# Updating the MFM1P Spiral

“Have you taught for 25 years? Or have you taught one year 25 times?”

I don’t think I’ve taught the same course the same way ever. Why would we? We don’t have the same kids in front of us. And especially with the resources at our finger tips from our colleagues inside and outside of our schools. I’ve wrote before about the power of #mtbos and it changes the way you teach.

I started spiralling the MFM1P course a few years ago with Kyle Pearce. Since then I’ve taught that course 3 or 4 semesters in row…..and never the same way. New amazing lessons and tools are springing up. For past lessons I wasn’t completely happy with I’ve got to see if this new lesson or that lesson will help my students understand the concepts more deeply.

One change I wanted to make was to include solving equations earlier in the course. In my old plan I waited to introduce it after introducing linear relations. But, after teaching solving equations using the Double Clothesline and the puzzle nature of learning it that way….I can introduce it now and continually practice our skills through warm ups.

If you want to follow along as my day-to-day plan unfolds follow this link! If any of you have been spiralling MFM1P I would love compare notes, or see your plans.

# Double Clothesline – Solving Equations

I have always taught solving 2-step linear equations by starting with a balance scale. Having students whittle their way down to see how many marbles were in each bag was always a win for me…..in most cases.

I valued this approach. It’s easy to visualize and it strengthens the “whatever you do to one side of an equation you do to the other” mantra we tell students when solving . However, I’ve always been left wanting more especially when we introduce solving equations with negative coefficients or even when the solution is a negative value. The balance scale kinda loses it’s effectiveness.

Using algebra tiles help fill this hole. And now…. thanks to Andrew Stadel, double clotheslines.

I was lucky enough to attend Andrew’s NCTM Annual session on Error Analysis this year. In his session he demonstrated how to use a double clothesline to solve equations. I later found this resource on his site. Watch his videos on how to use the clotheslines….they helped me piece this lesson together. Stop now and go and watch Andrew’s video on solving two step equations.

I stared as Andrew did at the NCTM session:

I put 0 on the top line and 0x on the bottom line.

I then held up the 3x card and asked where should this go? I asked if it should go on the left or the right of zero. The students overwhelming said it needed to go on the right. “3x is more than x, so it should go more to the right, just like a number line” (Always — Sometimes — Never was going through my head at this moment but i’ll wait to talk about this with the kids until a bit later in the lesson).

I then said “I’m going to place this 15 right above the 3x and that means equivalence. 3x is the same as 15”

Where should 9x go? You could see the some students spacing out where 9x should go. This is what I love about this method. It’s so visual and we’re forced to always think about how terms relate to each other.

I want to know what number should be above 9x. I had them draw the number lines on their desks and let them work on determining the value of 9x.

Going around the room there were a few different types of solutions. Some students said, “3 times 3x is 9x, so 3 times 15 is 45”

Some students said, “If 3x is 15 one x is 5, so 9x is 45.” Nice. We ensured the whole class understood both of these types.

Next puzzle: I asked where to place 3x + 4…then assigned it the value of 16.

Where should 3x be placed? It was easy to see that 3x is less than 3x + 4 so it should go to the left. Now for the amazing moment! What should be the number above?

from the class an overwhelmingly 12 was shouted. So now what must be the value of x?

Student: “The dividing is the easy part” We spent a few minutes here talking about why dividing 12 by 3 here makes sense.

Next Puzzle:

Where should the 5x go? At first some students had some difficulty deciding if it should go to the left or right of 5x – 2.

Once we settled to the right. They jumped to finishing it off to determine x.

Next Puzzle:

Where 3x should go was a discussion. We all agreed it should be 14 down…..and where would that be? This is where the clothesline (number line) feels superior and the balance scale visual falls short. We can use the bi direction of the number line to continue working with negative values.

What was awesome during this class was this wasn’t a big deal….the number lines seems natural!!

Also watch Andrew’s example with negative coefficients.

I had students practice solving a variety of equations by drawing the cards on their handout.

They finally demonstrated their understanding by creating their own equation where x had to equal 4. They put their creations up around the room for the group to solve.

I feel that the number line (clothesline) method builds a lot of great number sense. We get to reinforce our inverse operations as we build from conceptual understanding to abstract. Students’ strengthen their understanding of algebraic expressions and how those expressions relate to others.

I’m now going to investigate how to to demonstrate solving multi-step equations…. 3x + 5 = 2x + 7 using the clothesline. I’m thinking this might be a difficult task. Any ideas????

[UPDATE] – Solving equations with expression on both sides.

Since this lesson my class used the double number line to solve equations like 4x + 10 = 6x + 2. It was great to keep some continuity here while we solved harder equations.

We placed each side of the equation on separate clotheslines just like before.

We didn’t want to re-invent a new strategy….we were great at solving equations when one line was used for numbers and the other for expressions…..so we wanted that. How can we get one line to be just numbers and one to have the expression? We subtracted 4x from both lines.  Which left us exactly where we were last class!!

and then we subtracted 2 from both to isolate the “x-term”

Finally dividing by 2

Boom!

This will be our method too to solve a system of equations that are both in terms of y.

More clothesline:

I’ve had an amazing teacher candidate (@misschacon_7) paired with me for the last two weeks. Every day she comes excited to try and learn new things. Today she came in with a great lesson for solving multi-step equations. Here is her activity:

She paired up students by randomly assigning them a playing card……each person was to find their match. Each pair went to one of our vertical writing surfaces (blackboards and whiteboards) where she asked students to solve a series of problems like….

### How many marbles in each bag?

She then gave them all sets of cups and beads.

Player 1 is to create an equation by hiding the same number of beads in the cups. They also have to ensure each side (whiteboards) must balance (have the same number of beads total).

Player 2 is to “figure out” how many beads are in each cup.

After player 2 has determined how many, they switch roles and start again.

Here is a round:

Player 1 sets up this up (she decided to put cups inside cups so we couldn’t see how many beads)

Player 2 starts on it….

and she ended up with….

Lots of great thinking going on here…..especially for the creator of the equation. Here is a group who realized there was a problem…

While player 2 was solving, she realized that she was getting a negative answer for the number of beads. Weird!!! So we asked player 1 if she made sure they boards were balanced to begin with……ooops they weren’t. So they couldn’t have been equal to start with! Let’s re-do.

Miss Chacon worked the room awesomely….visiting each group and asking probing insightful questions! My students were sad to see her go….today was her last day. We’ll miss you Miss Chacon!

# Amazing Equation Race

For review students will run from room to room solving equations. Each clue, when solved will give the location of the next room to visit. Clues are hidden in QR barcodes. Students will have to scan the barcodes with a smartphone to see the clues.
Clues: