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”

War!(HUH!) What is it good for? (Absolutely lots in Math!)

With my family on the go so much during the non-lazy days of summer, we need easy games to entertain the tiny humans that don’t require mass amounts of attention from the adults (who are often in conversation). Enter the game of War. This versatile game can be used for all age groups and can really keep your child’s skills in arithmetic in check during the “summer slump”.

How to Play (Basic Version)

  1. Grab a deck of cards (I keep one in my purse and in the car at all times). You don’t have to, but I prefer to take out the face cards and jokers. Shuffle the rest and divvy out to all who are playing.
  2. All players shove all of their cards into a “deck” and keeps the deck face down.
  3. All players (at the same time to avoid cheating) flip the first card. The player with the largest value is the winner and takes all of the cards in the round.IMG_5434
  4. If there is a tie (that is the largest value), those players place 3 cards on their original face down and flip the fourth card. Whichever player NOW has the largest value gets all of the cards from the round. IMG_5435
  5. Continue playing until either a) one player has all of the cards; or b) you get sick of playing. The player with the most cards is the winner.

Additional Versions

  • For younger players: Use only 2-5 from the decks and play with those. The game dot cardsgoes faster and they are working only with 2, 3, 4, and 5. You can use the aces as 1. Even better, use number cards or dot cards (see below for links). Print on cardstock (4 cards per number) or go online and buy a set.
  • For any age: You can also play and whoever gets the smallest value wins. This is great for preK-1st graders!
  • For students who need review with addition: Play two cards at a time and add them. The player with the largest sum is the winner of the round.
  • For students who need review with multiplication: Play two cards at a time and multiply them. The player with the largest product is the winner of the round.
  • For grades 5-7: red cards are negative values; black cards are positive values. Flip over one card. If I have a red 6 and you have a black 2, you are the winner since positive values are always greater than negatives. IMG_5434
  • For grades 6-8 (or 7-8 if using Common Core): Play two cards and add them, using reds as negatives and blacks as positives. The player with the largest sum is the winner of the round.
  • For grades 6-8 (or 7-8 if using Common Core): Play two cards and multiply them, using reds as negatives and blacks as positives. The player with the largest product is the winner of the round.
  • For grades 6-8, use only values ace (for 1) through 5. Flip the first card; that is your base. Flip the second card; that is your exponent. The player with the highest value wins  the round.

Different Sets of Cards:

  • You could probably look on Amazon for different card types, but I love the sets at 52 Pickup. They are of high quality and there are many different types ranging from dot cards to ten frames to cards that go through the thousands (so you can work on place value!)



War!(HUH!) What is it good for? (Absolutely lots in Math!)

Taking The Distributive Property to Middle School: Making Multiplying Mixed Numbers Easier!

The last couple of blogs have highlighted the importance of the distributive property in multiplication for grades 3-8.  Let’s refresh ourselves…

Example: 6 x 9    Try it out using the distributive property! (There are many, but I have highlighted two.)

Method 1: Think about the 9 as (10 – 1).  Distribute (multiply everything in the parenthesis by) the 6.

 6 x (10 – 1) = (6 x 10) – (6 x 1) = 60 – 6 = 54

Method 2: Think about 9 as (4 + 5). Distribute the 6.

6 x (4 + 5) = (6 x 4) + (6 x 5) = 24 + 30 = 54

So you may think, “Jen, where the heck are you going with this?! This is a waste of my time and my child’s time.” Yet here was my son’s homework earlier this year.

multiplying mixed number

He was to multiply 17 2/5 by 5. Most of us learned you MUST convert the mixed number (17 2/5) to an improper fraction, then “multiply tops/multiply bottoms”. Finally change the improper fraction BACK TO A MIXED NUMBER. What a stinkin’ waste of time!

My oldest had the problem done in less than a minute. He got 87. Do you see it? What did he do? I have to admit I didn’t see it at first, and had to ask him. This is what he said (please insert a very monotone voice, since he wasn’t pleased with having to tell his mother his strategy…)

I thought about the 17 2/5 as two parts. 17 + 2/5. I mutliplied 5 times 17 and then 5 times the 2/5. Then I put it back together. (Duh, mom.)

So what does this look like mathematically?

5 x 17 2/5 = 5 x (17 + 2/5)

                                                 5 x (17 + 2/5) = (5 x 17) + (5 x 2/5) = 85 + 10/5 = 85 + 2 = 87.

In this case, using the distributive property is MUCH faster than the rules we were taught! And it makes sense! I am using partial products. I break up my number into easier parts, multiply them by the given value, and put it back together.

Here is another example. This came from one of my intervention students (students who are brilliant but don’t excel with the traditional book method in mathematics…) :

Import 9.3.14 709

You can admire all of his fabulousness, but the piece I want to focus on is 9 1/11 x 7. In red, notice he rewrote it for everyone to understand. Please note he had done it in his head (the blue). All red work was written after to let his class know what he was thinking.

                                  9 1/11 x 7 = 7 x 9 1/11 (commutative property)

7 x 9 1/11 = 7 x (9 + 1/11)

                                     7 x (9 + 1/11) = (7 x 9) + (7 x 1/11) (distributive property)

                         = 63 + 7/11 = 63 7/11%

Why am I showing you this? As parents, you WILL see the distributive property used with fractions, decimals, and percents. It makes it faster to compute, as students can do most of it in their heads. Most important, when students use the distribute property, I have noticed less “silly” errors. They get it right because it makes sense!

What can you do to support this kind of thinking? Play with numbers. Use a whiteboard. Use the sidewalk with chalk. Use the bathroom mirror with a dry erase pen. Give one problem a day and see how easy it gets to think about multiplication this way.

Here are some starter problems…

7 x 6, 7 x 7, 7 x 8, 7 x 9, 7 x 12 (break the 12 into 10 and 2)

8 x 4, 8 x 6, 8 x 7, 8 x 9, 8 x 12

9 x 4, 9 x 6, 9 x 7, 9 x 8, 9 x 12

3 x 16 (break into 10 + 6), 3 x 17, 3 x 18, …You could go on forever!

2 x 4 1/2, 2 x 5 1/2, 2 x 6 1/2, ….and so on

3 x 4 1/3, 3 x 4 2/3, 3 x 5 1/3, 3 x 5 2/3, …and so on

4 x 3 1/4, 4 x 3 1/2, 4 x 3 3/4, … and so on

If you need more, just look up free worksheets that multiply a mixed number by a whole number. Use one a day!

Taking The Distributive Property to Middle School: Making Multiplying Mixed Numbers Easier!