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I was at the beginning of my teaching career when the "turmoil" over the common core standards was in full force.  My state had changed their standards to the "Utah Core Standards," but they were really the common core standards in disguise.  They were trying to avoid the terminology "common core" and all the political problems it was causing, but the truth was that the state recognized that these were quality standards, and it would benefit the students in my state if they were implemented.  

Before the 1960's arithmetic was the majority of what was taught in math class.  Then the United States entered the space war with Russia...the USA was determined to become the best.  As a result, the "new math" was introduced.  The "new math" dove into matrices, trigonometry, geometry, and more all on a very conceptual level.  The "new math" eventually received a lot of push back as many people thought it would be more beneficial for students to learn a little about a lot of math.  The curriculum then changed to "a mile wide but an inch deep."  At this point, math became less conceptual and more algorithm based.  The students that naturally had good math reasoning were still pushed along and entered Calculus during high school, but every one else started to get left behind.  This became evident when they entered college. The basic level math classes at the the Universities were full and many students were struggling.  

It was clear that the standards were failing many students.  Purely conceptual wasn't a solution, and purely procedural didn't work either.  The new standards were created with the goal of valuing conceptual and procedural.  With these new standards, the hope of many educators is to not lot students get left behind.  To allow all students to succeed. First, teach at a conceptual level so students can reason through the mathematics and perhaps even discover an algorithm.  Encourage procedural fluency, but only after they have mastered conceptual understanding.  

With these ideas in mind is how I create all of my resources.  Valuing both conceptual understanding and procedural fluency.  If you are interested in some math assessments that assess both conceptual understanding and procedural fluency you can click on the links below. They are also editable for use year after year. 







Using bar models to teach percents of whole or to calculate the whole given the percent and part is a very effective way to teach percents conceptually. When introducing this way, I highly suggest starting off with percents that are divisible by 5.  You can eventually do any percent with combinations of 5% and 1% as a tenth of 10%, but start off simple.

Let's look at percents of a whole.


An entire bar will represent the whole.  Draw the bar and label the whole.



Then you need to divide up the bar based on the given percentage.  50% would be in half.  20% would be divided into fifths.  30% would be divided into tenths.  Use guiding questions to help your students think of how to divide the bar. If they are stuck, always dividing it up into tenths or fifths should work, as long as you are using percentages that are divisible by 5.  



Next you need to divide the whole up into that many sections.  Again, use guided questions to help your students figure out how to do this.  They should come to the conclusion that the whole divided by the number of parts is the amount per section.  Write that amount in each of the sections.




Lastly, determine how many sections you need for the given percentage.  The students should already know what percent is each section from the first step.  After the number of sections is determined, I like to color the sections in on my bar so students can visually see the percentage of the whole.  They can then determine the part by looking at the value of the total colored sections. 




Bar models can also be used with finding the whole given a part and a percent as well as finding the percent given the part and whole.  

Let me just put this disclaimer on this method:  Doing percents with bar models may take a lot of work at first.  Each problem will take longer than if you just gave your students an algorithm.  This is how it is with most conceptual teaching.  However, I know that as you teach conceptual at first, and take the time to have the math make sense to the students, they will retain the information, and in turn, you won't have to do all of the extra review at the end.

If you are interested in the task cards used in the photos you can get them HERE

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My first two years of teaching went ok.  I was learning the ropes of running a classroom and honestly, just trying to survive.  I didn't have my students write too much, because "Hey, I teach Math."  The summer after my second year of teaching I took an intense master's class about math pedagogy...homework included.  I noticed something that the professor always had us do on our homework, and that was to explain our reasoning.  I suddenly had as much writing on my homework as I did actual math.  I quickly learned the value of writing in a math classroom.  To be able to actually explain in words how to do the math, takes the math to a deeper level.  Students have to actually think about the why instead of just passing through meaningless algorithms.  Also, as a teacher, have you ever tried grading a student's work, and you are just not quite sure if they understand the concept?  Having students write their thinking can take your math in your classroom to a deeper level, and seriously, grading papers becomes an easier task.  No more second guessing if the student really understands or not.  If they can accurately explain their reasoning you know that they really understand the concept.  

There are a few ways you can have your students write in the math classroom.  One strategy is in a notebook.


You can give writing prompts on the board, a classroom screen, or on task cards and have the students write in their notebooks.  One suggestion that I enforced in my classroom was that they had to write in complete sentences.  I also put a minimum of three sentences.  Many students went beyond three sentences, but I learned I needed a minimum for some students in my classroom.  Also, another strategy I often incorporated was to have them include an example of what they were explaining.


You can also had out slips of paper and use them as exit slips. 

This is a great way to see how much your students understood the lesson.  Hand out the slips of paper with the writing prompt and have them complete the writing and hand it to you as you walk out the door.  Also, as a side note, if students know they have to complete these to leave class, they will be very engaged in the lesson ;). 

You can also use these slips of paper as openers.  Hand them to students as they walk in the classroom, and collect them when they are complete.  This also helps students not be tardy, because they have an assignment write when class begins.

Writing in my math classroom, seriously took my class to a new level.  If you have not yet tried writing, I highly suggest you do.  

If you want 50 prompts for your 7th grade or 8th grade math classrooms, or the product in the photos you can click on the links below.

CLICK HERE FOR 50 7TH GRADE PROMPTS, TASK CARDS AND WRITING SLIPS

CLICK HERE FOR 50 8TH GRADE PROMPTS, TASK CARDS AND WRITING SLIPS

GRAB YOUR FREEBIES HERE!




1) Make Sense of Multiplication
Students need to make sense of the meaning of multiplication in terms of groups.  Students may have previously learned multiplication as repeated addition, though this is accurate, extend their understanding to multiplication as meaning groups of.  For example, 2(x + 3)  means 2 groups of x plus 3.  Another example, 3(y - 1) means 3 groups of y minus 1.

2) Model the Expression with Manipulatives
Algebra tiles are great for modeling expressions, however, if you do not have Algebra tiles you can either make them with paper, or use objects to represent the variables and constants.  Have your students model the expression.  For example, they know that 2(x + 3) means  2 groups of x plus 3.  So now model the expression with manipulatives.  See photo for example.



Model with manipulatives for 3(y - 1)




After students model the expression have them write down what they see with combining like terms.


If your students see the connection between the initial expression and the simplified expression at this point, that is great.  If they don't, that is ok.  Your goal for this step is that they conceptually understand multiplying expressions. 

3) Model the Expression with Symbols
Now, instead of using manipulatives have your students write out the variables and the constants.  In the photo the expression 2(x+3) is modeled by writing out the groups. 



Here is the model for 3(y -1)


Again, have your students write the simplified version after modeling.  At this point, if your students have not already noticed the "shortcut" guide them through questioning. Ideally you want your students to make the connection so they retain the information.  


4) Multiply using the Distributive Property
Once your students have a strong conceptual understanding of the distributive property move on to actually using the property when multiplying.  Students should understand that every term from one expression needs to be multiplied by every term of the other expression.  Understanding this concept will greatly help them when multiplying binomials.  One strategy I use with my students are circling the terms including the signs. This helps students not miss the negative signs.  

Another strategy is drawing lines.  Lines become extremely helpful when multiplying binomials and beyond.  


If you don't have time to make your own notes about the distributive property, YOU CAN CLICK HERE TO USE MINE.  I've included two pages of notes to guide students in discovering the distributive property as well as FUN and ENGAGING stations so they can practice what they have learned. 

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