# 3.5: Let's Put it to Work - Mathematics

3.5: Let's Put it to Work - Mathematics

## Encouraging Students’ Independence in Math

Including emotion regulation activities in math lessons can help elementary students learn the content more effectively.

Have you ever poured your heart into preparing and teaching a lesson only to have most students raise their hand for help as soon as you asked them to work independently?

Math can evoke powerful emotions, such as embarrassment, anxiety, and apathy. At the beginning of my career, I failed to recognize the impact these feelings had on my elementary students’ learning. My teaching wasn’t helping them become critical thinkers, problem solvers, or self-managers. So I refocused my efforts on developing my students’ ability to regulate their emotions, thoughts, and behaviors in class by adding management components to our daily routines.

### Helping Students Improve Their Self-Management

Use a predictable, student-centered lesson structure: I wanted to allow my students opportunities during lessons to reason about the mathematics they were learning and guide them to develop critical thinking and problem-solving skills. I began structuring my lessons with four parts: a warm-up with some daily fluency practice, a launch problem (typically a word problem that would connect the previous day’s learning to the current day’s), exploration time, and a closing.

The four components of each lesson helped emphasize the importance of reasoning and making sense of math, culminating with a daily exit ticket that exemplified the lesson’s objective. As my students became familiar with this structure, they quickly learned that I expected them to do much of the thinking and speaking during each lesson.

The predictable nature of the lessons combined with the consistent expectations for student participation helped alleviate students’ stress.

Incorporate strategies for emotion regulation: Math anxiety—fear of making mistakes, of getting stuck, of not looking “smart”—can be crippling for a child. When I would jump to the rescue, my students learned to avoid working through the productive struggle that accompanies new learning.

To help my students learn to manage these intense emotions, I began incorporating some simple strategies designed to spark joy or reduce stress in our daily lessons. Our daily fluency practice was quick and high energy, combining movement or partner talk with the focus skill. Adding a counting activity with a quick movement and belly breathing or positive self-talk before we would explore a particularly challenging task helped students burn off some nervous energy.

After I introduced these strategies, I experienced fewer classroom disruptions, and far fewer students raised their hands to ask for help before attempting a task. Indeed, I often heard my students pause during their group explorations and independent work to say, “We can do this” or “Let’s go back and see where we made a mistake.”

Incorporate self-managed tasks daily: Clear classroom routines and behavioral expectations are a key component to student independence and self-management. In my math classes, one of the first routines students learned was working the launch problem: They would independently gather the necessary tools and work with a nearby partner to begin thinking about the problem displayed on the whiteboard.

To motivate students, I timed this transition. The partner work helped everyone find an approach to the problem before we discussed it as a class, and the sense of a clock ticking kept the pairs on task. We would often pause to reflect on what made the transition effective and what could be improved.

To prevent students from asking for help immediately in their independent work at the end of exploration time, I began using the “see three before me” rule: When students were confused or needed help, they had to seek out and document their use of at least three different resources before coming to me for assistance. In daily class discussions, we consistently referred to and revised a list of acceptable resources, including students’ math notebooks, any appropriate modeling tools, anchor charts, and their peers.

Support students in setting and tracking goals: I would periodically plan a math class that provided students a choice of practice centers. Each center activity was based on a major objective we’d been working toward. At the beginning of these class periods, I displayed the objectives and the associated activity on the board. Students decided which objectives they needed the most practice with and then moved to the corresponding center, where they would work for about 10 to 15 minutes.

After students had visited three centers, I asked them to complete a written reflection, stating a rationale for their chosen goals and how the center practice helped them. I collected, reviewed, and returned the reflections so that I could use the information formatively. Students stored and kept track of their reflections in a folder.

Setting and tracking goals gives students ownership of their learning, teaches them to regulate their behavior, and helps them prioritize their goals productively. At first, my students chose stations for reasons unrelated to their learning, picking one their friends chose or one they thought would be easiest for them. After a while, though, they became more reflective about their own learning needs, and their choices changed accordingly.

How did these strategies help my students? Developing independence and self-management skills during math class helped them achieve their math goals—and doing so also helped them learn key skills that were transferrable to any kind of learning. I noticed my students becoming more responsible, more motivated, and better able to work through emotional situations. By making self-management a cornerstone of my classroom environment, I helped my students become more effective learners.

## Looking For A New Math Puzzle Game? Try Sweet 16

I like sudoku as much as anyone, but a change of pace might be nice occasionally. Enter Sweet 16, a new math puzzle game. As the name suggests it involves the numbers 1 through 16, and it's a bit like sudoku in the sense that the player must put the numbers into a square array (4 × 4 in this case). The similarities end there, however.

Here's how it goes. We have to figure out how to place all the numbers from 1 to 16 into the boxes, using each only once. Sometimes we are given seeds, such as the placement of the number 1 in the top row. Also, circles can contain only odd numbers and squares can contain only even numbers. This narrows the possibilities significantly, which is a good thing since without any restrictions there are 16! ≈ 21 trillion configurations of the 16 digits into the spaces. Note the various arithmetic operations joining some of the boxes in this example all four appear (addition, subtraction, multiplication, division).

Each puzzle has a unique solution, and to find it you must use your knowledge of the arithmetic of small numbers to close in on it. Let's work through this one carefully. It's generally best to begin with multiplication and division since there are fewer possibilities. In this case, note the division problem involves only odd numbers. Many of the odd numbers less than 16 are prime (3,5,7,11,13), so this leaves only two possibilities for the second entry of the second row: 9 or 15. But 9 is out since it is divisible only by 3 and we can use each number only once. So we must put 15 in that spot, and then the entries to the right must be 3 and 5 in some order (15 ÷ 5 = 3 or 15 ÷ 3 = 5). Progress, but only a little.

The next place to try is the multiplication on the bottom row. It consists of even numbers, and so there are only six possibilities: 2 × 4 = 8, 2 × 6 = 12, 2 × 8 = 16, or these in reverse order. But we can quickly narrow these down by considering the vertical product in the third column. Since one of those numbers is either 3 or 5, the third entry in the fourth row must be 12 and our division must be 15 ÷ 3 = 5. That makes the entry above the 12 a 4.

Now, we still don't know if it's 2 × 6 or 6 × 2, so let's look at each possibility in turn. If we have 2 × 6, then we would have to have a 9 above the 6 (to make 15 - 9 = 6), but then that would force 9 - 4 = 5 in the third row and we've already used the 5. So we must have 6 × 2 = 12 in the bottom row. But then we can place the 13 above the 2 to make 15 - 13 = 2 and then we get 13 - 4 = 9 in the third row. We are then able to place 14 in the bottom right corner.

Almost there. What remains is the first column and the first row. In the first column we have two even numbers whose difference is 6. But we've already used 2, 4, and 14. This yields 16 - 10 = 6 in the first column. The lone remaining even number is 8, which must then go in the top right box. We then place the 7 in the second entry of the first row to complete the equation 7 + 1 = 8. Finally, 11 goes in the circle at the top left.

That's it. It's mostly logic (like sudoku) but there is the added twist of having to use arithmetic to make logical deductions. It took me about five minutes to solve this puzzle, but it is the first one in the book and so it is meant to be pretty easy. I can imagine later puzzles being quite challenging (I haven't gotten very far into the book yet).

There are other types of initial set-ups. For example, sometimes we are presented with inequalities among the entries, as in the puzzle below.

This creates a new set of challenges to work with, but it does help limit the number of possibilities for various spots in the puzzle. I'll leave this one unsolved.

Overall, I'd rate Sweet 16 as a solid addition to the mathematical puzzle literature. The game requires tenacity and careful deduction to reach the solution and provides a nice alternative to puzzles like sudoku. The book will be available soon and I suggest you grab a copy if these types of brain teasers appeal to you.

## The natural logarithm and the common logarithm

You can choose various numbers as the base for logarithms however, two particular bases are used so often that mathematicians have given unique names to them, the natural logarithm and the common logarithm.

Natural logarithm

If you want to compute a number&aposs natural logarithm, you need to choose a base that is approximately equal to 2.718281. Conventionally this number is symbolized by e, named after Leonard Euler, who defined its value in 1731. Accordingly, the logarithm can be represented as logₑx, but traditionally it is denoted with the symbol ln(x). You might also see log(x), which also refers to the same function, especially in finance and economics. Therefore, y = logₑx = ln(x) which is equivalent to x = eʸ = exp(y) .

One practical way to understand the function of the natural logarithm is to put in the context of compound interest. That is the interest that is calculated on both the principal and the accumulated interest.

The formula for annual compound interest is as follows:

• A is the value of the investment after t years
• P stands for the initial value
• r is the annual interest rate (in decimals)
• m represents the number of times the interest is compounded per year or compounding frequency and
• t refers to the numbers of years.

Let&aposs assume that you deposit some money for a year in a bank where compounding frequently occurs, thus m equal to a large number. It is easy to see how quickly the value of m is increasing if you compare yearly (m=1), monthly (m=12), daily (m=365), or hourly (m=8,760) frequencies. Now, let&aposs imagine that your money is recalculated every minute or second: the m became a considerably high number.

Now let&aposs check how the growing frequency affects your initial money:

m (1 + r/m)ᵐ
1 2
10 2.59374…
100 2.70481…
1000 2.71692…
10,000 2.71814…
100,000 2.71826…
1,000,000 2.71828…

You may notice that even though the frequency of compounding reaches an unusually high number, the value of (1 + r/m)ᵐ (which is the multiplier of your initial deposit) doesn&apost increase very much. Instead, it becomes somewhat stable: it&aposs approaching a unique value already mentioned above, e ≈ 2.718281 .

Since growth rates often follow a similar pattern as the above example, economics also heavily rely on natural logarithm. Two common variables involve natural logarithm: the GDP growth rate and the price elasticity of demand.

Common logarithm

The other popular form of logarithm is the common logarithm with the base of 10, log₁₀x, which is conventionally denoted as lg(x). It is also known as the decimal logarithm, the decadic logarithm, the standard logarithm, or the Briggsian logarithm, named after Henry Briggs, an English mathematician who developed its use.

As its name suggests, it is the most frequently used form of logarithm. It is used, for example, in our decibel calculator. Logarithm tables that aimed at easing computation in the olden times usually presented common logarithms, too.

The below table represents some frequent number common and natural logarithms.

x log₁₀x logₑx
0 undefined undefined
0+ -∞ -∞
0.0001 -4 -9.21034
0.001 -3 -6.907755
0.01 -2 -4.60517
0.1 -1 -2.302585
1 0 0
2 0.30103 0.693147
3 0.477121 1.098612
4 0.60206 1.386294
5 0.69897 1.609438
6 0.778151 1.791759
7 0.845098 1.94591
8 0.90309 2.079442
9 0.954243 2.197225
10 1 2.302585
20 1.30103 2.995732
30 1.477121 3.401197
40 1.60206 3.688879
50 1.69897 3.912023
60 1.778151 4.094345
70 1.845098 4.248495
80 1.90309 4.382027
90 1.954243 4.49981
100 2 4.60517
200 2.30103 5.298317
300 2.477121 5.703782
400 2.60206 5.991465
500 2.69897 6.214608
600 2.778151 6.39693
700 2.845098 6.55108
800 2.90309 6.684612
900 2.954243 6.802395
1000 3 6.907755
10000 4 9.21034

## Understanding Subtraction

Subtraction is a mathematical operation it is a process or action that you do with numbers. It is the process of finding the difference of 2 numbers.

The symbol used is: - (minus). In this process, we take a number and reduce it to a smaller number. That is, we take away one number from another.

The difference of 7 and 5 is 2.

Here are some ways to help your child understand how to subtract.

1. Using Real Objects
2. Number Bonds
3. Drawing Models
4. Understanding a Subtraction Statement
5. Using a Number Line
6. Subtraction vs Addition
7. Questions on More or Less
8. Learn to Subtract Bigger Numbers

### Using Real Objects

Give your child some counters (buttons, coins, paper clips or other small things).
Ask her to count the items. Refer to the number chart if your child forgets her numbers.
Count and remove some of the items.
Ask her to count the remaining items.
Remember to use Math terms like subtract, difference, left over, remainder and so on.

Examples: "Let's subtract 2 from 9" or "How many do we have left?"

Do this a few times with different numbers of objects.

1. First, count the number of items together.
2. Then secretly remove some of the items.
3. Let her count the remainder, then ask "How many . did I remove?"
4. After she gives the answer, let her count the items you removed to check if she is correct.
5. Remember to let your child test you too. Children feel more in control of their learning if you let them explore the subject by asking questions.

### Number Bonds

Number bonds is a basic Math concept that can help your child understand subtraction as well as addition.

A number bond shows two numbers being joined to form a bigger number (as if a number is made up of different parts).

The reverse is also true. A number can be broken into smaller numbers.

This number bond shows that two smaller numbers (3, 4) can combine to make a larger number (7).

This number bond shows that a larger number (7) can be 'split' into two smaller numbers (5, 2).

Use physical objects like buttons, toys and so on to demonstrate the idea of number bonds.

﻿Draw some number bonds on cards to help your child understand this concept:
(Or print them out here.)

1. Give your child a bunch of items.
2. Let your child count and write the number in the circle on the left.
3. Separate the items into 2 groups then count and write the numbers in the circles on the right.

Now let's put the ideas together.

### Drawing Models

You can also draw pictures or models to help your child understand subtraction. This is a very useful method as it can be used to help your child understand very complicated questions later on.
Here are 2 models that can be used to understand these two questions:

In drawing models, we draw boxes instead of circles to represent numbers.The main idea is still the same - numbers can combine to to form larger numbers or be broken into smaller numbers.

This method can be more accurate because we can draw bigger boxes to represent bigger numbers.  This will be really useful later on when dealing with large numbers.

### Understanding A Subtraction Statement

Once your child has understood the concept of finding the difference, it is time to learn to write a statement or equation.

Subtraction is the idea of 'taking away'.

7 - 5 means taking 5 away from 7.

The number you take away from is called the minuend. It is the biggest number in the equation.

Let your practice with sets of 3 numbers that can be used to form a subtraction statement and let him write the statement on his own.

Ask him to check his answer using real counters.  You can print some templates here.

### Using a Number Line

The process of subtracting is similar to counting backwards. You can show this on a number line.

1. Draw a number line.
2. Write the first number (minuend) in the subtraction statement on the line.
3. Count the number of steps backwards that correspond to the second number (subtrahend), writing the numbers on the line as you count.
4. The number that you land on is the answer (difference).

### Subtraction vs Addition

Subtraction is closely related to addition. Sometimes it is not so easy to tell which operation to use. See these examples:

• 3 + 5 = ___ (this is a typical addition question)
• 8 - 3 = ___ (this is a typical subtraction question)

Click here to use the template I've created to explore different ways of adding and subtracting.

### Questions on More or Less

Some children associate addition and subtraction with the terms 'more than' and 'less than'. This may not always be the case.

Click here for a worksheet on 'more than' and 'less than'. Use the template to create more questions for your child.

Try to figure out whether addition or subtraction is the correct operation to use to get the correct answer. Try to apply the strategies you've learned like drawing models or using a number line.

Remember to let your child set questions to test you. Make it a family math challenge.

Do you find this page useful?  Share it!

### Learn to Subtract Bigger Numbers

To subtract bigger numbers, think of them in this way:

37 = 3 tens and 7 ones
15 = 1 ten and 5 ones

Let's work out this question:

1.  Start with 3 tens and 7 ones to show 37.

2.  Remove 1 ten and 5 ones (for 15).

3.  We are left with 2 tens and 2 ones to form the number 22.

Let's try one with a carry over or regrouping:

We start with 42 (4 tens and 2 ones).

3. Change one of the tens to 10 ones. Now we have 3 tens and 12 ones.

4. Take away 2 tens and 5 ones.

5. We are left with 1 ten and 7 ones or the number 17.

### More Math Fun

Share this information with your family and friends who want to help their children build a strong foundation in Math.

## Problem of Helen's hair length

Problem. Helen has 2 inches of hair cut off each time she goes to the hair salon. If h equals the length of hair before she cuts it and c equals the length of hair after she cuts it, which equation would you use to find the length of Helen's hair after she visit the hair salon?
a. h = 2 &minus c c. c = h &minus 2
b. c = 2 &minus h d. h = c &minus 2

Solution. Ignoring the letters c and h for now, what are the quantities? What principle or relationship is there between them? Which possibility of the ones listed below is right? Which do you take away from which?

 1 cut hair &minus hair length before cutting = hair length after cutting 2 cut hair &minus hair length after cutting = hair length before cutting 3 hair length before cutting &minus cut hair = hair length after cutting 4 hair length after cutting &minus cut hair = hair length before cutting

SIMPLE, isn't it?? In the original problem, the equations are given with the help of h and c instead of the long phrases "hair length before cutting" and "hair length after cutting". You can substitute the c, h, and 2 into the relationships above, and then match the equations (1) - (4) with the equations (a) through (d).

## Introducing STEM is Easier Than You Think

You may be thinking, “This is great, but I…

• am squeezed for time.
• am not comfortable with math or STEM.
• am not sure how STEM activities fit into my curriculum.”

You are not alone! But all of these challenges can be overcome. Try using a math concept in your classroom that’s new to you, or find a real-world problem that can be explored through engineering. [14] When children’s own ideas actually work (or when their ideas don’t work at first, but children keep trying until they do), it can be transformative—for us and for them.

Through hands-on, minds-on challenges like these, in which children see math as a useful tool, we are on the right track to cultivating a love of math and STEM.

Alissa A. Lange, PhD, and her colleagues have worked with hundreds of preschool educators over the last 10 years, through two National Science Foundation-funded professional development projects and, more recently, through early childhood preservice teacher preparation programs in STEM. Lange and her colleagues started the Early Childhood STEM Lab at East Tennessee State University in 2019 as a hub for early childhood STEM.

[6] National Association for the Education of Young Children (NAEYC) position statements on early math.

[7] National Association for the Education of Young Children (NAEYC) position statements on early science.

[8] Standards for other grades (e.g., kindergarten standards from Common Core).

[9] Tennessee early childhood education early learning developmental standards for 4-year-olds.

[11] National Research Council. (2014). STEM Integration in K-12 Education: Status, Prospects, and an Agenda for Research.Washington, DC: The National Academies Press. https://doi.org/10.17226/18612.

[12] Robertson, L., Dunlap, E., Nivens, R., & Barnett, K. (2019). Sailing Into Integration: Planning and implementing integrated 5E learning cycles. Science and Children, 57(1), 61-67.

## Step-by-Step Math

Have you ever given up working on a math problem because you couldn’t figure out the next step? Wolfram|Alpha can guide you step by step through the process of solving many mathematical problems, from solving a simple quadratic equation to taking the integral of a complex function.

When trying to find the roots of 3x 2 +x–7=4x, Wolfram|Alpha can break down the steps for you if you click the “Show steps” button in the Result pod.

As you can see, Wolfram|Alpha can find the roots of quadratic equations. Wolfram|Alpha shows how to solve this equation by completing the square and then solving for x. Of course, there are other ways to solve this problem!

Wolfram|Alpha can demonstrate step-by-step solutions over a wide range of problems. This functionality will be expanded to include steps for solutions in other mathematical areas. Look through the following examples to see the abilities of the “Show steps” functionality.

If you need to learn how to do long division of polynomials, Wolfram|Alpha can show you the steps. Let’s try (x 5 –14x 4 +3x 2 –2x+17)/(2x 2 –x+1):

If you are stumped trying to find the limit of x x as x->0, consult Wolfram|Alpha:

When you need to find the derivative of (3x 2 +1)/(6x 3 +4x) for your calculus class, Wolfram|Alpha will find this derivative using the quotient rule.

Are you trying to integrate e 2 x cos(3x), but forgot the formula for integration by parts? Wolfram|Alpha will remind you how to integrate by parts.

Wolfram|Alpha can do virtually any integral that can be done by hand. Try the integral of x?[1–?[x]]:

Wolfram|Alpha also has the step-by-step functionality for partial fractions. Try partial fractions of 1/(x 3 –1):

The step-by-step programs in Wolfram|Alpha rely on a combination of basic algorithms and heuristics including Gaussian elimination, l’Hôpital’s rule, and Bernoulli’s algorithm for rational integration. These heuristics are a logical formulation of the natural methods used by humans for solving problems. By utilizing Mathematica’s powerful pattern-matching capabilities, Wolfram|Alpha’s developers have morphed these rules into a platform for breaking down and structuring the solutions to complicated problems, which closely mimics the ways by which a human would solve problems of these natures.

The “Show steps” feature allows you to learn basic mathematics on your own, or it can simply be a nice way to check your work! It can also give you insight on different ways to solve problems. So next time you find yourself ready to give up on a math problem, make sure to check with Wolfram|Alpha. Visit the Wolfram|Alpha Homework Day Gallery for examples of how you can use Wolfram|Alpha as a learning tool for other subjects.

## 20 Math Center Ideas for your Elementary Classroom

1.) Card Games. Most classrooms have at least one set of cards, but if not, you can find them at the dollar store. There are a variety of games you can have your students play and still practice their math concepts. For instance, remove the face cards and the aces and then have students play any variation of top-it. Divide the set of cards up evenly among the pair of students 2 students each flip a card at once and perform whichever operation you prefer (add, subtract, multiply). The first student to say the correct answer first gets both cards. Students keep going until there are no cards left. The student with the most cards wins!

You can also take a set of 3&Primex5&Prime index cards and cut them in half creating mini-cards. Then program them to be any set of numbers you wish. Then students can perform any operation, play a game of go fish, play rummy (place them in order), or any other fun game!

2.) Dice Games. I really can&rsquot mention card games without mentioning dice games as one of the math center ideas. Just like with card games, you can find dice at the dollar store, but you can also create your own using wooden blocks. I like to do this to program them with fractions or decimals. Then I have students play games such as adding the numbers and seeing how close they can get to one without going over.

3.) Bowling Math Centers. This math center idea is sure to delight your students! Using paper towel rolls, toilet paper rolls, or even plastic water bottles you can create bowling pins with numbers on them. If you cover them with packaging tape, you can wipe the numbers off for reuse again and again. Students can then bowl to add the pins knocked down. This makes a great center for fractions, large numbers, decimals, and you could use it for elapsed time. In the past, I have even placed task cards on them and students had to complete the pins they knocked down. It&rsquos definitely engaging! You can read more about it here.

4.) Journaling. What student couldn&rsquot use a little extra practice writing in math? I believe they all could. Provide students with a writing prompt related to whatever you are working on in math. It doesn&rsquot need to be fancy You could write it on an index card, or have it in strips for students to glue into their math journal. They then respond in their journal. Students could respond to a question such as, &ldquoWhat is a fraction?&rdquo It doesn&rsquot need to be complicated. You&rsquore basically having students explain math in their own words. You could even include a portion where they can illustrate it.

5.) Craftivities. Every once in a while, why not consider creating a craftivity for your center activity? This could be anything from the simple construction of a mobile or model of a product. Students enjoy hands-on activities and it&rsquos a great way to learn.

6.) Manipulatives. Manipulatives are a great way to make math more concrete, especially with difficult concepts. Place manipulatives in a center with some directions and students can practice over and over until they really have the concept down.

7.) Technology. There are a lot of great apps and websites out there that really help students practice both math skills and math test prep. I&rsquove mentioned many websites in my blog post, 5 Great Math Apps for Grades 3-5. There are also great websites like the Math Playground, IXL, MobyMax, and Reflex. There are even free sites that students can access from home such as AAA Math, Cool Math, and Fun Brain.

8.) Reading Literature. Create a center that integrates both reading and math. For instance, you could use any of Greg Tang&rsquos Books, such as The Grapes Of Math, or his problem-solving riddle book, Math-ter Pieces: The Art of Problem Solving. These are full of math riddles that students solve while enjoying reading too! Another fun one is The Math Curse. There are so many great options out there related to both literature and math!

9.) Math Sorts. These can be created easily with a computer or index cards. You can also just purchase them. Students can easily sort math based on similarities or attributes, such as in my Polygon Sort below.

10.) Practice Sheets. Of course, you can use regular worksheets to help students practice the math concepts you are teaching, but you can also use these sheets to remediate, reteach, review, or even enrich. There is nothing wrong with a &ldquoStay at Your Seat&rdquo kind of center.

11.) Problem Solving. Why not create a center that is solely based on problem-solving? My students struuuuuuugggggglllle with problem-solving. So they could definitely use some extra practice. This center provides the perfect opportunity for that. Do you know that appendix section in the common core standards? They provide you with examples of different types of problems you could create. Go from there. Or, start the first few weeks off with the students creating the problems first and trading papers to check each other&rsquos work. Then create some yourself. You can also mimic the ones in your textbook. Step it up a notch by throwing in some error analysis once in a while.

12.) Interactive Bulletin Boards. I love creating them, but I will admit, it sometimes requires a little time and thought on your part &ndash but you can definitely do them! Back around Halloween, I created this one made of a haunted house. On the outside of the windows, ghosts, and other objects were the math problems, on the inside were the self-checking answers. It was a great center (and is also perfect for early finishers). (Note: The image below was set up on a tri-board.)

Another interactive bulletin board I have created in the past was Math-no-poly. It was something I created for an end of the year review. My students really enjoyed it and it helped us focus on what we still needed to learn.

These math center ideas were definitely a blast!

13.) Math Stretchers. I talked a little about Math Stretchers in my Math Workshop Series. This is a form of a math warm-up that can be used for any of the math categories. It&rsquos a great way to prepare the student&rsquos brain for the math they are about to encounter and to start thinking about math more in their everyday life. This can be something as simple as collecting data each day regarding lunch or focusing on the number of the day.

14.) Calendar. I have always loved using the calendar portion in math workshop, even in the upper grades. Yes, even in fifth grade. Yes, even if I didn&rsquot have the official kit or book. I got creative. I just focused on what my students needed to know and found a way to integrate it in each day. You create small square pieces with patterns that change daily around a common theme, such as angles, polygons, fractions, etc. You can create nearly anything! The point is to gather students around and get them talking about math based on the math they are surrounded within their daily lives.

15.) Interactive Notebook Pieces. Interactive notebook pieces are not just a waste of time if you do them correctly. Students can actually use them to help remember important information and to practice solving problems. They can also be very engaging. For instance, Have students list all the multiples out on individual &ldquoFrench Fries&rdquo for each number and put them inside a French Fry box (envelope glued in their notebook).

16.) Regular Math Center Games . There are lots of games you can purchase in Teacher Stores, on Teachers Pay Teachers, or even make that can help reinforce a concept. I really help understand the importance of Game-Based Learning and provide 5 tips to using them in the classroom here.

17.) Vocabulary. Consider creating a math center that provides your students with the opportunity to practice the important vocabulary of the concepts you are working on. This can include a personal word wall, a class word wall, or just a list of words you have been learning. Students can complete any vocabulary activity. There are many blog posts here on this blog that provides you with engaging activities to do. Just search vocabulary over in the search bar on the right.

18.) Task Cards. Task cards do not have to be a whole group activity. You can provide students with a set of task cards to work on in stations or have them move around within their group. I have even placed task cards around the room that they quietly had to move to while the others worked. You don&rsquot necessarily have to do the entire set either. If you don&rsquot want to purchase any, don&rsquot. Take a worksheet, cut each individual question out and glue them to index cards. Viola! Task cards!

19.) Real World Math. Students really need to see math in the context of the real world. Why not make one of your math center ideas a real-world math center? This can be a range of math centers, from current events, to STEM. For current events, have students find examples in the current events that use whatever concept you are working on. This is not limited to social studies. You could have students complete a &ldquomini-stem&rdquo project, a project-based learning activity, or even just an activity related to real-life, such as planning for a family Bar-B-Que.

20.) Fact Fluency. Students really need to practice their basic facts in nearly every grade. They need to memorize them. The best way to memorize them is with constant practice. That is where this center comes in! Provide students with lots of opportunities to learn and practice their basic facts. This can be done in the form of activities (such as with my Math Workshop Shortcuts Unit), games, flashcards, math fact practice, or other engaging activity.

With these 20 math center ideas, you are sure to find something to keep your students from getting bored and to keep them hooked on math!

Are you looking for engaging math centers for your students that you can purchase to save yourself some planning time? Check out my centers and games here.

Want more valuable teaching tips and other ah-mazing perks, such as discounts, giveaways, flash freebies, dollar deals, and so much more? Become a VIP member by joining my email list! Click here to subscribe.

## 20 Grade-School Math Questions So Hard You'll Wonder How You Graduated

Unless you grew up to be an engineer, a banker, or an accountant, odds are that elementary and middle school math were the bane of your existence. You would study relentlessly for weeks for those silly standardized tests—and yet, come exam day, you'd still somehow have no idea what any of the equations or hard math problems were asking for. Trust us, we get it.

While logic might lead you to believe that your math skills have naturally gotten better as you've aged, the unfortunate reality is that, unless you've been solving algebra and geometry problems on a daily basis, the opposite is more likely the case.

Don't believe us? Then put your number crunching wisdom to the test with these tricky math questions taken straight from grade school tests and homework assignments and see for yourself.

### 1. Question: What is the number of the parking space covered by the car?

This tricky math problem went viral a few years back after it appeared on an entrance exam in Hong Kong… for six-year-olds. Supposedly the students had just 20 seconds to solve the problem!

Believe it or not, this "math" question actually requires no math whatsoever. If you flip the image upside down, you'll see that what you're dealing with is a simple number sequence.

### 2. Question: Replace the question mark in the above problem with the appropriate number.

This problem shouldn't be too difficult to solve if you play a lot of sudoku.

All of the numbers in every row and column add up to 15! (Also, 6 is the only number not represented out of numbers 1 through 9.)

### 3. Question: Find the equivalent number.

This problem comes straight from a standardized test given in New York in 2014.

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You're forgiven if you don't remember exactly how exponents work. In order to solve this problem, you simply need to subtract the exponents (4-2) and solve for 3 2 , which expands into 3 x 3 and equals 9.

### 4. Question: How many small dogs are signed up to compete in the dog show?

Image via Imgur/zakiamon

This question comes directly from a second grader's math homework. Yikes.

### Answer: 42.5 dogs.

In order to figure out how many small dogs are competing, you have to subtract 36 from 49 and then divide that answer, 13, by 2, to get 6.5 dogs, or the number of big dogs competing. But you're not done yet! You then have to add 6.5 to 36 to get the number of small dogs competing, which is 42.5. Of course, it's not actually possible for half a dog to compete in a dog show, but for the sake of this math problem let's assume that it is.

### 5. Question: Find the area of the red triangle.

This question was used in China to identify gifted 5th graders. Supposedly, some of the smart students were able to solve this in less than one minute.

In order to solve this problem, you need to understand how the area of a parallelogram works. If you already know how the area of a parallelogram and the area of a triangle are related, then adding 79 and 10 and subsequently subtracting 72 and 8 to get 9 should make sense—but if you're still confused, then check out this YouTube video for a more in-depth explanation.

### 6. Question: How tall is the table?

YouTuber MindYourDecisions adapted this mind-boggling math question from a similar one found on an elementary school student's homework in China.

### Answer: 150 cm.

Since one measurement includes the cat's height and subtracts the turtle's and the other does the opposite, you can essentially just act like the two animals aren't there. Therefore, all you have to do is add the two measurements—170 cm and 130 cm—together and divided them by 2 to get the table's height, 150 cm.

### 7. Question: If the cost of a bat and a baseball combined is $1.10 and the bat costs$1.00 more than the ball, how much is the ball?

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This problem, mathematically speaking, is very similar to one of the other ones on this list.

Think back to that problem about the dogs at the dog show and use the same logic to solve this problem. All you have to do is subtract $1.00 from$1.10 and then divide that answer, .10 by 2, to get your final answer, .05.

### 8. Question: When is Cheryl's birthday?

Image via Facebook/Kenneth Kong

If you're having trouble reading that, see here:

"Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.

August 14 August 15 August 17

Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.

Albert: I don't know when Cheryl's birthday is, but I know that Bernard doesn't not know too.

Bernard: At first I don't know when Cheryl's birthday is, but I know now.

Albert: Then I also know when Cheryl's birthday is.

So when is Cheryl's birthday?"

It's unclear why Cheryl couldn't just tell both Albert and Bernard the month and day she was born, but that's irrelevant to solving this problem.

### Answer: July 16.

Confused about how one could possibly find any answer to this question? Don't worry, so was most of the world when this question, taken from a Singapore and Asian Schools Math Olympiad competition, went viral a few years ago. Thankfully, though, the New York Times explains step-by-step how to get to July 16, and you can read their detailed deduction here.

### 9. Question: Find the missing letter.

Image via Facebook/The Holderness Family

This one comes from a first grader's homework.

### Answer: The missing letter is J.

When you add together the values given for S, B, and G, the sum comes out to 40, and making the missing letter J (which has a value of 14) makes the other diagonal's sum the same.

### 10. Question: Solve the equation.

This problem might look easy, but a surprising number of adults are unable to solve it correctly.

Start by solving the division part of the equation. In order to do that, in case you forgot, you have to flip the fraction and switch from division to multiplication, thus getting 3 x 3 = 9. Now you have 9 – 9 + 1, and from there you can simply work from left to right and get your final answer: 1.

### Answer: A line should be drawn on a "+" sign.

When you draw a slanted line in the upper left quadrant of a "+," it becomes the number 4 and the equation thusly becomes 5 + 545 + 5 = 555.

### 12. Question: Solve the unfinished equation.

Try to figure out what all of the equations have in common.

### Answer: 4 = 256.

The formula used in each equation is 4 x = Y. So, 4 1 = 4, 4 2 = 16, 4 3 = 64, and 4 4 = 256.

### 13. Question: How many triangles are in the image above?

Some people get stumped by the triangles hiding inside of the triangles and others forget to include the giant triangle housing all of the others. Either way, very few individuals—even math teachers—have been able to find the correct answer to this problem. And for more questions that will put your former education to the test, check out these 30 Questions You'd Need to Ace to Pass 6th Grade Geography.

### 14. Question: Add 8.563 and 4.8292.

Adding two decimals together is easier than it looks.

Don't let the fact that 8.563 has fewer numberrs than 4.8292 trip you up. All you have to do is add a 0 to the end of 8.563 and then add like you normally would.

### 15. Question: There is a patch of lily pads on a lake. Every day, the patch doubles in size…

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… If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

### Answer: 47 days.

Most people automatically assume that half of the lake would be covered in half the time, but this assumption is wrong. Since the patch of pads doubles in size every day, the lake would be half covered just one day before it was covered entirely.

### 16. Question: How many feet are in a mile?

This elementary school-level problem is a little less problem solving and a little more memorization.

This was one of the questions featured on the popular show Are You Smarter Than a 5th Grader?

### 17. Question: What value of "x" makes the equation below true?

Shutterstock

You'd be forgiven for thinking that the answer was 3. However, since the number alongside x is negative, we need x to be negative as well in order to get to 0. Therefore, x has to be -3.

### 18. Question: What is 1.92 divided by 3?

You might need to ask your kids for help on this one.

In order to solve this seemingly simple problem, you need to remove the decimal from 1.92 and act like it isn't there. Once you've divided 192 by 3 to get 64, you can put the decimal place back where it belongs and get your final answer of 0.64.