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

IMG_1857

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:

IMG_1858

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

Road Trippin’: Math Games for the Car

Special thanks for requesting this, Jen Duley!

Many of us are taking summer road trips with our tiny humans. Here are some ideas for keeping math in their brains and ditching the ‘summer slump’.

Counting Circle (sort-of)

This is something I have blogged about before. Kids must practice rote counting. Count up and down. Each person in the car takes a turn, counting by what the designated amount is. Below are a few ideas that I hope your kids will enjoy (And save your sanity!).

  1. Count by 1’s, first starting at 1, then building to start at a different value. Count up and down!
  2. Count by 2’s, 3’s or 5’s. Again, start with the value and practice skip counting, then start with different values. (Don’t forget to go backwards too!) One of the best things I ever did was make my clock in the car off by 5 min (too fast). My oldest had to figure out the time every time he got in the car. He had to regroup in his head almost every time!
  3. Count by 10’s, first starting at 10 to 100, then back down. Also start with different multiples of 10’s, different values other than tens, etc. Example: Start at 12 and count up by 10’s. Or start with 87 and count down by 10’s. (Super important for regrouping and subtraction!)
  4. Older kiddos: count by fractions. Start at 0 and add 1/2 each time. Start at 3 and count back by 1/3 each time (Gearing up for mixed number subtraction). Start at 1 2/3 and count up by 1/6.
  5. Older kiddos: count by integers. Start at 0 and add -2. You get the idea.

Guess My Number

Again, one that I have previously discussed, but super important.

  • I am thinking of a number between 11 and 13. What is my number?
  • I am thinking of a number that is less than 40 but greater than 35. It is odd. What is my number?
  • I am thinking of a number that is less than 100 and a multiple of 5. Now let them start asking yes/no questions to narrow it down.
  • I am thinking of a number between 11 and 12. Start them on fractions!!!!

Count the Cars

Choose a color, type, make, or model. Kiddos count all of that category of vehicles. Each child can count a different type (EX: Ev counts all the blue vehicles and Chris counts all the red ones) and whoever gets the most when you park wins!

Find the Number

Print out a 100 chart. Put it in a sheet protector and clip with a clipboard. With a dry erase pen (I attach the pen with yarn to the clipboard.) he/she crosses out every number he/she sees. Look at license plates, signs, billboards, etc. See who can cross out the most in a trip! You can also print out a partially filled in chart and they have to fill in the missing numbers before playing. For PDF 100 Charts: https://www.homeschoolmath.net/worksheets/number-charts.php

Three in a Row

Two options: print out a Three in a Row page for each child, put in sheet protector, and clip to clip board. Or print the blank, and allow them to fill in values 1-10 (they will have 1 missing since there are only 9 spots for 10 numbers).

Call out either addition or subtraction problems. You can just do number problems or put them as word problems. Example: Chris has 3 Stormtroopers. If he loses one in his car seat, how many does he have left? 3-1=2, so they cross off the 2 on their game board (if they have a 2). Once they get 3 in a row, they win! Link below for PDFs created for you.

3 in a Row

Target Value

Print out the Target Value sheet and put in the sheet protector. Clip to clipboard.

Give your child a target value (EX: 10). They write 10 in their Target Value box. They create as many addition problems that add up to 10. I honestly would rather just have the target box and let them make all kinds of equations (such as 1 + 4 + 5 or 20-10), but below is an example you are free to use.

Target Value

Shape Spotting

Great idea from my amazing friend and colleague, Kelli Wasserman! I found geometry cheat sheets if they need it. https://www.math-salamanders.com/geometry-cheat-sheet.html   Just put it in the handy-dandy sheet protector! Have your kids see who can spot each shape (square, rectangle, circle, etc) and they can cross it out on the sheet with a dry erase pen. Or, give them a shape to spot, and see who can spot the most. Love it!

 

Road Trippin’: Math Games for the Car

Relational Thinking to 10, More or Less

Our lives in kindergarten land are immersed in the idea of making 5’s and 10’s. Here is an activity you can do (After playing Make a 10…See previous blog!) to build relational thinking to 10.

Materials: Deck of Card, 3 post-its

Objective: To determine whether two addends (cards) make a sum (total) that is less than, more than, or the same as 10.

  1. Have your student write less than 10 on the first post-it, the same as 10, or just 10 on the second post-it, and more than 10 on the third post-it. more or less 10d(Note: You can also include the symbols <, =, >, but I prefer to work on the concept FIRST then introduce the symbolic notation later.) Place the post-its on a workspace that has lots of room.
  2. Shuffle the cards. Place deck face-down. I typically hold the decmore or less 10k and place two cards face-up for the child, but if students are playing in small groups they take turns taking the top two cards and placing them face-up. The child decides whether the sum is less than 10, the same as 10, or more than 10. If in small group, the others confirm or debate. Once the value is established, the student puts the cars face up as a pair under the correct post-it.
  3. Continue until all cards are used (That is A LOT of addition they are doing!).more or less 10c

Note: I totally stack my deck. I want to make sure some of the first pairs have a variety of sums so that the child (or children) see cards under each post-it. Here are a few of my favorite sets of cards to ‘stack’…

  • 1+2 (I like to start with a known fact and something a lot smaller than 10.)
  • 1+9 (Again, building on the “one more” facts, but this time it is 10.)
  • 3+9 (Relational to 1+9. If 1+9 is 10, then adding more makes more than 10. HUGE!!!!)
  • 10+4 (Any 10+ is great, as students really need to build to 10+ for first and second grade. It is amazing how many children do not see this as immediately more than 10, so it is a great one to have a conversation about!)
  • 2+3 (We have done so many that are greater than 10, nice to go back to a set less than 10.)
  • 5+5 (One of the first known facts for making 10.)
  • 5+8 (Similar to 1+9 above. If 5+5 makes 10, then adding more makes more than 10.)
  • 5+2 (Conversely, if 5+5 makes 10, then adding less makes less than 10.)

Alternative Games for Older Students

  • Use larger value cards and work less than, equal to, or greater than 20, 50, 100, etc.
  • Use cards with decimal values and play less than, equal to, or greater than 1.00.
  • Use cards with fraction values and play less than, equal to, or greater than 1.
  • Use black and red cards (reds are negative, blacks are positive) and play less than, equal to, or greater than 0.

 

Relational Thinking to 10, More or Less

Game: Making Ten

Kindergarten kiddos are immersed in addition and subtraction right now! They are exploring addition as adding more ‘stuff’ and subtraction as taking away (or removing) ‘stuff’. Many of the kids are in their Level 1 Counting All stage in which they rely on counting one-by-one to get the sum or difference.

For example: 3 + 4. A child at this level would count 1, 2, 3 then 1, 2, 3, 4; putting them together, 1, 2, 3, 4, 5, 6, 7.

This is acceptable for Kinder kiddos! This is awesome! This is the first step! But it isn’t where we want them to stay, particularly at the end of first grade. I tutor some students in grade 1 who haven’t moved past this level. So I took a game that has been around and edited to push kids into Level 2 Counting On.

Make a Ten!

Object: To find as many pairs of cards that add to 10 in your round.

Materials: Cards 0-10 (4 of each). Note: This is the most crucial component. I will talk more about the cards below.

Directions: (Below is a video clip. Sorry about the sniffling; it is allergy season here in TX!)IMG_9347

  1. Shuffle the cards. Lay out 4 rows of 4, face up.
  2.  Player 1 finds as many pairs of cards that add up to 10. He takes the cards and (I made them do this!) says, “________ and __________ make 10!” He continues until there are no more cards that pair up to make ten.
  3. Take the remaining cards (if any) and put them back in the pile. Reshuffle and lay out 4 more rows of 4 cards.
  4. Player 2 finds as many pairs of cards that add up to 10. She takes the cards and (I made them do this!) says, “________ and __________ make 10!” She continues until there are no more cards that pair up to make ten.

Continue alternating until there are not enough cards left to play. Player with the most cards wins.

Adaptations

Chris did struggle with 4 + 6 (or 6 + 4). I pulled out a ten frame to help with, “How many more do you need to make 10?”. img_9356.jpg

Chris refused to have any extra cards. In fact he got quite cheeky about it. This was his modification (he called it a ‘cheat’). I was perfectly fine with it, as I am sure you would be as well! I did not give him the word ‘altogether’ to use; that was a natural piece of the conversation. Woot! Woot!

 

Card Choices

  • If you are just starting out, only use 0-5 and make sets of 5. This is foundational and kids do not spend enough time on fact fluency to 5 before jumping in to 10.
  • The cards I used were from Eureka Math. I love them, as they are friendly shapes and are in sets of 5’s. So 10 is represented as two-fives. This link will get you to the cards I used as well as others they have (like ten-frames) http://eurekamath.didax.com/exclusive-items.html/
  • If students do not need the symbols (or you are pushing to counting on or fact fluency) I would suggest just write the numbers 0-10 in four colors on index cards. That would be cheap and easy.  You could also make your own cards with dots (if they need the dots to count) or ten frame cards this way as well.
  • You can mix/match as well. Use 2 of each number card 0-10 and 2 sets of each dot or ten-frame card. That way, students have to use counting on for some of the sums.
  • Another site for cards would be Sumboxes. They have number cards larger than 10 so you can play to other sums (like 20, 50, 100, etc.). sum boxesThey also have fantastic dot cards/ten frame cards together for some great exploration! https://sumboxes.com/collections/types?q=52+Pickup+Card+Decks&page=2

Whatever cards you choose to use, make sure they are appropriate for the level of the learner!!!

Game: Making Ten

It’s The Little Things…Writing Notes

When I took Chris to register for Kinder, he was terrified. He looked right at the Vice Principal and told her he would NEVER come to this school. I was mortified and heart broken. How could my child (coming from MEEEEEEE!!!!) be so fearful of school??? Was I in for days of tears and refusals to get up to go?

Fortunately, we were blessed with the most AMAZING Kindergarten teacher.  The first week of school she sent home a note to Chris. note 1 It was the first thing he handed me (all crumpled and loved on) that afternoon. He was so proud that his teacher wrote HIM a note. He asked me to read it again and again, and taped it to his wall near his bed.  This note takes him through the good and the bad; the ‘easy’ and the challenging. I have heard him read this note over and over (when he was busted and in time-out!). This note has carried him through the year.

We have since received numerous notes from his teacher, all as important to him as the first. This one hangs on our fridge as a celebration for his daily counting to learn up to 100! When he struggles with sight words, counting by 5’s, or any rote memorization, we look at that note as a reminder that all things take time to learn. Note 2

Note 3Is it just Chris that loves a little note? Nope. Fast forward to his recent eval for speech. I will be honest. It was a lot of pages expressing a lot of jargon that I forgot the minute I was done reading, except for the part on the back of the eval… I am still teary-eyed when I reread it.  At the end of the day, at the end of the struggles he has, my boy is a good person, and someone noticed.

In this electronic age, let us not forget the little things, like hand-written notes. Why are notes so important? It is unique; someone took time out of their day to physically express something to another person. It expresses that you matter so much that, instead of texting or emailing (which could be a cut/paste), someone took the time to individualize and express thoughts just for you. Wow.

So let’s each commit to writing one note this week:

  • Put a post-it note in your child’s (or students’) lunch or binder letting him/her know how much you care for them
  • Put a note in your significant other’s car specifically telling him/her one kind thought
  • With this being Teacher Appreciation Week for so many districts, send a note letting your child’s teacher know how much they mean to you and your child
  • Mother’s Day (wink…wink…nudge…nudge…) doesn’t have to be just the kids making handmade cards. Let a mama know how much they mean to YOU
  • Put a kind thought on a random door, car, locker, etc.

 

It’s The Little Things…Writing Notes

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)