Equations and “Flow Charts”

A group of seventh grade teachers and I were trying to figure out how to move from concrete representations of solving equations (some used chips/cups and some used tape diagrams) to the more symbolic procedural (traditional) representation. While students were able to model the “moves” with the concrete, some still struggled to move from that to solving on paper.

I had recalled a method a dear colleague, Bruce Grip, had shown me years ago using a flow chart. We decided to try it out ourselves.

Starting with expressions, we discussed what the “moves” are when simplifying. Order of operations made a showing, and we moved through the flow chart. We decided this was a valuable use of time, as it built understanding of the structure of numeric expressions and fluency with integers (Which, let’s be honest; they need LOTS of practice with!). IMG_1854   IMG_1856


From there, we decided to bust out a single step equation. We started the same way we did with expressions, using x as our starting value. “What moves am I making to x in this equation?” We then built our flow chart. HOWEVER, rather than simplifying (as in the expressions), we know the value we want to get. So the flow chart looks like this:


To solve for the value of x, we need to work backIMG_1859wards through our flow chart. If I had added 2 to a value to get -5, then I need to subtract that 2 to figure out what I started with. We could then parallel the flow chart with the more traditional algorithm for the students.

Below are several of our examples, limited to the structures seventh grade explores for CCSS.

We also explored some “messier” problems, as shown here.IMG_1863 It illustrates the fluency with the distributive property piece of “When do I need to distribute and when is it efficient to divide out the factor first?”. We liked that the students could show both ways and determine which route to take.



Our big commitments to this flow chart method:

  1. Start with the concrete/visual. This is not a substitute for chips/cups nor the tape diagram. This is the next step for students who need it.
  2. Next year, use the flow chart when exploring simplifying expressions so we can build on that understanding for solving equations.
  3. Use friendly numbers (NUMBER CHOICES MATTER!!!) first to build understanding.
  4. Bring in some messier problems to seal the deal and discuss different moves they can make based on the given numbers in the equation.
Equations and “Flow Charts”

Number Choices Matter

I had the privilege to work with fourth and fifth grade teachers this week. We explored multiplicative comparison problems in fourth and division with fifth (more on those in another blog). What I came away with is this: NUMBER CHOICES MATTER.

We don’t get much choice on the concepts we teach, nor often on the program we have to use. But we do get to choose what numbers we use with children. This could make all the difference for kiddos. If we choose our numbers wisely, we can build understanding through the patterns they see, the differences that appear, and talk about why those differences happened.

For example, consider the following set of problems:division setWhat is the same about each? If I am thinking about this as a partitive model and using base 10 blocks, I am sorting the amount I am given into 3 equal groups each time.

What is different about each? The amount I am giving to each of the 3 groups.

In the first example, 36 can be created by using 3-tens and 6-ones, and each can be fairly shared without any problems. Each would get a ten and two ones, or 12.

In the second example, I can still give out a ten to each of the 3 groups, but now I have to figure out what to do with the leftover ten and eight ones. This one builds off the first example, but pushes students to think about exchanging (regrouping).

The final example builds off the 48, but leaves 2 left that students have to consider. This allows to have a conversation about remainders.

These three build understanding of division, regrouping and remainders through strategically chosen problems to build from one to the next. Students have something to grasp on to when negotiating meaning with this tough tough subject.

So where can you build understanding through your number choices? I challenge you to think about what you want your students to learn next week and how your number choices can contribute to students understanding those goals!!!!



Number Choices Matter

Show Support (In 5 Words or Less)

It is Teacher Appreciation Week here in Texas. To show my support of teachers, I am dedicating this week’s blog to them. Now don’t get me wrong. Gift cards, flowers, and thoughtful cards from the kiddos (maybe wine…) are all great ways to show how much you and your tiny human appreciate the person who spends more time with them in the week than you do. But here are some ways in your daily interactions to show you care.

  1. I support you.   I cannot emphasize the importance of these three words. Knowing that a parent has your back is an amazing feeling. Educators go through a lot of schooling and training to teach our tiny humans. They know A LOT about children and how they learn. Let them do their job, and support them in whatever way you can.
  2. How can I help? Similar to #1, but requires action on your part. Come in and read to the kids. Help staple papers up on the bulletin boards. Donate materials/gift cards/Scholastic books. Clean the desks on a Friday afternoon. Not available during the day? Ask to have sent home items that need to be torn out, cut out, colored, etc. One year a parent asked to photocopy my papers (This was GOLD, people!!!!). These four words not only show you support your teacher, but you know they work their butts off as well. 
  3. What did he (or she) do? Favorite personal story. Picking up Ev (who is now a teenager) from preschool, I noticed a note in his cubby. I opened it as they were bringing him in from the playground. Ev: Is that from Mr. ____? Me: (Knowing it was a Birthday invite and not from his teacher…) You tell me. Ev: Yes. Me: Well then, what does it say? Ev: (spills his guts.)  Later that night I received a phone call from Mr. ____. I felt horrible for him, as he started by defending himself first. I cut him off and just asked, “What did he do?” And Mr. ____ sighed and said, “Wow. Thank you for believing what I have to say.”  People, these are tiny humans, and they are going to make mistakes. If you think your child will not cover up or lie to your face, then you must have the most angelic child ever, because that just does not happen in my world! Now, I recognize there are two sides to the story, but take into consideration that one side is an adult and one is a child. Do not immediately go on the defense. If you show you support your teacher, your child will behave as such.
  4. How can we help (insert child’s name here)? Every child needs to know that we are working together. It is the triangle of learning: the teacher, the student, AND the parent. Children need to know that learning occurs outside of the walls of their school. The more you can support their learning at home, the more they will see value in what they are learning, and the more they will engage in the learning process (and the more they learn!).
  5. Thank you. No two words are more important. Well, unless you say, “Thank you for…”. That may be even better. So here is mine. Thank you, teachers, for giving up your personal time and resources for my children. Thank you for loving learning so much you want to share that love with others. Thank you for putting up with my oldest’s sassiness and snarky remarks (Wonder where that comes from?!). Thank you for giving extra encouragement and love to my tiny human when he is frustrated or sad. Thank you for being your amazing selves and teaching others.

Much love to all the teachers out there! Jen

Show Support (In 5 Words or Less)