When You Can't Make Connections in Math

Recently, I visited Mexico for a short holiday.  While in Mexico, I experienced that feeling of not comprehending connections about money values.  When I would see a price in a menu, I had no connection to what that amount would be in American money.
I couldn't answer simple questions like:

Should I buy that ceramic glass... is it worth 320 pesos? 
Is this daily rate for my hotel room reasonable and in my budget?
Did that person give me the correct change?
How much is each person's bill at the restaurant and how much should I tip the server?


It was driving me crazy!!!
So I made up a simple chart for my family based on the current exchange rate that looked somewhat like this:
US            Mexico
$1              16
$5               80
$10            160
$15             240
$20             320
ETC.
I also wrote the American value on the end of each Mexican Bill until I started internalizing their value.  Suddenly I felt like I was in control. I had NUMBER SENSE again.

Do you have students in your class that can't make connections between mathematical concepts like fractions, percents, and decimals?    Please help them make those connections with concrete experiences, games, and charts.
     
It feels terrible when you just don't have any connections or background knowledge to help you understand....

Smiles,
Deborah

6's

Elsewhere on this site, I gave some general times tables tips. Here, we'll have a closer look at the six times tables. You can use this page to show your kids the hidden patterns in the nine times tables, and make it easier for them to learn. Keep in mind that these patterns will mean infinitely more to your children if you can somehow coax them to realize the rule for themselves, rather than just pointing it out to them.

Here's the six times table. By the way, you might like to download the printable six times table chart on this site, and stick it to the wall of your classroom, or your kid's bedroom!

6	x	1	=	6
6	x	2	=	12
6	x	3	=	18
6	x	4	=	24
6	x	5	=	30
6	x	6	=	36
6	x	7	=	42
6	x	8	=	48
6	x	9	=	54
6	x	10	=	60
6	x	11	=	66
6	x	12	=	72

If you have a look at every second row, you might notice an interesting pattern. Inspect carefully the ones digit.

6	x	1	=	6
6	x	2	=	12
6	x	3	=	18
6	x	4	=	24
6	x	5	=	30
6	x	6	=	36
6	x	7	=	42
6	x	8	=	48
6	x	9	=	54
6	x	10	=	60
6	x	11	=	66
6	x	12	=	72

Did you see the pattern?

The ones digit of six times something is the ones digit of the something, at least if the something is even.

What about odd 'somethings'?

6	x	1	=	6	(and 6-1=5)
6	x	2	=	12
6	x	3	=	18	(and 8-3=5)
6	x	4	=	24
6	x	5	=	30	(and 5-0=5)
6	x	6	=	36
6	x	7	=	42	(and 7-2=5)
6	x	8	=	48
6	x	9	=	54	(and 9-4=5)
6	x	10	=	60
6	x	11	=	66	(and 6-1=5)
6	x	12	=	72

There you have it!

For odd 'somethings', the ones digit of six times something is five more or less than the ones digit of the something.

Some other facts about the answers in the six times table :

• Since the answers are always even, the last digit must always be 0, 2, 4, 6 or 8.
• If you add the digits together, and do that again and again, you'll eventually get 3, 6 or 9. For example :
  • 6 x 1 = 6...
  • 6 x 2 = 12, and 1 + 2 = 3...
  • 6 x 3 = 18, and 1 + 8 = 9...
  • 6 x 4 = 24, and 2 + 4 = 6...
  • 6 x 5 = 30, and 3 + 0 = 3...
  • 6 x 6 = 36, and 3 + 6 = 9...
  • 6 x 7 = 42, and 4 + 2 = 6...
  • 6 x 8 = 48, and 4 + 8 = 12, and 1 + 2 = 3...
  • 6 x 9 = 54, and 5 + 4 = 9...
  • 6 x 10 = 60, and 6 + 0 = 6...
  • 6 x 11 = 66, and 6 + 6 = 12, and 1 + 2 = 3...
  • 6 x 12 = 72, and 7 + 2 = 9...
  Amazingly, this pattern, 6,3,9,6,3,9,... continues forever.
  • 6 x 5468 = 38208, and 3+8+2+0+8=21, and 2+1=3...
  • 6 x 5469 = 38214, and 3+8+2+1+4=18, and 1+8=9...
  • 6 x 5470 = 38220, and 3+8+2+2+0=15, and 1+5=6...
  • 6 x 5471 = 38226, and 3+8+2+2+6=21, and 2+1=3...
  • 6 x 5472 = 38232, and 3+8+2+3+2=18, and 1+8=9...

How's that for an interesting pattern?

Before I close this page, if your child can easily multiply by 5, or by 2 and 3, there are a couple of easy ways to multiply by 6.

• 6 times something is twice 3 times something. So if I want 6 times 9, I can say 3 times 9 is 27, then 27+27 is 54. Not easy for every kid, but maybe it'll work for yours.
• Probably, it's easier to use this trick : 6 times something is 5 times something, plus another something. So to work out 6 x 7, I'd remember 5 x 7 is 35, then add another 7 to get 42. Or to find 6 x 12, I'd remember 5 x 12 is 60, then add another 12 to get 72.

Opinion Writing as a Response to "Junie B. Jones Is Not a Crook"

Using the book, Junie B. Jone Is Not A Crook,  I wrote a reader response to the question: If you saw something in the Lost and Found at school would it be okay to take it?

You are welcome to copy my idea, or you can purchase all the needed pages so your students can create the fan book themselves, in my Teacher Pay Teacher store. They are having a 10% off sale today and tomorrow. This item is only $1.99 in my store.

The unit includes the fan template in two formats: lined and unlined. In addition, 2 activities are also included: 

1) Whole Group Oral Activity To Work on Adding Reasons to Support Their Opinion

 2) Supporting Your Opinion with Good Reasons from your Schema or Personal Experience. 

Check it out by CLICKING HERE

Created to address the following Common Core Standards:
CC.3.W.1  Write opinion pieces on familiar topics or texts, supporting a point of view with reasons.
CC.3. W. 1.a   Introduce the topic or text they are writing about, state an opinion, and create an organizational structure that lists reasons.
CC.3.W.1.b   Provide reasons that support the opinion
CC.3. W.1.c   Use linking words and phrases (e.g., because, therefore, since, for example) to connect opinions and reasons.
CC.3.W.1.d    Provide a concluding statement or section.

Let me know what you think about the fan book in the comment section of this post.  I read everyone of your comments and respond to your questions.

Smiles,

Deborah

Eureka Math Taught in Guided Math Groups

I'm going to a Eureka Math Training at the end of the month. 

Has anyone used this curriculum in a Guided Math Format? 

 Please leave me a comment as I would love to hear how it worked...

I've also been considering how to use it with

 Multi-grade/Split  Classes.

Smiles,

Deborah

Opinion Writing for a Multi-Grade Classroom: Conclusion

I've been working on a unit for 3rd, 4th, and 5th Graders about Opinion writing based on the Common Core Writing Standards.

  My goal is to create a unit that can be used in a

 Multi-Grade Classroom. 

 Here's an example of one of the activities.

To make it work for a Multi-Grade classroom, there are 3 different paragraphs that are appropriate for different grade levels. 

 The whole class has a mini-lesson about writing a conclusion to an opinion paragraph.   The teacher then divides students into small groups, and that group receives one paragraph based on their reading level.

Smiles,

Deborah

Clarifying the Differences Between Opinion, Persuasive, and Argument Writing

While developing materials to assist a 3rd Grader as she learns how to write an opinion piece that supports a point of view with reasons, I had to step back and clarify the difference between 

opinion, persuasive, and argument writing for myself.

Here is a simple chart from writestepswriting.com that helped me. Click on the chart to take you directly to their website.

Then I came upon this very informative  post written by 

the Six Traits Gurus, and I realized that the big difference between these different types of writing is evidence.

For the 3rd Grader that I am working with, I need to concentrate on just having her learn to state her opinion well but with her own voice.

Grade Level Differences: Opinion Pieces versus Arguments

Up through grade 5, the CCSS call for students to write opinion pieces, not arguments per se. The defining characteristics of an opinion piece are as follows:

• The writer makes a claim
• The writer offers reasons to support that claim (School uniforms are not a good idea because they are expensive)
• The writer offers facts or details to strengthen his/her reasons (School uniforms can cost over $100 each, and every student needs at least two of them)
• The writer uses transitions (For example, To illustrate, Consequently, On the other hand, In addition) to link reasons or details to the main claim
• The writer sets up the paper by making the issue clear and closes by reinforcing his/her position or otherwise guiding the reader toward a good decision

After reading this post, I also realized that I don't want my 3rd Grader to get sucked into the conclusion pit where they just restate the 3 reasons.  I want them to start to think about a powerhouse ending that contains those reasons without the step by step writing that is so boring (and yet so easy to do). Here's an excerpt from the blog post that started me thinking:

• A powerhouse ending. Endings matter. They need to stick in our minds, wrap up loose ends, give us new things to think about—and perhaps, in the case of argument, suggest new thinking or action. An ending must be more than a summary of what we’ve read. It is condescending to simply summarize what’s been said, as if the reader were inattentive or not very quick. It’s lazy to leave things dangling, or toss the choice of options to the reader—the old “What do you think?” way out. A good argument might close with a call to action, a summary of the consequences of inaction, or even with the most powerful piece of evidence—one the writer has held back until this moment. A good question to ask is, What doesn’t the reader know yet that will push him/her to a good conclusion?

 She also discusses implications for higher grades as they focus on argumentative writing and on using evidence to support those arguments.  Click HERE to read the whole article.  It's really worth your time! 

Lastly, I did more research on this topic using multiple sources and an old language arts textbook (dry reading)  and the definitions concurred with the above discussion. I didn't want you to think I just found one source on the internet and took that information as the "holy grail."

Smiles, 

Deborah 

 

Opinion Writing : Common Core Standard CC.3.W.1a

 I am working with a 3rd Grader on the

 Common Core Writing Standard CC.3.W.1.a

Introduce the topic or text they are writing about, state an opinion, and create an organizational structure that lists reasons.

We've been talking about the organizational structure that help us write a well-written opinion.  

Together we wrote down some of her thoughts that answer the topic: Is ballet a healthy activity?

Example of a completed Opinion Structure

Example of an individual Page

If you would like a fan pattern to use in your classroom, click HERE TO GO TO MY GOOGLE DOC FILE

 AND DOWNLOAD IT.

This document has both lined and unlined fan pages. It also includes the organizational graphic organizer that we used while talking about the structure of an opinion writing piece.

 (More on that in another post)

Leave a comment and let me know if you find it helpful or not. Share your OPINION

Smiles, 

Deborah
Update on 12/25/16: 
Fun post today on:  The Structure of Opinion Writing and FINGERPAINTING. (sensory experience)  http://splitmultigradeclassroom.blogspot.com/2016/12/the-structure-of-opinion-writing-for.html

Comparing Surface Area and Volume

A great activity for those "talented students" that you are urging to think about ideas on their own.

Give the student sentence strips that contain the above phrases.  In small groups, they create the Venn Diagram.  Then, challenge them to create their own Double Venn Diagrams that compare two mathematical concepts.  

Have the students generate a list of pairs of mathematical concepts that they could compare, and then confer with their teacher

 for final approval of their basic idea.

Lately, I've been thinking about this poem.

Perhaps we need to let our "Talented Learners"  FLY?

Smiles,

Deborah

Fractions: Why Numerator and Denominator?

Numerator comes from the Latin word meaning number.

Denominator comes from the Latin word meaning name.

 

Don't you just wish early mathematicians would have used 

number and name?

 It would have been so much easier to teach 3rd Graders

 to read and write number and name than numerator and denominator :)

Smiles,

Deborah 

Fraction Progress: Referring to the Same Whole

I've been doing some personal professional development about the fraction progression in the Common Core Standards.  

I was thinking about this standard:
 

CCSS.Math.Content.3.NF.A.3.d
Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

When I came upon this task at Illustrative Mathematics.

 

If you were to choose the two pictures that best compares 2/3 and 2/5, which two illustrations would you choose?

•Your first reaction would be that students could choose either illustration 3 or 4 to represent 2/3.  That is true, but why must you select illustrations 4 and 5 together?

•This question highlights the fact that in order to compare two fractions, they must refer to the same whole. 

Do we spend enough time on the fact that both fractions must refer to the same wholes?

     When we use pre-made worksheets that compare fractions, they usually have a pre-drawn "Whole."  

Using this task from Illustrative Mathematics, in a small group format, could lead to an amazing discussion of the importance of the "same whole."

Here at home, I think I will give my soon-to-be Third Grader 
a cookie like this and ask her to share 1/2 of the cookie
 with me. 

 

I will promise to share 1/2 of my (unseen ) cookie with her.

My cookie will look like this:

What do you think her reaction will be?  Will our discussion lead to the importance of the "same whole?"

P.S.  If I had a multi-grade 2/3 classroom, would this still be an appropriate lesson, using cookies instead of a worksheet?  What questions would I ask of a 2nd Grader versus a 3rd Grader? 

Smiles,

Deborah

Manipulatives that Assist Students in Thinking about Similar and Congruent

I have several sets of these  32 MiniRelational GeoSolids. 

 I mix them all together when we do an activity in which students sort the shapes into groups that illustrate the characteristics of similar and congruent.  There are very few concrete materials that help students think about the difference between the properties of these two concepts. 

In grades 4-6 you can also introduce the mathematical symbols for congruent and similar:

	Congruent (same shape and size)
	Similar (same shape, different size)

How about having that mathematically talented group of students in your class create an anchor chart for the class that compares the properties of congruent and similar.  

Here is an interesting anchor chart that I found on Google Images. Let student research the meanings of the markings on the triangles and explain the definitons of the words proportional and adjacent. (Don't worry about the calculations at the bottom, just use the illustrations.)

What do you think about this idea?  I had 3 very mathematically talented boys in my 3rd/4th grade class one year that kept me up at night thinking of challenging activities for them to do or investigate. (They loved geometry concepts, and I heard that one of the boys later became an engineer :) )When we would cover our "grade level" material, I would meet with them 8 minutes before class and discuss the lesson. Then using white boards or concrete materials, they would show me that they understood those concepts. 

 Next, I would give them their assignment, discuss the quality of work that I was looking for, and they would work and write in their math journals at the reading table in the back of the room.  I can only tell you that the quality of work that came from those boys was worth the sleepless nights and extra work.  

Smiles,

Deborah

3 -Dimensional Sort

This afternoon at our house was a lazy, snowy day. I decided to challenge my granddaughter, who is in 2nd Grade, to a 3-Dimensional Sort Challenge. 

This a great activity to do with a small group of students during Guided Math. 

 Using a set of 32 MiniRelational GeoSolids

and a set of cards labeled: cone, pyramid, prism, and  cylinder.

Earlier in the weekend, we built prisms and pyramids out of toothpicks and playdough.

We talked about what made 3-D shapes 

either a prism or pyramid.  

So when I gave her the bag of 32 shapes and asked her to categorize them, I did not review any concepts... 

she was just given the task. 

The first time around she made 2 common errors:

1) she categorized the hexagonal prisms as a cylinders.

2) she categorized the triangular prisms as a pyramids.

Why?  Probably because she was never exposed to these shapes before and did not deeply understand the "the characteristics" of the different types of 3-D shapes.  As we reviewed each category 

of 3- D shapes, we again talked about what she looked for when she was looking at each shape.

Cone- "Only 1 face at the bottom, and a vertex at the top of the shape."

Cylinder- "2 faces with curved sides. No vertex at all."

Prism - Faces- there can be a different number of them, and vertices."

Pyramid - " One vertex at the top, straight sides, and a bottom."

After our discussion I place all the shapes back into a plastic bag and asked her if she wanted to try again.  WITH THE INCENTIVE OF EARNING A DOLLAR IF SHE COULD CATEGORIZE THEM ALL CORRECTLY!

She is now $1.00 richer!

Smiles, 

Deborah