Introducing systems of equations: graphing

Many of my pre-algebra kids continually seek context, so I thought I’d head them off at the pass by introducing graphing systems of equations as an exploratory activity using a couple of word problems. Here’s one of them:

systems problem


We’ve been graphing linear equations for a while so I purposefully added the “How could graphing each equation help you?” question to guide their thinking.

I came to learn that some students would rather not have the context and just be given the systems to graph–it’s a lot easier. “I hate word problems,” said several students.

That’s my fault. I need to do a better job of applying these skills in context. After we talked about context it was time for some graphing practice. As I checked their work I was reminded that a few haven’t improved their “number sense” when it comes to graphing. I wish I had snapped a photo of one student’s work, but I’ve recreated it below.


My questions to her were: 1) While your table is accurate, why did you choose values that fall outside the graph?  2) How could the slope and graphing the y intercept help you?

This particular student doesn’t recognize how less accurate her lines will be if she picks points beyond the coordinate plane that’s provided. If students are graphing using a table, it’s a great opportunity to discuss when it’s appropriate to choose negative values in order to stay within the graph.

In previous conversations with this student I also know she doesn’t have a strong grasp of the meaning of rate of change. I need to get her into math lab.





Students learn to give feedback with a growth mindset

Yesterday Algebra’s Friend introduced me to a new professional development blog Read…Chat…Reflect…Learn! While I enjoy reading blogs for lesson ideas, I also love reading journal and magazine articles, books, research studies, etc. as another way to stay current. I’m glad I found this blog and I’m thrilled to see that it’s in its infancy, only because it makes me feel that I haven’t missed out on too much!

The current topic is feedback. The article for discussion was How Am I Doing? It offered a good overview, but what was missing was how to provide feedback using a growth mindset. This excerpt, Types of Feedback and Their Purposes,  gives clear examples. One type of feedback I tend to focus on is descriptive as opposed to judgmental feedback.

When examining student work for feedback I’ve collected their work in progress, and provided descriptive feedback. I won’t kid you. It’s time consuming. But an idea was floating in my head. What if I taught students to give each other feedback with a growth mindset?  I don’t want you to think I am shirking my responsibilities, but could math students offer feedback similar to students who participate in peer editing compositions?

After reading Algebra’s Friend post I thought I would give it a try. Yesterday and today I showed my students this Would You Rather problem. They were to work in groups of four to solve.

After the groups worked for about 10-15 minutes, I collected their work, redistributed them to different groups and asked the new group to provide feedback. Here’s an example:


Kids being kids, their feedback was somewhat judgmental and not of a growth mindset. Had they rephrased their wording to questions such as, “How could labeling help explain your thinking?” or a statement such as “The final answer is not clear,” would definitely put them on the track towards feedback with a growth mindset.

Doing this activity in groups made it quite manageable as only six responses were being critiqued. While I monitored each group I asked them what questions they had about the other group’s work. I also asked them to be positive with their feedback. Each problem was circulated twice. Doing so also gave groups the opportunity to see how others approached the problem and perhaps revamp their thinking.

Here’s how the above group implemented the feedback:



The next time I see them we’ll continue the conversation of how to offer feedback with a growth mindset. With more practice they’ll get better at it.

I should have thought of this at the start of the year.




Lost in the halls: distance between two points task

It’s been a great week, and a rough week. The highlight was having my pre-algebra students experience the Distance Between Two Points task.  I discovered it among a collection of tasks at the Complex Instruction Consortium website. I tweaked it a bit, and what I really liked about the task was that it got the students out of the classroom. They liked that too. In fact at some point I lost “visual contact” with every student but they never took advantage of the situation!

Students were anxious to get started, so many didn’t pay attention to the directions we read aloud.

distance task directions

Student: “I don’t get it!”

“Read the Resource Fact.”

“So we have to count the tiles?”

“Is that what the resource fact is suggesting?”

distance count

Student: “But how do we find AC and BD?”

To give you an idea of the layout of our building here’s what the kids were doing. I did NOT give them this diagram. They had to come up with it on their own, which was extremely helpful when describing the group’s procedure.


Here’s an example of the student work. They’re getting better at constructing an argument and explaining their reasoning.

distance student work

Letting them be kids with the Pythagorean Theorem

My slightly modified version of this Teaching Channel lesson on the Pythagorean Theorem started a bit slow but within 10 minutes it gained incredible momentum. By the end of the lesson students were set free to search the hallway for problems and gather anagram clues to solve a Who Done It.

Kids were in partners staring at this image from the SMARTboard file I created for the lesson. The rulers followed the directions on the left. The non-rulers followed the directions on the right.

pythag instructions

At first the kids didn’t want to play with the numbers. I wasn’t about to lose at this waiting game so I announced, “Let’s push the pencil and those calculator keys.” One pair began working, then another, then another. Soon the entire class was engaged.

A few minutes went by and one boy asked, “Can the numbers be used more than once?”

“Great question! Yes!”

We had a great discussion on the operations needed to get 10. Not one mention of a formula.

I then asked, “How could we apply what we just learned to golf. How far would I need to hit my tee shot if I wanted to get a hole in one?”

dogleg left

It’s at this point where I tweaked the lesson to my personality. Sidebar: I love the Muppets!

pythag kermit

what's different

Here is where the knowing the vocab came in handy. After pointing out the legs and hypotenuse, students then realized they would need to do something different to solve Question 3.

They practiced solving these:

right angle problems

Then it was time to set them free. The students got this handout.

pythag worksheet

The first two problems were done in class, but the other three created a mini scavenger hunt in the hallway. After correctly solving problems #3-5, each pair received a Who Done It clue. In the Teaching Channel lesson students received their clues in a text message. I chose human interaction because I prefer it.

Here are the anagram clues.

Mustard, kitchen, dagger.

The lesson continued the next day exploring right, acute and obtuse triangles using perfect square cutouts from AIMS Education.

Pre-assessment yields two random, but important, lessons learned

Lesson 1: be flexible when grouping students.

Earlier in the week I posted on my 180 math post-its blog that students were pre-assessed on fractions. I was disappointed to see that only two pre-tested out of many of the concepts. Given the caliber of this class there should have been at least five more ready for enrichment on the skill of adding and subtracting fractions.

I probably should have started the class immediately by differentiating for the two students, but I didn’t. I had enrichment ready for them,  yet I launched into a whole class whiteboard/quick check activity of adding and subtracting fractions. I admit I short changed those two girls, but I’m glad I did the whiteboard work because within 10 minutes five more students revealed to me that their pre-assessment was a signal that they simply forgot.

It troubles me that students do not retain their learning, but that’s for another post. In this instance I truly believe those five kids had, as they say, a brain fart.

For the rest of the block they worked at two tables on challenge problems. Which leads to…

Lesson 2: Students can be completely engaged with non-real world problems.



The kids loved the challenge. I know there is little context with these problems, yet they were engaged for the entire block. You can’t describe them as puzzles, but for 7th graders the problems were puzzling and they wanted to solve them.

By the way this same class is enjoying the NCTM palette of problems I’ve been using for determining importance.

determining importance2

Recently I’ve been presenting them at the start of class. Students work independently then we share solutions, discuss their “arguments” etc. Some students want me to do this everyday.

Trick photography asks: “What size is the missing wrench?”


Technically the image isn’t trick photography but the gap in positioning makes you think the size of the missing wrench must be between \frac{1}{2} and \frac{9}{16} of an inch. It may be a form of trickery but could a student argue the missing wrench size is  \frac{17}{32}? How would they justify the break in the pattern?

wrench fractions

One of the wrenches is indeed missing. Would students think to mentally close the gap and consider the possibility that the first or last wrench in the sequence is missing? What would be those sizes?

The idea is to let students explore common denominators and the patterns they discover in this task.

I’ll let you know how it goes.

I’ve never been out in the hall before! I feel so SMART!

My SMART goal to raise self efficacy was to implement four tasks in the first quarter. I only had two. My standard students experienced the Money Munchers and Variety Show tasks. My pre-algebra class had Molly’s Locker Combo task and the Boomerang task. While I didn’t reach my goal, I am making a wee bit of progress with one student’s self efficacy. I have to admit, it’s not me. Her teacher last year had her in math lab nearly every day.

On Wednesday, she was elated that she was part of a group that was allowed to work in the hall on an extension activity while I retaught a concept. It was the first time she ever experienced such a thing. “I’ve never been out in the hall before! I feel so smart!” Since the start of the year she’s only had to reassess one time. Attending math lab has paid off.

The extra time we devote to our students may or may not pay off this year, but it will in the future.

A list of go-to resources

There are so many great resources out there, but one of my favorite sites for lessons and tasks this year is from the Mathematics Assessment Project. I’ve already used their Money Munchers and Boomerang tasks as formative assessments. They’ve provided a window into student thinking as well as artifacts in terms of where students are at with constructing arguments.

Another source for MARS tasks can be found at the Sonoma County Office of Education. The list is organized by grade level and include links to Illustrative Mathematics and Inside Mathematics.

I can’t forget NRICH!

I also love the growing list of blogs in my Netvibes reader. However with time being so precious during the school year, I’ve been zeroing in on the post title and the first sentence displayed in the reader. I know I’m missing out on some great writing and ideas, but it’s the only way I can manage right now.

In my delicious account I found this list of tasks. I’m not sure of the source, but it includes rubrics that you can use as is or modify. I also found Cut the Knot. I haven’t used it but I must have bookmarked it for a reason!

Everybody loves Fawn. She goes without saying. The right rail of her blog links to several heavy hitters.

I’m sure I’ve missed plenty, but those are my first places to go for lessons, task, and activities.

What to do when you don’t know what to do

My students do not have much experience with non-routine problems. On occasion they’ve completed complex tasks, but certainly not enough, or a variety, for students to build self-efficacy. I’m trying to change that by living a SMART goal I created for myself. Last Thursday, day 8, I presented a formative lesson designed by the Mathematical Assessment Project.

The problem began: Emily doesn’t trust banks with her money. She has stored $24,000 in one dollar bills under her mattress. The rest of the problem can be seen below. This student’s attempt was typical.

money munchers4
Most students guessed and provided no mathematical reasoning to support their answer–revealing a lack of understanding of how to approach the problem.

After they worked on the task independently for fifteen minutes, I collected their work. I found myself providing the same feedback over and over. (My handwriting is atrocious. I know. )

As I reviewed their work, I came upon this response:

money munchers5
This student has recognized that mattresses come in a variety of sizes, but provides no math to justify her thinking.

One or two students thought to estimate the size of a mattress, but their dimensions weren’t reasonable. For them my feedback was, “How tall are you? What do you think the dimensions of your bed are?”

For estimating a stack of one dollar bills, nearly every student has this feedback: “How could a book help you? How could the pages in it help you estimate a stack of money?”

When I see the students tomorrow, they’ll take that feedback and work independently for about 10 minutes. They’ll then work in their groups to share their progress and come to a solution.

One thing I will add is criteria for success. Your work is: a reasonable estimate that is organized and clearly labeled.

I love this lesson. I get to see individual thinking. The feedback offers progress while not enabling. Plus, when they collaborate every student should be able to contribute to the solution.

Assessing the 8 mathematical practices

Like Fawn, I’ve been thinking about how to assess the 8 mathematical practices. Eventually I will want to “grade” them, but at the start the students will need opportunities to practice the practices. In addition I can’t just let them practice without giving them feedback. So I’m going to use this ASCD resource to drive my formative assessments, It provides a great framework and walks through the entire process using the Boomerangs lesson, a high school MARS task.

As you can see the feedback is mostly through written questions. When I first read the section on suggested questions I was thinking, “They sound a lot like the 8 mathematical practices.” For example the issue of difficulty in getting started could fall under the math practice of making sense of problems.


These questions and my comments will hopefully provide the necessary feedback for students to improve on the mathematical practices, but I’ll have to do it on a regular basis for the feedback to be effective. Plus I’ll have to provide them with a rubric. Our district created this rubric for students and teachers.

practice tool

In the fall I’m going to focus on only two or three practices to start with, then add other practices. I’m stealing that idea from language arts teachers who use Six Traits of Writing. From what I understand middle school language arts teachers only focus on a few traits at a time.

What I’m also planning on doing is using this recording tool for anecdotal, informal observations. Sometimes I feel like a mad scientist walking around with a clipboard and jotting down my observations but I’ll have to get over it. The tool and rubric were created last month so I have not had a chance to take them for a test drive.

Assessing the math practices will be new to me. If you have experience or see some red flags feel free to chime in.


I’m beginning to get incredible feedback. Your comments deserve attention so I’m placing them within the post. Please continue to share your thoughts.

  • Assessing them is not about “can they” do a specific SMP, but “do they” and can they eventually do so habitually?–Just one of Jessica’s suggestions.
  • Reading their reflections not only gave me great insight into how the students believed they were using the practices but also how they were beginning to think about solving problems in general–Jennifer shared her blogged about journaling.