Sneaking in Factoring

I started a series of new warm ups for my MPM2D class today. My goal is to sneak in factoring as warmups throughout the semester. By the time we need to learn it (like when we need to factor to solve equations) we will have mastered it already. I also previously snuck in multiplying binomials when we tackled quadratic patterns as Mary Bourassa did in her 2D class.

So today I gave them this slide and said I want you to solve a puzzle!

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They broke out their iPads and used the Algebra Tile app to put together the rectangle. The kids worked away and you could see them trying to put tiles in a way to make the rectangle

….and they soon found out that they had to fit a certain way!! 
On take up we made sure everyone had either my rectangle or a rotated version.

Then we did this one…..

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After we were done I asked the class: “If the combination of squares and rectangles makes up the area, what are the dimensions of the rectangle?” They had a little bit of a hard time here, but finally could see the x + 4 and the x + 2 as the length and the width. I then wrote …

 And then I heard some “aaah”s. We had previously seen both versions of the quadratic expressions and discussed why the factored form helped us out quite a bit if we wanted to find the x-intercepts.

We stopped there….It only took us 15 minutes. Tomorrow we will do a few more…..always writing the factored form after. I will also try to get students to notice efficient strategies to make the rectangles.

  • Why did you put 4 x terms along the width and 2 x terms along the length?
  • How does that relate to the number of singles?

Where I hope to go with these warm ups is to factor all types of trinomials:

  • Perfect Squares
    daum_equation_1443575324234

    This time…..make a square

… and get this…

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  • Trinomials of the Type ax^2 +bx + c


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  • Completing the square too!!!!

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This time…make a square




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We’ll be definitely working our way out of the app and onto paper with area diagrams…

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Factoring

 

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Completing the square


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Completing the square

I think working with these puzzles for the next few weeks first will give us a strong base when it’s time to factor to help solve equations and then complete the square. I think I’ll track all the warm ups we do like this and I’ll post them all!

Distance Formula without the Formula

Today in MPM2D our main goal was to discover how to find the distance between two points. But since I’m spiralling the 2D course I wanted to think big picture…I  wanted to tackle this overall expectation: verify geometric properties of triangles using analytic geometry.

We started with this beauty from Would You Rather  — www.wyrmath.wordpress.com

Students argued and discussed which ramp they would rather push that crate up. Most of the class picked A with their reason being it’s less steep and less work. One of the students who picked B said “I want muscles…..so I’m going to push that crate up the steepest slope“. Another student picked B because they wanted less distance and wanted to “get it over with“.

I left the discussion hanging here knowing I was going to come back and revisit this with more ammunition.

I showed them this video

and we completed the Corner to Corner problem (see the lesson plan here) to remind ourselves of the Pythagorean Theorem.

We came back to the Would You Rather problem from above and practiced finding the length of each hypotenuse to see how long each was.

I then presented them with this……and said our goal was to find the length of this line segment.

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Find the length of this line segment

I asked…”If I could help you out or provide you with more info what would you want?” Most students said they would want either a ruler or some sort of dimensions or units to look at.

So I  brought up the grid on Desmos and asked if this was enough.

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Most students thought it was…..I could see them drawing right triangles on their whiteboards and filling in the lengths of the legs. But one students yelled out “What is the scale?” ….Everyone paused! ….. I brought up the axis!

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Students finished drawing their right triangles and said that was easy! We did one more just like this (giving them the grid and axis) to practice.

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Here’s the next challenge: I took away the grid but gave them the coordinates of the endpoints. Find the length of this line segment.

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I let them struggle a bit here. The majority of the class prevailed and had a similar solution on take up:

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Student words: “One leg was the difference between the x-values and the other leg was the difference between the y-values”

We did another in the same format to practice this discovery.

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Then I took it up a notch…

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The three points shown represent vertices of a triangle. Classify the type of triangle.

And I saw a lot of this…

 

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I’ve been following Mary Bourassa’s Blog and I stole creating my own homework sets from her….so I left the class to complete this. Love how I can ask lagging questions in my homework. Students get multiple opportunities to master skills.

So we’ll take up those questions tomorrow and we’ll summarize the strategy to find the length of a line segment using this formula…daum_equation_1443477587316

Access: Pre-made Desmos graphs

 

Lollipop Lollipop oh la la Lollipop! — & Rates of Change

Last year on twitter I saw that Alex Overwijk and Janice Bernstein with their grade 12 advanced functions classes did this lollipop activity!

I knew that I wanted to give this a try for this semester! What I especially love about this activity other than students experiencing rates of change is that this is an activity that can span multi-grades!

Here is what we did,

Generating Curiosity

I found this video on YouTube and asked the class to think of great questions we could ask about what we see!

FullSizeRender-1Great questions from the kids and we all agreed to look at

  • How does the sucking time affect the radius, circumference, volume, and surface area?
  • How long will it take until the lollipop is all gone?

Let’s investigate those relationships starting with the easy to measure (circumference) and also estimate how long it will take until the lollipop is no more!

We had guesses : ranging from 10 minutes through to 35 minutes.

Gathering Data

I handed out one lollipop per pair of students, along with some dental floss for measuring circumference. We set our timer for 30 seconds and began sucking and capturing data!
We recorded the circumference every 30 seconds up to 7 minutes like Al’s and Janice’s instruct in their lesson Plan.
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They also have a great handout for tracking the circumference over the 30 second intervals. Screen Shot 2015-09-18 at 2.22.08 PM

Analyzing the Data

So we first looked at the Time vs. Circumference and Time vs. Radius relationship
Linear - Lollipop

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We discussed its linearity and why. Students predicted with more accuracy when their lollipop would run out.
Up to this point this task is great for grades 7, 8, 9, or 10!! (Just edit the file to exclude the average and instantaneous rates of change).

  • Grade 7 & 8: Practice plotting points and reading/interpreting graphs.
  • Grade 9 & 10: Find lines of best fit and first differences.

We found the average rate of change for each 30 second interval and discussed what this meant. We used the last column to talk about narrowing the interval down to estimate the instantaneous rate of change, and noticed that it’s about the same for all values. Why does this make sense???

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We moved on to looking at Time vs. Volume and Time vs. Surface Area

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Great talks around how Volume and Surface aren’t deceasing at a constant rate! It changes! Students can see these changes and see in their tables where the volume is changing the fastest.

Overall a great intro activity to get students thinking about narrowing intervals to approximate instantaneous rates of change.

Next up: We’ll relate what we did here with the tables to the graphical interpretation of rates of change (secant and tangent lines) and then on to the algebraic!

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Speedy Squares

Last week I attended the annual OAME (Ontario Association of Mathematics Educators) in Toronto. It was so great to finally meet some of the people I’ve been tweeting with.

I was pumped to attend Mary Bourassa’s double session on great classroom activities. One of the activities that I’ve seen on her blog, but not used in my own classroom was Speedy Squares. So when I had an opportunity to try it, I jumped on it!

There is something special about doing the lessons yourself while learning about a lesson at a conference.

Read about the lesson:

You can read about the lesson on her blog here part 1 and here part 2.

The big question: We want to determine how long it will take to build a 26 x 26 square out of link cubes.

More Curious

While actively building the squares I had a great idea to make the introduction to the activity a little more curious! So when I got back to my classroom I broke out the cubes and created this….

Maybe before the time trials of building the squares, we can dive into generating questions and wonderings first.

  • What is he making?
  • How many squares will he use?
  • How long will it take?

Now that we have generated questions….we can then move onto Mary’s awesome two day lesson.

Once students have got an answer to how long they would take to build the 26 x 26 square, you could show the video of me building it!

I’m really interested to see if elementary teachers can use this in their classes and what they come up with!