Cutting the holiday pie

A good opportunity for exploration and problem-solving for your students.

With one straight cut, you can cut a pie into two pieces. Cutting it again and having the second cut cross the first cut gives you four pieces. With three cuts, you might get up to seven pieces.  What is the maximum number of pieces you can get with 6 straight cuts?

Your cuts do not need to produce equal pieces.  Once students identify the solution, can they keep going with more cuts?  Is there a pattern that begins to emerge?

Photo by Alex Loup on Unsplash

What Does the Data Tell Us?

In a recent conversation with some representatives from a major nationwide business company, we were discussing the needs in elementary math education to help lead students to be productive workers in tomorrow's business world.

The big question they said that students need to learn to answer is, "What does the data tell us?"  We also discussed the importance of probability and statistics,  but in the end, they said that they are looking for workers who can answer this question.

What would this look like in an elementary classroom?   I think it just reframes our questioning.  In many of our activities, we already expose students to real-world information.  We just need to be sure to be more intentional in our questioning in order to get them to look at data.

For example, look at this picture.  I took it thinking that it would be a good one for students to determine what was the best deal. What does it tell us?  What are some possible reasons a person would be willing to pay more for 3 Peep trees when they can get 9 for such a better deal?  

Noticing and wondering is a classroom routine that really benefits our students.  When they notice and wonder, you can ask them what the picture tells them.  Make them infer from the data that they have.  Numberless graphs are a good method of providing data that the students have to make sense of.  Here is a good example of a 2nd grade lesson regarding them.  You can find lots of examples and ideas for using them by reading some of the posts listed here.

What do you notice about this data?  What do you wonder?

We want our students to be able to compute and do basic math, but not at the expense of good math thinking and discussion.  These are the skills that will carry them into the future--not only in their career, but also in their roles as consumers and citizens.

How can you incorporate statistics and probability into your classroom (whether it is a K classroom or a 5th classroom) by framing your questions and your students' thinking around the question, "What does the data tell us?"

Continuing Thoughts on Math Facts

Some of the biggest concerns I get from teachers revolve around math facts.  Oftentimes, when they hear my response, they think that I am saying that math fact fluency is not important.  That is not true.  I do believe math fact fluency is important; I just don't believe that most of our class time should be spent on rote math fact practice.

This is a topic of much discussion among math educators, and books continue to be written addressing the ways to build true math fluency.  Graham Fletcher and Tracy Zager are piloting a math fluency kit that should be available soon, and I can't wait to see it in action!

I have been recently reading No More Math Fact Frenzy by Davenport, Henry, Clements, and Sarama.    It reinforces the ideas that I continue to communicate to teachers.  Rote memorization is not a method for students to best learn their math facts.  It does not lead to a true understanding or flexibility of number that defines fluency.

Some ideas to consider:

Count.  Choral counting is a great way to build number fluency.  Counting forwards, backwards, by different multiples....This not only allows students to think about the strings of numbers, but it helps to build a mental number line for students which is so helpful for future success with mathematics.  Learn more about choral counting here.

Make your fact practice visual. Create structure that students are familiar with, and have them see the fact rather than just memorize the fact.  For example:

Seeing 7 x 6 this way helps students to visualize how knowing 7 x 5 can help them get the answer for 7 x 6.  Using subitizing is still important as students get older!

Rekenrek and ten frame visuals are also great for addition and subtraction facts.  There is nothing wrong with students using fingers for a while either.  They are a built-in tool. We hope that they eventually gain the confidence to know the sums and differences without their fingers, but if using their fingers helps it to make sense, let them!

Use number talks.  Asking students to use dot cards to make sense of math facts is important because it allows them to decompose numbers in different ways and to hear different people's strategies.  The above image could be an example of a dot card number strategy for older students, but more simple dot cards for primary will also allow for students to see the fact.  Traditional number talks with numbers written horizontally also build fact fluency through exposure to multiple strategies.  For example:

When students discuss the way that they solved a problem like this, they gain a better understanding of number.  Maybe one student added 7 + 7 and then added one more.  Another decomposed 7 into 2 and 5 so they could make 10 + 5.  Maybe another started at 7 and counted up to 8.  Number talks give students opportunities to make sense of problems in ways that make sense to them, but they also give you the opportunity to make connections between the strategy they used and those that their classmates used.  The goal is for students to not only think more flexibly but also to look for a more efficient method.

Make connections:  Help students to see the connection between operations.  How does addition help us to do subtraction?  How is multiplication related to addition?  How are subtraction and division related?  Not only asking these questions, but having students explore with manipulatives and discover these relationships will help students to have a better conceptual understanding of the relationship between operations.

Play games:  Games that practice math facts are always good not only because students get the opportunity to use their facts, but also because they get to practice important social skills like taking turns, good sportsmanship, and taking care of materials.  The best games include students having to use strategy besides knowledge of facts.  Encourage parents to play games at home with their children.  This is one of the best ways for parents to help their children become better mathematical thinkers.

Make it real:  Find as many examples as you can in the real world to help students see how the operations are used.  Adding the chairs at one table to the chairs at the other table, finding an array in a display of student work, and talking through the math of your lunch count are all real world ways for the students to make more sense of their math.

Trying some or all of these ideas should help students to gain more flexibility and fluency with their math facts.  I believe it is better use of time than pages of math facts, repeated rote practice on a computer program, or timed tests.  Knowing your math facts fluently does free up some brain space as you work through more complicated mathematical concepts, but conceptual understanding is much more important than rapid fire!

Reminder of math fact progessions:
Kindergarten:  Fluency of +/- facts within 5
1st grade: Fluency of +/- facts within 10
2nd grade: Fluency of +/- facts within 20
3rd grade: Fluency of x and / facts within 100

Playing with Numbers

After reading the book Math Recess by Sunil Singh and Dr. Christopher Brownell, I realized the importance of giving students time to explore numbers by playing with them.  This post shares a couple of ideas for these explorations.

This book is a great read and will make you rethink your instructional practices!

Abundant Numbers:  Have students search for ABUNDANT NUMBERS.  A number is considered abundant if the sum of its divisors is greater than the number.  For example, twelve is abundant because its divisors (1,2,3,4, and 6) is greater than 12.How many can your students find?

Circular Primes:  A circular prime is one that remains prime with the relocation of the first digit to the end.    So for example, 113 is a circular prime:  113 is prime.  When I move the 1 to the end of the number, my new number is 131, which is also prime.  When I again  move the first digit to the end, I get the number 311.  It is also prime; so it means that all 3 of those numbers are CIRCULAR PRIMES.

Happy Numbers: 19 is a HAPPY NUMBER.  How do I know?  To find a happy number, square each digit and find the sum. Continue doing until you find the final number.  If it is 1, then the number is happy.  

Click here to see how I know 19 is happy!

Make it a goal to give your students some time with these ideas.  Can they find more of any type of number?  How many can your class find this week?  before winter break?  this school year?  Can they prove that the numbers they found fit the definition provided?

Let your students spend time playing and thinking about numbers.  

Put Math in their Hands

I often talk to teachers about the importance of concrete representations for our students.  The use of concrete tools helps all students to gain a better understanding of the function at hand.  It also creates a visual representation that will be a reference for students to use as they advance into more advanced mathematical ideas.

In this post, I want to share some simple concrete representations that teachers can help students develop as they work to make sense of a new concept.

Doubles/Doubles +1: Both the rekenrek and ten frames help students to make more sense of what a double is and how we build from that to doubles + 1.

Doubles on ten-frame:  3 + 3 =6

Doubles + 1:  3 + 4 = (3 + 3) +1 =7

Doubles on rekenrek: 6 + 6 = 12

Doubles +1:  6 + 7 = (6 + 6) + 1 = 13

The rekenrek is a great tool for doubles facts between 10 and 20.  Student can see that all of the green beads have been pulled over, which is 10.  They only need to add on the white beads.


Addition/Subtraction within 100:  For this skill to take hold, I find beaded number lines and base ten blocks to be good concrete tools to use because they carry over so nicely into an open number line representation.  Open number lines are important for students to use to build number sense.

Addition on Beaded Number Line:  17 + 35 = 52

Addition with Base Ten:  17 + 35 = 52

Subtraction on Beaded Number Line:  44 - 18 = 26

Open number lines are important, but they are often too abstract for many of our students.  Combining the use of the beaded number line or base ten blocks with the creation of an open number line helps for it all to make more sense for the students.

Multiplication:  For learning multiplication facts, creating arrays is a nice way to start.  This should help students to see how repeated addition is connected to multiplication.  Any small object will work to help students create arrays.  Using the number card templates will allow them to create a visual more quickly at times when they need a visual but don't have time to build an array.

Multiplication with Base Ten Blocks:  22 x 23 = 506

The 22 and 23 were created at the top of this image and on the left side of the problem using base ten blocks.  Multiplying a blue ten and anothe blue ten gives you an orange 100.  The same continues until we end up with 4 hundreds, 10 tens, and 6 ones for a total of 506.

For multi-digit multiplication, building the array using base-ten blocks works well to connect the arrays that student made for facts under 100 with the longer problems that they are ready for now.  Besides--the base-ten blocks transition nicely into graph paper representations and finally into the area model of multiplication for multi-digit numbers.

Decimals:  The beaded number line can be used for addition and subtraction of decimals less than 1, and it can also be used to round and compare decimals.

Rounding on the beaded number line:  0.86 is closer to 0.9 than 0.8

Adding decimals on a beaded numer line:  0.6 + o.14 = 0.74

I hope that you find some of these ideas helpful and that you find ways to incorporate them into your explanation of numbers with your students.  There are many ways that can be used for different problems.  If you have a concrete tool that works well for teaching these skills, don't think you have to switch to one of these--do what works best for you and your students.  The key is to get the concrete math in their hands!  This way it will stick in their heads.  😊

THIS is the Message to Begin a New Year

What a great thing for us to do for our students: Help them understand the value of mistakes!  And now, at the beginning of the year, it sends a great message to all stakeholders:  This classroom celebrates mistakes!

"I have not failed.  I've just found 10,000 ways that won't work."  

Thomas A. Edison

Using Twitter to grow

If you are looking for a way to grow as an educator this year, I would recommend becoming more active on Twitter.

I'm sure you have other social media accounts, but this one will give you so much professionally!  Remember that there was a time when you didn't know how to upload a photo to Facebook or how to use a filter in Instagram or Snapchat?  You took them on and figured them out!  You can do the same with Twitter.  It doesn't take very long to become more comfortable with the platform.

I use Twitter now mainly as a math educator, but when I first became familiar with it, I used it to gain access to technology and classroom ideas.  It is all about following others.  You'll be amazed at the number of ideas and free resources you can find!

I would also recommend trying one of the many weekly chats that are on Twitter.  The first time or two, you may want to just read others' Tweets, but after a few times, you'll be ready to give it a try.  People on Twitter are very helpful and understanding.  Here are some weekly chats that happen throughout the school year that I would recommend.

#elemmathchat   :  Thursdays at 8:00 pm CST Math leaders from around the country lead discussions of different math concepts
#mathstratchat   :  This chat is not a live chat like the others.  Pam Harris posts a problem on Thursdays, and people tweet the way that they solved it.  It is amazing to see so many flexible thinkers!
#mathconceptions   : Mondays at 8:30 pm CST  half-hour long chat with great people around fun topics
#kinderchat, #1stchat, #2ndchat...:  Search these hashtags to learn more about each grade levels chats.  They are usually weekly.

At the time of the chat, just search the hashtag and then go to Latest.  It will help you see the conversation unfold...

If you are looking for some people to follow, here are some people (mainly math people) that will get you started.  Also, most of your schools have a Twitter account.  You can follow them, too.
@pearse_margie
@joboaler
@mraspinall
@educatormomof3
@gcouros
@Trianglemancsd
@JoyKirr
@Mathgarden
@eriksonmath
@dbc_inc

Give it a try!  I am happy to help you however I can. Before you know it, you'll be tweeting, retweeting, and using hashtags! #goals