Introducing systems of equations: graphing

I’ve run out of gas with my daily postings. Here’s what we did in pre-algebra on Friday.

Curiouser and Curiouser

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.

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Students learn to give feedback with a growth mindset

I’ve lost track of the days, but here’s what I did today.

Curiouser and Curiouser

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…

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Days 96-115: Finding my way out of the doldrums

Between needing to recharge my battery and having nothing extraordinary to share, my desire to blog has been limited.


From Lizzie Nichols Concept Design

Most of my energy has been depleted by reteaching a small group of at-grade-level students. This group cannot remember integer rules and the numerous steps involved with multi-step equations. I thought continual practice would help, but for these students something else is blocking their progress. Earlier in the week I spoke with our school psychologist who said, “They’re just not ready for the concepts. Have them make note cards, or use their notes.” So on Friday they made color coded note cards.

We’ve moved on to inequalities, but the group that needs to reassess two step equations will be using their note cards when they reassess.


The kids have been learning slope. I used a couple of Dan’s Graphing Stories to pique interest and launch the unit. They’ll be assessing this week.

Days 93-95: revisiting one step equations

In my previous post, Monday, I noted the results of a formative assessment on one step equations (adding and subtracting) and shared how I differentiated for the students. I’m sorry to report that for my A day class (the class with the widest disparity) the same students who struggle continue to be baffled. It’s clear these students lack conceptual understanding.  For many, I wonder if a contributing factor is inattention. It seems to be getting in the way of moving concepts into long term memory. When they are focused, they dwell on procedure without regards to whether their process and answer make mathematical sense.

Here, one student combines like terms, then combines unlike terms, then maybe divides by 10? However the work from that point on is unclear.

one step

I thought I had been breaking down the lessons into manageable parts, but something’s amiss.

I’m hoping my other standard class will have a better conceptual understanding.

Day 92: differentiating after one-step formative assessment

Students in my “A” day standard class have MAP scores ranging from the 15th to 90th percentile. My “B” class has a narrower distribution. After reviewing the formative assessment on adding and subtracting one step equations, I wasn’t surprised to discover that I need to address the concepts again in the following way:


After I returned the assessments I sent the 6 enrichment students out in the hall to work on a set of problems from our pre-algebra series. The rest of the class stayed with me for another mini-lesson. I chose to begin with the distributive property because all of those students needed to see it again. Plus, those who already got one-step shouldn’t have to sit in on another one step mini-lesson. After reviewing the distributive property, those six students received additional practice problems I made using Kuta software. They stayed in the classroom and periodically checked their answers against some keys I had posted in the room. As they finished they worked on the enrichment.

The final 10 were with me as I once again walked through adding and subtracting one-step equations. I got them going on their one step worksheets (again Kuta software generated) and had keys posted for them as well.  As they finished, they moved on to practicing the distributive property.

This took about 40 minutes. Four of the 10 students who had the most remediation came to see me in math lab today. The rest is homework.

The rest of the block was devoted to multiplying and dividing one step equations.

My “B” day standard class will experience the same lesson format tomorrow. Though fewer students will need remediation.

Day 91: Distance formula continued, reviewing one step assessment

In Pre-algebra we continued discovering the distance formula. Yesterday I stretched the students a bit too much, thinking they would catch on quicker than what they did. It’s my first time teaching the topic and today was much better. While their discovery of the formula was more teacher led, it was to their benefit.

I created a SMART board lesson and walked them through this problem:



I then gave them four problems as a handout and guided them through the first problem.


Some were still confused by which coordinates to use so I added a table with headers for them to keep track.


By the last problem the kids were sailing on their own, without the aid of the coordinate plane. Their ticket out the door was to find the distance of (1, 3) (5, 8). Only one student struggled.

The standard classes today reviewed their one step equations (add/subtract) assessment. I was pleased with the results for most of the students, but I want to add more problems with simplifying and solving. I’m looking forward to next week where we’ll be doing more complex problems.

Days 89-90 one step equations and distance formula

We had two snow days on Monday and Tuesday.

My standard classes on Wednesday and Thursday had an assessment on one-step equations. In pre-algebra students students explored the distance formula using this resource along with help from portions of this video. The students had some difficulty getting started so I showed the first few minutes to get them thinking about using x coordinates to find the length of the parallel line segment along the x axis.

distance formula

When I gave them snippets from the video, it was just enough to get them going. Then they would get stuck. I showed a bit more of the video and they would be able to continue. We’ll keep working at it.

Days: 86-88 MLK short week

Only three days of class this week. Students were off Monday for Martin Luther King day and Friday was Institute.


After the long weekend, students needed a refresher on combining like terms. Actually they needed more instruction on applying integer rules. I was hoping to give an assessment this week on writing expressions, the distributive property, and simplifying expressions but they needed more practice.


We explored the Pythagorean Theorem. I wanted to try something different so I followed this lesson idea from the Teaching Channel. I really liked it. The kids loved it. Part of the lesson included a scavenger hunt and unscrambling anagram clues to solve a Who Done It. Instead of using technology to get the clues after solving each problem I substituted it for human contact. They had to see me to get a strip of paper for each clue. The write up is on the Curiouser and Curiouser blog.

Days 81- 85 I forget…oh, yeah, MAP testing

I had to look in my plan book to remind me what we did during this time. That’s right, it was MAP testing. The rest of the week focused on:


Writing expressions, identifying constants, coefficients, like terms and combining like terms.


An assessment on multi-step equations, variables on both sides, and special cases. When we go over the assessment, I’m continuing the practice of students first meeting in small groups to discuss answers. I then hand out copies of the key for further discussion. I’ll then take specific questions from the class. I’m finding this approach allows for much more mathematical conversation, gives students more time to carefully review their work and identify their own mistakes instead of me pointing them out to them.

Days: 79-80 Writing expressions, special cases

I’m dabbling in math workshop model and I see a glimmer of hope that I’ll be able to eventually make the transition. There are some elements of the workshop model that I already do, but other aspects need work. A more complete post can be found here. Below is a brief overview of the past two days.


I introduced writing expressions and equations using a warm-up. I handed out four expressions.

writing expressions

Students shared their answers under the document camera. This led to the mini-lesson which included using a term with a coefficient instead of the multiplication symbol x and interpreting the division symbol as a fraction bar. (And the kids thought they we were done with fractions!) For the rest of the period the students worked in pairs completing this puzzle:

expressions puzzle

They didn’t have enough time to finish the puzzle, so we’ll pick it up again next week. When we debrief we’ll reflect on both strategies examine the translations. As I checked on the pairs I heard one student say, “Look for ‘equals 12’ to see if it matches” Another said, “Subtract means minus. Where are all the minuses?” Even though those strategies help the student complete the puzzle, the point of the puzzle is to analyze the translations. I can’t overlook that when we debrief.


Students entered the class with this warm-up:

variables both sides2

Most chose to solve by using their newly discovered method of clearing fractions. One student chose to solve it by keeping the fractions.