of gravity of a triangle….we can average the 3 x values and 3 y values of the corner points, and the result will be the same as if we go through the process of finding medians and then substituting.

Grade 9s wrote their first summative in class today.  We had cake to celebrate all that we’d learned.

We also noticed that our beans have a scare crow now! 

We started class today with a close look at what our portfolio task is.  We read through the prompt together and talked about what evidence would be good to include, and what writing is expected.  We talked about the rubric, and reviewed how to submit work online.  

After that we followed up on a formative we did yesterday.  Yesterday we did 3 questions and indicated where we’d like more practice, and what we could maybe help others with.
Today we got into groups to work on our skills.  Some of us wanted to work on communication, others on using the pythagorean theorem, others on using various formulae for surface area and volume.  Each group had at least one “expert” in that category, and we worked through the solutions to yesterday’s questions, and did a bit of work on the model test for chapter 8.

After some time working, we got up at the boards to show what we know.

We worked together and checked our work for a couple complicated area and volume questions that involved the pythagorean theorem.  

Tomorrow is our “celebration of knowledge” (a.k.a. Summative assessment).  We’re ready!

Spheres are good at any point in the year!

We looked today at the amount of empty space that exists in these packages.

Given minimal information (one side length of the prism) we determined the other dimensions of the prism, and the spheres.  We practiced our volume calculations that we derived yesterday, and we subtracted to find the empty space.

The cups on the desk are our signal that we need help.  If there’s a red cup stacked on top it means “urgent help needed”, if there’s a yellow cup it means “we’re running into trouble, we need some help”, and the green cup on top means “smooth sailing, we’re doing well”.

This will hopefully eliminate some loud attempts to get help.  

After some practice with finding the equation of a line that goes through 2 given points we tackled an application of this process.  We determined where the medians were for the triangle, by eye.  We know a median cuts the area in half.  The way we do that is to join up a corner with the midpoint of the opposite side.  We were pretty good at eyeballing the medians.

We noticed medians all intersect together.  We know we can determine intersections by using substitution and elimination.

We got started on a problem involving numbers.

The fire drill interrupted the flow of things, but we found a numerical way to determine the midpoint of a segment.  The next step (which is for homework) is to determine the equation of the line that goes through point A (3,-5) and M(-4.5,2), the midpoint we calculated.