Open Middle© problems are challenging, fun problems for students of any age. The idea behind Open Middle© problems is that there are multiple solutions for finding the answer. To solve these and to prove that you have a valid solution, you need to find different possible solutions. Here’s one that I tried:

I tried Multiplying Two-Digit Numbers – Closest to 7,000. For my first attempt, I tried using the digits 1, 2, 3, and 4. I was not really thinking about the impact of place value. I was just wanting to see what happened. On my next attempt, I knew I had to have a significantly larger number so I tried 91 x 52. While that was better, it was still a bit away. My last attempt was 91 x 82. It was closer but I am wondering if I could get closer.

These Open Middle© problems are available by grade level and by different domains, in different languages. Each problem has a hint and a solution. With each of these problems, student can record their different attempts in finding solutions. The Open Middle worksheet provides opportunities for students to record their thinking and ideas for their next attempt.

**Where’s the math? ** For these problems, you can choose the math that you want students to focus on. The Open Middle© problems encourage students to make multiple attempts to find the best solution for the problem, so students are making sense of problems and persevering in solving them.

**What grade levels?** With all the choices, you can do K through 12+. Adults even have fun with these.

- Synchronous Technology – If you are wanting to attempt these with synchronous technology, I would suggest using a template from Alice Keeler. She has created one for equivalent fractions to give you an idea. Each student would be assigned a slide to enter in their attempt. You can find her template here.
- Asynchronous Technology – Students could fill out the Open Middle worksheet showing their attempts and then share what they came up with for their final attempt.
- Paper– Students could fill out the Open Middle worksheet and submit that.
- Homework – Give students one of these to attempt at home and have their family members try to solve it. Have them interview their family members as to why they used the numbers they did. And then ask the students what did they learn from interviewing their family members.

Did you try this activity? What happened? We would love to hear how it went.

]]>I love this game for helping students with their multiplication facts! Rectangular Fit, taken from the work of Jennifer Bay-Williams and Gina Kling in *Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention*, can be adapted for different ages and situations.

For Rectangular Fit, you will need two dice (or you could use playing cards), pencil, and a game board for each player. You will need at least two players. I am sharing how to adapt the game for at least a two player game. The attached version is for a whole class game. The goal is to be able to have the most rectangles on your game board. Here are the directions:

- Player 1 rolls the two dice.
- Each player decides where on their game board to place a rectangle of the dimensions rolled. For example, if 4 and 5 were rolled, each player decides where and in what orientation they can best fit a 4 x 5 rectangle on their game board. (So they could draw the rectangle as 4 x 5 or 5 x 4.)
- Each player writes the related multiplication fact inside the rectangle.
- The next player rolls, and each student fills in a rectangle with the dimensions on their game board.
- When a player cannot fit a rectangle with the dimensions rolled, they are out of the game.
- The last player in the game in the winner.

The reason I love this game is because it allows students to explore their multiplication facts with a visual representation and without the focus on speed. *Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention* shares the research around learning math facts. In our current culture of teaching math facts, we tend to go from Phase 1 of counting to Phase 3 of mastery and do not allow students to spend time in Phase 2 of deriving. Games like Rectangular Fit allow students to play with their math facts, helping them develop connections to what they know. When students have the time to be in the deriving phase, they are able to retain their facts.

Another reason I love this game is all the variations that students are able to create from it. After teaching the game whole class, students play in partners. From playing with partners, students adapt the rules. One variation that students created was to play on one game board where the goal is to be able to place the last rectangle. Another variation would be to play as teams of two. If you want to challenge students, use a 10 sided dice or playing cards. But be careful, you will want a game board with more squares.

**Where’s the math?** For Rectangular Fit, students are practicing their multiplication facts. They are also using a visual structure to help them make sense of the problem. This game does involve strategy as well.

**What grade levels? ** 2nd to 12th+ Because of the amount of strategy involved and the ways you can adapt this game, you could play this game with almost any age.

- Synchronous Technology – You could have students “roll the dice” and then fill in a rectangle on a virtual whiteboard. The goal could be to see how many rectangles you could fit onto the game board. This might be a good way to teach the game.
- Asynchronous Technology – You could post the game for students to do. They could take a picture of their game board and share how many rectangles they were able to fit. You could have them compare with another students to see why one person had more rectangles on their game board.
- Paper – You could have students play this game at home with family – at least twice. Is there a strategy that works well to get the most amount of rectangles on the game board?
- Homework – Is there a strategy that works well to get the most amount of rectangles on the game board?

Did you play this game? How did you adapt it? Let us know how it went.

]]>One cut geometry is a fun challenge based on a theorem that every pattern (plane graph) of straight-line cuts can be made by folding and one complete straight cut. So it should be possible to make every single polygon from **one** cut on a folded piece of paper.

You need scissors and a LOT of paper for this challenge. The recommendation is that the paper is square but I think you could use a regular piece of paper. Draw a scalene triangle in the middle of the paper. Can you cut the triangle with one cut? What shapes did you cut? What mistakes did you make? Below are the directions for this activity taken from Youcubed:

We decided that this challenge would be a great problem for our Summer Math Camp students because it fosters a growth mindset. Students have to try multiple times (you can see above the mistakes poster) to figure how to make a shape. When students did figure out how to cut out a shape, other students wanted to know how they did it. We celebrated mistakes as well, realizing that they were learning to learn from them.

You can adapt this idea to younger grade levels by having them do a Fold and Cut Challenge (no one cut here). This idea comes from Games for Young Minds.

Students take a piece of paper and fold it in half. Can you cut a rectangle? Can you cut a square? Can you cut a triangle? Can you cut the letter X? This version is great for students who are learning their shapes and letters.

**Where’s the math?** For both versions, students have to persevere to solve the problem. For the version for younger students, students learn about symmetry and shapes. For the one cut activity, students learn about properties of different polygons.

**What grade level?** K-12+ (The two versions, Fold and Cut and One Cut Geometry, make this activity great for all ages.

- Synchronous Technology – Give students the challenge and have them share what they made in the first 2 or 3 attempts. I would be cautious in using this in a synchronous meeting because I think it could use a lot of time.
- Asynchronous Technology – Give students this challenge and have them post photos of ones that worked and ones that did not. You could do a GoogleSheet with different shapes on different sheets and separate the sheet into two columns – worked and did not work. Have them post their photos in the correct column.
- Paper – Send this home with students to try with family members.
- Homework – I think this activity could go on and on and on…so not sure about homework for this one.

Did you try this activity? How did it go? Do you have some suggestions. Leave a note.

]]>Which One Doesn’t Belong is one of my favorite routines to use with students. Students choose which one they think does not belong from a group of images. The most important thing is that they explain why they believe that particular image does not belong. And anything goes. An example from the image above would be I think that the bottom right does not belong because it is the only one that is green. Students also need to be thoughtful that their explanation only works for that particular image. If I said that the bottom right was outlined in red, that would also apply to two other images so that I could not be why just that image does not belong.

The reason I like this routine so much is because all students have the opportunity to share how they see the images and I learn just as much from them sharing. Often, I make assumptions about what students know and what they do not know. By using Which One Doesn’t Belong, those assumptions are usually proven incorrect and I am surprised by what they do remember. Often, vocabulary that they do know comes out and leads to a discussion about what that particular word really does mean. In the end, students realize that each one of the images does not belong for some reason as well as realizing that they all have something in common.

When I introduction Which One Doesn’t Belong, I like to start with ones that are not mathematical in focus so that students are worried about whether or not they have the correct answer. Once they have the idea of Which One Doesn’t Belong, then we start to do ones that are focused on the math that they are studying. Here’s one that I did in a classroom where you can see all the student thinking:

You can find more ideas for Which One Doesn’t Belong here.

**Where’s the math?** For Which One Doesn’t Belong, student have to construct viable arguments and critique the reasoning of others. Once students understand the routine, then you can choose the math that is relevant to them. For first grade, that might be coins. For high school, they could compare graphs of different polynomials.

**What grade level? ** K-12+

- Synchronous Teaching – This idea was shared by Sara Van Der Werf at NCSM’s Virtual Conference. You can have students go and grab a household item for which one they think doesn’t belong. Using the Which One Doesn’t Belong from the top, if you think that the heart doesn’t belong, then go and grab a fork. If you think the star doesn’t belong, go and grab a knife…and so on. And then have students share reasons why.
- Asynchronous Teaching – Students could share on a GoogleDoc why each one does not belong.
- Paper – Have students ask members of the family why they think each one does not belong and have them record it.
- Homework – Students could create their own Which One Doesn’t Belong focused on a particular topic.

Leave a note and let us know what you think.

]]>The idea behind this game is that it helps younger students learn about patterns in the 100’s chart. For older students, there are a lot of strategies that can be used that makes this game interesting. Since I did not have any pictures of the game being played, I asked my high school sophomore to play with me. When I explained the rules, he thought that this was going to be an easy game. Once we started playing, he realized that the strategy needed to win this game could be quite complex.

For the game, you need at least two players, two color pencils (or a pen and pencil), and a game board. Here are the rules for the game:

- Choose a player to go first and choose a color for each player. Player 1 chooses a blank space on the gameboard and writes in the accurate number for that space. Player 2 confirms the accuracy of that number.

- Player 2 continues, choosing a blank space and writing in the accurate number for that space. Players continue alternating turns.

- When a player gets 4 numbers in a row (horizontally, vertically, diagonally)
**of their own color**, they connect the numbers with a line, and score a tally mark at the top of the board under their name.

- Play continues until the board is filled or no more 4-in-a-rows are possible.

- The player with the highest number of tally marks wins.

You can also play variations for this game. For younger students, you could use 1 to 50 and do connect 3. You could change the chart to be 401 to 500 or 321 to 460. Or you could try using a multiplication chart.

Here are two different game boards that you can use:

**Where’s the math? ** For math content, students are learning numbers 1 to 100. Student engage with the mathematical practices of looking for and making use of structure through looking for patterns when placing their number on the 100’s chart.

**What grade level?** This game is ideal for kindergarten to 2nd grade. However, all ages can enjoy this game since it requires a lot of strategy.

- Synchronous technology – For this game, you could have the class play against the teacher. The teacher could have the video on the game board and students could take turns placing numbers on the 100’s chart.
- Asynchronous technology – Students could play this game with a sibling or family member and share what strategy worked for them (or didn’t work).
- Paper – Students could play this game with a sibling or family member and share what strategy worked for them (or didn’t work).
- Homework – You could ask students questions about the game such as is there a best place to start and why or is it better to play offensive (trying to score) or defensive (trying to block) and why.

We had fun playing this game. Leave us a note about playing 100 Chart Connect 4. What did you like? How did it go? What would you change?

This puzzle, or challenge, is all about making squares. Sarah from Math = Love have adapted the original puzzle from a Netherlands brain teaser website where you use all five pieces to create a square. I love how she has scaffold this activity to allow students to build their stamina to figure out the original challenge. Know that if you do puzzles like this to remind students that finding the answer to the challenges could take days. We are still working on Challenge #3. Here’s her adaptation:

Can you make a square using only one piece? (I think the idea behind this question is to warm up the brain to making squares with pieces.)

Can you make a square using exactly 4 pieces?

Can you make a square using exactly 5 pieces?

Sarah wrote about her experience using this puzzle challenge. You can read more about her experiences here. Below are the directions and the pieces.

**Where’s the math?** For younger students, the math content is focused on geometry and what a square is and is not. For all students, this puzzle challenge requires students to persevere in solving problems. Remember that we want mathematical thinkers who have the stamina to solve difficult problems.

**What grade level? **K-12+ In kindergarten, students are learning about shapes, including squares.

I**deas for Distance Delivery Options**

- Synchronous Technology – You might want to send them the pieces prior to a synchronous lesson. You can pose the first challenge and have them share what the solution is. For the second challenge, you could pose it and give them time to solve it. I would advise to send them home with the third challenge.
- Asynchronous Technology – Give students the challenge and have them send you pictures of the solution. I would not have them share the pictures to a group discussion/chat since there is only one way to solve this problem.
- Paper – This puzzle challenge could be sent home in a packet for students.
- Homework – Here are two ideas for homework/extension. 1. Have them find another puzzle online that is similar to this one. 2. Have them create their own puzzle similar to this one.

We are having fun trying this one at home and are still looking for a solution to Challenge 3.

I would love to hear how it goes and if you would like more of these kinds of activities.

How are these pictures the same? How are these pictures different? This routine of same or different allows students to compare two things, looking for how they are similar and how they are different.

Students start by thinking and studying this picture – noticing and wondering how these images are the same and how they are different. Then students share, verbalizing what the see being the same and different. Students learn how to share and talk about math in a way that is fun and engaging. They also see how different students see the world by listening to what they see as being the same or different.

Here’s where you can find more of these activities:

**Where’s the math?** The routine of Same or Different has students construct viable arguments and critique the reasoning of others which is a standard of mathematical practice. Depending upon the images that are used, different areas of math can be addressed. Here’s an example that can be used with algebra:

**What grade level?** The routine of Same or Different can be used K-12.

**Ideas for Distance Delivery Options**

- Synchronous tech – Ask students to write on a GoogleDoc or in the chat box what they see as being the same or different
- Asynchronous tech – Use a GoogleDoc where students could put their responses or try a GoogleForm where they do not get to see what others are thinking until the end
- Paper – Send images home and have them write about what is the same and what is different
- Possible homework – Have them find two pictures and ask people in their household what they see as being the same or different. Ask them what they learned from the people in their house.

What do you think? Leave me a note and let me know how it goes.

]]>The 5 x 5 game from Sara Van Der Werf is great for any age. Using cards, student place numbers on a 5 x 5 grid. When numbers are adjacent, they can count those numbers for their score. Numbers can be adjacent either vertically or horizontally. The goal of the game is to get the highest total points. To see a better explanation of the rules, you can find them here.

Every time that students have played this game, it takes at least two attempts before they start to strategize where to put the numbers. And students of all ages enjoy playing it.

**What grade level? **2nd grade and up

**Where’s the math?** The math in the 5 x 5 game involves knowing at least doubles of 1 to 10 and where you want to strategically place the numbers to earn the most points. Students learn to make sense of problems and persevere to solve them. They also look for and make use of structure with the way the game works.

**Distance delivery options**

- You could play online and then have students share their scores in the chat box.
- You could send this home in a packet for the student to do with their family.
- You could give students numbers and have them place them in two different ways. They could share their two different ways in a post and explain why one board had a higher score than the other.
- What other ways could you adapt this game?

What did you think of this game? Leave me a note and let me know what you think.

]]>Puzzles are a great way to develop mathematical habits of mind and logical reasoning. SolveMe puzzles focuses on supporting algebraic and mathematical thinking in a fun way. With each of these puzzles, *Mobiles, Who Am I?*, and *Mystery Grid*, students are able to explore different ways to solve them using logic and reasoning. These puzzles allow students to track their progress in solving them.

Who Am I? puzzles have students figure out a mystery number. Through a series of clues, student figure out the mystery number. These clues reinforce the math vocabulary students have learned.

MysteryGrid puzzles are similar to KenKen. Using clues, students need to figure out where numbers go in the grid.

**Where’s the math?** These puzzles encourage mathematical habits of mind. The SolveMe puzzles help students develop algebraic reasoning. For Who Am I? puzzles, students use mathematical vocabulary to find a mystery number. In MysteryGrid, students use logic and reasoning to determine where different values belong in the grid.

**What grade level?** 2nd to 12th+ and some younger students might enjoy these as well

Here are more puzzles that from EDC’s Transition to Algebra puzzle sampler that you can send hom.

What did you think? Leave me a note and let me know what you would like to see more of.

]]>What do you notice? What do you wonder?

Here are some other questions that Youcubed has included: What questions do you have? What information does this graph provide? Other questions might be which emoji is the least cute? Do you agree with that? Which emoji does this person like the most? How do you know that? Where would you put cat emojis? Why are the emojis with hearts on the top of the graph? If an emoji had hearts, where would you put it on the graph?

Now that you are familiar with the graph, try this with your own set of emojis. Or change the axes to something else like funny and not funny or ugly and not ugly. Where would you put the emojis? Here is the activity from Youcubed for you to try – WIM Emoji Activity.

**Where’s the math?** For this particular activity, students are learning to make sense of what the axes mean in graphs. They are also learning to make sense of the problem by studying where the emojis are on the graph.

**What grade level?** The notice and wonder could be used K-12. Youcubed has taken this activity and adapted it for all grade levels. You can find the different grade level versions below:

If you try this activity, please let me know how it goes. What did you like? What would you do differently? What questions do you have?

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