Think About It!: The Equals Sign

Ask yourself and answer these three questions:

• What do you understand the equals sign to represent?
• What do you teach your students about the equals sign?
• What do your students understand about what the equals sign represents?

8 + 3 = 11

12 - 5 = 7

33 x 10 = 330

Most people understand the equals sign to show where to put an answer. When they look at the above number sentences they will likely interpret the left-hand side as the problem and the right-hand side as the answer.

A few weeks ago, I started having my classes work on basic number sense (since they still don't have it in 4th grade--a future "Think About It! discussion!). The very first thing we talked about was the equals sign. I asked them what they knew about it and all of them told me something along the lines of "it means the answer" or "it means 'is'; where you put the answer." I was a little taken aback and suddenly realized why they struggle with very basic problems such as 12 + __ = 20 or 7 x __ = 140. (Do your students struggle with those too?)

I have seen several recent studies that involve asking students the same questions about the equals sign. Let's, for instance, use 4 + 7 = __ + 2. On problems like these students would oftentimes add 4 and 7 and place the sum in the blank so it reads 4 + 7 = 11 + 2, completely ignoring the 2. I'm sure you as an adult (and teacher) look at that and instantly understand how the number sentence and thinking is wrong. Unfortunately, most students will not.

The equals sign indicates equality. It is placed between two things of equal value. Think of it as a balance. So when you say 8 + 3 = 11, you are saying that 8 + 3 has the same value as 11.

Can you see how knowing this in elementary school will help students better understand algebraic equations?? When students are learning about evaluating algebraic equations such as 3x + 7 = 28 they MUST understand that both sides are equal or they will never understand how to solve for x. The time to teach this is NOT in middle school but in the early grades.  

Students should also understand that one side of the equal sign does not always have to have a single number. The equals sign can show two expressions that have the same value, like 9 x 4 = 12 x 3.

I want to address one more misconception--the running equals sign. Here is a word problem and example:

Sally had 4 marbles. She cleaned her room and found 3 more marbles. Her friend then gave her 8 marbles. How many marbles does Sally have now?

Number sentence: 4 + 3 = 7 + 8 = 15

How is that wrong? Remember, both sides have to be equal. The correct way to write it would either include multiple number sentences or parentheses:

(4 + 3) + 8 = 15

OR

4 + 3 = 7

7 + 8 = 15

I wouldn't say repairing this misconception is a quick fix, but it is definitely doable. (And, of course, it would be better if instruction about the equals sign is correct from the beginning.)

• You can start by having a discussion about it with your kiddos. Bring in a balance to help you explain (for instance, put two pennies on one side, then add 6 more to it. Record that as 2 + 6. Then add pennies to the other side (8) and let your students witness how the trays become balanced. Record it as 2 + 6 = 8.)
• Another activity would be to have students use equality cards. You can make some simply by putting a number sentence with one missing number on one card and have the students match it to a number sentence in the same fact family with a different number missing (e.g., 6 + __ = 8 on one card and 8 = __ + 2 on the other.) Look for a free Equality Card activity in my Teachers Notebook store!
• You could also make a mat with an equals sign and have students place cards of equal value on either side of the symbol. The cards could contain pictorial models, standard form, expanded form, etc.

Sidenote: I'm not sure if you are familiar with the National Library of Virtual Manipulatives--if you aren't, you NEED to be! It is exactly what it states. The site has every manipulative imaginable in virtual form for FREE. It also has activities to go along with each.

To those that teach the higher grades--check out this balance that provides visual representation of algebraic equations:

Awesome, right?!?

Even if you don't teach the higher grades I encourage you to check out the algebraic equation scale and ponder about how easy it would be for our current students who do not understand an equals sign.

Think about it...

I always enjoy incorporating literature into math class. I have not actually read this book but it looks like a cute little story about creating equal sides. Check it out by clicking on the picture.

Thanks for tuning in. I WOULD LOVE TO HEAR YOUR THOUGHTS! THROW 'EM AT ME! :D

Also, I would LOVE to tell you to look for these posts on a certain day and time but I can't right now--life is not that predictable! Installment number 2 will be before school starts back though! :)

Think About It!: The Purpose

Before I jump into the first post (which will be tomorrow) I wanted to give you a little background about why I'm doing this...

When I was in college ELAR (English Language Arts and Reading) instruction was emphasized SO much; the majority of my classes focused on reading, reading, reading. We developed lesson cycles and thematic units, tutored and gave reading inventories to students, and participated in numerous reading labs, i.e. field experiences. All were great experiences and taught me a lot! My first real teaching job, however, was 4th grade math (and still is). Teaching math was not intimidating to me--after all, math has always been my best subject. I didn't realize though until after getting started that I was NOT as prepared to teach math as I was to teach reading (which I didn't do). In college, math did not receive the same emphasis reading did (at least at my alma mater). I had a few classes about math instruction but not nearly as many as ELAR. The only field experience we participated in for math was student teaching (IF you were placed in a classroom that taught math).

In bloggy blog world there are SO many ELAR experts--you all make the neatest products and have the neatest ideas to incorporate into instruction. It is really encouraging to know that I could get help with anything ELAR from numerous sources should I need it (since I do not teach ELAR at this time). However, there is slim-pickin' when it comes to math experts. I think being a math expert is more than just creating fun games and neat activities. To me, a math expert understands math well and understands how children learn math; he/she knows how to build a foundation AND knows/understands more than just processes, procedures, and rules. He/she knows the "how" and "why" behind such processes, procedures, and rules, and uses this knowledge to help children understand them. Most importantly, a math expert understands that mathematical concepts should be explored and NOT taught solely using a set of rules and procedures (note that I am emphasizing a point by using rules, procedures and processes several times!). Effective mathematics instruction is paramount in the early years as it serves as the building blocks for higher level concepts. Building a strong mathematical foundation in the early grades is vital for success in math as building reading foundational skills is crucial in learning to read. (For instance, algebra is not something reserved for a middle school math class, rather a concept that is introduced and built upon in the early years in life.) 4th grade teachers should not have to teach 1st grade math as 6th grade teachers should not have to teach 4th grade math.

My hope in launching this series is to encourage you to reflect upon your own teaching and thinking (as I have and still do) in order to enhance mathematics instruction in your classroom. Please note that I KNOW that I do not have all of the answers which is why I encourage you to actively participate in this series with me (one of the many benefits of being a part of an online community of teachers!). I hope to challenge you to examine your methods and ways of thinking. Though we all come from different places, backgrounds and may have different teaching styles, our goal is the same--to EDUCATE children.

Thanks for your interest and I SOO look forward to your input! :D

Winter Wonderland: Equivalent Fractions

I am in the process of making a "Winter Wonderland" package of math games! I am releasing it one piece at a time and here is the first piece:

Click on the picture for more information! Thanks :D

New Series: Think About It!

I am excited to announce that I am starting a new blog series entitled "Think About It!" The purpose of this series is to encourage you to think about/reflect upon how you are teaching math. I have learned so much about teaching math this past year (mostly due to working on a masters in math curriculum and instruction). I want to challenge you in the same way I have been challenged. The first post will be:

Expect to catch part 1 either tonight or tomorrow--thanks for tuning in! I hope you are as excited as I am!

Fraction Fun!

Please check out the newest addition to my Teachers Notebook store--"Fractions as Part of the Whole".

I used this activity with my students our last week in school and they LOVED it! It really challenged them and helped me realize how much they understand (or don't understand) about parts and wholes. To complete the activities students will need Cuisenaire rods, pattern blocks and thinkers!

I know it is an odd time to post but hopefully SOMEONE is out there reading this--first three to comment will receive a free copy of the activity!

MERRY CHRISTMAS!!

Decimal Journal Pages

Since I have NO time to make notebook packages right now (or anytime soon) I have decided to post individual journal pages as I can. The first ones I have available are for adding and subtracting decimals and comparing and ordering decimals. Click on the pictures below to check out the items.

I am anxious to finally start Christmas break and post a REAL blog post! It's been a while since I have! We're in school until Wednesday afternoon so look for it sometime after then!

As usual, the first 3 people to comment will receive a free copy (of each). Don't forget to leave your email!!

TWO new math games!

I just added two math games to my Teachers Notebook store. Click on the pictures for more information!

I'm Hosting Day 6 on Oh' Boy 4th Grade!

Hop on over to enter my Christmas giveaway at Oh' Boy 4th Grade! You will also have the chance to enter to win some other FAB prizes from some other totally awesome blog divas! Click on the join the parrrrtty!

12 Days of Christmas Giveaway!!

It's time for a giveaway--make that 12 to be exact! Farley at Oh Boy is hosting an oh-so-fab giveaway for 12 days! Each day includes a new prize and a different guest blogger. I'll be hopping over on the 6th and I'm super excited about it! Click on her button to visit her blog and join in on the fun!