Playing with Math: The Function Machine

I love this game, because you can tailor it to the level of your child. Any child, preK through adult, can play and find it challenging and fun!

The Function Machine

Grade Levels: Any, depending on how complex you make the function.

Materials: paper and pencil (whiteboards or Magna Doodles are great, too!  We just used the Magna Doodle in the car on a 3-hour drive and it worked great!!)

  1. Draw a function machine. It need not be fancy, but if you google image function machines you will get tons of ideas. function machie twoMine (as I am on the lazy side) typically look like a rectangle with an opening at the top (for the input) and an opening at the bottom (for the output). They are just very boring and sad. The one to the right is super cute and soooo not me.
  2. Choose a rule (function) to use. So my rule for my PreK could be “one more” or “+1”.
  3. Your child gives an input number. You give the output number. Example: My son loves the number 5 (after all, he IS 5).  So he would put in the number 5 and I would tell him the output is 6. [It is better if you make cool sounds as if the function machine is DOING something. This is supposed to be FUN, remember???]
  4. Your child keeps guessing numbers, as you fill in the outputs each time, until they realize the pattern and can express it either in words or as a ‘math rule’ (expression). If they guess the rule wrong, keep playing until they get the correct rule.
  5. This is really fun when you have more than one kiddo playing. However, they may only guess when it is their turn. That way, everyone has a chance to play and learn.

Let’s Try One! 

  1. You enter 5 into the function machine. beep boop beep boop! Out comes 7.
  2. You enter 2 into the function maching. beep boop beep boop! Out comes 4. (Got a guess for what the function rule is????)
  3. You enter 10 into the function machine. beep boop beep boop! Out come 12.
  4. You say the rule is “plus two each time”. Yup! The rule is the machine adds two to each input (or y=x+2 for your middle schoolers).

Suggestions for Different Grade Levels:

Grades K-2: add 1, add 2, add 5, add 10, add 100, subtract 1, subtract 2, subtract 10, subtract 20, etc

Grades 3-5: same as above (Fluency in arithmetic is important!) and multiply by 2, multiply by 5, multiply by 10, take half, divide by 10

Grades 6-up: combine operations multiply by 2 and add 1 (2x+1), triple a number minus 1 (3x-1),  a number times itself (square it), take half of a number and add 1 (1/2x+ 1), etc. You can also include negative rules as well (like multiply by -3).

You can also create a function table and just fill in as input/outputs as well, if they are above having a cute function machine. I have inserted one below as an example. function table

These are so great for trips and such, because all it takes is a napkin and a pen (and your rule). Finding relationships is imperative for algebra. This is a great way for kids to play with the ideas they need for later mathematics!

 

Playing with Math: The Function Machine

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!)

https://sumboxes.com/collections/types?q=52%20Pickup%20Card%20Decks

 

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

How Many Are Hiding?

This is a great game for tiny humans in need of some entertainment while waiting at a restaurant. Just make sure to ask for an extra children’s cup to play.

Age Level: 3-6 year olds

Materials: children’s plastic cup (any cup will do, just not transparent), a set of objects (I used goldfish for the example, as that was what was in my purse! Other options are pennies, beans, tiny annoying toys, etc.)

How to Play: Place out a certain number of ‘stuff’. Normally for me, it is however many I have in my purse! I would recommend starting with 5 or less, see how they do, and adjust as needed. The first time I played with my 5 year old, we started with 10 and it was quite frustrating for him. They catch on and you can up the number as they grow!

  1. Have your child count how many there are. IMG_5269
  2. Have your child close his/her eyes. Hide some of the objects under the cup.IMG_5276      IMG_5275
  3. Ask your child to open his/her eyes. Ask the following:
    1. How many do you see?
    2. If there were _________ to start with, how many are hiding?
  4. Allow your child to check their answer by lifting the cup.
  5. Ask the following:IMG_5274
    1. How many were hiding?
    2. How many were out?
    3. How many in all? (Woah! It is the same as what we started with! Weird!)
  6. Switch who hides and who plays.

Why play? Aside from counting one-by-one and ‘holding’ that number in their heads, students need lots of practice understanding that a number can represent an amount. That amount can be broken into parts (decomposed), but when we put them together (add them) they make the original number we had. This is part of the idea of conservation, which is critical for young mathematicians to understand (not the word but the idea) in order to add and subtract numbers.

Special thanks to my tiny human for playing this morning! Love ya, bud!

 

 

How Many Are Hiding?

Playing With Math: Circles and Stars

Ahhh…summer. For many of us, that means more time with the kids…waiting. Waiting at a restaurant, doctor’s office, airport, etc. For many kids, it may also mean waiting to use their brain. Research suggests that students can lose as much as 2 months of learning skills during the summer months (Oxford, 2017).  So how can we use the waiting times (or times at home when they are claiming boredom) to retain and advance their learning in mathematics? Play games/activities.

While there are many great apps for kids, I would request less screen time and more interaction with your children.  For the next three months, I will suggest a game/activity that you can use with your child. I will suggest different levels, so that you can play it often and in different ways. I use these same games with my own children, and find the time waiting goes much quicker, with less outbursts and meltdowns. Further, I am modelling playing with math, which is truly the way I feel our children learn and understand math best.

Circles and Stars (Marilyn Burns)

Grade Levels: Though used in grade 3, if all you are doing is counting the number of stars I would recommend grades 1-5. My preK has played it and just counts one by one. He cannot make the stars, so he draws x’s.

Materials: die (number cube or dots; doesn’t matter), paper or napkin, pen or pencil (I prefer a travel size Magna Doodle or whiteboard with dry erase marker)

LEVEL 1

  1. Roll the die. Draw that many circles.circles
  2. Roll a second time. Draw that many stars in EACH circle. stars
  3. Total the stars. Whoever has the most stars wins the round. (Play as many rounds as you want. The winner could be the one with the most stars total. Woo hoo! More math!)
  4. Alternative: The winner is the player with the least amount of stars.

LEVEL 2

  1. Roll 2 dice (or the die twice in a row). Player chooses which die represents the number of circles and the number of stars in each circle.
  2. Total the stars. Whoever has the most (or least) stars wins the round.

Questions to ask:

  • If I was to switch which die represented the number of circles and stars, what would happen? (The picture would look different, but the total stars would stay the same. This is the beginning of understanding the commutative property for multiplication.)
  • How could we represent what we did in words? (Example: 4 groups of 3 stars is 12 total stars.)
  • How could we represent what we did as an expression?                                   (Example: 4 x 3 = 12)

Link for Summer Learning: https://www.oxfordlearning.com/summer-learning-loss-statistics/

 

 

Playing With Math: Circles and Stars

Cool Tools for Kids in Math

Happy Mother’s Day AND Teacher Appreciation! Here are my gifts to you: FREE APPS and Sites to help your children (and students) learn math!!! Read on!

These past three days I got to geek it up at the NCTM annual conference in San Francisco. I have gone to several annual conferences, but this was the first time I worked in an exhibitor booth rather than attending as a participant. I was excited to be on the other side of the conference scene, but sad that I wasn’t sitting on the carpet (like so many) scouring the magazine o’ options for the perfect sessions.

As a teacher, I would ditch the exposition hall (except to get the free Legos and swag for my boys!) and attend every session I could. I would take copious notes, trying hard not to miss anything that was said in case THAT was my take away for the trip. Those fabulous notebooks that I poured my 72 hours of the conference into gather dust in a box in the garage. Don’t get me wrong; I would typically use 3-5 ideas/worksheets/tasks/quotes per conference. But was that worth the hours I sat in the back of a crowded room? Was there more to the conference than the sessions?

YES! This year I attended a single 60 min session and got several great ideas for a district I work with. The rest of my time was spent in the exhibition hall talking to reps (and long-time friends!), discussing mathematics, and truly learning from one another in a more intimate setting. I learned so much in these conversations, AND spent time at many vendor booths playing with the technology that I believe can truly make a difference in how students view mathematics. Though I am still grappling with the lack of notes in my handy-dandy notebook, I feel I am leaving with far more applicable ideas and tools than ever before!

The links below are (free!) sites you and your child can explore to really learn mathematics. They allow students to truly see what is going on and why the math “is what it is”. I hope this summer you are able to spend some time on these sites and give your students an opportunity to open up mathematics in amazing ways.  

Note to teachers: These are open source and free to use on your devices at school. You are welcome!

NumFu: http://www.origoeducation.com/num-fu/?mageloc=USnum fu

Currently their Mastering Addition Facts app is free. Get it now before they change this! Students work on their math facts in a developmental way, understanding as they gain mastery. (They do have a multiplication app as well, but it is not free.)

 

DESMOS: https://www.desmos.com/Desmos 2

This is a site that allows you to graph functions, plot tables of data, evaluate equations, explore transformations, and much more! Desmos

 

GeogebraGeogebra: http://www.geogebra.org/

Geogebra makes a link between geometry and algebra using visual representations students can manipulate and finally see what is going on mathematically.Geogebra 2

 

math Learning centerMath Learning Center: http://mathlearningcenter.org/apps

These apps are amazing! So many to choose from to help students conceptually understand mathematics. There are number lines, geoboards, money pieces with a number rack, rekenreks, ten frames, pattern blocks and more! Just allowing your child to play with these apps will enhance their understanding of number! Here are just a few from the site:math learning center 2

 Finally, this is a book that came highly recommended. Though not free, it can be a support for parents in navigating Common Core mathematics. (It does come with videos as well!)CCSS Math

http://www.amazon.com/Common-Parents-Dummies-Videos-Online/dp/1119013933

Cool Tools for Kids in Math

Everything I Learned, I Learned From Teaching: The Resignation I Wish I Could Have Written

Just an FYI: This is not a blog for learning math. This one is to thank the many individuals who have shaped my being, both in and out of school.

So I am super bummed. I had to write a letter of resignation for a job I love. With that, it felt like I was writing a resignation for the CA portion of my lifetime. Though I still do some important work with amazing teachers in CA (Cuca peeps you aren’t getting rid of me yet!), I am no longer affiliated with a school, a set of students, or that place to call my teaching home base.

A bit of recap. My husband was given this amazing opportunity to make a difference in the lives of our Wounded Warriors. This is not something our family takes lightly, as I am pretty sure between the two of us we have had family in every branch of the military. So we took it on for two years. My district was generous in giving me a two-year leave of absence. However, we have fallen in love with our current location and the work we both are being able to do. So….We are taking a HUGE leap into the world of unknown and seeing where these opportunities take us.

Teaching is something I was born to do. I love learning and I love watching that ‘spark’ children get when they have an ‘ah-ha’ moment. Yet it is through the village who raised and sculpted me that I feel confident today in my teaching. Here are the top 5 lessons I have learned through teaching that have transcended my professional life and helped make me who I am today.

  1.  “You will understand once you have your own children.” I had been teaching a whopping year when this got thrown at me. I was a middle school math teacher and thought I was a bad ass. I had a great rapport with most of the students, loved the curriculum, and was a fav with my colleagues. When parent-teacher conferences came about during my second year, I thought I had it down. A mother sat with me, listened to all I said about her child, and then responded with a knowing look this phrase. I was floored and was ticked off. How would I learn more just from having my own kids?! Wasn’t I great now? No, I was not. And yes, she was right. I am far more patient now. I understand when students don’t have their work done. I get it that there is (gasp!) more to life than math. And from the moment I had my first, I worked harder at creating young learners who cared about learning and each other rather than simply good seventh grade math kids.
  2. “If you show you truly care, the mistakes you make will be forgiven.” After about 12 years of teaching, I became a curriculum specialist for our County Office of Ed. My mentor, Carol Cronk, is amazing. I could write novels about the lessons I take back to my classroom from her. The first few weeks I was able to just shadow and watch how she worked with teachers and administrators, and we would debrief after. After a Framework PD session, she told me some of the mistakes she had made, and that she knew the teachers knew she had made them as well. When I questioned her about it, this was her reply. And it’s true. If I made a mistake with my students (or teachers), they forgave me. Just as I have forgiven them. It’s the level of respect you have for one another. And I think it works in life as well. If you show you care (and really mean it), people are far more accepting. This lesson just hit me again as I watched so many different journalists report on Super Tuesday… How many of the candidates have my respect? How many of them can I forgive? How about you?
  3. “You need to see it the way students do.” Ahhhh….Dr. Fischman. She has been the toughest on me, and I have probably learned the most from her. Let’s backtrack. I love numbers. See them everywhere and do fabulous things with them. When I was working with Dr. Fischman, procedural methods and ‘playing with numbers and formulas’ were indeed my strong suits. But that is not what we were teaching. I learned that there are other ways of thinking about mathematics. I learned to listen to my students and use what they knew to tap into the mathematics I needed to teach them. I learned to LISTEN, and not just be heard. (Still working on this one, though!)Thank you for that.
  4. “Fake it Til You Mean it” So there are many things I love to learn about, technology NOT being one of them. I feel so helpless with technology, so I typically steer clear. However, after going back into the classroom, I was pushed kicking and screaming into SMART boards, responders, laptops, etc. My dear friend, Aly, would say to us during our tech trainings, to suck it up and literally, fake it til you get it. And I would. I would fake that I knew what was going on. And it totally worked! As I faked it, my confidence level rose, I tried new “stuff”, and so on. It really helped me get out of my phobia (well…still a bit fearful…). So thank you, Aly, for pushing me onward and not letting me drown in my fears.
  5. “I Will End My Days Teaching” One of the teachers at my first middle school position was an Immersion teacher for English Learners, grades 3-8. Linda Chace is amazing (and I had her husband for fifth grade- small world!). However, I knew that she had been an administrator for several years prior to this position. I asked why she left the administrative side. She said she wanted to finish her time teaching. I still take this to heart. Do what you love. Do other things and learn from them, but always go back to what you are passionate about. Make a positive difference in the world, while enjoying what you do.
There are so many others I could add to this list. Yet these five nuggets of truth stick with me. Thank you for supporting me. Thank you for helping me grow. Thank you for making me be better for our children.
Jen

 

 

 

Everything I Learned, I Learned From Teaching: The Resignation I Wish I Could Have Written

Response to Confusion 43-13

So today a friend tagged me in a FB post regarding the “frightening” method that students MUST solve subtraction problems. I have posted the link below, and I believe the link is at the bottom of this post as well! Take a look.

So let’s summarize the tutor’s concerns. 1. That we are writing problems horizontally rather than vertically. 2. That students are using a strategy of “adding up” rather than “stack and subtract”. 3. They MUST use this strategy and no other.

I would like to address each of these and provide some comments.

  1. Most of us are used to seeing math problems vertically. Why? Well, for one, it makes the problem ready to go for  the algorithm  of “stack and subtract” (which is not the ONLY algorithm in the world, mind you). I would contend it also saves space for publishing companies in their workbooks. If the problem is already written for the algorithm, publishers do not have to provide additional space and therefore can fit more problems on a page, and save money. (Yeah, I went there. Bring on the comments!) However, when you write a subtraction problem vertically, you lose the essence of the numbers.   IMG_7211Here I only see single digit subtracted by single digit. I do not see 40 – 20, but 4 – 2 and 5 – 1. This is okay for problems where we do not have regrouping. However, some students get so stuck in the process of regrouping that they no longer see the value of the places and just write a very ‘random’ value as the difference. When I write the problem horizontally (45-21), it allows me to view it from a place value perspective. My eyes look first at the tens and then the ones, versus the horizontal example where I start with the ones and then look at the tens. Also, writing it horizontally does not constrict me to a “stack and subtract” method. (See prior blog for more info on great subtraction strategies to help kiddos.) Really, both are fine; no need to get all uppity about it. And if a teacher says they HAVE  to write it that way, it is not true, but what is true is that they learn to think about relationships and different strategies using place value and properties of numbers BEFORE learning the standard algorithm. In fact, the standard algorithm for multi-digit subtraction should not be mastered until grade 3: CCSS.Math.Content.3.NBT.A.2
    Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. So students have lots and lots of time to process what it means to subtract and its relationship to addition.
  2. (and 3.) Adding up is one strategy students can use. Makes sense to me! If I already know how to add, I can simply use that to help me figure out the missing addend. This is all a subtraction problem really is: a missing addend problem! Consider the following: 43 – 16. This is really finding out 16 + ____ = 43. Now typically, students do not need to go to the next five as the tutor suggests. And really a number line is FABULOUS for modeling adding up. Here is one way to get the value. IMG_7213Notice this is a great strategy for students who struggle with regrouping, because there IS NO REGROUPING!!!  I went up 4 to the nearest ten (20), added 20 more (40), and ‘hopped’ 3 more to get to my end point (43). 4 + 20 + 3 is 27. Therefore, 43 -16 = 27.                                                               Using the tutor’s problem (43-13), I think adding up is efficient, if you move up by tens. I can simply add by tensIMG_7214 until I reach 43. 3-tens is 30. Not sure why it is so convoluted in her explanation and NO! Students do not have to add up the same way the tutor did. In fact, that is the wonderful thing about Common Core. CCSS.Math.Content.2.NBT.B.7
    Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. Notice that students can use any method, with or without drawings or concrete models!!! Super awesome! Hooray for creativity! I no longer have to use one method (that I don’t really get and just memorized because my teacher told me to); I can use any method so long as I keep in mind place value and my rules (properties).
“Our classrooms are filled with students and adults who think of mathematics as rules and procedures to memorize without understanding the numerical relationships that provide the foundation for these rules. The teaching of mathematics has been viewed as a discrete set of rules and procedures to be implemented with speed and accuracy but without necessarily understanding mathematical logic. For the majority of our nation, knowledge of mathematical rules has not allowed them to use math confidently in their daily lives. With almost two-thirds of the nation’s adult population fearful of mathematics, they have simply said “NO” to math and closed the doors to careers that require higher math” (Burns, 1998; Parrish, 2010).
Let’s allow students to make sense of number relationships, what the operations MEAN, and figure out what makes sense and how to approach the mathematics based on the numbers given.

 

Response to Confusion 43-13