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Ten Black Dots - Linking Math and Literature

I am an avid reader, and I love books that integrate math and literature. Occasionally, my blog will feature a book that links the two.  I will summarize the book, give its overall mathematical theme, as well as list various activities you can use in your classroom.

Today's featured book is Ten Black Dots by Donald Crews (Greenwillow Books, 1986).  This picture book is for grades PreK-2 and deals with numbers and operations. 

The book asks the question, What can you do with ten black dots?  Then the question is answered throughout the book by using  illustrations of everyday objects beginning with one dot and continuing up to ten. Simple rhymes accompany the pictures such as:

"Two dots can make the eyes of a fox, Or the eyes of keys that open locks."

Materials Needed: 
  • Unifix cubes or Snap Cubes (multi-link cubes) as seen on the right
  • Black circles cut from construction paper or black circle stickers
  • Crayons
  • Pencils
  • Story paper
  • Calculators -simple ones like you purchase for $1.00 at Walmart


1)  Read the book a number of times to your class.  Let the students count the dots in each picture. On about the third reading, have the children use the snap cubes to build towers that equal the number of dots in each picture.

2)  Have the children think of different ways to make combinations, such as: How could we arrange four black dots?  (e.g. 1 and 3, 4 and 0, 2 and 2)  Have the children use black dots or snap cubes to make various combinations for each numeral from 2-10.

3)  This is a perfect time to work on rhyming words since the book is written in whimsical verse. Make lists of words so that the students will have a Word Wall of Rhyming Words for activity #4.
  • How many words can we make that rhyme with:  sun?  fox?  face?  grow?  coat?  old?  rake?  rain?  rank?  tree?
  • Except for the first letter, rhyming words do not have to be spelled the same.  Give some examples (fox - locks or see - me)
4)  Have the children make their own Black Dot books  (Black circle stickers work the best although you can use black circles cut from construction paper. I'm not a big fan of glue!)  Each child makes one page at a time.  Don't try to do this all in one day.  Use story paper so that the children can illustrate how they used the dots as well as write a rhyme about what they made.  Collate each book, having each child create a cover.

5)  Have the children figure out how many black dots are needed to make each book. (The answer is 55.)  This is a good time to introduce calculators and how to add numbers using the calculator.

If you can't find Ten Black Dots in your library, it is still available on Amazon.

The "Lure" of Fish - The Benefits of Keeping an Aquarium in Your Classroom

I am a critter "wife"; not by choice, but since my husband teaches science, creatures of all kinds enter our home. We have been blessed with hissing cockroaches from Madagascar, (they are huge) tree frogs, meal worms, (they turn into black bugs) red worms, etc. Some have lived in our guest room while others have found a special place in my refrigerator. We've even had horse dung soaking in water so the bacteria could grow. Oh, that was delightful and aromatic! Thank goodness for the invention of "Oust". So after all of those creatures, what kind of a story could I possibly write to make critters attractive? Well, I do have a fish tale, but up front I must disclose that it is not your typical "fish" story.

When I taught third grade in an inner city school, I knew the children needed something to love, but being a city girl, my love of animals was deficient. That is when my husband helped me to set up an aquarium. I purchased a water heater, a bubblier, chemicals, plants, fish food, and of course the fish! Little did I know what effect this would have on my students.
Every day, the children would enter the room, go over to the fish tank, and talk to the fish. Each fish had a name, and being the fish keeper became the prized chore. Even though we couldn't pet the fish, they were loved by every child, and they brought a sense of family to my classroom. Naturally, one of the fish died, but it allowed us, as a class, to mourn together.

When a guest entered our room, s/he had to be formally introduced to the fish. They became the focal point of the classroom. But there was something else that transpired that truly surprised me. The fish had a calming effect on my student who had a behavior disorder. If his desk were moved near the aquarium, he would sit quietly and actually do some of his work without disruption.

At the end of the year, the remaining fish made their way to my home where they spent the summer with my husband's critters. Unfortunately, they failed to calm the cockroaches into silence! But all six survived to be in third grade again!

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Algebraic Terms - Finding the Greatest Common Factor and Least Common Multiple Using a Venn Diagram

I tutor math at the college where I teach. Many of those students have been confused on how to find the greatest common factor for a set of algebraic terms. Having an elementary background, I introduce them to a factor tree which, believe it or not, many have never seen.

When just a rule is given by an instructor, often times, students get lost in the mathematical process. I have found that utilizing a visual can achieve an understanding of a concept better than just a rule. A Venn Diagram is such a visual and helps students to follow the process and understand the connection and relationship between each step of finding the GCF and LCM.

It's important to always begin with the definitions for the
words factor, greatest common factor and least common multiple. If a student doesn't know the vocabulary, they can't do the work! I continue by explaining and illustrating what a factor tree is (on your left) and how to construct and use a Venn Diagram as a graphic organizer.

Let's suppose we have the algebraic terms of 75xy and 45xyz. I have the students construct factor trees for each of the numbers as illustrated on the left.

Then all the common factors are placed in the intersection of the two circles. In this case, it would be the 5 and the xy. 

The students then put the remaining factors and variables in the correct big circle. Five and three would go in the left hand circle and the three 2’s and the z would be placed in the right hand circle.

The intersection is the GCF; so, the GCF for 75xy and 40xyz is 5xy.   To find the LCM, multiply the number(s) in the first big circle by the GCF (numbers in the intersection) times the number (s) in the second big circle.

5 × 3 × GCF × 2 × 2 × 2 × z = 15 × 5xy × 8z = 240. The LCM is 600xyz

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This method is applicable and helpful in algebra when students are asked to find the LCM or GCF of a set of algebraic terms such as: 25xy, 40xyz. (LCM = 200xyz; GCF = 5xy) or when they must factor out the GCF from a polynomial such as 6x2y+ 9xy2. Using a Venn Diagram is also an effective and valuable tool when teaching how to reduce fractions. 

Are you interested in finding out more about this method?  Then download my newest free resource entitled: Algebraic Terms and Fractions - Finding the Greatest Common Factor and the Lowest Common Multiple Using a Venn Diagram. 

Why is 'x' Usually the Unknown in Algebra?

Ted Talk
Again, it's time for some math information you might have missed in school. (Don't worry, I missed a great deal as well.)  Today's question is: Why is the letter "x" the symbol usually used for an unknown?

Even though the letter "x" is commonly used in mathematics, its use often appears in non-numerical areas within different industries such as The X Files or Project X. Terry Moore clears up this mathematical mystery in a TED Talk presentation at Long Beach, California.  In a short and funny four minute talk, he gives an unexpected answer to "why." Just click under the illustration to find out the reason!

What Is Your Mindset?

In the fall, I taught a new course entitled Conquering College. We have found that many students entering college are not prepared, lack study skills as well as the soft skills of being on time, regularly doing homework, turning in assignments - on time, etc. This class is required for every student who tests into developmental math and/or reading.  It has three purposes:
  1. To enable students to learn and use Advancement Via Individual Determination (AVID) strategies necessary for persistence and success at the college level, 
  2. To develop a learning plan based upon personal abilities and goals, and 
  3. To become more self-reliant in fulfilling academic goals.
During the sixteen weeks, we spent time focusing on the the growth and fixed mindset. If you haven't heard of the fixed mindset and the growth mindset, I'll summarize it this way - it is our attitudes, thoughts and beliefs about something.

Study the two charts below. In which category do you fit? Where would you place many of your students?

Carol Dweck has done a great deal of research on Growth Mindset. There is a ten minute video on You Tube which is well worth watching and sharing with your fellow teachers. It is called How To Help Every Child Fulfill Their Potential. (click on title) If for no other reason, watch it to discover what 15 years of praising children for their intelligence has done to our students. It is definitely eye opening!

Magically Squaring Numbers

My college math students lack confidence (I classify them as mathphobics.); so, I like to show them math "tricks" which they can use to impress their peers.  I encourage them to know their squares through 25. (Yes, I know they can use a calculator, but the mind is so much quicker!)  When we get to solving equations using the Pythagorean Theorem, I introduce this trick. Please note: For the trick to work, it must be a two digit number that ends in 5.
Suppose we have 352.  (This means will be making a square.)
  • First, look at the number in the hundred’s place. In this case, it is the “3”. 
  •  Next think of the number that comes directly after 3. That would be “4”. 
  •  Now, in your head, multiply 3 × 4. The answer is 12. 
  • Finally, multiply 5 × 5 which is 25. 
  •  Place 12 in front of 25 to get the answer. Thirty-five squared is 1,255.
  • 3 × 4 = 12      5 × 5 = 25       
  • The answer is 1,225.
This means that we can build a square that is 35 by 35, and it will contain 1,225 squares or have an area of 1225 squares.

Now let's try 652
  • One more than 6 is 7; so, 6 x 7 is 42. 
  • Place 42 in front of 25 (5 x 5) and so 65 squared is 4,225.
  • 6 × 7 = 42      5 × 5 = 25      
  • The answer is 4,225.
How about finding the square root? We begin by looking at the numbers in the thousands and hundreds place. In the answer of 1,225, we would use the 12. Think of the factors of 12 that are consecutive numbers. In this case, they would be 3 and 4. Use the smaller of the two which, in this case, is 3. Now place a five after it. You now know the square root of 1,225 is 35.
Thirty-five represents the length of one of the sides of a square that contains 1,225 squares.

Now, try some numbers on your own. When you get comfortable with the "trick", try it with your students. They will find out that math can be magical!

Homework - Is It an Essential Part of School? defines homework as "schoolwork assigned to be done outside the classroom (distinguished from classwork)", but is homework beneficial? Teaching on the college level, I see many benefits to those students who have been required to complete real homework in high school. Here are just a few.

1) Homework can improve student achievement. Studies show that homework improves student achievement in terms of better grades, test results, and the likelihood of attending college.

2) Homework helps to reinforce learning and to develop good study habits and life skills. Homework assists students in developing key skills that they will use throughout their lives, such as accountability, self-sufficiency, discipline, time management, self-direction, critical thinking, and independent problem-solving. Homework assignments given to students actually help students prepare for getting a higher education degree. In fact, the more time a student spends honing his skills, the higher his chances are to enter the University of his dreams or later acquire the work he always wanted to do.

3) Homework can make students more responsible. Knowing that each homework assignment has a specific deadline that cannot be postponed makes students more responsible. It requires grit (perseverance), teaches them time management and causes them to prioritize their time for academic lessons.

As you read this list, I know there are many of you, especially those who have small children or teach younger children, who disagree. I am not here to argue about whether homework is appropriate in the lower grades, but I do want to advocate real homework on the high school level. When I say real homework here is what I mean.

In high school, students might finish their homework in the hall right before class and still earn a good grade; that just isn't possible in college. Homework may be due on a certain day, but it is acceptable if it is turned late. This typically doesn’t float on the college level. In high school, a student gets to the end of a semester and needs a few more points to pull up a grade because of missing or incomplete assignments; so, the student asks the teacher for extra credit work. Extra credit does not exist on the college level! You do the work you are given when you are given it!

I teach college freshmen, many who are woefully unprepared for the academic rigors and demands that are expected. For every one hour students take in college, they should expect two hours of outside work. In other words, if a student is taking 12 hours, they should expect to spend 24 hours on homework (12 x 2).  Of course this formula doesn't always work perfectly, but it is a good starting point. Usually, college freshmen are in disbelief that they are expected to spend so much time on work outside of class. In reality, they should expect to spend as much time on homework in college as they would at a job because college is a full time job!

When I hand out my syllabus, many of my freshmen are astonished when they discover the amount of homework I expect and require them to do (readings, papers, on-line research, projects, etc.) AND to compound the problem, many instructors (including me) expect it to be done and handed in on time! Unfortunately, several students have to test the waters to find out that late papers are not accepted.

For those college students who've had little real homework in high school compounded by teachers who have allowed it to be turned in late, those students are aboard a sinking ship that is leaking fast! Sadly, those are the 2-3 students who fail my required class and have to retake it the next semester.

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So, as you can see, the decision to agree with or disagree with assignments is really up to the student, but also they need to remember that the learning institution they attend has rules in place regarding assignments. And if homework is assigned, then it will need to be completed and handed in on time, or the impact on the final semester grade will certainly be negative.

Want a quicker and easier way to grade math homework? Try one of these two math rubrics. I still use them on the college level, and they save me a great deal of time!