math

Equations and “Flow Charts”

Jen Moffett, , , , Leave a comment

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

Jen Moffett, , , Leave a comment

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

Jen Moffett, , Leave a comment

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

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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

Quick Shows With Ten-Frames

Jen Moffett, , , , 1 Comment

I was asked to come in and work with small groups (4-5 students) in Kindergarten today using ten-frames. The teacher wanted students to unitize by 5, 10, and 15, counting on the rest by ones. For example, if I asked a student, “How many do you see? How do you see them?”, she wanted the students to understand that you could find the value in a variety of ways. Here are a few of the anticipated answers she wants them to give by the end of the year:14

  • I counted them all. One, two, …twelve, thirteen, fourteen (Level 1)
  • I saw two- 5’s, so 5, 10, 11, 12, 13, 14 (Level 2)
  • I saw a ten, so 10, 11, 12, 13, 14 (Level 2)
  • I saw 5’s and 1 missing. 5, 10, 15, (counting backwards) 14 (Level 3)

I had a deck of ten- and double ten-frame cards, so I decided to do some “quick shows”.  I would show a card to the kids for about 5 seconds, and they had to ‘think’ about their value (versus just shouting out the number). We rotated who gave the value first, but every child had to give the value they thought was on the card. I chose a different student to explain how they got the value, then gave every other student a chance to share their thinking. We did this for about 15 minutes per group of 4-5 students.

Here are our ah-ha’s:

  • Out of the 4 groups, only one group stayed within the single 10-frame. I was getting answers from this group that were bigger than 10 every time. For example, when I showed them a card with 8 dots, one said 8, one said 12, and the other two were still counting by ones. I quickly drew a double (or triple) 10-frame and grabbed some counters (plastic circle thingies) and would show them their answer, then the original card. That worked for all but one student. For him, I kept on the table the card with the ten-frame filled in, then did the quick show. That clicked for today, but I need to do some hands-on work with this group. I also need to go back to a 5-frame and really focus on 5+ values before moving beyond 10.
  • Students needed to be convinced that the two cards below each showed a value of 5. Great for starting the discussion about the commutative property! We rotated the card over and over until someone said, “It is just the same thing! You didn’t put more on or take any off. Geez!”

5 different ways

  • The sequencing of the quick show was instrumental in students building strategies beyond counting one-by-one. The order that seemed to work the best today was as follows: 3, 5, 5 (again, upside down), 4 (to see it was 1 less than 5), 6, 8, 10, 9, 11. Notice we kept them seeing 1 more/less so they could use that strategy as well.
  • For the groups that could “just see” the ten-frame, I worked up to 20. Here is the orde18r we used with those groups: 3, 5, 5 (upside down), 8, 10, 12, 15, 14, 20, 18. 18 was tough (see the number of dots), as students really needed to push to 5’s versus  counting 10 then by ones.
  • One group finished about 5 minutes early, so we played war. That way, they each had a different card and had to tell me their value before determining who had the most dots. This was interesting, as they had the cards to touch and many reverted back to one-to-one counting. We will need to think about that for next time.

What I loved about this activity was that I had 15 solid minutes to informally assess each child. I heard what they understood and where they struggled. I was able to note for the teacher which cards each child got quickly, and which he/she reverted back to counting by ones (or guessed). Every child was engaged and had to listen to their friends as each shared out their strategy. And most important to me, every child left my group smiling, asking when I was coming back to do more “quick thinking”.

For large ten-frame cards: https://lrt.ednet.ns.ca/PD/BLM/pdf_files/five_and_ten_frames/ten_frames_large_with_dots.pdf

For double ten-frame cards: We made them by cutting/pasting two ten-frames together. I am sure you can buy the cards, but this was cheapest for us.

Quick Shows With Ten-Frames

Cross-Out: Sums to 12

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Chris asked for a new game yesterday, and I didn’t have one ready (Gasp!) So we made one up together called “cross-out”. This was quick, easy to organize, and he had fun playing it and ‘cheating’.

Materials: white board, dry-erase marker, two dice (we used dot dice, but you can use number cubes to up the level of thinking)

Objective: We played as a team. The goal is to cross-out every sum when rolling two dice (2-12).

How to Play

  1. Have your child write the numbers 2-12 on the white board. This is great fine-motor practice! cross out 9
  2. Player 1 rolls the dice and adds up the values. Player 2 crosses out the sum on the board. I rolled a 9, so Chris had to find the 9 and cross it out (see below).cross out 9
  3. Player 2 rolls the dice and adds up the values. Player 1 crosses out the sum on the board. If a sum is already crossed out, continue rolling (and therefore practicing addition and counting on) until you get a sum that you can cross out. No losing turns here!
  4. Once your team has crossed-out every sum, you won! Do a silly dance to celebrate your success!

Fun Note:
When we only had the 3 to cross-out, Chris asked if we could change dice to be 0-5 instead of 1-6. “Why?” I asked. “So that I have a better chance of rolling a 3! The only way I can get it is with a 1 and a 2 and that’s tough!” If I had the 0-5 dice at my fingertips, I would have totally given in. This is a great statistics insight for such a tiny human!

He rolled a few more times, got sick of rolling and decided to just roll one die. BAM! First roll he got a 3. He was very proud of his ‘cheating’ scheme!3

Differentiation Ideas:

  • Use a number cube and a dot die to work on counting on (Level 2).
  • Use two number cubes to work on addition rather than one-to-one counting with dots.
  • Use cubes that have larger values and work on the teens/twenties. I buy square wooden cubes at a hobby/craft shop and use a Sharpie to make whatever dice I want to use. Easy and cheap!
  • Play against each other. Each person could write 2-12 and see who can cross-out their board first.
Cross-Out: Sums to 12

Tiny Human Perspectives: What About 0?

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What is up with 0? It is nothing, nada, zilch. So why spend time thinking about nothing?

While playing a game with dice (labeled 0-5 each), pre-schoolers had no trouble thinking about zero as nothing.

Student (rolls a 0 and 5): 0 and 5 is still 5!

Me: Why is it 5?

Student (now rolling eyes): Because you added nothing to 5, so it stays 5. You didn’t do anything to it! (Duh…Mrs. M!)

Playing the same game in Kindergarten. Out of 20 students, only 2 (one being my son, since he already struggled with it at home and had made some headway with clarifying what happens when you add 0) students were okay not changing the value of the addend when added to 0. The others added at least one more to their addend, or just sat there and said they lost a turn because they got a 0. ????

Why the struggle?

Students use their instincts when learning. While playing (without formal teaching), the preschoolers made sense of the zero. When you add nothing to a number, it stays the same (AKA Additive Identity Property). However, this ‘sense-making’ was left behind once (I will use my son as the example) Chris started learning addition. He figured out that when you add numbers, the value changes. Every problem he did resulted in a larger value than the two addends. Mama, it gets bigger as you add. When he rolled a 0, he couldn’t make sense of that with his understanding of what addition IS. We had to roll LOTS of 0’s before he finally clicked that adding nothing doesn’t change the other addend.

What Can You Do?

Allow you child to play with a die that has a 0. Allow them to make sense of this new phenomena and open their eyes to new learnings about addition. This will help them later, when adding different kinds of numbers (like negatives) results in smaller sums.

Remember, Zero really is a Hero!

Tiny Human Perspectives: What About 0?

Race to the Top: Kinder Observations

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Last week I posted a game called “Race to the Top”. I was able to play the game with all of the students in my son’s kindergarten class. I was one of the stations during center time, and had between 4-6 students each rotation. We kept them 15 min per station, which seemed enough time for their engagement. (One group ran at 20 min and it was too long.) Each person at my station had their game card (see links below for options) and each pair had one number cube (labeled either 0-5, 1-6, or 5-10) and one dot die (1-6 dots).

Here are a few observations and patterns of misunderstandings.

  1. Counting different representations is tough! The students were used to having to ‘add’ (i.e. count one-by-one) two dot dice, but hadn’t yet used a number cube WITH a dot die. When we started to play, at least one in each group said their sum was 2, no matter what was on the dice. This is because there were 2 dice. It only took a few rolls for all of them to get the hang of it.
  2. For this time of year, the 0-5 number cube was too easy (with the exception of 1 pair of students). I would use the 6-10, as most of our misconceptions surfaced once they passed the number 10. Here are the misconceptions after 10.

Example: 9 + 5 dots

  • Misconception 1 (counted backwards from their number):  “9…8, 7, 6, 5, 4.”
  • Misconception 2 (started counting by 10’s):  “9…10, 20, 30, 40.”
  • Misconception 3 (counted them separately and got stuck): “9 and 5…pause…”
  • Misconception 4 (wanted to put 14 squares on the game board, not see it as one-14)

After about 2 minutes of playing with me direct instructing, the students were successfully independent, taking turns and excited to see what their value was. In 15 minutes, I was able to watch, assess, and intervene every child in my group several times. They were engaged and excited to be playing. We averaged 18 sums per player in the 15 min, which is a LOT of addition!

Favorite Quotes of the Day

  • You cheated! You can’t keep rolling until you get your number! (This would have been me as a child!)
  • Wait. We can’t play anymore? Awe.
  • I only need two more 12’s to win! You need two more 7’s! It’s a tie! (I get giddy when students do more math than asked!)
  • Thanks for playing games with us, Chris’ Mom!

Game Board for 0-10 (Change as needed depending on the number choices for your dice)

Race to the Top Game Board

 

Race to the Top: Kinder Observations

5 Easy Daily Math Ideas

Jen Moffett, , , , 2 Comments

I was walking Chris (5 yo) to school yesterday (as we do most days) and was stopped by his Principal. He commented, “You always are counting when you come to school. I wish more parents would do that with their kids.” So it got me thinking; what are some easy-peasy ways parents can support their kiddos with math in the early years? Here are 5 ideas that you could start TODAY! Choose 1 or 2, and use them every day for a few weeks to really see their number sense and mad math skills bloom! Please comment with other ideas as well so we can have a huge vat of fantastic learning opportunities to use with our babes!!!

  1. Counting, Counting, Counting! We do count on the way to school. EVERY. SINGLE. DAY. At first, it was to 20, then to 50, and now we are up to 150! We count by 1’s most days, but sometimes we go rogue and count by 10’s (gasp!). Just like singing the alphabet, rote counting is a must to learn numbers. Start with 1’s, then 10’s for Kinders and younger. Move to 2’s, 5’s, 3’s, etc… and you have provided a solid foundation for initial multiplication! Move to counting by 1/2’s, 1/4’s, 1/3’s (you get the idea) and you are rocking initial fractions! Start with a different value that 1 and you are moving mountains!
  2. Counting Forwards AND Backwards: Stairs are great for this. Count up when you go up stairs, down when you go down. Don’t know how many stairs? Start at 20 and count down. If you don’t hit 0, oh well!
  3. I have...: In the produce section of the grocery store, you need 6 potatoes. Ask you tiny human, “I have 2. How many more do I need to have 6?” This is AMAZINGNESS! I cannot tell you how many teachers have students who struggle with missing addend problems. This will help so much!!! Do it at home with the silverware. “We need 4 plates. I have 1. How many more do I need?” The opportunities are endless for this!
  4. Sort! Sort! Sort! Kids can sort the silverware that comes out of the dishwasher (Take out sharp objects first, please!), socks, toys, coins (when you are at a restaurant or doctor’s office), mail, school work, books, buttons, etc. Sorting is super important, as it builds the idea of structure and patterns as well as organizing and classifying information.
  5. Guess my Number: A car-time fav in our family. “I am thinking of a number. It is bigger than 5 and smaller than 10. What is my number?” And you can amp this up for older students as well. Use multiples, even/odd, negative values, fractions, square roots, etc. And once they get the hang of it, each person in the car gets a turn to be the number-chooser. Super easy, and builds magnitude of numbers, place value, and relational thinking.
5 Easy Daily Math Ideas

Race To The Top! Counting On

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So my tiny human, Chris, loves to add, but always has to start from one (no matter what the values are). I finally figured out a game to play to push into Level 2: Counting On (see previous blog for more info on the levels of single digit addition/subtraction).

Objective: To get to the top of one of the columns (doesn’t matter which one) first!

Materials: 2- Race to the Top Game Boards (click on link) Race to the Top Game Board 2-dice:  Since Chris is working on making sums to 10, I created my dice. You do not have to, but it is super easy and cheap! Go to a hobby store or home improvement store and buy wooden cubes. On the first one, label with the numbers 0-5. On the second one, draw dots on it (like a normal die), but use 0-5. Notice my mad dot-drawing skills! img_8144.jpg

How to Play: First player rolls the dice. He ‘traps’ the number (in the case above, the 3) and counts on one-by-one if need-be the dots. He puts an object above the sum’s number on the game board. Please view the example. Note the time it takes for Chris to figure it out, but he still gets it. This is tough when children are used to adding with objects and we replace one set of objects with a number!

Notice how he is also doing some mental work with figuring out “how many more” he needs to win the game.

IMG_8148Player 2: Roll dice, count on, and place the object on her game board. Continue until someone reaches the top of a column.

Look at how many addition problems he had to complete to finish the game! He never even noticed all the fantastic work he did! (Insert evil laugh.)

For Differentiation

Level 1 (Counting All): Use dot dice for both instead of one dot and one number cube. That way he is working on his one-to-one counting and adding all together! You can also make a 0-6 page and use 0, 1, 1, 2, 2, 3 dots for each cube.

Level 3: Use both number cubes instead of dots. You could also use number cubes 1-6 and create a game board from 2-12.

 

 

 

Race To The Top! Counting On