Show What You Know

One of my favorite math activities is to ask students to show me why math works they way it does. There is no better way to get to conceptual understanding – and beyond procedural knowledge – than having students explain and show how math works. This is a great idea to keep in mind during school closings. Students could show what they know using a variety of online tools.

My daughter is in fifth grade. She’s been working on multiplying fractions. So I asked her today to show me why the product of two fractions is less than either of the factors. I purposefully used academic vocabulary in the prompt – product and factor.

This was the prompt I wrote on the whiteboard in my daughter’s room today: Show me why the product of 2 fractions is less than each of the 2 factors.

I let her think about the prompt first. Then I asked her if there was anything she needed clarify or if there were any questions she had. Her first question: Do I need to use any particular fractions? No. Do I have to show it a certain way? Can you tell me more about “a certain way?” Can I use fraction circles OR fraction bars? You can use any visual model that works for you.

There were other questions like this as she worked through the problem. This one activity gave me more insight into her understanding of fractions than if I had given her 10 multiple choice questions related to multiplying fractions.

It’s such a simple activity but the results are so powerful.

Number Sense Activities

There are two things I love in math – activities which build number sense and puzzles. There is a great need to promote number sense in our classrooms – especially at the elementary level.

British researchers studied low, middle, and high achieving math students solving problems. What was the difference between the low and high achieving students? High achieving students used number sense to understand and solve problems.

Kwame Sarfo-Mensah shared three great numeracy activities in a recent Edutopia post which are great.

  1. The Hundred Challenge – Students use four given numbers (i.e. 1, 4, 6, 8) to create equations which equal each of the numbers from 0 to 100.
  2. Do Now: Equation or Expression of the Day – Students solve an equation which represents that day’s date.
  3. Cryptarithmetic Puzzles – Digits are replaced by letters, and students have to solve the letter/number puzzle.

I’m excited to try some of these with my class. They might need to be adjusted a bit for elementary students, but they’re too good not to try.

Open Middle Problems

That’s the problem I presented to my students today for our math warm-up. It seems like a basic problem, but the conversation it started was wonderful.

I got the idea for the problemfrom Open Middle. As his website states, most of these problems have:

  1. a closed beginning meaning that they all start with the same initial problem;
  2. a closed end meaning they all end with the same answer;
  3. an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.

We started with this basic problem today. Students were given time to work on their whiteboard and talk with a partner before we shared as a class. One student, Jimmy, pointed to his partner and asked, “Mr. Rashid, we have two different answers. Can we have different answers and both be right?”

My answer was, “Can you prove that you’re both correct?”

Jimmy and his partner took turns explaining why each answer was correct. Then I asked, “Jimmy, can you have different answers and both be right?”

He smiled and said yes.

Students were excited to share their thinking with the class. Each student shared their answer and explained how they solved the problem. We will increase the complexity of the problems as we move forward, but this was a great way to introduce the idea to students and help them understand the process.

This was Jimmy’s solution to our Open Middle problem.
This was his partner’s solution.

There are many great examples on the Open Middle website. They are organized by grade level and standard. This resource makes diving into the process a little easier. I hope it works as well for you as it did for my students.

Kids Love Problem Solving

Kids love problem solving. I really believe this. They like to figure things out. If given the right environment, students like to challenge their thinking and struggle with concepts.

I was reminded of this last weekend when I checked my “Shared With Me” file on Google Drive. I had a number of files from students titled “Math Puzzles.” A few of my students created puzzles to challenge their classmates.

The student-created puzzles really showcased students’ individual interests.

Everyday we start our math lesson with a math puzzle and some mental math problems to build number sense and improve our problem solving. My students love completing these tasks, so some of them they made their own puzzles.

Some students used adorable pictures of cuddly animals. Others used their favorite sports teams or video game characters.

This got me wondering if students really don’t like math, or do they just not like how math is typically presented. I had six students who spent their own time creating puzzles. I didn’t tell them they had to do it or assign it for work. They enjoyed the challenge and wanted to show their creativity. It doesn’t get much better than that – creativity, engagement, and problem solving. Now I have to keep that excitement going.

I know… So I… To find… Therefore…

Teaching problem solving is a tough task. Students have to read a word problem, understand what to do, complete the computation, give an answer, and then determine if the answer is reasonable. So how do teachers best support students in that endeavor?

There is no magic bullet to help students become better problem solvers. They need to be exposed to different problem types. They need to see, hear, and learn how other mathematicians around them are solving problems. Some students also need a structure to help them solve problems and explain their thinking.

One strategy I’ve use is called “I know… So I… To find… Therefore…” This helps students work through a problem and explain their thinking. Here’s a breakdown of each of those parts.

  • I know… What important information was given in the problem?
  • So I… What did you do to solve the problem?
  • To find… What answer did you get?
  • Therefore… What is the answer with the correct label?

Let’s look at the following problem to walk through the process.

David started his coin collection with 14 coins. He added 3 coins to his collection at the end of each month for 5 months. How many coins are in David’s collection after 5 months?

Students would answer the question with: I know David has 14 coins in his collection. He adds 3 coins each month, and this happens for 5 months. So I added 14 + 3 + 3 + 3 + 3 + 3 because he gets 3 coins after each of the 5 months to find 29. Therefore, David has 29 coins in collection after 5 months. (Students could also say they added 14 + (3 x 5) instead of using repeated addition.)

This takes some modeling at the beginning of the year, and I always supply a graphic organizer with the “I know.. So I… To find… Therefore…” components already listed. I created a copy of the graphic organizer on Google Docs.

Students pick up on the process pretty quickly. Using the graphic organizer slows their thinking, which improves focus during the problem solving process. It also helps improve a student’s explanation by giving them a structure to follow.

Last week, my students were working on problem involving fractions. The problem was: A quarter is 1/4 dollar. Noah has 20 quarters. How much money does he have? Explain.

I gave my students graphic organizers. (We’re not quite ready to take the training wheels off yet.) Here are two examples of how an average student completed the graphic organizer.

This student used repeated addition to solve the problem.
This student used division to determine how many dollars Noah had.

I have adjusted this approach over the years. I’m sure I’ll continue to make slight changes in the future to try and improve the supports I’m providing for my students. What strategies and supports do you use with your students to help them be better problem solvers?

When Altogether Doesn’t Mean Add

I read an article a few years ago that really changed the way I talk to my students in math class. The article is called 13 Rules That Expire. It was published inNCTM’s August 2014 issue of Teaching Children Mathematics.

The gist of the article is that there are math “rules” used in classrooms. These are often presented as rules that always work, but in the case of these 13 rules, there is an expiration date. The article goes into greater detail about each of the rules, including the expiration date (or expiration grade). My goal is not to thoroughly discuss all 13 rules in this post, but wanted to at least list them.

  1. When you multiply a number by a ten, just add a zero to the end of the number.
  2. Use keywords to solve word problems.
  3. You cannot take a bigger number from a small number.
  4. Addition and multiplication make numbers bigger.
  5. Subtraction and division make numbers smaller.
  6. You always divide the larger number by the smaller number.
  7. Two negatives make a positive.
  8. Multiply everything inside the parentheses by the number outside the parentheses.
  9. Improper fractions should always be written as a mixed number.
  10. The number you say first in counting is always less than the number that comes next.
  11. The longer the number, the larger the number.
  12. Please Excuse My Dear Aunt Sally.
  13. The equal sign means Find the answer or Write the answer.

There is one rule I wanted to discuss a little further. Before reading this article, I was a big fan of teaching students to solve problems with the help of keywords. This often included making an anchor chart with the keyword on one side and the meaning on the other side (i.e. in all means add, how many more means subtract).

One common keyword strategy to teach students is that the word altogether means you should add. While that does work most of the time, it doesn’t always work. Take for instance the following problem:

Jimmy read 11 books in fourth grade, and Brian read 8 books. Jimmy, Brian, and Elyse, read 37 books altogether. How many books did Elyse read? 

If students attack word problem by simply looking for keywords to help them solve problems, they will most likely see altogether and add 11 + 8 + 37. If students are thinking about the context of problem, and not just the keywords, they will see that they have to subtract 19 (11 + 8) from 37. This requires students to understand what is happening in the problem and not simply use the keyword as their only problem solving tool.

Teaching problem solving to our students is not an easy endeavor. It takes many tools in the mathematician’s toolbox. It takes constant repetition and exposure to many problem types. Teaching problem solving is not easy, but it is one of the most important skills we teach in any subject area.

Fun with Numbers

It’s always great to show students the wonder of math and numbers in the real world. Today, February 2, 2020 is one of those opportunities to have fun with numbers. Not only is today Super Bowl Sunday and Groundhog Day, but it is also an eight-digit palindrome. A palindrome is a word or phrase that reads the same forward and backward – Anna, nurses run, or my daughter’s favorite book Taco Cat.

Today is 02/02/2020 or 02022020. It is symmetry at its finest. How rare is a global eight-digit palindrome? The last time it happened was 909 years ago – November 11, 1111 or 11/11/1111. The good news is you won’t have to wait another 909 years. There will be another in a mere 101 years on December 12, 2121… 12/12/2121.

But wait, the math fun isn’t done yet. Today is the 33 day of the year and there are 333 days left in 2020. Thank you leap year for that. With all this math fun, it’s like Christmas in February!

Numberless Word Problems

A few years ago I was doing a lot of one-on-one work with a student who struggled in math. We often met before class to do some mental math and try to improve his number sense. After school, we would revisit the skill we discussed in class that day. He made a lot of progress and was starting to feel much more confident about math in general and problem solving in particular. Then we introduced fractions.

Shortly after we started our unit on fractions, he put his head down in frustration. I asked him what was wrong, and he replied, “I was just starting to get math and then… then fractions happened.” It was as if he had contracted a deadly virus named fractions. We talked about how good problem solving doesn’t change just because we moved from whole numbers to fractions. Over the next month or so, we continued to work and his confidence with fractions improved as well.

My approach with that student would be slightly different today. Instead of moving him right into word problems with fractions, I would use numberless word problems to help ease some of the anxiety. The numberless word problems would be a sort of scaffold to the problems with fractions. This would help him see that we’re still using the same problem solving strategies we used with whole numbers.

Numberless Word Problem in 4th Grade

Numberless word problems are just that – word problems without numbers. Let’s look at the following problem.

Brian eats 1/8 of the pizza and Julie eats 4/8 of the same pizza. How much of the pizza did Brian and Julie both eat?

This is a pretty basic word problem involving fractions. Some students might struggle with this problem simply because they are intimidated by the fractions. So, why not take the fractions out all together?

Brian eats some pizza and Julie eats some pizza. How much of the pizza did Brian and Julie both eat?

Now we can have a conversation about what is happening in the problem. We can talk about how one person has some amount of pizza and another person has another amount. We want to know how much they have together. That hopefully leads students to see that this is an addition problem. Once they understand the context of the problem and have determined the operations they need to use (addition), then they can start looking at specific numbers.

Depending on the students and the problem, I might put some whole numbers in the problem before moving to fractions. This will give the student another opportunity to see the context of the problem before the fractions are introduced. I did this today with my class.

I took a problem from our math book and covered up the fractions to create a numberless word problem. This is how I introduced the problem.
After we discussed the context of the problem, and determined the necessary operation, I put whole numbers into the problem.
Then, we looked at the original problem with fractions and talked about how the structure of the problem didn’t change because of the fractions.

This is a wonderful strategy because it helps students focus on the context of the problem. The focus on the narrative portion of the problem before worrying about the numbers. They can’t worry about the numbers, because they aren’t there yet.

Math Clothesline

We just wrapped up a unit on fractions which covered equivalent fractions, common denominators, simplest form, comparing fractions, and ordering fractions. Comparing and ordering fractions gave me an opportunity to use a strategy which I absolutely love: clothesline math.

As the name suggests, I put a clothesline up in my room (really just a piece of yarn) and students use it to create a number line. It’s a great strategy because it works with whole numbers, fractions, decimals, algebra, and on and on.

I fold a notecard in half and write the fraction on one half. This allows the fold of the card to hold the number on the clothesline. Since I’m writing the numbers myself, I can make the numbers fit whatever lesson or standard I need. For example, in Pennsylvania fourth graders are only comparing fractions with denominators limited to 2, 3, 4, 5, 6, 8, 10, 12, and 100. So, I create fractions cards which fit those parameters.

There are times when I will place cards on a table and let students choose their own number. Other times, I want to be a little more strategic with who gets each card, so I will distribute them to students. This allows me to ensure that students are getting fractions that match their math abilities. I give students who have a better grasp of the concepts fractions which are in the sixths, while struggling students might only get fourths. If a student has to draw a model of a fraction, it is easier to draw fourths as a visual representation than sixths.

Once all students have the cards, we discuss which numbers should be placed on the clothesline first. Yesterday, my students determined that 0 and 1 should be the first two numbers. Then they decided that 1/2 should go next because it is the benchmark we used in our lessons on comparing and ordering fractions.

Once we have 0, 1/2, and 1 placed, we start to have a discussion about the other fraction cards. I allow students to talk with a partner as we move through the numbers. When students go to place their fraction, they have to share three things:

  1. Say the fraction
  2. Share which two numbers their fraction goes between
  3. Explain, using math vocabulary why you placed it there (common denominators, common numerators, relative location to the benchmark, etc.)

Once we get started, my role is to simply facilitate the conversation. I don’t tell students whether they are right or wrong when they place their fraction. Yesterday, three of the fractions were placed in the wrong spot. Once every student placed their card, I asked if there were any changes we needed to make. Then the conversation moved into a bit of error analysis. After some discussion and debate, the cards were placed in the correct order. Once again, I let my students talk about what they saw and what needed to be changed.

Students were drawing models on whiteboards, comparing fractions using common denominators, and one student even asked to use the fraction tiles we have in our room to model his fraction.

One student models 5/6 and 8/10 with fraction tiles.

This is such a wonderful activity because it checks so many boxes. In addition to comparing and ordering, students are experiencing many of the Standards for Mathematical Practice. Students are constructing viable arguments and critiquing the reasoning of others, modeling with math, attending to precision, and looking for and making use of structure. I am a stickler about precision in this activity. One student said, “The number on the top in 4/8 is smaller than the number on the top in 5/8.” I will ask them what we call the number on the top and ask them to restate their reasoning. “The numerator in 4/8 is smaller than the numerator in 5/8.” If the students stops there, I will prompt them. “What does the numerator represent?” I will prompt the students until they explain that the numerator in 4/8 represents 4 parts, and the numerator in 5/8 represents 5 parts. The parts are the same size since the denominators are the same, so 5/8 is greater than 4/8.”

I also make sure there are some equivalent fractions included in the cards. When students have an equivalent fraction, they use a paperclip to stack the numbers to show they fall on the same point on the number line, or clothesline.

Each year I introduce this as a whole-group lesson. This allows students to experience those rich conversations with students of all math abilities. Then I use it as a small group activity. Students work in groups of 3-4 and deal cards to one another. They take turns placing their cards and have a similar discussion. As an exit slip, I’ll have students choose three fractions and write how they ordered them. This can also be done by taking a picture and using an app or other digital sharing tool.

This is a great activity that can be used in so many applications throughout the grade levels. The conversations are fantastic, and students are engaged in the activity. Hopefully you find it as valuable as I do.

Three for the Week

Saturdays are my time to reflect on what I read, heard, and discussed throughout the week. It is my “exit ticket” for the last seven days. So here are three ideas that made me think this week. (They are not in any particular order.)

1 – The overall health of teachers is so important. We work in a very emotionally, physically, and mentally demanding field. It is critical that we take care of our bodies and minds in order to better take care of our students. So I was doing some digging for the best exercises and workouts. I love running but know that the impact is not great for your body. I found an interesting article which outlines exercises to do at every age group. For example, they recommend boot camp in your 20’s, high-intensity interval training in your 30’s, and running in your 40’s. Guess I don’t have to stop running quite yet!

2 – An article from The University of Virginia really caught my attention. It is a Q&A with NCTM President and UVA professor Dr. Samuel Braley Gray. He outlines what effective math teaching looks like in our schools, touches on some inequities in math education, and even talks about why children should use their fingers in math. (That last point alone got me wondering why we would encourage students to use printed ten frames, but discourage them from using their fingers – which are ten frames.) What really struck me was what Dr. Gray said about the effective ways to teach math. “These ideas are a shift from focusing on memorization. Mathematics is more than getting an answer quickly. Effective mathematics teaching engages students in explaining why their answers make sense and why the strategy they used is appropriate.” Well said, Dr. Gray!

3 – Last week I chose something lighthearted as my third point for the week. I’ll keep that trend going this week. Two ridiculously cute boys show up to a Canadian airport to pick up their grandmother. The boys decide to play a trick on grandma and dress up in full T-Rex costumes. Grandma, as grandmas always seem to do, was one step ahead of the boys. She appeared wearing… a full T-Rex costume of her own. The video is well worth the 2:29 of your time and will definitely put a smile on your face.